All rules related to fractions. Rules for arithmetic operations on ordinary fractions

This article examines 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, numerical expressions or expressions with variables. In conclusion, examples of solutions with detailed descriptions will be considered.

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Rules for performing operations with general numerical fractions

General fractions have a numerator and a denominator that contain natural numbers or numerical expressions. If we consider fractions such 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 various types.

Definition 1

There are rules by which operations with ordinary fractions are carried out. It is also suitable for general fractions:

• When subtracting fractions with like denominators, only the numerators are added, and the denominator remains the same, namely: a d ± c d = a ± c d, the values ​​a, c and d ≠ 0 are some numbers or numerical expressions.
• When adding or subtracting a fraction with different denominators, it is necessary to reduce it to a common denominator, and then add or subtract the resulting fractions with the same exponents. 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, the action is performed with numerators, after which with denominators, then we get a b · c d = 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 inverse, that is, we swap the numerator and denominator: a b: c d = a b · d c.

Rationale for the rules

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

• the slash means the division sign;
• division by a number is treated as multiplication by its reciprocal value;
• application of the property of operations with real numbers;
• application of the basic property of fractions and numerical inequalities.

With their help, you can perform 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 operations with fractions. It is after this that the fraction needs to be simplified. This topic was discussed in detail in the paragraph on converting fractions.

First, let's look at an example of adding and subtracting fractions with the same denominator.

Example 1

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

Solution

Then we get a fraction of the form 8 + 1 2, 7. After performing the addition, we obtain 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 solution. To begin with, we switch 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's subtract from 1 - 2 3 · log 2 3 · log 2 5 + 1 a fraction 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 reduction to a common denominator. Without this, we will not be able to perform further operations with fractions.

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

If the fractions being added do not have common factors, then their product can become one.

Example 3

Let's look at the example of adding fractions 2 3 5 + 1 and 1 2.

Solution

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 for the first fraction it is equal to 2, and for the second it is 3 5 + 1. After multiplication, the fractions are reduced to the form 4 2 · 3 5 + 1. The general reduction of 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 general fractions, then we usually do not talk about the lowest common denominator. It is unprofitable to take the product of the numerators as the denominator. First you need to check if there is a number that is less in value than their product.

Example 4

Let's consider the example of 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 the common denominator.

Let's look at examples of multiplying general fractions.

Example 5

To do this, you need to multiply 2 + 1 6 and 2 · 5 3 · 2 + 1.

Solution

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

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

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

Then they must multiply and simplify the resulting fraction. If necessary, get rid of 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 entry will look like multiplying two fractions of the form 1 6 · 7 4 - 1 · 3 = 1 6 · 7 4 - 1 · 3 1.

Performing Operations on 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 permissible 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 d 0. Substitution of the form A D ± C D results in a difference of the form a 0 d 0 ± c 0 d 0 , where, using 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 here we conclude that the selected 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 a provable equality of the form A D ± C D = A ± C D .

Examples of adding and subtracting fractions with variables

When you have the same denominators, you only need 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 .

Solution

1. To make the calculation, you need to subtract fractions that have the same denominator. Then we get that x 2 + 1 x + x - 2 - 5 - x x + x - 2 = x 2 + 1 - 5 - x x + x - 2 . After which you can expand the brackets and add 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, all that remains is 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 it is possible to reduce the fraction. Its numerator can be folded using the formula for the square of the sum, then we get (l g x + 2) 2 from 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 move on to addition.

Let's consider a twofold solution.

The first method is that the denominator of the first fraction is factorized using squares, 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 method is to multiply the numerator and denominator of the second fraction by the expression x - 1. Thus, we get rid of irrationality and move on to adding fractions 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. When adding or subtracting, you always need to look for a common denominator, which looks like the product of the denominators with additional factors added to the numerators.

