This article deals with operations on fractions. Rules for addition, subtraction, multiplication, division or exponentiation of fractions of the form A B will be formed and justified, where A and B can be numbers, numeric expressions or expressions with variables. In conclusion, examples of solutions with a detailed description will be considered.

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Rules for performing operations with numerical fractions of a general form

Numerical fractions of a general form have a numerator and a denominator, in which there are natural numbers or numerical expressions. If we consider such fractions as 3 5 , 2 , 8 4 , 1 + 2 3 4 (5 - 2) , 3 4 + 7 8 2 , 3 - 0 , 8 , 1 2 2 , π 1 - 2 3 + π , 2 0 , 5 ln 3 , then it is clear that the numerator and denominator can have not only numbers, but also expressions of a different plan.

Definition 1

There are rules by which actions are performed with ordinary fractions. It is also suitable for fractions of a general form:

• When subtracting fractions with the same denominators, only the numerators are added, and the denominator remains the same, namely: a d ± c d \u003d a ± c d, the values ​​a, c and d ≠ 0 are some numbers or numerical expressions.
• When adding or subtracting fractions with different denominators, it is necessary to reduce to a common one, and then add or subtract the resulting fractions with the same indicators. Literally, it looks like this a b ± c d = a p ± c r s , where the values ​​a , b ≠ 0 , c , d ≠ 0 , p ≠ 0 , r ≠ 0 , s ≠ 0 are real numbers, and b p = d r = s. When p = d and r = b, then a b ± c d = a d ± c d b d.
• When multiplying fractions, an action is performed with numerators, after which with denominators, then we get a b c d \u003d a c b d, where a, b ≠ 0, c, d ≠ 0 act as real numbers.
• When dividing a fraction by a fraction, we multiply the first by the second reciprocal, that is, we swap the numerator and denominator: a b: c d \u003d a b d c.

Rationale for the rules

Definition 2

There are the following mathematical points that you should rely on when calculating:

• a fractional bar means a division sign;
• division by a number is treated as a multiplication by its reciprocal;
• application of the property of actions with real numbers;
• application of the basic property of a fraction and numerical inequalities.

With their help, you can make transformations of the form:

a d ± c d = a d - 1 ± c d - 1 = a ± c d - 1 = a ± c d ; a b ± c d = a p b p ± c r d r = a p s ± c e s = a p ± c r s ; a b c d = a d b d b c b d = a d a d - 1 b c b d - 1 = = a d b c b d - 1 b d - 1 = a d b c b d b d - 1 = = (a c) (b d) - 1 = a c b d

Examples

In the previous paragraph, it was said about actions with fractions. It is after this that the fraction needs to be simplified. This topic was discussed in detail in the section on converting fractions.

First, consider the example of adding and subtracting fractions with the same denominator.

Example 1

Given fractions 8 2 , 7 and 1 2 , 7 , then according to the rule it is necessary to add the numerator and rewrite the denominator.

Decision

Then we get a fraction of the form 8 + 1 2 , 7 . After performing the addition, we get a fraction of the form 8 + 1 2 , 7 = 9 2 , 7 = 90 27 = 3 1 3 . So 8 2 , 7 + 1 2 , 7 = 8 + 1 2 , 7 = 9 2 , 7 = 90 27 = 3 1 3 .

Answer: 8 2 , 7 + 1 2 , 7 = 3 1 3

There is another way to solve. To begin with, a transition is made to the form of an ordinary fraction, after which we perform a simplification. It looks like this:

8 2 , 7 + 1 2 , 7 = 80 27 + 10 27 = 90 27 = 3 1 3

Example 2

Let us subtract from 1 - 2 3 log 2 3 log 2 5 + 1 fractions of the form 2 3 3 log 2 3 log 2 5 + 1 .

Since equal denominators are given, it means that we are calculating a fraction with the same denominator. We get that

1 - 2 3 log 2 3 log 2 5 + 1 - 2 3 3 log 2 3 log 2 5 + 1 = 1 - 2 - 2 3 3 log 2 3 log 2 5 + 1

There are examples of calculating fractions with different denominators. An important point is the reduction to a common denominator. Without this, we will not be able to perform further actions with fractions.

The process is remotely reminiscent of reduction to a common denominator. That is, a search is made for the least common divisor in the denominator, after which the missing factors are added to the fractions.

If the added fractions have no common factors, then their product can become one.

Example 3

Consider the example of adding fractions 2 3 5 + 1 and 1 2 .

