From 11 all operations with fractions. Problems and examples for all operations with ordinary fractions

Multiplying and dividing fractions.

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

This operation is much nicer than addition-subtraction! Because it's easier. As a reminder, to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). That is:

For example:

Everything is extremely simple. And please don't look for a common denominator! There is no need for him here...

To divide a fraction by a fraction, you need to reverse second(this is important!) fraction and multiply them, i.e.:

For example:

If you come across multiplication or division with integers and fractions, it’s okay. As with addition, we make a fraction from a whole number with one in the denominator - and go ahead! For example:

In high school, you often have to deal with three-story (or even four-story!) fractions. For example:

How can I make this fraction look decent? Yes, very simple! Use two-point division:

But don't forget about the order of division! Unlike multiplication, this is very important here! Of course, we will not confuse 4:2 or 2:4. But it’s easy to make a mistake in a three-story fraction. Please note for example:

In the first case (expression on the left):

In the second (expression on the right):

Do you feel the difference? 4 and 1/9!

What determines the order of division? Either with brackets, or (as here) with the length of horizontal lines. Develop your eye. And if there are no brackets or dashes, like:

then divide and multiply in order, from left to right!

And another very simple and important technique. In actions with degrees, it will be so useful to you! Let's divide one by any fraction, for example, by 13/15:

The shot has turned over! And this always happens. When dividing 1 by any fraction, the result is the same fraction, only upside down.

That's it for operations with fractions. The thing is quite simple, but it gives more than enough errors. Note practical advice, and there will be fewer of them (errors)!

Practical tips:

1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not general words, not good wishes! This is a dire necessity! Do all calculations on the Unified State Exam as a full-fledged task, focused and clear. It’s better to write two extra lines in your draft than to mess up when doing mental calculations.

2. In examples with different types fractions - go to ordinary fractions.

3. We reduce all fractions until they stop.

4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).

5. Divide a unit by a fraction in your head, simply turning the fraction over.

Here are the tasks that you must definitely complete. Answers are given after all tasks. Use the materials on this topic and practical tips. Estimate how many examples you were able to solve correctly. The first time! Without a calculator! And draw the right conclusions...

Remember - the correct answer is received from the second (especially the third) time does not count! Such is the harsh life.

So, solve in exam mode ! This is already preparation for the Unified State Exam, by the way. We solve the example, check it, solve the next one. We decided everything - checked again from first to last. But only Then look at the answers.

Calculate:

Have you decided?

We are looking for answers that match yours. I deliberately wrote them down in disarray, away from temptation, so to speak... Here they are, the answers, written with semicolons.

0; 17/22; 3/4; 2/5; 1; 25.

Now we draw conclusions. If everything worked out, I’m happy for you! Basic calculations with fractions are not your problem! You can do more serious things. If not...

So you have one of two problems. Or both at once.) Lack of knowledge and (or) inattention. But this solvable Problems.

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.

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.

Yandex.RTB R-A-339285-1

Rules for performing operations with general numerical fractions

Numerical fractions general view have a numerator and 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 actions are performed with ordinary fractions. 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

Definition 2

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 the number or numeric expression can be presented as a fraction with a denominator equal to 1, then the action with such a fraction is considered as 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 .

If you notice an error in the text, please highlight it and press Ctrl+Enter

496. Find X, If:

497. 1) If you add 10 1/2 to 3/10 of an unknown number, you get 13 1/2. Find the unknown number.

2) If you subtract 10 1/2 from 7/10 of an unknown number, you get 15 2/5. Find the unknown number.

498 *. If you subtract 10 from 3/4 of an unknown number and multiply the resulting difference by 5, you get 100. Find the number.

499 *. If you increase an unknown number by 2/3 of it, you get 60. What number is this?

500 *. If you add the same amount to the unknown number, and also 20 1/3, you get 105 2/5. Find the unknown number.

501. 1) The potato yield with square-cluster planting averages 150 centners per hectare, and with conventional planting it is 3/5 of this amount. How much more potatoes can be harvested from an area of ​​15 hectares if potatoes are planted using the square-cluster method?

2) An experienced worker produced 18 parts in 1 hour, and an inexperienced worker produced 2/3 of this amount. How many more parts can an experienced worker produce in a 7-hour day?

