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In the middle and high school course, students studied the topic "Fractions". However, this concept is much broader than given in the learning process. Today, the concept of a fraction is encountered quite often, and not everyone can calculate any expression, for example, multiplying fractions.
What is a fraction?
It so happened historically that fractional numbers appeared due to the need to measure. As practice shows, there are often examples for determining the length of a segment, the volume of a rectangular rectangle.
Initially, students are introduced to such a concept as a share. For example, if you divide a watermelon into 8 parts, then each will get one-eighth of a watermelon. This one part of eight is called a share.
A share equal to ½ of any value is called a half; ⅓ - third; ¼ - a quarter. Entries like 5/8, 4/5, 2/4 are called common fractions. An ordinary fraction is divided into a numerator and a denominator. Between them is a fractional line, or fractional line. A fractional bar can be drawn as either a horizontal or a slanted line. AT this case it stands for the division sign.
The denominator represents how many equal shares the value, object is divided into; and the numerator is how many equal shares are taken. The numerator is written above the fractional bar, the denominator below it.
It is most convenient to show ordinary fractions on a coordinate ray. If you divide a single segment into 4 equal parts, designate each part with a Latin letter, then as a result you can get an excellent visual aid. So, point A shows a share equal to 1/4 of the entire unit segment, and point B marks 2/8 of this segment.
Varieties of fractions
Fractions are common, decimal, and mixed numbers. In addition, fractions can be divided into proper and improper. This classification is more suitable for ordinary fractions.
A proper fraction is a number whose numerator is less than the denominator. Accordingly, an improper fraction is a number whose numerator is greater than the denominator. The second kind is usually written as a mixed number. Such an expression consists of an integer part and a fractional part. For example, 1½. 1 - integer part, ½ - fractional. However, if you need to perform some manipulations with the expression (dividing or multiplying fractions, reducing or converting them), the mixed number is converted into an improper fraction.
A correct fractional expression is always less than one, and an incorrect one is always greater than or equal to 1.
As for this expression, they understand a record in which any number is represented, the denominator of the fractional expression of which can be expressed through one with several zeros. If the fraction is correct, then the integer part in the decimal notation will be zero.
To burn decimal, you must first write the integer part, separate it from the fractional part with a comma, and then write the fractional expression. It must be remembered that after the comma the numerator must contain as many numeric characters as there are zeros in the denominator.
Example. Represent the fraction 7 21 / 1000 in decimal notation.
Algorithm for converting an improper fraction to a mixed number and vice versa
It is incorrect to write down an improper fraction in the answer of the problem, so it must be converted to a mixed number:
- divide the numerator by the existing denominator;
- in specific example incomplete quotient - whole;
- and the remainder is the numerator of the fractional part, with the denominator remaining unchanged.
Example. Convert improper fraction to mixed number: 47 / 5 .
Decision. 47: 5. The incomplete quotient is 9, the remainder = 2. Hence, 47 / 5 = 9 2 / 5.
Sometimes you need to represent a mixed number as an improper fraction. Then you need to use the following algorithm:
- the integer part is multiplied by the denominator of the fractional expression;
- the resulting product is added to the numerator;
- the result is written in the numerator, the denominator remains unchanged.
Example. Express the number in mixed form as an improper fraction: 9 8 / 10 .
Decision. 9 x 10 + 8 = 90 + 8 = 98 is the numerator.
Answer: 98 / 10.
Multiplication of ordinary fractions
You can perform various algebraic operations on ordinary fractions. To multiply two numbers, you need to multiply the numerator with the numerator, and the denominator with the denominator. Moreover, the multiplication of fractions with different denominators does not differ from the product of fractional numbers with the same denominators.
It happens that after finding the result, you need to reduce the fraction. AT without fail the resulting expression should be simplified as much as possible. Of course, it cannot be said that an improper fraction in the answer is a mistake, but it is also difficult to call it the correct answer.
Example. Find the product of two ordinary fractions: ½ and 20/18.
As can be seen from the example, after finding the product, a reducible fractional notation is obtained. Both the numerator and the denominator in this case are divisible by 4, and the result is the answer 5 / 9.
Multiplying decimal fractions
The product of decimal fractions is quite different from the product of ordinary fractions in its principle. So, multiplying fractions is as follows:
- two decimal fractions must be written under each other so that the rightmost digits are one under the other;
- you need to multiply the written numbers, despite the commas, that is, as natural numbers;
- count the number of digits after the comma in each of the numbers;
- in the result obtained after multiplication, you need to count as many digital characters on the right as are contained in the sum in both factors after the decimal point, and put a separating sign;
- if there are fewer digits in the product, then so many zeros must be written in front of them to cover this number, put a comma and assign an integer part equal to zero.
Example. Calculate the product of two decimals: 2.25 and 3.6.
Decision.
Multiplication of mixed fractions
To calculate the product of two mixed fractions, you need to use the rule for multiplying fractions:
- convert mixed numbers to improper fractions;
- find the product of numerators;
- find the product of the denominators;
- write down the result;
- simplify the expression as much as possible.
Example. Find the product of 4½ and 6 2 / 5.
Multiplying a number by a fraction (fractions by a number)
In addition to finding the product of two fractions, mixed numbers, there are tasks where you need to multiply by a fraction.
So, to find the product of a decimal fraction and a natural number, you need:
- write the number under the fraction so that the rightmost digits are one above the other;
- find the work, despite the comma;
- in the result obtained, separate the integer part from the fractional part using a comma, counting to the right the number of characters that is after the decimal point in the fraction.
To multiply an ordinary fraction by a number, you should find the product of the numerator and the natural factor. If the answer is a reducible fraction, it should be converted.
Example. Calculate the product of 5 / 8 and 12.
Decision. 5 / 8 * 12 = (5*12) / 8 = 60 / 8 = 30 / 4 = 15 / 2 = 7 1 / 2.
Answer: 7 1 / 2.
As you can see from the previous example, it was necessary to reduce the resulting result and convert the incorrect fractional expression into a mixed number.