Example 7

Calculate the values ​​of the 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

Solution

1. The denominator does not require any complex calculations, so you need to choose their product of the form 3 x 7 + 2 · 2, then choose x 7 + 2 · 2 for the first fraction as an additional factor, and 3 for 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 in the form of a product, which means that additional transformations are unnecessary. The common denominator will be considered to be a product of the form x 5 · ln 2 x + 1 · 2 x - 4 . Hence 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 fraction denominators. It is necessary to apply the formulas for the difference of squares and the square of the sum, since they will make it possible to move on 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 + x 2

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 the 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.

Solution

Multiplication needs to be done. 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 moved 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 it 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 considering operations with general fractions with exponentiation. If there is a power with a natural exponent, then the action is considered as multiplication of equal fractions. But it is recommended to use a general approach based on the properties of degrees. 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 valid. 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

Procedure for performing operations with fractions

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

Example 9

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

Solution

Since we have the same denominator, then 1 - x cos x and 1 c o s x, but subtractions cannot be performed according to the rule; first, the actions in parentheses are performed, then multiplication, and then 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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Arithmetic operations with ordinary fractions

1. Addition.

To add fractions with the same denominators, you need to add their numerators and leave the denominator the same.

Example. .

To add fractions with different denominators, you need to reduce them to the lowest common denominator, and then add the resulting numerators and write the common denominator under the sum.

Example.

In short it is written like this:

To add mixed numbers, you need to separately find the sum of the integers and the sum of the fractions. The action is written like this:

2. Subtraction.

To subtract fractions with like denominators, you need to subtract the numerator of the subtrahend from the numerator of the minuend and leave the same denominator. The action is written like this:

To subtract fractions with different denominators, you must first reduce them to the lowest common denominator, then subtract the numerator of the minuend from the numerator of the minuend and sign the common denominator under their difference. The action is written like this:

If you need to subtract one mixed number from another mixed number, then, if possible, subtract a fraction from a fraction, and a whole from a whole. The action is written like this:

If the fraction of the subtracted is greater than the fraction of the minuend, then take one unit from the whole number of the minuend, split it into the appropriate shares and add it to the fraction of the minuend, after which they proceed as described above. The action is written like this:

Do the same thing when you need to subtract a fraction from a whole number.

Example. .

3. Extension of the properties of addition and subtraction to fractions.All laws and properties of addition and subtraction of natural numbers are also valid for fractional numbers. Their use in many cases greatly simplifies the calculation process.

4. Multiplication.

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 product the denominator.

When multiplying, you should make (if possible) reduction.

Example. .

If we take into account that an integer is a fraction with a denominator of 1, then multiplying a fraction by an integer and an integer by a fraction can be followed by the same rule.

Examples.

5. 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 fractions.

Example. .

6. Dividing a fraction by a fraction.

To divide a fraction into a fraction, you need to multiply the numerator of the first fraction by the denominator of the second, and the denominator of the first by the numerator of the second, and write the first product as the numerator, and the second as the denominator.

Example. .

Using the same rule, you can divide a fraction by a whole number and a whole number by a fraction, if you represent the whole number as a fraction with a denominator of 1.

Examples.

7. Division of mixed numbers.

To divide mixed numbers, they are first converted to improper fractions and then divided according to the rule for dividing fractions.

Example. .

8. Replacing division with multiplication.

If you swap the numerator and denominator in a fraction, you get a new fraction, the inverse of the given one. For example, for a fractionthe reciprocal fraction will be.

Obviously, the product of two mutually inverse fractions is equal to 1.

1. Finding a fraction from a number.

There are many problems that require you to find a part or fraction of a given number. Such problems are solved by multiplication.

Task. The hostess had 20 rubles;She spent them on shopping. How much do the purchases cost?

Here you need to findnumber 20. You can do it like this:

Answer. The hostess spent 8 rubles.

Examples. Find from 30. Solution. .

Find from . Solution. .

1. Finding a number from the known magnitude of its fraction.

Sometimes it is necessary to determine the entire number using a known part of a number and a fraction expressing this part. Such problems are solved by division.

Task. There are 12 Komsomol members in the class, which isparts of all students in the class. How many students are there in the class?

Solution. .

Answer. 20 students.

Example. Find the numberwhich is 34.

Solution. .

Answer. The required number is.