502. 1) The pioneers collected 56 kg of different seeds over three days. On the first day, 3/14 of the total amount was collected, on the second, one and a half times more, and on the third day, the rest of the grain. How many kilograms of seeds did the pioneers collect on the third day?

2) When grinding the wheat, the result was: flour 4/5 of the total amount of wheat, semolina - 40 times less than flour, and the rest was bran. How much flour, semolina and bran separately were produced when grinding 3 tons of wheat?

503. 1) Three garages can accommodate 460 cars. The number of cars that fit in the first garage is 3/4 of the number of cars that fit in the second, and the third garage has 1 1/2 times as many cars as the first. How many cars fit in each garage?

2) A factory with three workshops employs 6,000 workers. In the second workshop there are 1 1/2 times fewer workers than in the first, and the number of workers in the third workshop is 5/6 of the number of workers in the second workshop. How many workers are there in each workshop?

504. 1) First 2/5, then 1/3 of the total kerosene was poured from a tank with kerosene, and after that 8 tons of kerosene remained in the tank. How much kerosene was in the tank initially?

2) The cyclists raced for three days. On the first day they covered 4/15 of the entire journey, on the second - 2/5, and on the third day the remaining 100 km. How far did the cyclists travel in three days?

505. 1) The icebreaker fought its way through the ice field for three days. On the first day he walked 1/2 of the entire distance, on the second day 3/5 of the remaining distance and on the third day the remaining 24 km. Find the length of the path covered by the icebreaker in three days.

2) Three groups of schoolchildren planted trees to green the village. The first detachment planted 7/20 of all trees, the second 5/8 of the remaining trees, and the third the remaining 195 trees. How many trees did the three teams plant in total?

506. 1) A combine harvester harvested wheat from one plot in three days. On the first day, he harvested from 5/18 of the entire area of ​​the plot, on the second day from 7/13 of the remaining area, and on the third day from the remaining area of ​​30 1/2 hectares. On average, 20 centners of wheat were harvested from each hectare. How much wheat was harvested in the entire area?

2) On the first day, the rally participants covered 3/11 of the entire route, on the second day 7/20 of the remaining route, on the third day 5/13 of the new remainder, and on the fourth day the remaining 320 km. How long is the route of the rally?

507. 1) On the first day the car covered 3/8 of the entire distance, on the second day 15/17 of what it covered on the first, and on the third day the remaining 200 km. How much gasoline was consumed if a car consumes 1 3/5 kg of gasoline for 10 km?

2) The city consists of four districts. And 4/13 of all residents of the city live in the first district, 5/6 of the residents of the first district live in the second, 4/11 of the residents of the first live in the third; two districts combined, and 18 thousand people live in the fourth district. How much bread does the entire population of the city need for 3 days, if on average one person consumes 500 g per day?

508. 1) The tourist walked on the first day 10/31 of the entire journey, on the second 9/10 of what he walked on the first day, and on the third the rest of the way, and on the third day he walked 12 km more than on the second day. How many kilometers did the tourist walk on each of the three days?

2) The car covered the entire route from city A to city B in three days. On the first day the car covered 7/20 of the entire distance, on the second 8/13 of the remaining distance, and on the third day the car covered 72 km less than on the first day. What is the distance between cities A and B?

509. 1) The Executive Committee allocated land to the workers of three factories for garden plots. The first plant was allocated 9/25 of the total number of plots, the second plant 5/9 of the number of plots allocated for the first, and the third - the remaining plots. How many total plots were allocated to the workers of three factories, if the first factory was allocated 50 fewer plots than the third?

2) The plane delivered a shift of winter workers to the polar station from Moscow in three days. On the first day he flew 2/5 of the entire distance, on the second - 5/6 of the distance he covered on the first day, and on the third day he flew 500 km less than on the second day. How far did the plane fly in three days?

510. 1) The plant had three workshops. The number of workers in the first workshop is 2/5 of all workers in the plant; in the second workshop there are 1 1/2 times fewer workers than in the first, and in the third workshop there are 100 more workers than in the second. How many workers are there in the factory?