Also, the multiplication of fractions also applies to finding the product of a number in mixed form and a natural factor. To multiply these two numbers, you should multiply the integer part of the mixed factor by the number, multiply the numerator by the same value, and leave the denominator unchanged. If necessary, you need to simplify the result as much as possible.
Example. Find the product of 9 5 / 6 and 9.
Decision. 9 5 / 6 x 9 \u003d 9 x 9 + (5 x 9) / 6 \u003d 81 + 45 / 6 \u003d 81 + 7 3 / 6 \u003d 88 1 / 2.
Answer: 88 1 / 2.
Multiplication by factors 10, 100, 1000 or 0.1; 0.01; 0.001
The following rule follows from the previous paragraph. To multiply a decimal fraction by 10, 100, 1000, 10000, etc., you need to move the comma to the right by as many digit characters as there are zeros in the multiplier after one.
Example 1. Find the product of 0.065 and 1000.
Decision. 0.065 x 1000 = 0065 = 65.
Answer: 65.
Example 2. Find the product of 3.9 and 1000.
Decision. 3.9 x 1000 = 3.900 x 1000 = 3900.
Answer: 3900.
If you need to multiply natural number and 0.1; 0.01; 0.001; 0.0001, etc., you should move the comma to the left in the resulting product by as many digit characters as there are zeros before one. If necessary, a sufficient number of zeros are written in front of a natural number.
Example 1. Find the product of 56 and 0.01.
Decision. 56 x 0.01 = 0056 = 0.56.
Answer: 0,56.
Example 2. Find the product of 4 and 0.001.
Decision. 4 x 0.001 = 0004 = 0.004.
Answer: 0,004.
So, finding the product of various fractions should not cause difficulties, except perhaps the calculation of the result; In this case, you simply cannot do without a calculator.
Last time we learned how to add and subtract fractions (see the lesson "Addition and subtraction of fractions"). The most difficult moment in those actions was bringing fractions to a common denominator.
Now it's time to deal with multiplication and division. The good news is that these operations are even easier than addition and subtraction. To begin with, consider the simplest case, when there are two positive fractions without a distinguished integer part.
To multiply two fractions, you need to multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.
To divide two fractions, you need to multiply the first fraction by the "inverted" second.
Designation:
From the definition it follows that the division of fractions is reduced to multiplication. To flip a fraction, just swap the numerator and denominator. Therefore, the entire lesson we will consider mainly multiplication.
As a result of multiplication, a reduced fraction can arise (and often does arise) - of course, it must be reduced. If, after all the reductions, the fraction turned out to be incorrect, the whole part should be distinguished in it. But what definitely won’t happen with multiplication is reduction to a common denominator: no crosswise methods, maximum factors and least common multiples.
By definition we have:
Multiplication of fractions with an integer part and negative fractions
If there is an integer part in the fractions, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.
If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the limits of multiplication or removed altogether according to the following rules:
- Plus times minus gives minus;
- Two negatives make an affirmative.
Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was required to get rid of the whole part. For a product, they can be generalized in order to “burn” several minuses at once:
- We cross out the minuses in pairs until they completely disappear. In an extreme case, one minus can survive - the one that did not find a match;
- If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, since it did not find a pair, we take it out of the limits of multiplication. You get a negative fraction.
Task. Find the value of the expression:
We translate all fractions into improper ones, and then we take out the minuses outside the limits of multiplication. What remains is multiplied according to the usual rules. We get:
Let me remind you once again that the minus that comes before a fraction with a highlighted integer part refers specifically to the entire fraction, and not just to its integer part (this applies to the last two examples).
Also pay attention to negative numbers: When multiplied, they are enclosed in parentheses. This is done in order to separate the minuses from the multiplication signs and make the whole notation more accurate.
Reducing fractions on the fly
Multiplication is a very laborious operation. The numbers here are quite large, and to simplify the task, you can try to reduce the fraction even more before multiplication. Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be reduced using the basic property of a fraction. Take a look at the examples:
Task. Find the value of the expression:
By definition we have:
In all examples, the numbers that have been reduced and what is left of them are marked in red.
Please note: in the first case, the multipliers were reduced completely. Units remained in their place, which, generally speaking, can be omitted. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.
However, in no case do not use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:
You can't do that!
The error occurs due to the fact that when adding a fraction, the sum appears in the numerator of a fraction, and not the product of numbers. Therefore, it is impossible to apply the main property of a fraction, since this property deals specifically with the multiplication of numbers.
There is simply no other reason to reduce fractions, so correct solution the previous task looks like this:
Correct solution:
As you can see, the correct answer turned out to be not so beautiful. In general, be careful.
In this article, we will analyze multiplication of mixed numbers. First, we will voice the rule for multiplying mixed numbers and consider the application of this rule when solving examples. Next, we will talk about the multiplication of a mixed number and a natural number. Finally, we will learn how to multiply a mixed number and an ordinary fraction.
Page navigation.
Multiplication of mixed numbers.
Multiplication of mixed numbers can be reduced to multiplying ordinary fractions. To do this, it is enough to convert mixed numbers to improper fractions.
Let's write down multiplication rule for mixed numbers:
- First, the mixed numbers to be multiplied must be replaced by improper fractions;
- Secondly, you need to use the rule of multiplying a fraction by a fraction.
Consider examples of applying this rule when multiplying a mixed number by a mixed number.
Perform mixed number multiplication and .
First, we represent the multiplied mixed numbers as improper fractions: 
and 
. Now we can replace the multiplication of mixed numbers with the multiplication of ordinary fractions: 
. Applying the rule of multiplication of fractions, we get 
. The resulting fraction is irreducible (see reducible and irreducible fractions), but it is incorrect (see regular and improper fractions), therefore, to get the final answer, it remains to extract the integer part from the improper fraction: .
Let's write the whole solution in one line: .
.
To consolidate the skills of multiplying mixed numbers, consider the solution of another example.
Do the multiplication.
Funny numbers and are equal to the fractions 13/5 and 10/9, respectively. Then 
. At this stage, it's time to remember about fraction reduction: we will replace all the numbers in the fraction with their expansions into prime factors, and we will perform the reduction of the same factors.