Decision

In this case, the common denominator is the product of the denominators. Then we get that 2 · 3 5 + 1 . Then, when setting additional factors, we have that to the first fraction it is equal to 2, and to the second 3 5 + 1. After multiplication, the fractions are reduced to the form 4 2 3 5 + 1. The general cast 1 2 will be 3 5 + 1 2 · 3 5 + 1 . We add the resulting fractional expressions and get that

2 3 5 + 1 + 1 2 = 2 2 2 3 5 + 1 + 1 3 5 + 1 2 3 5 + 1 = = 4 2 3 5 + 1 + 3 5 + 1 2 3 5 + 1 = 4 + 3 5 + 1 2 3 5 + 1 = 5 + 3 5 2 3 5 + 1

Answer: 2 3 5 + 1 + 1 2 = 5 + 3 5 2 3 5 + 1

When we are dealing with fractions of a general form, then the least common denominator is usually not the case. It is unprofitable to take the product of numerators as a denominator. First you need to check if there is a number that is less in value than their product.

Example 4

Consider the example 1 6 2 1 5 and 1 4 2 3 5 when their product is equal to 6 2 1 5 4 2 3 5 = 24 2 4 5 . Then we take 12 · 2 3 5 as a common denominator.

Consider examples of multiplications of fractions of a general form.

Example 5

To do this, it is necessary to multiply 2 + 1 6 and 2 · 5 3 · 2 + 1.

Decision

Following the rule, it is necessary to rewrite and write the product of numerators as a denominator. We get that 2 + 1 6 2 5 3 2 + 1 2 + 1 2 5 6 3 2 + 1 . When the fraction is multiplied, reductions can be made to simplify it. Then 5 3 3 2 + 1: 10 9 3 = 5 3 3 2 + 1 9 3 10 .

Using the rule of transition from division to multiplication by a reciprocal, we get the reciprocal of the given one. To do this, the numerator and denominator are reversed. Let's look at an example:

5 3 3 2 + 1: 10 9 3 = 5 3 3 2 + 1 9 3 10

After that, they must perform multiplication and simplify the resulting fraction. If necessary, get rid of the irrationality in the denominator. We get that

5 3 3 2 + 1: 10 9 3 = 5 3 3 9 3 10 2 + 1 = 5 2 10 2 + 1 = 3 2 2 + 1 = = 3 2 - 1 2 2 + 1 2 - 1 = 3 2 - 1 2 2 2 - 1 2 = 3 2 - 1 2

Answer: 5 3 3 2 + 1: 10 9 3 = 3 2 - 1 2

This paragraph is applicable when a number or numerical expression can be represented as a fraction with a denominator equal to 1, then the operation with such a fraction is considered a separate paragraph. For example, the expression 1 6 7 4 - 1 3 shows that the root of 3 can be replaced by another 3 1 expression. Then this record will look like a multiplication of two fractions of the form 1 6 7 4 - 1 3 = 1 6 7 4 - 1 3 1 .

Performing an action with fractions containing variables

The rules discussed in the first article are applicable to operations with fractions containing variables. Consider the subtraction rule when the denominators are the same.

It is necessary to prove that A , C and D (D not equal to zero) can be any expressions, and the equality A D ± C D = A ± C D is equivalent to its range of valid values.

It is necessary to take a set of ODZ variables. Then A, C, D must take the corresponding values ​​a 0 , c 0 and d0. A substitution of the form A D ± C D results in a difference of the form a 0 d 0 ± c 0 d 0 , where, according to the addition rule, we obtain a formula of the form a 0 ± c 0 d 0 . If we substitute the expression A ± C D , then we get the same fraction of the form a 0 ± c 0 d 0 . From this we conclude that the chosen value that satisfies the ODZ, A ± C D and A D ± C D are considered equal.

For any value of the variables, these expressions will be equal, that is, they are called identically equal. This means that this expression is considered to be a provable equality of the form A D ± C D = A ± C D .

Examples of addition and subtraction of fractions with variables

When there are the same denominators, it is only necessary to add or subtract the numerators. This fraction can be simplified. Sometimes you have to work with fractions that are identically equal, but at first glance this is not noticeable, since some transformations must be performed. For example, x 2 3 x 1 3 + 1 and x 1 3 + 1 2 or 1 2 sin 2 α and sin a cos a. Most often, a simplification of the original expression is required in order to see the same denominators.

Example 6

Calculate: 1) x 2 + 1 x + x - 2 - 5 - x x + x - 2 , 2) l g 2 x + 4 x (l g x + 2) + 4 l g x x (l g x + 2) , x - 1 x - 1 + x x + 1 .