2) The collective farm includes residents of three neighboring villages. The number of families in the first village is 3/10 of all families on the collective farm; in the second village the number of families is 1 1/2 times greater than in the first, and in the third village the number of families is 420 less than in the second. How many families are there on the collective farm?

511. 1) The artel used up 1/3 of its stock of raw materials in the first week, and 1/3 of the rest in the second week. How much raw material is left in the artel if in the first week the consumption of raw materials was 3/5 tons more than in the second week?

2) Of the imported coal, 1/6 of it was spent for heating the house in the first month, and 3/8 of the remainder in the second month. How much coal is left to heat the house if 1 3/4 more was used in the second month than in the first month?

512. 3/5 of the total land of the collective farm is allocated for sowing grain, 13/36 of the remainder is occupied by vegetable gardens and meadows, the rest of the land is forest, and the sown area of ​​the collective farm is 217 hectares larger than the forest area, 1/3 of the land allocated for sowing grain is sown with rye, and the rest is wheat. How many hectares of land did the collective farm sow with wheat and how many with rye?

513. 1) The tram route is 14 3/8 km long. Along this route, the tram makes 18 stops, spending on average up to 1 1/6 minutes per stop. The average speed of the tram along the entire route is 12 1/2 km per hour. How long does it take for a tram to complete one trip?

2) Bus route 16 km. Along this route the bus makes 36 stops of 3/4 minutes each. on average each. The average bus speed is 30 km per hour. How long does a bus take for one route?

514*. 1) It’s 6 o’clock now. evenings. What part is the remaining part of the day from the past and what part of the day is left?

2) A steamer travels the distance between two cities with the current in 3 days. and back the same distance in 4 days. How many days will the rafts float downstream from one city to another?

515. 1) How many boards will be used to lay the floor in a room whose length is 6 2/3 m, width 5 1/4 m, if the length of each board is 6 2/3 m, and its width is 3/80 of the length?

2) A rectangular platform has a length of 45 1/2 m, and its width is 5/13 of its length. This area is bordered by a path 4/5 m wide. Find the area of ​​the path.

516. Find the average arithmetic numbers:

517. 1) The arithmetic mean of two numbers is 6 1/6. One of the numbers is 3 3/4. Find another number.

2) The arithmetic mean of two numbers is 14 1/4. One of these numbers is 15 5/6. Find another number.

518. 1) The freight train was on the road for three hours. In the first hour he covered 36 1/2 km, in the second 40 km and in the third 39 3/4 km. Find the average speed of the train.

2) The car traveled 81 1/2 km in the first two hours, and 95 km in the next 2 1/2 hours. How many kilometers did he walk on average per hour?

519. 1) The tractor driver completed the task of plowing the land in three days. On the first day he plowed 12 1/2 hectares, on the second day 15 3/4 hectares and on the third day 14 1/2 hectares. On average, how many hectares of land did a tractor driver plow per day?

2) A group of schoolchildren, making a three-day tourist trip, were on the road for 6 1/3 hours on the first day, 7 hours on the second. and on the third day - 4 2/3 hours. How many hours on average did schoolchildren travel every day?

520. 1) Three families live in the house. The first family has 3 light bulbs to illuminate the apartment, the second has 4 and the third has 5 light bulbs. How much should each family pay for electricity if all the lamps were the same, and the total electricity bill (for the whole house) was 7 1/5 rubles?

2) A polisher was polishing the floors in an apartment where three families lived. The first family had a living area of ​​36 1/2 square meters. m, the second is 24 1/2 sq. m, and the third - 43 sq. m. For all the work, 2 rubles were paid. 08 kop. How much did each family pay?

521. 1) In the garden plot, potatoes were collected from 50 bushes at 1 1/10 kg per bush, from 70 bushes at 4/5 kg per bush, from 80 bushes at 9/10 kg per bush. How many kilograms of potatoes are harvested on average from each bush?

2) The field crew on an area of ​​300 hectares received a harvest of 20 1/2 quintals of winter wheat per 1 hectare, from 80 hectares to 24 quintals per 1 ha, and from 20 hectares - 28 1/2 quintals per 1 ha. What is the average yield in a brigade with 1 hectare?