Multiplication of a mixed number and a natural number
After replacing the mixed number with an improper fraction, multiplying a mixed number and a natural number is reduced to the multiplication of an ordinary fraction and a natural number.
Multiply the mixed number and the natural number 45 .
A mixed number is a fraction, then 
. Let's replace the numbers in the resulting fraction with their expansions into prime factors, make a reduction, after which we select the integer part: .
.
Multiplication of a mixed number and a natural number is sometimes conveniently done using the distributive property of multiplication with respect to addition. In this case, the product of a mixed number and a natural number is equal to the sum of the products of the integer part by the given natural number and the fractional part by the given natural number, that is, 
.
Compute the product.
We replace the mixed number with the sum of the integer and fractional parts, after which we apply the distributive property of multiplication: .
Multiplying a mixed number and a common fraction it is most convenient to reduce to the multiplication of ordinary fractions, representing the multiplied mixed number as an improper fraction.
Multiply the mixed number by the common fraction 4/15.
Replacing the mixed number with a fraction, we get 
.
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Multiplication of fractional numbers
§ 140. Definitions. 1) The multiplication of a fractional number by an integer is defined in the same way as the multiplication of integers, namely: to multiply some number (multiplier) by an integer (multiplier) means to make a sum of identical terms, in which each term is equal to the multiplicand, and the number of terms is equal to the multiplier.
So multiplying by 5 means finding the sum:
2) To multiply some number (multiplier) by a fraction (multiplier) means to find this fraction of the multiplicand.
Thus, finding a fraction of a given number, which we considered before, we will now call multiplication by a fraction.
3) To multiply some number (multiplier) by a mixed number (factor) means to multiply the multiplicand first by the integer of the factor, then by the fraction of the factor, and add the results of these two multiplications together.
For example:
The number obtained after multiplication is in all these cases called work, i.e., in the same way as when multiplying integers.
From these definitions it is clear that the multiplication of fractional numbers is an action that is always possible and always unambiguous.
§ 141. Expediency of these definitions. To understand the expediency of introducing the last two definitions of multiplication into arithmetic, let us take the following problem:
Task. The train, moving evenly, travels 40 km per hour; how to find out how many kilometers this train will travel in a given number of hours?
If we had remained with that one definition of multiplication, which is indicated in the arithmetic of integers (addition of equal terms), then our problem would have three various solutions, namely:
If the given number of hours is an integer (for example, 5 hours), then to solve the problem, 40 km must be multiplied by this number of hours.
If a given number of hours is expressed as a fraction (for example, hours), then you will have to find the value of this fraction from 40 km.
Finally, if the given number of hours is mixed (for example, hours), then it will be necessary to multiply 40 km by an integer contained in the mixed number, and add to the result such a fraction from 40 km as is in the mixed number.
The definitions we have given allow us to give one general answer to all these possible cases:
40 km must be multiplied by the given number of hours, whatever it may be.
Thus, if the task is presented in general view So:
A train moving uniformly travels v km per hour. How many kilometers will the train cover in t hours?
then, whatever the numbers v and t, we can express one answer: the desired number is expressed by the formula v · t.
Note. Finding some fraction of a given number, by our definition, means the same thing as multiplying a given number by this fraction; therefore, for example, to find 5% (i.e. five hundredths) of a given number means the same as multiplying the given number by or by; finding 125% of a given number is the same as multiplying that number by or by , etc.
§ 142. A note about when a number increases and when it decreases from multiplication.
From multiplying by a proper fraction, the number decreases, and from multiplying by an improper fraction, the number increases if this improper fraction more than one, and remains unchanged if it is equal to one.
Comment. When multiplying fractional numbers, as well as integers, the product is taken equal to zero if any of the factors is equal to zero, so,.
§ 143. Derivation of multiplication rules.
1) Multiplying a fraction by an integer. Let the fraction be multiplied by 5. This means to increase by 5 times. To increase a fraction by 5, it is enough to increase its numerator or decrease its denominator by 5 times (§ 127).
So:
Rule 1. To multiply a fraction by an integer, you must multiply the numerator by this integer, and leave the denominator the same; instead, you can also divide the denominator of the fraction by the given integer (if possible), and leave the numerator the same.
Comment. The product of a fraction and its denominator is equal to its numerator.
So:
Rule 2. To multiply an integer by a fraction, you need to multiply the integer by the numerator of the fraction and make this product the numerator, and sign the denominator of the given fraction as the denominator.
Rule 3. To multiply a fraction by a fraction, you need to multiply the numerator by the numerator and the denominator by the denominator and make the first product the numerator and the second the denominator of the product.
Comment. This rule can also be applied to the multiplication of a fraction by an integer and an integer by a fraction, if only we consider the integer as a fraction with a denominator of one. So:
Thus, the three rules now stated are contained in one, which can be expressed in general terms as follows:
4) Multiplication of mixed numbers.
Rule 4. To multiply mixed numbers, you need to convert them to improper fractions and then multiply according to the rules for multiplying fractions. For example:
§ 144. Reduction in multiplication. When multiplying fractions, if possible, a preliminary reduction should be done, as can be seen from the following examples:
Such a reduction can be done because the value of the fraction will not change if the numerator and denominator are reduced in the same number once.
§ 145. Change of product with change of factors. When the factors change, the product of fractional numbers will change in exactly the same way as the product of integers (§ 53), namely: if you increase (or decrease) any factor several times, then the product will increase (or decrease) by the same amount .
So, if in the example:
in order to multiply several fractions, it is necessary to multiply their numerators among themselves and the denominators among themselves and make the first product the numerator and the second the denominator of the product.
Comment. This rule can also be applied to such products in which some factors of the number are integer or mixed, if only we consider the whole number as a fraction whose denominator is one, and we turn mixed numbers into improper fractions. For example:
§ 147. Basic properties multiplication. Those properties of multiplication that we have indicated for integers (§ 56, 57, 59) also belong to the multiplication of fractional numbers. Let's specify these properties.
1) The product does not change from changing the places of the factors.
For example:
Indeed, according to the rule of the previous paragraph, the first product is equal to the fraction, and the second is equal to the fraction. But these fractions are the same, because their terms differ only in the order of the integer factors, and the product of integers does not change when the places of the factors change.