Decision

1. To make a calculation, you need to subtract fractions that have the same denominators. Then we get that x 2 + 1 x + x - 2 - 5 - x x + x - 2 = x 2 + 1 - 5 - x x + x - 2 . After that, you can open the brackets with the reduction of similar terms. We get that x 2 + 1 - 5 - x x + x - 2 = x 2 + 1 - 5 + x x + x - 2 = x 2 + x - 4 x + x - 2
2. Since the denominators are the same, it remains only to add the numerators, leaving the denominator: l g 2 x + 4 x (l g x + 2) + 4 l g x x (l g x + 2) = l g 2 x + 4 + 4 x (l g x + 2)
   The addition has been completed. It can be seen that the fraction can be reduced. Its numerator can be folded using the sum square formula, then we get (l g x + 2) 2 from the abbreviated multiplication formulas. Then we get that
   l g 2 x + 4 + 2 l g x x (l g x + 2) = (l g x + 2) 2 x (l g x + 2) = l g x + 2 x
3. Given fractions of the form x - 1 x - 1 + x x + 1 with different denominators. After the transformation, you can proceed to addition.

Let's consider a two way solution.

The first method is that the denominator of the first fraction is subjected to factorization using squares, and with its subsequent reduction. We get a fraction of the form

x - 1 x - 1 = x - 1 (x - 1) x + 1 = 1 x + 1

So x - 1 x - 1 + x x + 1 = 1 x + 1 + x x + 1 = 1 + x x + 1 .

In this case, it is necessary to get rid of irrationality in the denominator.

1 + x x + 1 = 1 + x x - 1 x + 1 x - 1 = x - 1 + x x - x x - 1

The second way is to multiply the numerator and denominator of the second fraction by x - 1 . Thus, we get rid of irrationality and proceed to adding a fraction with the same denominator. Then

x - 1 x - 1 + x x + 1 = x - 1 x - 1 + x x - 1 x + 1 x - 1 = = x - 1 x - 1 + x x - x x - 1 = x - 1 + x x - x x - 1

Answer: 1) x 2 + 1 x + x - 2 - 5 - x x + x - 2 = x 2 + x - 4 x + x - 2, 2) l g 2 x + 4 x (l g x + 2) + 4 l g x x (l g x + 2) = l g x + 2 x, 3) x - 1 x - 1 + x x + 1 = x - 1 + x x - x x - 1.

In the last example, we found that reduction to a common denominator is inevitable. To do this, you need to simplify the fractions. To add or subtract, you always need to look for a common denominator, which looks like the product of the denominators with the addition of additional factors to the numerators.

Example 7

Calculate the values ​​of fractions: 1) x 3 + 1 x 7 + 2 2, 2) x + 1 x ln 2 (x + 1) (2 x - 4) - sin x x 5 ln (x + 1) (2 x - 4) , 3) ​​1 cos 2 x - x + 1 cos 2 x + 2 cos x x + x

Decision

1. The denominator does not require any complicated calculations, so you need to choose their product of the form 3 x 7 + 2 2, then to the first fraction x 7 + 2 2 is chosen as an additional factor, and 3 to the second. When multiplying, we get a fraction of the form x 3 + 1 x 7 + 2 2 = x x 7 + 2 2 3 x 7 + 2 2 + 3 1 3 x 7 + 2 2 = = x x 7 + 2 2 + 3 3 x 7 + 2 2 = x x 7 + 2 2 x + 3 3 x 7 + 2 2
2. It can be seen that the denominators are presented as a product, which means that additional transformations are unnecessary. The common denominator will be the product of the form x 5 · ln 2 x + 1 · 2 x - 4 . From here x 4 is an additional factor to the first fraction, and ln (x + 1) to the second. Then we subtract and get:
   x + 1 x ln 2 (x + 1) 2 x - 4 - sin x x 5 ln (x + 1) 2 x - 4 = = x + 1 x 4 x 5 ln 2 (x + 1 ) 2 x - 4 - sin x ln x + 1 x 5 ln 2 (x + 1) (2 x - 4) = = x + 1 x 4 - sin x ln (x + 1) x 5 ln 2 (x + 1) (2 x - 4) = x x 4 + x 4 - sin x ln (x + 1) x 5 ln 2 (x + 1) (2 x - 4) )
3. This example makes sense when working with denominators of fractions. It is necessary to apply the formulas of the difference of squares and the square of the sum, since they will make it possible to pass to an expression of the form 1 cos x - x · cos x + x + 1 (cos x + x) 2 . It can be seen that the fractions are reduced to a common denominator. We get that cos x - x cos x + x 2 .