522. 1) The sum of two numbers is 7 1/2. One number is 4 4/5 greater than the other. Find these numbers.

2) If we add the numbers expressing the width of the Tatar and Kerch Straits together, we get 11 7/10 km. The Tatar Strait is 3 1/10 km wider than the Kerch Strait. What is the width of each strait?

523. 1) The sum of three numbers is 35 2 / 3. The first number is greater than the second by 5 1/3 and greater than the third by 3 5/6. Find these numbers.

2) Islands New Earth, Sakhalin and Severnaya Zemlya together occupy an area of ​​196 7/10 thousand square meters. km. The area of ​​Novaya Zemlya is 44 1/10 thousand square meters. km larger than the area of ​​Severnaya Zemlya and 5 1/5 thousand square meters. km larger than the area of ​​Sakhalin. What is the area of ​​each of the listed islands?

524. 1) The apartment consists of three rooms. The area of ​​the first room is 24 3/8 sq. m and is 13/36 of the entire area of ​​the apartment. The area of ​​the second room is 8 1/8 square meters. m more than the area of ​​the third. What is the area of ​​the second room?

2) A cyclist during a three-day competition on the first day was on the road for 3 1/4 hours, which was 13/43 of the total travel time. On the second day he rode 1 1/2 hours more than on the third day. How many hours did the cyclist travel on the second day of the competition?

525. Three pieces of iron weigh together 17 1/4 kg. If the weight of the first piece is reduced by 1 1/2 kg, the weight of the second by 2 1/4 kg, then all three pieces will have the same weight. How much did each piece of iron weigh?

526. 1) The sum of two numbers is 15 1/5. If the first number is reduced by 3 1/10, and the second is increased by 3 1/10, then these numbers will be equal. What is each number equal to?

2) There were 38 1/4 kg of cereal in two boxes. If you pour 4 3/4 kg of cereal from one box into another, then there will be equal amounts of cereal in both boxes. How much cereal is in each box?

527 . 1) The sum of two numbers is 17 17 / 30. If you subtract 5 1/2 from the first number and add it to the second, then the first will still be greater than the second by 2 17/30. Find both numbers.

2) There are 24 1/4 kg of apples in two boxes. If you transfer 3 1/2 kg from the first box to the second, then in the first there will still be 3/5 kg more apples than in the second. How many kilograms of apples are in each box?

528 *. 1) The sum of two numbers is 8 11/14, and their difference is 2 3/7. Find these numbers.

2) The boat moved along the river at a speed of 15 1/2 km per hour, and against the current at 8 1/4 km per hour. What is the speed of the river flow?

529. 1) There are 110 cars in two garages, and in one of them there are 1 1/5 times more than in the other. How many cars are in each garage?

2) The living area of ​​an apartment consisting of two rooms is 47 1/2 sq. m. m. The area of ​​one room is 8/11 of the area of ​​the other. Find the area of ​​each room.

530. 1) An alloy consisting of copper and silver weighs 330 g. The weight of copper in this alloy is 5/28 of the weight of silver. How much silver and how much copper is in the alloy?

2) The sum of two numbers is 6 3/4, and the quotient is 3 1/2. Find these numbers.

531. The sum of three numbers is 22 1/2. The second number is 3 1/2 times, and the third is 2 1/4 times the first. Find these numbers.

532. 1) The difference of two numbers is 7; quotient of division more for less 5 2/3. Find these numbers.

2) The difference between two numbers is 29 3/8, and their multiple ratio is 8 5/6. Find these numbers.

533. In a class, the number of absent students is 3/13 of the number of students present. How many students are in the class according to the list if there are 20 more people present than absent?

534. 1) The difference between two numbers is 3 1/5. One number is 5/7 of another. Find these numbers.

2) The father is 24 years older than his son. The number of the son's years is equal to 5/13 of the father's years. How old is the father and how old is the son?

535. The denominator of a fraction is 11 units greater than its numerator. What is the value of a fraction if its denominator is 3 3/4 times the numerator?

No. 536 - 537 orally.

536. 1) The first number is 1/2 of the second. How many times is the second number greater than the first?

2) The first number is 3/2 of the second. What part of the first number is the second number?