2) The product will not change if any group of factors is replaced by their product.
For example:
The results are the same.
From this property of multiplication, one can deduce the following conclusion:
to multiply some number by a product, you can multiply this number by the first factor, multiply the resulting number by the second, and so on.
For example:
3) The distributive law of multiplication (with respect to addition). To multiply the sum by some number, you can multiply each term by this number separately and add the results.
This law has been explained by us (§ 59) as applied to whole numbers. It remains true without any changes for fractional numbers.
Let us show, in fact, that the equality
(a + b + c + .)m = am + bm + cm + .
(the distributive law of multiplication with respect to addition) remains true even when the letters mean fractional numbers. Let's consider three cases.
1) Suppose first that the factor m is an integer, for example m = 3 (a, b, c are any numbers). According to the definition of multiplication by an integer, one can write (limited for simplicity to three terms):
(a + b + c) * 3 = (a + b + c) + (a + b + c) + (a + b + c).
On the basis of the associative law of addition, we can omit all brackets on the right side; applying the commutative law of addition, and then again the associative one, we can obviously rewrite right side So:
(a + a + a) + (b + b + b) + (c + c + c).
(a + b + c) * 3 = a * 3 + b * 3 + c * 3.
Hence, the distributive law in this case is confirmed.
Multiplication and division of fractions
Last time we learned how to add and subtract fractions (see the lesson "Adding and subtracting fractions"). The most difficult moment in those actions was bringing fractions to a common denominator.
Now it's time to deal with multiplication and division. The good news is that these operations are even easier than addition and subtraction. To begin with, consider the simplest case, when there are two positive fractions without a distinguished integer part.
To multiply two fractions, you need to multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.
To divide two fractions, you need to multiply the first fraction by the "inverted" second.
From the definition it follows that the division of fractions is reduced to multiplication. To flip a fraction, just swap the numerator and denominator. Therefore, the entire lesson we will consider mainly multiplication.
As a result of multiplication, a reduced fraction can arise (and often does arise) - of course, it must be reduced. If, after all the reductions, the fraction turned out to be incorrect, the whole part should be distinguished in it. But what definitely won’t happen with multiplication is reduction to a common denominator: no crosswise methods, maximum factors and least common multiples.
By definition we have:
Multiplication of fractions with an integer part and negative fractions
If there is an integer part in the fractions, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.
If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the limits of multiplication or removed altogether according to the following rules:
- Plus times minus gives minus;
- Two negatives make an affirmative.
Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was required to get rid of the whole part. For a product, they can be generalized in order to “burn” several minuses at once:
- We cross out the minuses in pairs until they completely disappear. In an extreme case, one minus can survive - the one that did not find a match;
- If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, since it did not find a pair, we take it out of the limits of multiplication. You get a negative fraction.
Task. Find the value of the expression:
We translate all fractions into improper ones, and then we take out the minuses outside the limits of multiplication. What remains is multiplied according to the usual rules. We get:
Let me remind you once again that the minus that comes before a fraction with a highlighted integer part refers specifically to the entire fraction, and not just to its integer part (this applies to the last two examples).
Also pay attention to negative numbers: when multiplied, they are enclosed in brackets. This is done in order to separate the minuses from the multiplication signs and make the whole notation more accurate.
Reducing fractions on the fly
Multiplication is a very laborious operation. The numbers here are quite large, and to simplify the task, you can try to reduce the fraction even more before multiplication. Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be reduced using the basic property of a fraction. Take a look at the examples:
Task. Find the value of the expression:
By definition we have:
In all examples, the numbers that have been reduced and what is left of them are marked in red.
Please note: in the first case, the multipliers were reduced completely. Units remained in their place, which, generally speaking, can be omitted. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.
However, in no case do not use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:
You can't do that!
The error occurs due to the fact that when adding a fraction, the sum appears in the numerator of a fraction, and not the product of numbers. Therefore, it is impossible to apply the main property of a fraction, since this property deals specifically with the multiplication of numbers.
There is simply no other reason to reduce fractions, so the correct solution to the previous problem looks like this:
As you can see, the correct answer turned out to be not so beautiful. In general, be careful.
Multiplication of fractions.
To correctly multiply a fraction by a fraction or a fraction by a number, you need to know simple rules. We will now analyze these rules in detail.
Multiplying a fraction by a fraction.
To multiply a fraction by a fraction, you need to calculate the product of the numerators and the product of the denominators of these fractions.
Consider an example:
We multiply the numerator of the first fraction with the numerator of the second fraction, and we also multiply the denominator of the first fraction with the denominator of the second fraction.
Multiplying a fraction by a number.
Let's start with the rule any number can be represented as a fraction \(\bf n = \frac \) .
Let's use this rule for multiplication.
The improper fraction \(\frac = \frac = \frac + \frac = 2 + \frac = 2\frac \\\) was converted to a mixed fraction.
In other words, When multiplying a number by a fraction, multiply the number by the numerator and leave the denominator unchanged. Example:
Multiplication of mixed fractions.
To multiply mixed fractions, you must first represent each mixed fraction as an improper fraction, and then use the multiplication rule. The numerator is multiplied with the numerator, the denominator is multiplied with the denominator.
Multiplication of reciprocal fractions and numbers.
Related questions:
How to multiply a fraction by a fraction?
Answer: the product of ordinary fractions is the multiplication of the numerator with the numerator, the denominator with the denominator. To get the product of mixed fractions, you need to convert them to an improper fraction and multiply according to the rules.
How to multiply fractions with different denominators?
Answer: it doesn't matter whether they are the same or different denominators for fractions, multiplication occurs according to the rule of finding the product of the numerator with the numerator, the denominator with the denominator.
How to multiply mixed fractions?
Answer: first of all, you need to convert the mixed fraction to an improper fraction and then find the product according to the rules of multiplication.
How to multiply a number by a fraction?
Answer: We multiply the number with the numerator, and leave the denominator the same.
