To solve various problems from mathematics and physics courses, you have to divide fractions. It's very easy to do if you know certain rules perform this mathematical operation.
Before we move on to formulating the rule for dividing fractions, let's remember some mathematical terms:
- The top part of the fraction is called the numerator, and the bottom part is called the denominator.
- When dividing, numbers are called as follows: dividend: divisor = quotient
How to divide fractions: simple fractions
To divide two simple fractions, multiply the dividend by the reciprocal of the divisor. This fraction is also called inverted because it is obtained by swapping the numerator and denominator. For example:
3/77: 1/11 = 3 /77 * 11 /1 = 3/7
How to divide fractions: mixed fractions
If we have to divide mixed fractions, then everything here is also quite simple and clear. First, we convert the mixed fraction to a regular improper fraction. To do this, multiply the denominator of such a fraction by an integer and add the numerator to the resulting product. As a result, we received a new numerator of the mixed fraction, but its denominator will remain unchanged. Further, the division of fractions will be carried out in exactly the same way as the division of simple fractions. For example:
10 2/3: 4/15 = 32/3: 4/15 = 32/3 * 15 /4 = 40/1 = 40
How to divide a fraction by a number
In order to divide a simple fraction by a number, the latter should be written as a fraction (irregular). This is very easy to do: this number is written in place of the numerator, and the denominator of such a fraction is equal to one. Further division is performed in the usual way. Let's look at this with an example:
5/11: 7 = 5/11: 7/1 = 5/11 * 1/7 = 5/77
How to divide decimals
Often an adult has difficulty dividing a whole number or a decimal fraction by a decimal fraction without the help of a calculator.
So, to divide decimals, you just need to cross out the comma in the divisor and stop paying attention to it. In the dividend, the comma must be moved to the right exactly as many places as it was in the fractional part of the divisor, adding zeros if necessary. And they continue to produce regular division by an integer. To make this more clear, consider the following example.
You can do everything with fractions, including division. This article shows the division of ordinary fractions. Definitions will be given and examples will be discussed. Let us dwell in detail on dividing fractions by natural numbers and vice versa. Dividing a common fraction by a mixed number will be discussed.
Dividing fractions
Division is the inverse of multiplication. When dividing, the unknown factor is found with the known product of another factor, where its given meaning is preserved with ordinary fractions.
If it is necessary to divide a common fraction a b by c d, then to determine such a number you need to multiply by the divisor c d, this will ultimately give the dividend a b. Let's get a number and write it a b · d c , where d c is the inverse of the c d number. Equalities can be written using the properties of multiplication, namely: a b · d c · c d = a b · d c · c d = a b · 1 = a b, where the expression a b · d c is the quotient of dividing a b by c d.
From here we obtain and formulate the rule for dividing ordinary fractions:
Definition 1
To divide a common fraction a b by c d, you need to multiply the dividend by the reciprocal of the divisor.
Let's write the rule in the form of an expression: a b: c d = a b · d c
The rules of division come down to multiplication. To stick with it, you need to have a good understanding of multiplying fractions.
Let's move on to considering the division of ordinary fractions.
Example 1
Divide 9 7 by 5 3. Write the result as a fraction.
Solution
The number 5 3 is the reciprocal fraction 3 5. It is necessary to use the rule for dividing ordinary fractions. We write this expression as follows: 9 7: 5 3 = 9 7 · 3 5 = 9 · 3 7 · 5 = 27 35.
Answer: 9 7: 5 3 = 27 35 .
When reducing fractions, separate out the whole part if the numerator is greater than the denominator.
Example 2
Divide 8 15: 24 65. Write the answer as a fraction.
Solution
To solve, you need to move from division to multiplication. Let's write it in this form: 8 15: 24 65 = 2 2 2 5 13 3 5 2 2 2 3 = 13 3 3 = 13 9
It is necessary to make a reduction, and this is done as follows: 8 65 15 24 = 2 2 2 5 13 3 5 2 2 2 3 = 13 3 3 = 13 9
We select the whole part and get 13 9 = 1 4 9.
Answer: 8 15: 24 65 = 1 4 9 .
Dividing an extraordinary fraction by a natural number
We use the rule of dividing a fraction by natural number: to divide a b by a natural number n, you only need to multiply the denominator by n. From here we get the expression: a b: n = a b · n.
The division rule is a consequence of the multiplication rule. Therefore, representing a natural number as a fraction will give an equality of this type: a b: n = a b: n 1 = a b · 1 n = a b · n.
Consider this division of a fraction by a number.
Example 3
Divide the fraction 16 45 by the number 12.
Solution
Let's apply the rule for dividing a fraction by a number. We obtain an expression of the form 16 45: 12 = 16 45 · 12.
Let's reduce the fraction. We get 16 45 12 = 2 2 2 2 (3 3 5) (2 2 3) = 2 2 3 3 3 5 = 4 135.
Answer: 16 45: 12 = 4 135 .
Dividing a natural number by a fraction
The division rule is similar O the rule for dividing a natural number by an ordinary fraction: in order to divide a natural number n by an ordinary fraction a b, it is necessary to multiply the number n by the reciprocal of the fraction a b.