537. 1) 1/2 of the first number is equal to 1/3 of the second number. What part of the first number is the second number?

2) 2/3 of the first number is equal to 3/4 of the second number. What part of the first number is the second number? What part of the second number is the first?

538. 1) The sum of two numbers is 16. Find these numbers if 1/3 of the second number is equal to 1/5 of the first.

2) The sum of two numbers is 38. Find these numbers if 2/3 of the first number is equal to 3/5 of the second.

539 *. 1) Two boys collected 100 mushrooms together. 3/8 of the number of mushrooms collected by the first boy is numerically equal to 1/4 of the number of mushrooms collected by the second boy. How many mushrooms did each boy collect?

2) The institution employs 27 people. How many men work and how many women work if 2/5 of all men are equal to 3/5 of all women?

540 *. Three boys bought a volleyball. Determine the contribution of each boy, knowing that 1/2 of the contribution of the first boy is equal to 1/3 of the contribution of the second, or 1/4 of the contribution of the third, and that the contribution of the third boy is 64 kopecks more than the contribution of the first.

541 *. 1) One number is 6 more than the other. Find these numbers if 2/5 of one number are equal to 2/3 of the other.

2) The difference of two numbers is 35. Find these numbers if 1/3 of the first number is equal to 3/4 of the second number.

542. 1) The first team can complete some work in 36 days, and the second in 45 days. In how many days will both teams, working together, complete this job?

2) A passenger train covers the distance between two cities in 10 hours, and a freight train covers this distance in 15 hours. Both trains left these cities at the same time towards each other. In how many hours will they meet?

543. 1) A fast train covers the distance between two cities in 6 1/4 hours, and a passenger train in 7 1/2 hours. How many hours later will these trains meet if they leave both cities at the same time towards each other? (Round answer to the nearest 1 hour.)

2) Two motorcyclists left simultaneously from two cities towards each other. One motorcyclist can travel the entire distance between these cities in 6 hours, and another in 5 hours. How many hours after departure will the motorcyclists meet? (Round answer to the nearest 1 hour.)

544. 1) Three vehicles of different carrying capacity can transport some cargo, working separately: the first in 10 hours, the second in 12 hours. and the third in 15 hours. In how many hours can they transport the same cargo, working together?

2) Two trains leave two stations simultaneously towards each other: the first train covers the distance between these stations in 12 1/2 hours, and the second in 18 3/4 hours. How many hours after leaving will the trains meet?

545. 1) Two taps are connected to the bathtub. Through one of them the bath can be filled in 12 minutes, through the other 1 1/2 times faster. How many minutes will it take to fill 5/6 of the entire bathtub if you open both taps at once?

2) Two typists must retype the manuscript. The first driver can complete this work in 3 1/3 days, and the second 1 1/2 times faster. How many days will it take both typists to complete the job if they work simultaneously?

546. 1) The pool is filled with the first pipe in 5 hours, and through the second pipe it can be emptied in 6 hours. After how many hours will the entire pool be filled if both pipes are opened at the same time?

Note. In an hour, the pool is filled to (1/5 - 1/6 of its capacity.)

2) Two tractors plowed the field in 6 hours. The first tractor, working alone, could plow this field in 15 hours. How many hours would it take the second tractor, working alone, to plow this field?

547 *. Two trains leave two stations simultaneously towards each other and meet after 18 hours. after his release. How long does it take the second train to cover the distance between stations if the first train covers this distance in 1 day 21 hours?

548 *. The pool is filled with two pipes. First they opened the first pipe, and then after 3 3/4 hours, when half of the pool was filled, they opened the second pipe. After 2 1/2 hours of working together, the pool was full. Determine the capacity of the pool if 200 buckets of water per hour poured through the second pipe.

549. 1) A courier train left Leningrad for Moscow and travels 1 km in 3/4 minutes. 1/2 hour after this train left Moscow, a fast train left Moscow for Leningrad, the speed of which was equal to 3/4 the speed of the express train. At what distance will the trains be from each other 2 1/2 hours after the courier train leaves, if the distance between Moscow and Leningrad is 650 km?