Example #1:
Calculate the product: a) \(\frac \times \frac \) b) \(\frac \times \frac \)
Example #2:
Calculate the product of a number and a fraction: a) \(3 \times \frac \) b) \(\frac \times 11\)
Example #3:
Write the reciprocal of the fraction \(\frac \)?
Answer: \(\frac = 3\)
Example #4:
Calculate the product of two reciprocals: a) \(\frac \times \frac \)
Example #5:
Can mutually inverse fractions be:
a) both proper fractions;
b) simultaneously improper fractions;
c) natural numbers at the same time?
Decision:
a) Let's use an example to answer the first question. The fraction \(\frac \) is correct, its reciprocal will be equal to \(\frac \) - an improper fraction. Answer: no.
b) in almost all enumerations of fractions, this condition is not met, but there are some numbers that fulfill the condition of being an improper fraction at the same time. For example, the improper fraction is \(\frac \) , its reciprocal is \(\frac \). We get two improper fractions. Answer: not always under certain conditions, when the numerator and denominator are equal.
c) natural numbers are the numbers that we use when counting, for example, 1, 2, 3, .... If we take the number \(3 = \frac \), then its reciprocal will be \(\frac \). The fraction \(\frac \) is not a natural number. If we go through all the numbers, the reciprocal is always a fraction, except for 1. If we take the number 1, then its reciprocal will be \(\frac = \frac = 1\). The number 1 is a natural number. Answer: they can be simultaneously natural numbers only in one case, if this number is 1.
Example #6:
Perform the product of mixed fractions: a) \(4 \times 2\frac \) b) \(1\frac \times 3\frac \)
Decision:
a) \(4 \times 2\frac = \frac \times \frac = \frac = 11\frac \\\\ \)
b) \(1\frac \times 3\frac = \frac \times \frac = \frac = 4\frac \)
Example #7:
Can two reciprocal numbers be simultaneously mixed numbers?
Let's look at an example. Let's take a mixed fraction \(1\frac \), find its reciprocal, for this we translate it into an improper fraction \(1\frac = \frac \) . Its reciprocal will be equal to \(\frac \) . The fraction \(\frac \) is a proper fraction. Answer: Two mutually inverse fractions cannot be mixed numbers at the same time.
Multiplying a decimal by a natural number
Presentation for the lesson
Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested this work please download the full version.
- In a fun way, introduce students to the rule of multiplying a decimal fraction by a natural number, by a bit unit and the rule of expressing a decimal fraction as a percentage. Develop the ability to apply the acquired knowledge in solving examples and problems.
- Develop and activate logical thinking students, the ability to identify patterns and generalize them, strengthen memory, the ability to cooperate, provide assistance, evaluate their work and the work of each other.
- To cultivate interest in mathematics, activity, mobility, ability to communicate.
Equipment: interactive board, a poster with a cyphergram, posters with mathematicians' statements.
- Organizing time.
- Oral counting is a generalization of previously studied material, preparation for the study of new material.
- Explanation of new material.
- Homework assignment.
- Mathematical physical education.
- Generalization and systematization of the acquired knowledge in a playful way with the help of a computer.
- Grading.
2. Guys, today our lesson will be somewhat unusual, because I will not spend it alone, but with my friend. And my friend is also unusual, now you will see him. (A cartoon computer appears on the screen.) My friend has a name and he can talk. What's your name, friend? Komposha replies: "My name is Komposha." Are you ready to help me today? YES! Well then, let's start the lesson.
Today I received an encrypted cyphergram, guys, which we must solve and decipher together. (A poster is posted on the board with an oral account for adding and subtracting decimal fractions, as a result of which the guys get the following code 523914687. )
Komposha helps to decipher the received code. As a result of decoding, the word MULTIPLICATION is obtained. Multiplication is the keyword of the topic of today's lesson. The topic of the lesson is displayed on the monitor: “Multiplying a decimal fraction by a natural number”
Guys, we know how the multiplication of natural numbers is performed. Today we will consider the multiplication of decimal numbers by a natural number. The multiplication of a decimal fraction by a natural number can be considered as the sum of terms, each of which is equal to this decimal fraction, and the number of terms is equal to this natural number. For example: 5.21 3 = 5.21 + 5, 21 + 5.21 = 15.63 So 5.21 3 = 15.63. Representing 5.21 as an ordinary fraction of a natural number, we get
And in this case, we got the same result of 15.63. Now, ignoring the comma, let's take the number 521 instead of the number 5.21 and multiply by the given natural number. Here we must remember that in one of the factors the comma is moved two places to the right. When multiplying the numbers 5, 21 and 3, we get a product equal to 15.63. Now, in this example, we will move the comma to the left by two digits. Thus, by how many times one of the factors was increased, the product was reduced by so many times. Based on the similar points of these methods, we draw a conclusion.
To multiply a decimal by a natural number, you need:
1) ignoring the comma, perform the multiplication of natural numbers;
2) in the resulting product, separate with a comma on the right as many characters as there are in a decimal fraction.
The following examples are displayed on the monitor, which we analyze together with Komposha and the guys: 5.21 3 = 15.63 and 7.624 15 = 114.34. After I show multiplication by a round number 12.6 50 \u003d 630. Next, I turn to the multiplication of a decimal fraction by a bit unit. I show the following examples: 7.423 100 \u003d 742.3 and 5.2 1000 \u003d 5200. So, I introduce the rule for multiplying a decimal fraction by a bit unit:
To multiply a decimal fraction by bit units 10, 100, 1000, etc., it is necessary to move the comma to the right in this fraction by as many digits as there are zeros in the bit unit record.
I end the explanation with the expression of a decimal fraction as a percentage. I enter the rule:
To express a decimal as a percentage, multiply it by 100 and add the % sign.
I give an example on a computer 0.5 100 = 50 or 0.5 = 50%.
4. At the end of the explanation, I give the guys homework, which is also displayed on the computer monitor: № 1030, № 1034, № 1032.
5. In order for the guys to rest a little, to consolidate the topic, we do a mathematical physical education session together with Komposha. Everyone stands up, shows the class the solved examples and they must answer whether the example is correct or incorrect. If the example is solved correctly, then they raise their hands above their heads and clap their palms. If the example is not solved correctly, the guys stretch their arms to the sides and knead their fingers.