1. Finding the ratio of two numbers.

Consider the problem: A worker produced 40 parts in a day. What part of the monthly task has the worker completed if the monthly plan is 400 parts?

Solution. .

Answer. The worker completedpart of the monthly plan.

In this case, a part (40 parts) is expressed as a fraction of the whole (400 parts). They also say that the ratio of the number of parts manufactured per day to the monthly plan has been found.

1. Converting a decimal fraction into a common fraction.

To convert a decimal fraction to a common fraction, write it with the denominator and, if possible, abbreviate it:

Examples.

1. Converting a fraction to a decimal.

There are several ways to convert a fraction to a decimal.

First way. To convert a fraction to a decimal, you divide the numerator by the denominator.

Examples. .

Second way. To turn a fraction into a decimal, you need to multiply the numerator and denominator of the fraction by such a number that the denominator ends up being one with zeros (if possible).

Example.

1. Comparing decimals by magnitude. To find out which of two decimal fractions is larger, you need to compare their whole parts, tenths, hundredths, etc. When the whole parts are equal, the fraction that has more tenth parts is greater; if integers and decimals are equal, the one with more hundredths is greater, etc.

Example. Of the three fractions 2.432; 2.41 and 2.4098 is the largest first, since it has the most hundredths, and the whole and tenths are the same in all fractions.

Operations with decimals

1. Multiplying and dividing decimals by 10, 100, 1000, etc.

To multiply a decimal by 10, 100, 1000, etc. you need to move the comma to one, two, three, etc., respectively. sign to the right. If there are not enough signs in the number, then zeros are assigned.

Example. 15.45 10 = 154.5; 32.3 · 100 = 3230.

To divide a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point to one, two, three, etc., respectively. sign to the left. If there are not enough characters to move the comma, their number is supplemented with the corresponding number of zeros on the left.

Examples. 184.35: 100 = 1.8435; 3.5: 100 = 0.035.

1. Adding and subtracting decimals.

Decimals are added and subtracted in much the same way as natural numbers are added and subtracted. The digit is written under the digit, the comma is written under the comma.

Examples.

1. Multiplying decimals.

To multiply two decimal fractions, it is enough, without paying attention to commas, to multiply them as whole numbers and in the product to separate as many decimal places with a comma on the right as there were in the multiplicand and the multiplier together.

Example 1. 2.064 · 0.05.

We multiply the integers 2064 · 5 = 10320. The first factor had three decimal places, the second had two. The product must have five decimal places. We separate them on the right and get 0.10320. The zero at the end can be discarded: 2.064 · 0.05 = 0.1032.

Example 2. 1.125 · 0.08; 1125 · 8 = 9000.

The number of decimal places should be 3 + 2 = 5. We add zeros to 9000 on the left (009000) and separate five decimal places on the right. We get 1.125 · 0.08 = 0.09000 = 0.09.

1. Dividing decimals.

Two cases of dividing decimal fractions without a remainder are considered: 1) dividing a decimal fraction by an integer; 2) dividing a number (integer or fraction) by a decimal fraction.

Dividing a decimal by a whole number is done in the same way as dividing integers; the resulting residues are split sequentially into smaller decimal parts and division continues until the remainder is zero.

Examples.

Dividing a number (integer or fraction) by a decimal fraction in all cases results in division by a whole number. To do this, increase the divisor by 10, 100, 1000, etc. times, and so that the quotient does not change, the dividend is increased the same number of times, and then divided by a whole number (as in the first case).

Example. 47.04: 0.0084 = 470400: 84 = 5600;

1. Examples of joint actions with ordinary and decimal fractions.

Let's first consider an example of all operations with decimal fractions.

Example 1. Calculate:

Here they use the reduction of the dividend and divisor to an integer, taking into account the fact that the quotient does not change. Then we have:

When solving examples of joint actions with ordinary and decimal fractions, some actions can be performed in decimal fractions, and some in ordinary ones. It must be borne in mind that a common fraction cannot always be converted into a final decimal fraction. Therefore, writing as a decimal fraction can only be done when it has been verified that such a conversion is possible.