2) From the collective farm to the city 24 km. A truck leaves the collective farm and travels 1 km in 2 1/2 minutes. After 15 min. After this car left the city, a cyclist drove out to the collective farm, at a speed half as fast as the speed of the truck. How long after leaving will the cyclist meet the truck?

550. 1) A pedestrian came out of one village. 4 1/2 hours after the pedestrian left, a cyclist rode in the same direction, whose speed was 2 1/2 times the speed of the pedestrian. How many hours after the pedestrian leaves will the cyclist overtake him?

2) A fast train travels 187 1/2 km in 3 hours, and a freight train travels 288 km in 6 hours. 7 1/4 hours after the freight train leaves, an ambulance departs in the same direction. How long will it take the fast train to catch up with the freight train?

551. 1) From two collective farms through which the road to the regional center passes, two collective farmers rode out to the district at the same time on horseback. The first of them traveled 8 3/4 km per hour, and the second was 1 1/7 times more than the first. The second collective farmer caught up with the first after 3 4/5 hours. Determine the distance between collective farms.

2) 26 1/3 hours after the departure of the Moscow-Vladivostok train, the average speed of which was 60 km per hour, a TU-104 plane took off in the same direction, at a speed 14 1/6 times the speed of the train. How many hours after departure will the plane catch up with the train?

552. 1) The distance between cities along the river is 264 km. The steamer covered this distance downstream in 18 hours, spending 1/12 of this time stopping. The speed of the river is 1 1/2 km per hour. How long would it take a steamship to travel 87 km without stopping in still water?

2) A motor boat traveled 207 km along the river in 13 1/2 hours, spending 1/9 of this time on stops. The speed of the river is 1 3/4 km per hour. How many kilometers can this boat travel in still water in 2 1/2 hours?

553. The boat covered a distance of 52 km across the reservoir without stopping in 3 hours 15 minutes. Further, going along the river against the current, the speed of which is 1 3/4 km per hour, this boat covered 28 1/2 km in 2 1/4 hours, making 3 stops of equal duration. How many minutes did the boat wait at each stop?

554. From Leningrad to Kronstadt at 12 o'clock. The steamer left in the afternoon and covered the entire distance between these cities in 1 1/2 hours. On the way, he met another ship that left Kronstadt for Leningrad at 12:18 p.m. and walking at 1 1/4 times the speed of the first. At what time did the two ships meet?

555. The train had to cover a distance of 630 km in 14 hours. Having covered 2/3 of this distance, he was detained for 1 hour 10 minutes. At what speed should he continue his journey in order to reach his destination without delay?

556. At 4:20 a.m. morning a freight train left Kyiv for Odessa with average speed 31 1/5 km per hour. After some time, a mail train came out of Odessa to meet him, the speed of which was 1 17/39 times higher than the speed of a freight train, and met the freight train 6 1/2 hours after its departure. At what time did the postal train leave Odessa, if the distance between Kiev and Odessa is 663 km?

557*. The clock shows noon. How long will it take for the hour and minute hands to coincide?

558. 1) The plant has three workshops. The number of workers in the first workshop is 9/20 of all workers of the plant, in the second workshop there are 1 1/2 times fewer workers than in the first, and in the third workshop there are 300 fewer workers than in the second. How many workers are there in the factory?

2) There are three secondary schools in the city. The number of students in the first school is 3/10 of all students in these three schools; in the second school there are 1 1/2 times more students than in the first, and in the third school there are 420 fewer students than in the second. How many students are there in the three schools?

559. 1) Two combine operators worked in the same area. After one combiner harvested 9/16 of the entire plot, and the second 3/8 of the same plot, it turned out that the first combiner harvested 97 1/2 hectares more than the second. On average, 32 1/2 quintals of grain were threshed from each hectare. How many centners of grain did each combine operator thresh?

2) Two brothers bought a camera. One had 5/8, and the second 4/7 of the cost of the camera, and the first had 2 rubles worth. 25 kopecks more than the second one. Everyone paid half the cost of the device. How much money does everyone have left?

560. 1) A passenger car leaves city A for city B, the distance between them is 215 km, at a speed of 50 km per hour. At the same time, a truck left city B for city A. How many kilometers did the passenger car travel before meeting the truck, if the truck's speed per hour was 18/25 the speed of the passenger car?