6. And now you have a little rest, you can solve the tasks. Open your textbook to page 205, № 1029. in this task it is necessary to calculate the value of expressions:
Tasks appear on the computer. As they are solved, a picture appears with the image of a boat, which, when complete assembly floats away.
Solving this task on a computer, the rocket gradually develops, solving the last example, the rocket flies away. The teacher gives a little information to the students: “Every year from the Kazakh land from the Baikonur cosmodrome take off to the stars spaceships. Near Baikonur, Kazakhstan is building its new Baiterek cosmodrome.
How far will a car travel in 4 hours if the speed of the car is 74.8 km/h.
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Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, to come to a common opinion about the essence of paradoxes scientific community has not yet succeeded ... mathematical analysis, set theory, new physical and philosophical approaches; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.
If we turn the logic we are used to, everything falls into place. Achilles runs with constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells of a flying arrow:
A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia logical paradox it is overcome very simply - it is enough to clarify that at each moment of time the flying arrow rests at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (naturally, you still need additional data for calculations, trigonometry will help you). What do I want to focus on Special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for exploration.
Wednesday, July 4, 2018
Very well the differences between set and multiset are described in Wikipedia. We look.
As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Applicable mathematical theory sets to the mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.
First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms for each coin is unique ...
And now I have the most interest Ask: where is the boundary beyond which elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science here is not even close.
Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but they are shamans for that, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics, the task sounds like this: "Find the sum of graphic symbols representing any number." Mathematicians cannot solve this problem, but shamans can do it elementarily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let's say we have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number to a number graphic symbol. This is not a mathematical operation.
2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic characters to numbers. This is not a mathematical operation.
4. Add up the resulting numbers. Now that's mathematics.
The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that is not all.
From the point of view of mathematics, it does not matter in which number system we write the number. So, in different number systems, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With a large number of 12345, I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have already done that. Let's look at the result.
As you can see, in different number systems, the sum of the digits of the same number is different. This result has nothing to do with mathematics. It's like finding the area of a rectangle in meters and centimeters would give you completely different results.
Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement of numbers. Because we can't compare numbers with different units measurements. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then it has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical action does not depend on the value of the number, the unit of measure used, and on who performs this action.
Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?
Female... A halo on top and an arrow down is male.
If you have such a work of design art flashing before your eyes several times a day,
Then it is not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.
1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.
We will consider the multiplication of ordinary fractions in several possible ways.
Multiplying a fraction by a fraction
This is the simplest case, in which you need to use the following fraction multiplication rules.
To multiply a fraction by a fraction, necessary:
- multiply the numerator of the first fraction by the numerator of the second fraction and write their product into the numerator of the new fraction;
- multiply the denominator of the first fraction by the denominator of the second fraction and write their product into the denominator of the new fraction;
- Adding fractions with the same denominators
- Adding fractions with different denominators
Before multiplying numerators and denominators, check if the fractions can be reduced. Reducing fractions in calculations will greatly facilitate your calculations.
Multiplying a fraction by a natural number
To fraction multiply by a natural number you need to multiply the numerator of the fraction by this number, and leave the denominator of the fraction unchanged.
If the result of multiplication is an improper fraction, do not forget to turn it into a mixed number, that is, select the whole part.
Multiplication of mixed numbers
To multiply mixed numbers, you must first convert them into improper fractions and then multiply according to the rule for multiplying ordinary fractions.
Another way to multiply a fraction by a natural number
Sometimes in calculations it is more convenient to use a different method of multiplying an ordinary fraction by a number.
To multiply a fraction by a natural number, you need to divide the denominator of the fraction by this number, and leave the numerator the same.
As can be seen from the example, it is more convenient to use this version of the rule if the denominator of the fraction is divisible without a remainder by a natural number.
Actions with fractions
Adding fractions with the same denominators
Adding fractions is of two types:
Let's start with adding fractions with the same denominators. Everything is simple here. To add fractions with the same denominators, you need to add their numerators, and leave the denominator unchanged. For example, let's add the fractions and . We add the numerators, and leave the denominator unchanged:
This example can be easily understood if we think of a pizza that is divided into four parts. If you add pizza to pizza, you get pizza:
Example 2 Add fractions and .
Again, add the numerators, and leave the denominator unchanged:
The answer is an improper fraction. If the end of the task comes, then it is customary to get rid of improper fractions. To get rid of an improper fraction, you need to select the whole part in it. In our case, the integer part is allocated easily - two divided by two is equal to one:
This example can be easily understood if we think of a pizza that is divided into two parts. If you add more pizzas to the pizza, you get one whole pizza:
Example 3. Add fractions and .
This example can be easily understood if we think of a pizza that is divided into three parts. If you add more pizzas to pizza, you get pizzas:
Example 4 Find the value of an expression 
This example is solved in exactly the same way as the previous ones. The numerators must be added and the denominator left unchanged:
Let's try to depict our solution using a picture. If you add pizzas to a pizza and add more pizzas, you get 1 whole pizza and more pizzas.
As you can see, adding fractions with the same denominators is not difficult. It is enough to understand the following rules:
- To add fractions with the same denominator, you need to add their numerators, and leave the denominator the same;
- If the answer turned out to be an improper fraction, then you need to select the whole part in it.
- Find the LCM of the denominators of fractions;
- Divide the LCM by the denominator of each fraction and get an additional multiplier for each fraction;
- Multiply the numerators and denominators of fractions by their additional factors;
- Add fractions that have the same denominators;
- If the answer turned out to be an improper fraction, then select its whole part;
- Subtraction of fractions with the same denominators
- Subtraction of fractions with different denominators
Adding fractions with different denominators
Now we will learn how to add fractions with different denominators. When adding fractions, the denominators of those fractions must be the same. But they are not always the same.
For example, fractions can be added because they have the same denominators.
But fractions cannot be added at once, because these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.
There are several ways to reduce fractions to the same denominator. Today we will consider only one of them, since the rest of the methods may seem complicated for a beginner.
The essence of this method is that the least common multiple (LCM) of the denominators of both fractions is first searched for. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained. They do the same with the second fraction - the NOC is divided by the denominator of the second fraction and the second additional factor is obtained.