Based on the rule, we have n: a b = n · b a, and thanks to the rule of multiplying a natural number by an ordinary fraction, we get our expression in the form n: a b = n · b a. It is necessary to consider this division with an example.
Example 4
Divide 25 by 15 28.
Solution
We need to move from division to multiplication. Let's write it in the form of the expression 25: 15 28 = 25 28 15 = 25 28 15. Let's reduce the fraction and get the result in the form of the fraction 46 2 3.
Answer: 25: 15 28 = 46 2 3 .
Dividing a fraction by a mixed number
When dividing a common fraction by a mixed number, you can easily begin to divide common fractions. You need to convert a mixed number to an improper fraction.
Example 5
Divide the fraction 35 16 by 3 1 8.
Solution
Since 3 1 8 is a mixed number, let's represent it as an improper fraction. Then we get 3 1 8 = 3 8 + 1 8 = 25 8. Now let's divide fractions. We get 35 16: 3 1 8 = 35 16: 25 8 = 35 16 8 25 = 35 8 16 25 = 5 7 2 2 2 2 2 2 2 (5 5) = 7 10
Answer: 35 16: 3 1 8 = 7 10 .
Dividing a mixed number is done in the same way as ordinary numbers.
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Last time we learned how to add and subtract fractions (see lesson “Adding and subtracting fractions”). The most difficult part of 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 simpler than addition and subtraction. First, let's consider the simplest case, when there are two positive fractions without a separated integer part.
To multiply two fractions, you must 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 fraction.
Designation:
From the definition it follows that dividing fractions reduces to multiplication. To “flip” a fraction, just swap the numerator and denominator. Therefore, throughout the lesson we will mainly consider multiplication.
As a result of multiplication, a reducible fraction can arise (and often does arise) - it, of course, must be reduced. If after all the reductions the fraction turns out to be incorrect, the whole part should be highlighted. But what definitely won't happen with multiplication is reduction to a common denominator: no criss-cross methods, greatest factors and least common multiples.
By definition we have:
Multiplying fractions with whole parts and negative fractions
If fractions contain an integer part, 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 multiplication or removed altogether according to the following rules:
- Plus by 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 necessary to get rid of the whole part. For a work, they can be generalized in order to “burn” several disadvantages at once:
- We cross out the negatives in pairs until they completely disappear. In extreme cases, one minus can survive - the one for which there was no mate;
- If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, because there was no pair for it, we take it out of the limits of multiplication. The result is a negative fraction.
Task. Find the meaning of the expression:
We convert all fractions to improper ones, and then take the minuses out of the multiplication. We multiply what is left according to the usual rules. We get:
Let me remind you once again that the minus that appears in front of a fraction with a highlighted whole part refers specifically to the entire fraction, and not just to its whole part (this applies to the last two examples).
Also pay attention to negative numbers: when multiplying, they are enclosed in parentheses. This is done in order to separate the minuses from the multiplication signs and make the entire notation more accurate.
Reducing fractions on the fly
Multiplication is a very labor-intensive operation. The numbers here turn out to be quite large, and to simplify the problem, you can try to reduce the fraction further 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 meaning of the expression:
By definition we have:
In all examples, the numbers that have been reduced and what remains of them are marked in red.
Please note: in the first case, the multipliers were reduced completely. In their place there remain units that, generally speaking, need not be written. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.
However, never 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 because when adding, the numerator of a fraction produces a sum, not a product of numbers. Consequently, it is impossible to apply the basic property of a fraction, since this property deals specifically with the multiplication of numbers.
There are simply no other reasons for reducing 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.
Ordinary fractional numbers first meet schoolchildren in the 5th grade and accompany them throughout their lives, since in everyday life it is often necessary to consider or use an object not as a whole, but in separate pieces. Start studying this topic - shares. Shares are equal parts, into which this or that object is divided. After all, it is not always possible to express, for example, the length or price of a product as a whole number; parts or shares of some measure should be taken into account. Formed from the verb “to split” - to divide into parts, and having Arabic roots, the word “fraction” itself arose in the Russian language in the 8th century.
Fractional expressions have long been considered the most difficult branch of mathematics. In the 17th century, when first textbooks on mathematics appeared, they were called “broken numbers,” which was very difficult for people to understand.
Modern look simple fractional remainders, the parts of which are separated by a horizontal line, were first promoted by Fibonacci - Leonardo of Pisa. His works are dated to 1202. But the purpose of this article is to simply and clearly explain to the reader how mixed fractions are multiplied with different denominators.
Multiplying fractions with different denominators
Initially it is worth determining types of fractions:
- correct;
- incorrect;
- mixed.
Next, you need to remember how fractional numbers with the same denominators are multiplied. The very rule of this process is not difficult to formulate independently: the result of multiplying simple fractions with identical denominators is a fractional expression, the numerator of which is the product of the numerators, and the denominator is the product of the denominators of these fractions. That is, in fact, the new denominator is the square of one of the initially existing ones.
When multiplying simple fractions with different denominators for two or more factors the rule does not change:
a/b * c/d = a*c / b*d.
The only difference is that the resulting number under the fractional line will be the product of different numbers and, naturally, the square of one numerical expression it is impossible to name it.