Then we get that

1 cos 2 x - x + 1 cos 2 x + 2 cos x x + x = = 1 cos x - x cos x + x + 1 cos x + x 2 = = cos x + x cos x - x cos x + x 2 + cos x - x cos x - x cos x + x 2 = = cos x + x + cos x - x cos x - x cos x + x 2 = 2 cos x cos x - x cos x + x2

Answer:

1) x 3 + 1 x 7 + 2 2 = x x 7 + 2 2 x + 3 3 x 7 + 2 2, 2) x + 1 x ln 2 (x + 1) 2 x - 4 - sin x x 5 ln (x + 1) 2 x - 4 = = x x 4 + x 4 - sin x ln (x + 1) x 5 ln 2 (x + 1) ( 2 x - 4) , 3) ​​1 cos 2 x - x + 1 cos 2 x + 2 cos x x + x = 2 cos x cos x - x cos x + x 2 .

Examples of multiplying fractions with variables

When multiplying fractions, the numerator is multiplied by the numerator and the denominator by the denominator. Then you can apply the reduction property.

Example 8

Multiply fractions x + 2 x x 2 ln x 2 ln x + 1 and 3 x 2 1 3 x + 1 - 2 sin 2 x - x.

Decision

You need to do the multiplication. We get that

x + 2 x x 2 ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x) = = x - 2 x 3 x 2 1 3 x + 1 - 2 x 2 ln x 2 ln x + 1 sin (2 x - x)

The number 3 is transferred to the first place for the convenience of calculations, and you can reduce the fraction by x 2, then we get an expression of the form

3 x - 2 x x 1 3 x + 1 - 2 ln x 2 ln x + 1 sin (2 x - x)

Answer: x + 2 x x 2 ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x) = 3 x - 2 x x 1 3 x + 1 - 2 ln x 2 ln x + 1 sin (2 x - x) .

Division

Division of fractions is similar to multiplication, since the first fraction is multiplied by the second reciprocal. If we take, for example, the fraction x + 2 x x 2 ln x 2 ln x + 1 and divide by 3 x 2 1 3 x + 1 - 2 sin 2 x - x, then this can be written as

x + 2 x x 2 ln x 2 ln x + 1: 3 x 2 1 3 x + 1 - 2 sin (2 x - x) , then replace with a product of the form x + 2 x x 2 ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x)

Exponentiation

Let's move on to consider the action with fractions of a general form with exponentiation. If there is a degree with a natural exponent, then the action is considered as a multiplication of identical fractions. But it is recommended to use a general approach based on the properties of powers. Any expressions A and C, where C is not identically equal to zero, and any real r on the ODZ for an expression of the form A C r, the equality A C r = A r C r is true. The result is a fraction raised to a power. For example, consider:

x 0 , 7 - π ln 3 x - 2 - 5 x + 1 2 , 5 = = x 0 , 7 - π ln 3 x - 2 - 5 2 , 5 x + 1 2 , 5

The order of operations with fractions

Actions on fractions are performed according to certain rules. In practice, we notice that an expression can contain several fractions or fractional expressions. Then it is necessary to perform all actions in a strict order: raise to a power, multiply, divide, then add and subtract. If there are brackets, the first action is performed in them.

Example 9

Calculate 1 - x cos x - 1 c o s x · 1 + 1 x .

Decision

Since we have the same denominator, then 1 - x cos x and 1 c o s x , but it is impossible to subtract according to the rule, first the actions in brackets are performed, after which the multiplication, and then the addition. Then, when calculating, we get that

1 + 1 x = 1 1 + 1 x = x x + 1 x = x + 1 x

When substituting the expression into the original one, we get that 1 - x cos x - 1 cos x · x + 1 x. When multiplying fractions, we have: 1 cos x x + 1 x = x + 1 cos x x . Having made all the substitutions, we get 1 - x cos x - x + 1 cos x · x . Now you need to work with fractions that have different denominators. We get:

x 1 - x cos x x - x + 1 cos x x = x 1 - x - 1 + x cos x x = = x - x - x - 1 cos x x = - x + 1 cos x x

Answer: 1 - x cos x - 1 c o s x 1 + 1 x = - x + 1 cos x x .