Then the numerators and denominators of the fractions are multiplied by their additional factors. As a result of these actions, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to add such fractions.
Example 1. Add fractions and
These fractions have different denominators, so you need to bring them to the same (common) denominator.
First of all, we find the least common multiple of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 2. The least common multiple of these numbers is 6
LCM (2 and 3) = 6
Now back to fractions and . First, we divide the LCM by the denominator of the first fraction and get the first additional factor. LCM is the number 6, and the denominator of the first fraction is the number 3. Divide 6 by 3, we get 2.
The resulting number 2 is the first additional factor. We write it down to the first fraction. To do this, we make a small oblique line above the fraction and write down the found additional factor above it:
We do the same with the second fraction. We divide the LCM by the denominator of the second fraction and get the second additional factor. LCM is the number 6, and the denominator of the second fraction is the number 2. Divide 6 by 2, we get 3.
The resulting number 3 is the second additional factor. We write it to the second fraction. Again, we make a small oblique line above the second fraction and write the found additional factor above it:
Now we are all set to add. It remains to multiply the numerators and denominators of fractions by their additional factors:
Look closely at what we have come to. We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to add such fractions. Let's complete this example to the end:
Thus the example ends. To add it turns out.
Let's try to depict our solution using a picture. If you add pizzas to a pizza, you get one whole pizza and another sixth of a pizza:
Reduction of fractions to the same (common) denominator can also be depicted using a picture. Bringing the fractions and to a common denominator, we get the fractions and . These two fractions will be represented by the same slices of pizzas. The only difference will be that this time they will be divided into equal shares (reduced to the same denominator).
The first drawing shows a fraction (four pieces out of six) and the second picture shows a fraction (three pieces out of six). Putting these pieces together we get (seven pieces out of six). This fraction is incorrect, so we have highlighted the integer part in it. The result was (one whole pizza and another sixth pizza).
Note that we have painted this example in too much detail. AT educational institutions it is not customary to write in such a detailed manner. You need to be able to quickly find the LCM of both denominators and additional factors to them, as well as quickly multiply the additional factors found by your numerators and denominators. While at school, we would have to write this example as follows:
But there is also the other side of the coin. If detailed notes are not made at the first stages of studying mathematics, then questions of the kind “Where does that number come from?”, “Why do fractions suddenly turn into completely different fractions? «.
To make it easier to add fractions with different denominators, you can use the following step-by-step instructions:
Example 2 Find the value of an expression 
.
Let's use the diagram above.
Step 1. Find the LCM for the denominators of fractions
We find the LCM for the denominators of both fractions. The denominators of the fractions are the numbers 2, 3 and 4. You need to find the LCM for these numbers:
Step 2. Divide the LCM by the denominator of each fraction and get an additional multiplier for each fraction
Divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 2. Divide 12 by 2, we get 6. We got the first additional factor 6. We write it over the first fraction:
Now we divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 3. Divide 12 by 3, we get 4. We got the second additional factor 4. We write it over the second fraction:
Now we divide the LCM by the denominator of the third fraction. LCM is the number 12, and the denominator of the third fraction is the number 4. Divide 12 by 4, we get 3. We got the third additional factor 3. We write it over the third fraction:
Step 3. Multiply the numerators and denominators of fractions by your additional factors
We multiply the numerators and denominators by our additional factors:
Step 4. Add fractions that have the same denominators
We came to the conclusion that fractions that had different denominators turned into fractions that have the same (common) denominators. It remains to add these fractions. Add up:
The addition didn't fit on one line, so we moved the remaining expression to the next line. This is allowed in mathematics. When an expression does not fit on one line, it is carried over to the next line, and it is necessary to put an equal sign (=) at the end of the first line and at the beginning of a new line. The equal sign on the second line indicates that this is a continuation of the expression that was on the first line.
Step 5. If the answer turned out to be an improper fraction, then select its integer part
Our answer is an improper fraction. We must single out the whole part of it. We highlight:
Got an answer
Subtraction of fractions with the same denominators
There are two types of fraction subtraction:
First, let's learn how to subtract fractions with the same denominators. Everything is simple here. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same.
For example, let's find the value of the expression . To solve this example, it is necessary to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same. Let's do this:
This example can be easily understood if we think of a pizza that is divided into four parts. If you cut pizzas from a pizza, you get pizzas:
Example 2 Find the value of the expression .
Again, from the numerator of the first fraction, subtract the numerator of the second fraction, and leave the denominator the same:
This example can be easily understood if we think of a pizza that is divided into three parts. If you cut pizzas from a pizza, you get pizzas:
Example 3 Find the value of an expression 
This example is solved in exactly the same way as the previous ones. From the numerator of the first fraction, you need to subtract the numerators of the remaining fractions:
The answer is an improper fraction. If the example is complete, then it is customary to get rid of the improper fraction. Let's get rid of the wrong fraction in the answer. To do this, select its whole part:
As you can see, there is nothing complicated in subtracting fractions with the same denominators. It is enough to understand the following rules:
Subtraction of fractions with different denominators
For example, a fraction can be subtracted from a fraction, since these fractions have the same denominators. But a fraction cannot be subtracted from a fraction, since these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.
The common denominator is found according to the same principle that we used when adding fractions with different denominators. First of all, find the LCM of the denominators of both fractions. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained, which is written over the first fraction. Similarly, the LCM is divided by the denominator of the second fraction and a second additional factor is obtained, which is written over the second fraction.
The fractions are then multiplied by their additional factors. As a result of these operations, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to subtract such fractions.
Example 1 Find the value of an expression:
First, we find the LCM of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 4. The least common multiple of these numbers is 12
LCM (3 and 4) = 12
Now back to fractions and
Let's find an additional factor for the first fraction. To do this, we divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 3. Divide 12 by 3, we get 4. We write the four over the first fraction:
We do the same with the second fraction. We divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 4. Divide 12 by 4, we get 3. We write the triple over the second fraction:
Now we are all set for subtraction. It remains to multiply the fractions by their additional factors:
We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to subtract such fractions. Let's complete this example to the end:
Got an answer
Let's try to depict our solution using a picture. If you cut pizzas from a pizza, you get pizzas.