Example 2. Calculate:

Interest

The concept of percentage.A percentage of a number is a hundredth part of that number. For example, instead of saying “54 hundredths of all the inhabitants of our country are women,” one might say “54 percent of all the inhabitants of our country are women.” Instead of the word "percentage" they also write the % sign, for example 35% means 35 percent.

Since a percentage is a hundredth part, it follows that a percentage is a fraction with a denominator of 100. Therefore, the fraction is 0.49, or, can be read as 49 percent and written without a denominator as 49%. In general, having determined how many hundredths there are in a given decimal fraction, it is easy to write it as a percentage. To do this, use the rule: to write a decimal fraction as a percentage, you need to move the decimal point in this fraction two places to the right.

Examples. 0.33 = 33%; 1.25 = 125%; 0.002 = 0.2%; 21 = 2100%.

And vice versa: 7% = 0.07; 24.5% = 0.245; 0.1% = 0.001; 200% = 2.

1. Finding the percentage of a given number

Task. According to the plan, a team of tractor drivers must consume 9 tons of fuel. Tractor drivers have made a social commitment to save 20% of fuel. Determine fuel savings in tons.

If in this problem, instead of 20%, we write the number 0.2 equal to it, we get a problem to find the fraction of a number. And such problems are solved by multiplication. This is the solution:

20% = 0.2; 9 · 0.2 = 1.8 (m).

The calculations can be written like this:

(m)

To find several percent of a given number, it is enough to divide the given number by 100 and multiply the result by the number of percent.

Task. A worker in 1963 received 90 rubles a month, and in 1964 he began to receive 30% more. How much did he earn in 1964?

Solution (first method).

1) How many more rubles did the worker receive?

(rub.)

90 + 27 = 117 (rub).

Second way.

1) What percentage of previous earnings did the worker begin to receive in 1964?

100% + 30% = 130%.

2) What was the monthly salary of a worker in 1964?

(rub.)

2. Finding a number from a given value of its percentage.

Task. The collective farm planted corn on an area of ​​280 hectares, which is 14% of the total sown area. Determine the sown area of ​​the collective farm.

If in this problem instead of 14% we write 0.14 or, then we get the task of finding a number from the known value of its fraction. And such problems are solved by division.

Solution. 14% = 0.14; 280: 0.14 = 2000 (ha). This solution can also be formulated like this:

(ha)

To find a number based on a given value of several percent of it, it is enough to divide this value by the number of percent and multiply the result by 100.

Task. In March the plant smelted 125.4 T metal, exceeding the plan by 4.5%. How many tons of metal was the plant supposed to smelt in March according to plan?

Solution.

1) By what percentage did the plant fulfill the plan in March?

100% + 4,5% = 104,5%.

2) How many tons of metal should the plant smelt?

(ha)

1. Finding the percentage relationship between two numbers.

Task. We need to plow 300 hectares of land. On the first day, 120 hectares were plowed. What percentage of the task was plowed on the first day?

Solution.

First way. 300 hectares is 100%, which means that 1% accounts for 3 hectares. By determining how many times 3 hectares, constituting 1%, are contained in 120 hectares, we find out what percentage of the task the land was plowed on the first day

120: 3 = 40(%).

Second way. Having determined what part of the land was plowed on the first day, we express this fraction as a percentage.

Let's write down the calculation:

To calculate the percentage of a number a to number b , you need to find a relationship a to b and multiply it by 100.

Actions with fractions.

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

So, what are fractions, types of fractions, transformations - we remembered. Let's get to the main issue.

What can you do with fractions? Yes, everything is the same as with ordinary numbers. Add, subtract, multiply, divide.

All these actions with decimal working with fractions is no different from working with whole numbers. Actually, that’s what’s good about them, decimal ones. The only thing is that you need to put the comma correctly.

Mixed numbers, as I already said, are of little use for most actions. They still need to be converted to ordinary fractions.