It is worth considering the multiplication of fractions with different denominators using examples:
- 8/ 9 * 6/ 7 = 8*6 / 9*7 = 48/ 63 = 16/2 1 ;
- 4/ 6 * 3/ 7 = 2/ 3 * 3/7 <> 2*3 / 3*7 = 6/ 21 .
The examples use methods for reducing fractional expressions. You can only reduce numerator numbers with denominator numbers; adjacent factors above or below the fraction line cannot be reduced.
Along with simple fractions, there is the concept of mixed fractions. A mixed number consists of an integer and a fractional part, that is, it is the sum of these numbers:
1 4/ 11 =1 + 4/ 11.
How does multiplication work?
Several examples are provided for consideration.
2 1/ 2 * 7 3/ 5 = 2 + 1/ 2 * 7 + 3/ 5 = 2*7 + 2* 3/ 5 + 1/ 2 * 7 + 1/ 2 * 3/ 5 = 14 + 6/5 + 7/ 2 + 3/ 10 = 14 + 12/ 10 + 35/ 10 + 3/ 10 = 14 + 50/ 10 = 14 + 5=19.
The example uses multiplication of a number by ordinary fractional part, the rule for this action can be written as:
a* b/c = a*b /c.
In fact, such a product is the sum of identical fractional remainders, and the number of terms indicates this natural number. Special case:
4 * 12/ 15 = 12/ 15 + 12/ 15 + 12/ 15 + 12/ 15 = 48/ 15 = 3 1/ 5.
There is another solution to multiplying a number by a fractional remainder. You just need to divide the denominator by this number:
d* e/f = e/f: d.
This technique is useful to use when the denominator is divided by a natural number without a remainder or, as they say, by a whole number.
Convert mixed numbers to improper fractions and obtain the product in the previously described way:
1 2/ 3 * 4 1/ 5 = 5/ 3 * 21/ 5 = 5*21 / 3*5 =7.
This example involves a way of representing a mixed fraction as an improper fraction, it can also be represented as general formula:
a bc = a*b+ c / c, where the denominator of the new fraction is formed by multiplying the whole part with the denominator and adding it with the numerator of the original fractional remainder, and the denominator remains the same.
This process also works in reverse side. To separate the whole part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator using a “corner”.
Multiplication improper fractions produced in a generally accepted way. When writing under a single fraction line, you need to reduce fractions as necessary in order to reduce numbers using this method and make it easier to calculate the result.
There are many helpers on the Internet to solve even complex mathematical problems in various variations of programs. A sufficient number of such services offer their assistance in counting multiplication of fractions with different numbers in denominators - so-called online calculators for calculating fractions. They are able not only to multiply, but also to perform all other simple arithmetic operations with ordinary fractions and mixed numbers. It’s not difficult to work with; you fill in the appropriate fields on the website page, select the sign of the mathematical operation, and click “calculate.” The program calculates automatically.
The topic of arithmetic operations with fractions is relevant throughout the education of middle and high school students. In high school, they no longer consider the simplest species, but integer fractional expressions, but the knowledge of the rules for transformation and calculations obtained earlier is applied in its original form. Well-mastered basic knowledge gives complete confidence in a successful solution to the most complex tasks.
In conclusion, it makes sense to quote the words of Lev Nikolaevich Tolstoy, who wrote: “Man is a fraction. It is not in the power of a person to increase his numerator - his merits - but anyone can reduce his denominator - his opinion about himself, and with this decrease come closer to his perfection.
A fraction is one or more parts of a whole, usually taken to be one (1). As with natural numbers, you can perform all basic arithmetic operations (addition, subtraction, division, multiplication) with fractions; to do this, you need to know the features of working with fractions and distinguish between their types. There are several types of fractions: decimal and ordinary, or simple. Each type of fraction has its own specifics, but once you thoroughly understand how to handle them, you will be able to solve any examples with fractions, since you will know the basic principles of performing arithmetic calculations with fractions. Let's look at examples of how to divide a fraction by a whole number using different types fractions
How to divide a simple fraction by a natural number?Ordinary or simple fractions are fractions that are written in the form of a ratio of numbers in which the dividend (numerator) is indicated at the top of the fraction, and the divisor (denominator) of the fraction is indicated at the bottom. How to divide such a fraction by a whole number? Let's look at an example! Let's say we need to divide 8/12 by 2.
To do this we must perform a number of actions:
Thus, if we are faced with the task of dividing a fraction by a whole number, the solution diagram will look something like this: 
In a similar way, you can divide any ordinary (simple) fraction by an integer.
How to divide a decimal by a whole number?
A decimal is a fraction that is obtained by dividing a unit into ten, a thousand, and so on parts. Arithmetic operations with decimal fractions are quite simple.
Let's look at an example of how to divide a fraction by a whole number. Let's say we need to divide the decimal fraction 0.925 by the natural number 5.
To summarize, let us dwell on two main points that are important when performing the operation of dividing decimal fractions by an integer: - for separation decimal Column division is used for a natural number;
- A comma is placed in a quotient when the division of the whole part of the dividend is completed.