This is the detailed version of the solution. Being at school, we would have to solve this example in a shorter way. Such a solution would look like this:
Reduction of fractions and to a common denominator can also be depicted using a picture. Bringing these fractions to a common denominator, we get the fractions and . These fractions will be represented by the same pizza slices, but this time they will be divided into the same fractions (reduced to the same denominator):
The first drawing shows a fraction (eight pieces out of twelve), and the second picture shows a fraction (three pieces out of twelve). By cutting off three pieces from eight pieces, we get five pieces out of twelve. The fraction describes these five pieces.
Example 2 Find the value of an expression 
These fractions have different denominators, so you first need to bring them to the same (common) denominator.
Find the LCM of the denominators of these fractions.
The denominators of the fractions are the numbers 10, 3 and 5. The least common multiple of these numbers is 30
LCM(10, 3, 5) = 30
Now we find additional factors for each fraction. To do this, we divide the LCM by the denominator of each fraction.
Let's find an additional factor for the first fraction. LCM is the number 30, and the denominator of the first fraction is the number 10. Divide 30 by 10, we get the first additional factor 3. We write it over the first fraction:
Now we find an additional factor for the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 30, and the denominator of the second fraction is the number 3. Divide 30 by 3, we get the second additional factor 10. We write it over the second fraction:
Now we find an additional factor for the third fraction. Divide the LCM by the denominator of the third fraction. LCM is the number 30, and the denominator of the third fraction is the number 5. Divide 30 by 5, we get the third additional factor 6. We write it over the third fraction:
Now everything is ready for subtraction. It remains to multiply the fractions by their additional factors:
We came to the conclusion that fractions that had different denominators turned into fractions that have the same (common) denominators. And we already know how to subtract such fractions. Let's finish this example.
The continuation of the example will not fit on one line, so we move the continuation to the next line. Don't forget about the equal sign (=) on the new line:
The answer turned out to be a correct fraction, and everything seems to suit us, but it is too cumbersome and ugly. We should make it simpler and more aesthetically pleasing. What can be done? You can reduce this fraction. Recall that the reduction of a fraction is the division of the numerator and denominator by the largest common divisor numerator and denominator.
To correctly reduce a fraction, you need to divide its numerator and denominator by the greatest common divisor (GCD) of the numbers 20 and 30.
Do not confuse GCD with NOC. The most common mistake many beginners make. GCD is the greatest common divisor. We find it for fraction reduction.
And LCM is the least common multiple. We find it in order to bring fractions to the same (common) denominator.
Now we will find the greatest common divisor (gcd) of the numbers 20 and 30.
So, we find the GCD for the numbers 20 and 30:
GCD (20 and 30) = 10
Now we return to our example and divide the numerator and denominator of the fraction by 10:
Got a nice answer
Multiplying a fraction by a number
To multiply a fraction by a number, you need to multiply the numerator of the given fraction by this number, and leave the denominator the same.
Example 1. Multiply the fraction by the number 1.
Multiply the numerator of the fraction by the number 1
The entry can be understood as taking half 1 time. For example, if you take pizza 1 time, you get pizza
From the laws of multiplication, we know that if the multiplicand and the multiplier are interchanged, then the product will not change. If the expression is written as , then the product will still be equal to . Again, the rule for multiplying an integer and a fraction works:
This entry can be understood as taking half of the unit. For example, if there is 1 whole pizza and we take half of it, then we will have pizza:
Example 2. Find the value of an expression
Multiply the numerator of the fraction by 4
The expression can be understood as taking two quarters 4 times. For example, if you take pizzas 4 times, you get two whole pizzas.
And if we swap the multiplicand and the multiplier in places, we get the expression. It will also be equal to 2. This expression can be understood as taking two pizzas from four whole pizzas:
Multiplication of fractions
To multiply fractions, you need to multiply their numerators and denominators. If the answer is an improper fraction, you need to select the whole part in it.
Example 1 Find the value of the expression .
Got an answer. It is desirable to reduce this fraction. The fraction can be reduced by 2. Then the final solution will take the following form:
The expression can be understood as taking a pizza from half a pizza. Let's say we have half a pizza:
How to take two-thirds from this half? First you need to divide this half into three equal parts:
And take two from these three pieces:
We'll get pizza. Remember what a pizza looks like divided into three parts:
One slice from this pizza and the two slices we took will have the same dimensions:
In other words, we are talking about the same pizza size. Therefore, the value of the expression is
Example 2. Find the value of an expression
Multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:
The answer is an improper fraction. Let's take a whole part of it:
Example 3 Find the value of an expression
The answer turned out to be a correct fraction, but it will be good if it is reduced. To reduce this fraction, it must be divided by the gcd of the numerator and denominator. So, let's find the GCD of the numbers 105 and 450:
GCD for (105 and 150) is 15
Now we divide the numerator and denominator of our answer to the GCD:
Representing an integer as a fraction
Any whole number can be represented as a fraction. For example, the number 5 can be represented as . From this, the five will not change its meaning, since the expression means “the number five divided by one”, and this, as you know, is equal to five:
Reverse numbers
Now we will get acquainted with interesting topic in mathematics. It's called "reverse numbers".
Definition. Reverse to number a is the number that, when multiplied by a gives a unit.
Let's substitute in this definition instead of a variable a number 5 and try to read the definition:
Reverse to number 5 is the number that, when multiplied by 5 gives a unit.
Is it possible to find a number that, when multiplied by 5, gives one? It turns out you can. Let's represent five as a fraction:
Then multiply this fraction by itself, just swap the numerator and denominator. In other words, multiply the fraction by itself, only inverted:
What will be the result of this? If we continue to solve this example, we get one:
This means that the inverse of the number 5 is the number, since when 5 is multiplied by one, one is obtained.
The reciprocal can also be found for any other integer.
- the reciprocal of 3 is a fraction
- the reciprocal of 4 is a fraction
You can also find the reciprocal for any other fraction. To do this, it is enough to turn it over.