2) Between cities A and B 210 km. A passenger car left city A for city B. At the same time, a truck left city B for city A. How many kilometers did the truck travel before meeting the passenger car, if the passenger car was traveling at a speed of 48 km per hour, and the speed of the truck per hour was 3/4 of the speed of the passenger car?

561. The collective farm harvested wheat and rye. 20 hectares more were sown with wheat than with rye. The total rye harvest amounted to 5/6 of the total wheat harvest with a yield of 20 c per 1 ha for both wheat and rye. The collective farm sold 7/11 of the entire harvest of wheat and rye to the state, and left the rest of the grain to satisfy its needs. How many trips did the two-ton trucks need to make to remove the bread sold to the state?

562. Rye and wheat flour were brought to the bakery. The weight of wheat flour was 3/5 of the weight of rye flour, and 4 tons more rye flour was brought than wheat flour. How much wheat and how much rye bread will the bakery bake from this flour if the baked goods make up 2/5 of the total flour?

563. Within three days, a team of workers completed 3/4 of the entire work on repairing the highway between the two collective farms. On the first day, 2 2/5 km of this highway were repaired, on the second day 1 1/2 times more than on the first, and on the third day 5/8 of what was repaired in the first two days together. Find the length of the highway between collective farms.

564. Fill in the empty spaces in the table, where S is the area of ​​the rectangle, A- the base of the rectangle, a h-height (width) of the rectangle.

565. 1) The length of a rectangular plot of land is 120 m, and the width of the plot is 2/5 of its length. Find the perimeter and area of ​​the site.

2) The width of the rectangular section is 250 m, and its length is 1 1/2 times the width. Find the perimeter and area of ​​the site.

566. 1) The perimeter of the rectangle is 6 1/2 inch, its base is 1/4 inch greater than its height. Find the area of ​​this rectangle.

2) The perimeter of the rectangle is 18 cm, its height is 2 1/2 cm less than the base. Find the area of ​​the rectangle.

567. Calculate the areas of the figures shown in Figure 30 by dividing them into rectangles and finding the dimensions of the rectangle by measurement.

568. 1) How many sheets of dry plaster will be required to cover the ceiling of a room whose length is 4 1/2 m and width 4 m, if the dimensions of the plaster sheet are 2 m x l 1/2 m?

2) How many boards 4 1/2 m long and 1/4 m wide are needed to lay a floor that is 4 1/2 m long and 3 1/2 m wide?

569. 1) A rectangular plot 560 m long and 3/4 of its length wide was sown with beans. How many seeds were required to sow the plot if 1 centner was sown per 1 hectare?

2) A wheat harvest of 25 quintals per hectare was collected from a rectangular field. How much wheat was harvested from the entire field if the length of the field is 800 m and the width is 3/8 of its length?

570 . 1) A rectangular plot of land, 78 3/4 m long and 56 4/5 m wide, is built up so that 4/5 of its area is occupied by buildings. Determine the area of ​​land under the buildings.

2) On a rectangular plot of land, the length of which is 9/20 km and the width is 4/9 of its length, the collective farm plans to lay out a garden. How many trees will be planted in this garden if an average area of ​​36 sq.m. is required for each tree?

571. 1) For normal daylight illumination of the room, it is necessary that the area of ​​all windows be at least 1/5 of the floor area. Determine whether there is enough light in a room whose length is 5 1/2 m and width 4 m. Does the room have one window measuring 1 1/2 m x 2 m?

2) Using the condition of the previous problem, find out whether there is enough light in your classroom.

572. 1) The barn has dimensions of 5 1/2 m x 4 1/2 m x 2 1/2 m. How much hay (by weight) will fit in this barn if it is filled to 3/4 of its height and if 1 cu. m of hay weighs 82 kg?

2) The woodpile has the shape of a rectangular parallelepiped, the dimensions of which are 2 1/2 m x 3 1/2 m x 1 1/2 m. What is the weight of the woodpile if 1 cubic. m of firewood weighs 600 kg?

573. 1) A rectangular aquarium is filled with water up to 3/5 of its height. The length of the aquarium is 1 1/2 m, width 4/5 m, height 3/4 m. How many liters of water are poured into the aquarium?