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Fraction- a form of representation of a number in mathematics. The slash indicates the division operation. numerator fractions is called the dividend, and denominator- divider. For example, in a fraction, the numerator is 5 and the denominator is 7.

correct A fraction is called if the modulus of the numerator is greater than the modulus of the denominator. If the fraction is correct, then the modulus of its value is always less than 1. All other fractions are wrong.

Fraction is called mixed, if it is written as an integer and a fraction. This is the same as the sum of this number and a fraction:

Basic property of a fraction

If the numerator and denominator of a fraction are multiplied by the same number, then the value of the fraction will not change, that is, for example,

Bringing fractions to a common denominator

To bring two fractions to a common denominator, you need:

1. Multiply the numerator of the first fraction by the denominator of the second
2. Multiply the numerator of the second fraction by the denominator of the first
3. Replace the denominators of both fractions with their product

Actions with fractions

Addition. To add two fractions, you need

1. Add new numerators of both fractions, and leave the denominator unchanged

Example:

Subtraction. To subtract one fraction from another,

1. Bring fractions to a common denominator
2. Subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged

Example:

Multiplication. To multiply one fraction by another, multiply their numerators and denominators:

Division. To divide one fraction by another, multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second:

Now that we have learned how to add and multiply individual fractions, we can consider more complex structures. For example, what if addition, subtraction, and multiplication of fractions occur in one problem?

First of all, you need to convert all fractions to improper ones. Then we sequentially perform the required actions - in the same order as for ordinary numbers. Namely:

1. First, exponentiation is performed - get rid of all expressions containing exponents;
2. Then - division and multiplication;
3. The last step is addition and subtraction.

Of course, if there are brackets in the expression, the order of actions changes - everything that is inside the brackets must be considered first. And remember about improper fractions: you need to select the whole part only when all other actions have already been completed.

Let's translate all the fractions from the first expression into improper ones, and then perform the following actions:

Now let's find the value of the second expression. There are no fractions with an integer part, but there are brackets, so we first perform addition, and only then division. Note that 14 = 7 2 . Then:

Finally, consider the third example. There are brackets and a degree here - it is better to count them separately. Given that 9 = 3 3 , we have:

Pay attention to the last example. To raise a fraction to a power, you must separately raise the numerator to this power, and separately the denominator.

You can decide differently. If we recall the definition of the degree, the problem will be reduced to the usual multiplication of fractions:

Multistoried fractions

So far, we have considered only "pure" fractions, when the numerator and denominator are ordinary numbers. This is consistent with the definition of a numerical fraction given in the very first lesson.

But what if a more complex object is placed in the numerator or denominator? For example, another numerical fraction? Such constructions occur quite often, especially when working with long expressions. Here are a couple of examples:

There is only one rule for working with multi-storey fractions: you must immediately get rid of them. Removing "extra" floors is quite simple, if you remember that the fractional bar means the standard division operation. Therefore, any fraction can be rewritten as follows:

Using this fact and following the procedure, we can easily reduce any multi-storey fraction to a regular one. Take a look at the examples:

Task. Convert multistory fractions to common ones:

In each case, we rewrite the main fraction, replacing the dividing line with a division sign. Also remember that any integer can be represented as a fraction with a denominator of 1. That is, 12 = 12/1; 3 = 3/1. We get:

In the last example, the fractions were reduced before the final multiplication.

The specifics of working with multi-storey fractions

There is one subtlety in multi-storey fractions that must always be remembered, otherwise you can get the wrong answer, even if all the calculations were correct. Take a look:

1. In the numerator there is a separate number 7, and in the denominator - the fraction 12/5;
2. The numerator is the fraction 7/12, and the denominator is the single number 5.

So, for one record, we got two completely different interpretations. If you count, the answers will also be different:

To ensure that the record is always read unambiguously, use a simple rule: the dividing line of the main fraction must be longer than the nested line. Preferably several times.

If you follow this rule, then the above fractions should be written as follows:

Yes, it's probably ugly and takes up too much space. But you will count correctly. Finally, a couple of examples where multi-level fractions really occur:

Task. Find expression values:

So, let's work with the first example. Let's convert all the fractions to improper ones, and then perform the operations of addition and division:

Let's do the same with the second example. Convert all fractions to improper and perform the required operations. In order not to bore the reader, I will omit some obvious calculations. We have:

Due to the fact that the numerator and denominator of the main fractions contain sums, the rule for writing multi-storey fractions is observed automatically. Also, in the last example, we deliberately left the number 46/1 in the form of a fraction in order to perform the division.