But the actions with ordinary fractions they will be more cunning. And much more important! Let me remind you: all actions with fractional expressions with letters, sines, unknowns, and so on and so forth are no different from actions with ordinary fractions! Operations with ordinary fractions are the basis for all algebra. It is for this reason that we will analyze all this arithmetic in great detail here.

Adding and subtracting fractions.

Everyone can add (subtract) fractions with the same denominators (I really hope!). Well, let me remind those who are completely forgetful: when adding (subtracting), the denominator does not change. The numerators are added (subtracted) to give the numerator of the result. Type:

In short, in general terms:

What if the denominators are different? Then, using the basic property of a fraction (here it comes in handy again!), we make the denominators the same! For example:

Here we had to make the fraction 4/10 from the fraction 2/5. For the sole purpose of making the denominators the same. Let me note, just in case, that 2/5 and 4/10 are the same fraction! Only 2/5 are uncomfortable for us, and 4/10 are really okay.

By the way, this is the essence of solving any math problems. When we from uncomfortable we do expressions the same thing, but more convenient for solving.

Another example:

The situation is similar. Here we make 48 from 16. By simple multiplication by 3. This is all clear. But we came across something like:

How to be?! It's hard to make a nine out of a seven! But we are smart, we know the rules! Let's transform every fraction so that the denominators are the same. This is called “reduce to a common denominator”:

Wow! How did I know about 63? Very simple! 63 is a number that is divisible by 7 and 9 at the same time. Such a number can always be obtained by multiplying the denominators. If we multiply a number by 7, for example, then the result will certainly be divisible by 7!

If you need to add (subtract) several fractions, there is no need to do it in pairs, step by step. You just need to find the denominator common to all fractions and reduce each fraction to this same denominator. For example:

And what will be the common denominator? You can, of course, multiply 2, 4, 8, and 16. We get 1024. Nightmare. It’s easier to estimate that the number 16 is perfectly divisible by 2, 4, and 8. Therefore, from these numbers it’s easy to get 16. This number will be the common denominator. Let's turn 1/2 into 8/16, 3/4 into 12/16, and so on.

By the way, if you take 1024 as the common denominator, everything will work out, in the end everything will be reduced. But not everyone will get to this end, because of the calculations...

Complete the example yourself. Not some kind of logarithm... It should be 29/16.

So, the addition (subtraction) of fractions is clear, I hope? Of course, it is easier to work in a shortened version, with additional multipliers. But this pleasure is available to those who worked honestly in the lower grades... And did not forget anything.

And now we will do the same actions, but not with fractions, but with fractional expressions. New rake will be revealed here, yes...

So, we need to add two fractional expressions:

We need to make the denominators the same. And only with the help multiplication! This is what the main property of a fraction dictates. Therefore, I cannot add one to X in the first fraction in the denominator. (that would be nice!). But if you multiply the denominators, you see, everything grows together! So we write down the line of the fraction, leave an empty space at the top, then add it, and write the product of the denominators below, so as not to forget:

And, of course, we don’t multiply anything on the right side, we don’t open the parentheses! And now, looking at the common denominator on the right side, we realize: in order to get the denominator x(x+1) in the first fraction, you need to multiply the numerator and denominator of this fraction by (x+1). And in the second fraction - to x. This is what you get:

Note! Here are the parentheses! This is the rake that many people step on. Not parentheses, of course, but their absence. The parentheses appear because we are multiplying all numerator and all denominator! And not their individual pieces...

In the numerator of the right side we write the sum of the numerators, everything is as in numerical fractions, then we open the brackets in the numerator of the right side, i.e. We multiply everything and give similar ones. There is no need to open the parentheses in the denominators or multiply anything! In general, in denominators (any) the product is always more pleasant! We get:

So we got the answer. The process seems long and difficult, but it depends on practice. Once you solve the examples, get used to it, everything will become simple. Those who have mastered fractions in due time do all these operations with one left hand, automatically!