2) A pool in the shape of a rectangular parallelepiped is 6 1/2 m long, 4 m wide and 2 m high. The pool is filled with water up to 3/4 of its height. Calculate the amount of water poured into the pool.

574. A fence needs to be built around a rectangular piece of land, 75 m long and 45 m wide. How many cubic meters of boards should go into its construction if the thickness of the board is 2 1/2 cm and the height of the fence should be 2 1/4 m?

575. 1) What is the angle between the minute hand and the hour hand at 13 o'clock? at 15 o'clock? at 17 o'clock? at 21 o'clock? at 23:30?

2) How many degrees will the hour hand rotate in 2 hours? 5 o'clock? 8 o'clock? 30 min.?

3) How many degrees does an arc equal to half a circle contain? 1/4 circle? 1/24 of a circle? 5/24 circles?

576. 1) Using a protractor, draw: a) a right angle; b) an angle of 30°; c) an angle of 60°; d) angle of 150°; e) an angle of 55°.

2) Using a protractor, measure the angles of the figure and find the sum of all the angles of each figure (Fig. 31).

577. Follow these steps:

578. 1) The semicircle is divided into two arcs, one of which is 100° larger than the other. Find the size of each arc.

2) The semicircle is divided into two arcs, one of which is 15° less than the other. Find the size of each arc.

3) The semicircle is divided into two arcs, one of which is twice as large as the other. Find the size of each arc.

4) The semicircle is divided into two arcs, one of which is 5 times smaller than the other. Find the size of each arc.

579. 1) The diagram “Population Literacy in the USSR” (Fig. 32) shows the number of literate people per hundred people of the population. Based on the data in the diagram and its scale, determine the number of literate men and women for each of the indicated years.

Write the results in the table:

2) Using the data from the diagram “Soviet envoys into space” (Fig. 33), create tasks.

580. 1) According to the pie chart “Daily routine for a fifth grade student” (Fig. 34), fill out the table and answer the questions: what part of the day is allocated to sleep? for homework? to school?

2) Construct a pie chart about your daily routine.

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

Reducing a fraction

Mathematical operation. Reducing a fraction

To reduce the fraction \frac(m)(n) you need to find the greatest common divisor of its numerator and denominator: gcd(m,n), and 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 seems to be a difficult task, and in practice, a fraction is reduced in several stages, step by step isolating 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)

Reducing fractions to a common denominator

Mathematical operation. Reducing fractions to a common denominator

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

• find the least common multiple of the denominators: M=LMK(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 transform 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 a simple task. Therefore, a number equal to the product of the denominators of the original fractions is chosen as the 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)

Comparison of fractions

Mathematical operation. Comparison of fractions

To compare two ordinary fractions you need:

• 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 fraction whose numerator is greater 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 simultaneously 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 (those with both the numerator and denominator greater than zero).

Adding and subtracting fractions

Mathematical operation. Adding and subtracting 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 from one fraction, you need:

• reduce 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 step 1 (reduction to a common denominator) is skipped.

Converting a mixed number to improper fraction and back

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

To convert a mixed fraction to an improper fraction, simply sum the whole part of the mixed fraction with the fraction 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 whole part by the denominator of the fraction with the numerator of the mixed fraction, and the denominator will remain 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 whole part 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 Fraction

Mathematical operation. Converting a Decimal to a Fraction

In order to convert a decimal fraction to a common fraction, you need to:

1. take the nth power of ten as the denominator (here n is the number of decimal places);
2. as the numerator, take the number after the decimal point (if the integer part of the original number is not equal to zero, then take all the 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) (there are 4 decimal places, so the denominator has 10 4 =10000, since the integer part is 0, the numerator contains 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 before it we add the whole part of the original number “31”)

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

3.014=3\frac(14)(100)

To convert a fraction to a decimal, simply divide the numerator by the denominator. Sometimes it will be endless 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

Multiplying and dividing fractions

Mathematical operation. Multiplying and dividing fractions

To multiply two ordinary 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 fraction- a fraction in which the numerator and denominator are swapped.

\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 natural number, then the above rules of multiplication and division remain in force. You just need to take into account 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