I also note that in both examples, the fractional bar actually replaces the brackets: first of all, we found the sum, and only then - the quotient.

Someone will say that the transition to improper fractions in the second example was clearly redundant. Perhaps that is the way it is. But this way we insure ourselves against mistakes, because the next time the example may turn out to be much more complicated. Choose for yourself what is more important: speed or reliability.

This section deals with operations with ordinary fractions. If it is necessary to perform a mathematical operation with mixed numbers, then it is enough to convert the mixed fraction into an extraordinary one, perform the necessary operations and, if necessary, present the final result as a mixed number again. This operation will be described below.

Fraction reduction

mathematical operation. Fraction reduction

To reduce the fraction \frac(m)(n) you need to find the greatest common divisor of its numerator and denominator: gcd(m,n), then divide the numerator and denominator of the fraction by this number. If gcd(m,n)=1, then the fraction cannot be reduced. Example: \frac(20)(80)=\frac(20:20)(80:20)=\frac(1)(4)

Usually, immediately finding the greatest common divisor is a difficult task, and in practice the fraction is reduced in several stages, step by step highlighting obvious common factors from the numerator and denominator. \frac(140)(315)=\frac(28\cdot5)(63\cdot5)=\frac(4\cdot7\cdot5)(9\cdot7\cdot5)=\frac(4)(9)

Bringing fractions to a common denominator

mathematical operation. Bringing fractions to a common denominator

To reduce two fractions \frac(a)(b) and \frac(c)(d) to a common denominator, you need:

• find the least common multiple of the denominators: M=LCM(b,d);
• multiply the numerator and denominator of the first fraction by M / b (after which the denominator of the fraction becomes equal to the number M);
• multiply the numerator and denominator of the second fraction by M/d (after which the denominator of the fraction becomes equal to the number M).

Thus, we convert the original fractions to fractions with the same denominators (which will be equal to the number M).

For example, the fractions \frac(5)(6) and \frac(4)(9) have LCM(6,9) = 18. Then: \frac(5)(6)=\frac(5\cdot3)(6 \cdot3)=\frac(15)(18);\quad\frac(4)(9)=\frac(4\cdot2)(9\cdot2)=\frac(8)(18) . Thus, the resulting fractions have a common denominator.

In practice, finding the least common multiple (LCM) of denominators is not always an easy task. Therefore, a number equal to the product of the denominators of the original fractions is chosen as a common denominator. For example, the fractions \frac(5)(6) and \frac(4)(9) are reduced to a common denominator N=6\cdot9:

\frac(5)(6)=\frac(5\cdot9)(6\cdot9)=\frac(45)(54);\quad\frac(4)(9)=\frac(4\cdot6)( 9\cdot6)=\frac(24)(54)

Fraction Comparison

mathematical operation. Fraction Comparison

To compare two common fractions:

• compare the numerators of the resulting fractions; a fraction with a larger numerator will be larger.

For example, \frac(9)(14)

When comparing fractions, there are several special cases:

1. From two fractions with the same denominators the greater is the fraction whose numerator is greater. For example \frac(3)(15)
2. From two fractions with the same numerators the larger is the fraction whose denominator is smaller. For example, \frac(4)(11)>\frac(4)(13)
3. That fraction, which at the same time larger numerator and smaller denominator, more. For example, \frac(11)(3)>\frac(10)(8)

Attention! Rule 1 applies to any fractions if their common denominator is a positive number. Rules 2 and 3 apply to positive fractions (which have both numerator and denominator greater than zero).

Addition and subtraction of fractions

mathematical operation. Addition and subtraction of fractions

To add two fractions, you need:

• bring them to a common denominator;
• add their numerators and leave the denominator unchanged.

Example: \frac(7)(9)+\frac(4)(7)=\frac(7\cdot7)(9\cdot7)+\frac(4\cdot9)(7\cdot9)=\frac(49 )(63)+\frac(36)(63)=\frac(49+36)(63)=\frac(85)(63)

To subtract another fraction from one, you need:

• bring fractions to a common denominator;
• subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged.

Example: \frac(4)(15)-\frac(3)(5)=\frac(4)(15)-\frac(3\cdot3)(5\cdot3)=\frac(4)(15) -\frac(9)(15)=\frac(4-9)(15)=\frac(-5)(15)=-\frac(5)(3\cdot5)=-\frac(1)( 3)

If the original fractions initially have a common denominator, then point 1 (reduction to a common denominator) is skipped.