And one more note. Many smartly deal with fractions, but get stuck on examples with whole numbers. Like: 2 + 1/2 + 3/4= ? Where to fasten the two-piece? You don’t need to fasten it anywhere, you need to make a fraction out of two. It's not easy, but very simple! 2=2/1. Like this. Any whole number can be written as a fraction. The numerator is the number itself, the denominator is one. 7 is 7/1, 3 is 3/1 and so on. It's the same with letters. (a+b) = (a+b)/1, x=x/1, etc. And then we work with these fractions according to all the rules.

Well, the knowledge of addition and subtraction of fractions was refreshed. Converting fractions from one type to another was repeated. You can also get checked. Shall we settle it a little?)

Calculate:

Answers (in disarray):

71/20; 3/5; 17/12; -5/4; 11/6

Multiplication/division of fractions - in the next lesson. There are also tasks for all operations with fractions.

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

Now that we have learned how to add and multiply individual fractions, we can look at more complex structures. For example, what if the same problem involves adding, subtracting, and multiplying fractions?

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

1. Exponentiation is done first - 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 parentheses in the expression, the order of operations changes - everything that is inside the parentheses must be counted first. And remember about improper fractions: you need to highlight the whole part only when all other actions have already been completed.

Let's convert all the fractions from the first expression to improper ones, and then perform the following steps:

Now let's find the value of the second expression. There are no fractions with an integer part, but there are parentheses, so first we 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. Considering 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 a degree, the problem will be reduced to the usual multiplication of fractions:

Multistory fractions

Until now, we have considered only “pure” fractions, when the numerator and denominator are ordinary numbers. This is quite consistent with the definition of a number fraction given in the very first lesson.

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

There is only one rule for working with multi-level fractions: you must get rid of them immediately. Removing “extra” floors is quite simple, if you remember that the slash 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-story fraction to an ordinary one. Take a look at the examples:

Task. Convert multistory fractions to ordinary 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 canceled before the final multiplication.

Specifics of working with multi-level fractions

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

1. The numerator contains the single number 7, and the denominator contains the fraction 12/5;
2. The numerator contains the fraction 7/12, and the denominator contains the separate number 5.

So, for one recording 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 line of the nested fraction. Preferably several times.

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

Yes, it's probably unsightly and takes up too much space. But you will count correctly. Finally, a couple of examples where multi-story fractions actually arise:

Task. Find the meanings of the expressions:

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

Let's do the same with the second example. Let's convert all fractions to improper ones 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 basic fractions contain sums, the rule for writing multi-story fractions is observed automatically. Also, in the last example, we intentionally left 46/1 in fraction form to perform division.

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

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

Convenient and simple online fraction calculator with detailed solutions Maybe:

• Add, subtract, multiply and divide fractions online,
• Receive a ready-made solution to fractions with a picture and conveniently transfer it.

The result of solving fractions will be here...

0 1 2 3 4 5 6 7 8 9
Fraction sign "/" + - * :
_erase Clear
Our online fraction calculator has quick input. To solve fractions, for example, simply write 1/2+2/7 into the calculator and press the " Solve fractions". The calculator will write to you detailed solution of fractions and will issue an easy-to-copy image.

Signs used for writing in a calculator

Features of the online fraction calculator

If you need to solve negative fractions, just use the properties of minus. When multiplying and dividing negative fractions, minus by minus gives plus. That is, the product and division of negative fractions is equal to the product and division of the same positive ones. If one fraction is negative when multiplying or dividing, then simply remove the minus and then add it to the answer. When adding negative fractions, the result will be the same as if you were adding the same positive fractions. If you add one negative fraction, then this is the same as subtracting the same positive one.
When subtracting negative fractions, the result will be the same as if they were swapped and made positive. That is, minus by minus in this case gives a plus, but rearranging the terms does not change the sum. We use the same rules when subtracting fractions, one of which is negative.

To solve mixed fractions (fractions in which the whole part is isolated), simply fit the whole part into the fraction. To do this, multiply the whole part by the denominator and add to the numerator.

If you need to solve 3 or more fractions online, you should solve them one by one. First, count the first 2 fractions, then solve the next fraction with the answer you get, and so on. Perform the operations one by one, 2 fractions at a time, and eventually you will get the correct answer.