Converting a mixed number to an improper fraction and vice versa

mathematical operation. Converting a mixed number to an improper fraction and vice versa

To convert a mixed fraction to an improper one, it is enough to sum the whole part of the mixed fraction with the fractional part. The result of such a sum will be an improper fraction, the numerator of which is equal to the sum of the product of the integer part and the denominator of the fraction with the numerator of the mixed fraction, and the denominator remains the same. For example, 2\frac(6)(11)=2+\frac(6)(11)=\frac(2\cdot11)(11)+\frac(6)(11)=\frac(2\cdot11+ 6)(11)=\frac(28)(11)

To convert an improper fraction to a mixed number:

• divide the numerator of a fraction by its denominator;
• write the remainder of the division into the numerator, and leave the denominator the same;
• write the result of the division as an integer part.

For example, the fraction \frac(23)(4) . When dividing 23:4=5.75, that is, the integer part is 5, the remainder of the division is 23-5*4=3. Then the mixed number will be written: 5\frac(3)(4) . \frac(23)(4)=\frac(5\cdot4+3)(4)=5\frac(3)(4)

Converting a Decimal to a Common Fraction

mathematical operation. Converting a Decimal to a Common Fraction

To convert a decimal to a common fraction:

1. take the n-th power of ten as a denominator (here n is the number of decimal places);
2. as a numerator, take the number after the decimal point (if the integer part of the original number is not equal to zero, then take all leading zeros as well);
3. the non-zero integer part is written in the numerator at the very beginning; the zero integer part is omitted.

Example 1: 0.0089=\frac(89)(10000) (4 decimal places, so the denominator 10 4 =10000, since the integer part is 0, the numerator is the number after the decimal point without leading zeros)

Example 2: 31.0109=\frac(310109)(10000) (in the numerator we write the number after the decimal point with all zeros: "0109", and then we add the integer part of the original number "31" before it)

If the integer part of a decimal fraction is different from zero, then it can be converted to a mixed fraction. To do this, we translate the number into an ordinary fraction as if the integer part were equal to zero (points 1 and 2), and simply rewrite the integer part before the fraction - this will be the integer part of the mixed number. Example:

3.014=3\frac(14)(100)

To convert an ordinary fraction to a decimal, it is enough to simply divide the numerator by the denominator. Sometimes you get an infinite decimal. In this case, it is necessary to round to the desired decimal place. Examples:

\frac(401)(5)=80.2;\quad \frac(2)(3)\approx0.6667

Multiplication and division of fractions

mathematical operation. Multiplication and division of fractions

To multiply two common fractions, you need to multiply the numerators and denominators of the fractions.

\frac(5)(9)\cdot\frac(7)(2)=\frac(5\cdot7)(9\cdot2)=\frac(35)(18)

To divide one common fraction by another, you need to multiply the first fraction by the reciprocal of the second ( reciprocal is a fraction in which the numerator and denominator are reversed.

\frac(5)(9):\frac(7)(2)=\frac(5)(9)\cdot\frac(2)(7)=\frac(5\cdot2)(9\cdot7)= \frac(10)(63)

If one of the fractions is a natural number, then the above multiplication and division rules remain in force. Just keep in mind that an integer is the same fraction, the denominator of which is equal to one. For example: 3:\frac(3)(7)=\frac(3)(1):\frac(3)(7)=\frac(3)(1)\cdot\frac(7)(3)= \frac(3\cdot7)(1\cdot3)=\frac(7)(1)=7

Actions with fractions. In this article, we will analyze examples, everything is detailed with explanations. We will consider ordinary fractions. In the future, we will analyze decimals. I recommend to watch the whole and study sequentially.

1. Sum of fractions, difference of fractions.

Rule: when adding fractions with equal denominators, the result is a fraction - the denominator of which remains the same, and its numerator will be equal to the sum of the numerators of the fractions.

Rule: when calculating the difference of fractions with the same denominators, we get a fraction - the denominator remains the same, and the numerator of the second is subtracted from the numerator of the first fraction.

Formal notation of the sum and difference of fractions with equal denominators:

Examples (1):

It is clear that when ordinary fractions are given, then everything is simple, but if they are mixed? Nothing complicated...

Option 1- you can convert them into ordinary ones and then calculate them.

Option 2- you can separately "work" with the integer and fractional parts.

Examples (2):

More:

And if the difference of two mixed fractions is given and the numerator of the first fraction is less than the numerator of the second? It can also be done in two ways.

Examples (3):

* Converted to ordinary fractions, calculated the difference, converted the resulting improper fraction to a mixed one.

* Divided into integer and fractional parts, got three, then presented 3 as the sum of 2 and 1, with the unit presented as 11/11, then found the difference between 11/11 and 7/11 and calculated the result. The meaning of the above transformations is to take (select) a unit and present it as a fraction with the denominator we need, then from this fraction we can already subtract another.

Another example:

Conclusion: there is a universal approach - in order to calculate the sum (difference) of mixed fractions with equal denominators, they can always be converted into improper ones, then perform the necessary action. After that, if as a result we get an improper fraction, we translate it into a mixed one.

Above, we looked at examples with fractions that have equal denominators. What if the denominators differ? In this case, the fractions are reduced to the same denominator and the specified action is performed. To change (transform) a fraction, the main property of the fraction is used.

Consider simple examples:

In these examples, we immediately see how one of the fractions can be converted to get equal denominators.

If we designate ways to reduce fractions to one denominator, then this one will be called METHOD ONE.

That is, immediately when “evaluating” the fraction, you need to figure out whether such an approach will work - we check whether the larger denominator is divisible by the smaller one. And if it is divided, then we perform the transformation - we multiply the numerator and denominator so that the denominators of both fractions become equal.

Now look at these examples:

This approach does not apply to them. There are other ways to reduce fractions to a common denominator, consider them.

Method SECOND.

We multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first:

*In fact, we bring fractions to the form when the denominators become equal. Next, we use the rule of adding timid with equal denominators.

Example:

*This method can be called universal, and it always works. The only negative is that after the calculations, a fraction may turn out that will need to be further reduced.

Consider an example:

It can be seen that the numerator and denominator are divisible by 5:

Method THIRD.

Find the least common multiple (LCM) of the denominators. This will be the common denominator. What is this number? This is the smallest natural number that is divisible by each of the numbers.

Look, here are two numbers: 3 and 4, there are many numbers that are divisible by them - these are 12, 24, 36, ... The smallest of them is 12. Or 6 and 15, 30, 60, 90 are divisible by them .... Least 30. Question - how to determine this least common multiple?

There is a clear algorithm, but often this can be done immediately without calculations. For example, according to the above examples (3 and 4, 6 and 15), no algorithm is needed, we took large numbers (4 and 15), doubled them and saw that they are divisible by the second number, but pairs of numbers can be others, such as 51 and 119.

Algorithm. In order to determine the least common multiple of several numbers, you must:

- decompose each of the numbers into SIMPLE factors

- write out the decomposition of the BIGGER of them

- multiply it by the MISSING factors of other numbers

Consider examples:

50 and 60 50 = 2∙5∙5 60 = 2∙2∙3∙5

in the expansion of a larger number, one five is missing

=> LCM(50,60) = 2∙2∙3∙5∙5 = 300

48 and 72 48 = 2∙2∙2∙2∙3 72 = 2∙2∙2∙3∙3

in the expansion of a larger number, two and three are missing

=> LCM(48,72) = 2∙2∙2∙2∙3∙3 = 144

* The least common multiple of two prime numbers is equal to their product

Question! And why is it useful to find the least common multiple, because you can use the second method and simply reduce the resulting fraction? Yes, you can, but it's not always convenient. Look at the denominator for the numbers 48 and 72, if you just multiply them 48∙72 = 3456. Agree that it is more pleasant to work with smaller numbers.

Consider examples:

*51 = 3∙17 119 = 7∙17

in the expansion of a larger number, a triple is missing

=> LCM(51,119) = 3∙7∙17

And now we apply the first method:

* Look at the difference in the calculations, in the first case there is a minimum of them, and in the second you need to work separately on a piece of paper, and even the fraction that you got needs to be reduced. Finding the LCM simplifies the work considerably.

More examples:

* In the second example, it is already clear that the smallest number that is divisible by 40 and 60 is 120.

TOTAL! GENERAL CALCULATION ALGORITHM!

- we bring fractions to ordinary ones, if there is an integer part.

- we bring the fractions to a common denominator (first we look to see if one denominator is divisible by another, if it is divisible, then we multiply the numerator and denominator of this other fraction; if it is not divisible, we act using the other methods indicated above).

- having received fractions with equal denominators, we perform actions (addition, subtraction).

- if necessary, we reduce the result.

- if necessary, select the whole part.

2. Product of fractions.

The rule is simple. When multiplying fractions, their numerators and denominators are multiplied:

Examples: