Complex fractions. Actions with fractions

AT this section actions are considered with ordinary fractions. In case it is necessary to mathematical operation with mixed numbers, it suffices to translate mixed fraction into extraordinary, carry out the necessary operations and, if necessary, present the final result again as a mixed number. This operation will be described below.

Fraction reduction

mathematical operation. Fraction reduction

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

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

Bringing fractions to common denominator

mathematical operation. Bringing fractions to a common denominator

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

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

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

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

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

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

Fraction Comparison

mathematical operation. Fraction Comparison

To compare two common fractions:

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

For example, \frac(9)(14)

When comparing fractions, there are several special cases:

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

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

Addition and subtraction of fractions

mathematical operation. Addition and subtraction of fractions

To add two fractions, you need:

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

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

To subtract another fraction from one, you need:

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

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

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

Converting a mixed number to not proper fraction and back

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

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

To convert an improper fraction to mixed number necessary:

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 Common Fraction

mathematical operation. Converting a Decimal to a Common Fraction

To convert a decimal to a common fraction:

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

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

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

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

3.014=3\frac(14)(100)

To convert an ordinary fraction to a decimal, it is enough to simply divide the numerator by the denominator. Sometimes it gets 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

Multiplication and division of fractions

mathematical operation. Multiplication and division of fractions

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

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

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

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

If one of the fractions is a natural number, then the above multiplication and division rules remain in force. Just keep in mind that an integer is the same fraction, the denominator of which 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

Fraction expansion. Fraction reduction. Fraction comparison.
Reduction to a common denominator. Addition and subtraction of fractions.
Multiplication of fractions. Division of fractions.
Fraction expansion. The value of a fraction does not change if its numerator and denominator are multiplied by the same non-zero number. This transformation is called fraction expansion. For example,

Fraction reduction. The value of a fraction does not change if its numerator and denominator are divided by the same non-zero number. This transformation is called fraction reduction. For example,

Fraction comparison. Of two fractions with the same numerator, the larger one is the one with the smaller denominator:

Of two fractions with the same denominators, the one with the larger numerator is greater:

To compare fractions that have different numerators and denominators, you need to expand them to bring them to a common denominator.
EXAMPLE Compare two fractions:

The transformation used here is called reducing fractions to a common denominator.
Addition and subtraction of fractions. If the denominators of fractions are the same, then in order to add fractions, you need to add their numerators, and in order to subtract fractions, you need to subtract their numerators (in the same order). The resulting sum or difference will be the numerator of the result; the denominator will remain the same. If the denominators of the fractions are different, you must first reduce the fractions to a common denominator. When adding mixed numbers, their integer and fractional parts are added separately. When subtracting mixed numbers, we recommend that you first convert them to the form of improper fractions, then subtract from one another, and then again reduce the result, if necessary, to the form of a mixed number.
EXAMPLE

Multiplication of fractions. To multiply a number by a fraction means to multiply it by the numerator and divide the product by the denominator. Hence we have general rule multiplication of fractions: to multiply fractions, you need to multiply their numerators and denominators separately and divide the first product by the second.
EXAMPLE

Division of fractions. To divide a number by a fraction, you need to multiply that number by its reciprocal. This rule follows from the definition of division (see section “Arithmetic operations”).
EXAMPLE

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Decimal system Decimal number system - positional number system based on 10. The most common number system in the world. To write numbers, the most commonly used characters are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, called Arabic numerals. Base 10 is thought to be related to the number of fingers a person has. .

Mathematics. Grade 1 - 4 In this section you will get acquainted with such concepts and terms as addition, subtraction, multiplication and division. You will also get acquainted with mathematical operations and the order in which they are performed, mathematical fairy tales and much, much more. .

for-schoolboy.ru

Adding ordinary fractions is done like this:

a) if the denominators of the fractions are the same, then the numerator of the second fraction is added to the numerator of the first fraction and the same denominator is left, i.e.

b) if the denominators of the fractions are different, then the fractions are first reduced to a common denominator, preferably to the smallest one, and then rule a) is applied.

Example 1. Add fractions and Solution. We have:

Subtraction of ordinary fractions is performed as follows:

a) if the denominators of the fractions are the same, then

b) if the denominators are different, then first the fractions are reduced to a common denominator, and then rule a) is applied.

Multiplication of ordinary fractions is performed as follows:

that is, they multiply the numerators separately, the denominators separately, the first product is made the numerator, the second the denominator.

For example,

The division of ordinary fractions is performed as follows:

i.e. the dividend is multiplied by the reciprocal of the divisor

For example, .

Example 2. Find the value of a numeric expression

Decision. 1) Having reduced the numerator and denominator by 3 (it is useful to do this before performing the multiplication operations in the numerator and denominator), we get i.e. So

3) When finding the value of the expression, the actions of addition and subtraction can be performed simultaneously. The least common multiple of the numbers 15, 20, 30 is the number 60. We bring all three fractions to the denominator 60 using additional factors: for the first fraction 4, for the second - 3, for the third - 2. We get:

Example 3. Perform actions: a)

Solution, a) The first way. Let's turn each of these mixed numbers into an improper fraction, and then perform the addition:

Let's turn the improper fraction into a mixed number:

The second way. We have

b) In the case of multiplication and division of mixed numbers, they always go to improper fractions:

So at 7

Operations with common fractions

Sections: Mathematics

1) control and systematization of students' knowledge on the topic;

2) develop computational skills, logic, mathematical vigilance;

3) to cultivate independence, interest in the subject, a conscientious attitude to educational work.

EQUIPMENT: computer class, PC - 9 pcs.

1) student-centered learning;

2) level differentiation;

3) gaming technology;

2. STATEMENT OF THE GOAL OF THE LESSON.

Today on the eve control work we will have the opportunity to analyze our learning activities and work out the computational skills of performing all actions with ordinary fractions on an electronic simulator.

Students write down the number and name of the work on specially prepared sheets.

3. UPDATING THE BASIC KNOWLEDGE

To get permission to individual work you must verbally answer the questions (everyone on the table didactic material A.P. Ershova, V.V. Goloborodko " oral mathematics»):

1. Formulate the main property of a fraction.

2. The rule for finding the least common denominator of two fractions.

3. Add up

4. What numbers are called mutually inverse?

5. How to divide a fraction into a fraction?

Students frontally repeat the rules for performing actions with ordinary fractions and complete the task with commenting.

4. INSTRUCTIONS for completing the steps of the lesson

Today you have the opportunity to test yourself in 3 categories: computer scientists, mathematicians and analysts. Students are divided into 3 groups and receive self-analysis cards (Appendix 1), according to which they go through all the stages. (The teacher fixes the grades of all three stages and sets the arithmetic mean in the team cards Appendix 2)

On a computer, on grade sheets, on correction cards or creative assignments

5. Stage 1 ELECTRONIC SIMULATOR (Appendix 3) - computer science

First of all, your success at this stage depends on how carefully you follow the rules of the Biathlon game.

The training consists of three stages, differing from each other in the complexity of the tasks. Each stage includes a "ski race" and a "firing line". In the "cross-country skiing" mode, you need to determine whether the proposed statement is true or false and click on the appropriate button on the screen.

In the "on the firing line" mode, you must complete four (stage 1) or three (stages 2 and 3) tasks for calculating the sum, difference, product, or private of two fractions. Your answer is a shot at the target. You hit the bull's-eye if your answer is an irreducible fraction.

The teacher records the grades given by the computer. On the team map.

Oral independent work study.

Students verbally answer questions, perform actions and record the result on a computer. And in the self-analysis map they fix their mistakes.

(each student of the group at the computer)

At the end of the game, the computer evaluates the student.

6. Stage 2 THEORY CREDIT ( A.P. Ershova "Oral Mathematics"):— analysts

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Ordinary fractions. Actions on ordinary fractions

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Topic number 1.

Arithmetic calculations. Interest.

Ordinary fractions. Operations on ordinary fractions.

1º. Integers are the numbers used in counting. The set of all natural numbers is denoted by N, i.e. N= .

Shot is called a number consisting of several fractions of one. Common fraction is called a number of the form , where natural number n shows how much equal parts unit is divided, and a natural number m shows how many such equal parts are taken. Numbers m and n are called respectively numerator and denominator fractions.

If the numerator is less than the denominator, then the fraction is called correct; If the numerator is equal to or greater than the denominator, then the fraction is called wrong. A number that consists of an integer and a fractional part is called mixed number.

For example, - proper ordinary fractions, - improper ordinary fractions, 1 - mixed number.

2º. When performing operations on ordinary fractions, remember the following rules:

1) Basic property of a fraction. If the numerator and denominator of a fraction are multiplied or divided by the same natural number, then a fraction equal to the given one will be obtained.

For example, a) ; b) .

Dividing the numerator and denominator of a fraction by their common divisor, which is different from one, is called fraction reduction.

2) To represent a mixed number in the form improper fraction, you need to multiply its integer part by the denominator of the fractional part and add the numerator of the fractional part to the resulting product, write the resulting amount as the numerator of the fraction, and leave the denominator the same.

Similarly, any natural number can be written as an improper fraction with any denominator.

For example, a) , because ; b) etc.

3) In order to write an improper fraction as a mixed number (i.e., select an integer part from an improper fraction), you need to divide the numerator by the denominator, take the quotient as the integer part, the remainder as the numerator, leave the denominator the same.

For example, a), since 200: 7 = 28 (remaining 4);
b), since 20: 5 = 4 (remaining 0).

4) To bring fractions to the lowest common denominator, you need to find the least common multiple (LCM) of the denominators of these fractions (it will be their least common denominator), divide the least common denominator by the denominators of these fractions (i.e. find additional factors for fractions) , multiply the numerator and denominator of each fraction by its additional factor.

For example, let's reduce fractions to the lowest common denominator:

630: 18 = 35, 630: 10 = 63, 630: 21 = 30.

Means, ; ; .

5) rules arithmetic operations over ordinary fractions:

a) Addition and subtraction of fractions with the same denominators is performed according to the rule:

b) Adding and subtracting fractions with different denominators is carried out according to rule a), having previously reduced the fractions to the least common denominator.

c) When adding and subtracting mixed numbers, you can convert them to improper fractions, and then follow the rules a) and b),

d) When multiplying fractions, use the rule:

e) To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor:

f) When multiplying and dividing mixed numbers, they are first converted to improper fractions, and then rules d) and e) are used.

Presentation on the subject "Mathematics" on the topic: "Presentation for the lesson "Actions with ordinary fractions" Performed by the teacher of mathematics Kolbina Evgenia Viktorovna.". Download for free and without registration. - Transcript:

1 Presentation for the lesson "Actions with ordinary fractions" Made by the teacher of mathematics Kolbina Evgenia Viktorovna

2 objectives of the lesson. Educational: repetition of the rules of comparison, addition, subtraction, multiplication and division of ordinary fractions; generalization and systematization of knowledge about ordinary fractions, consolidation and improvement of skills of actions with ordinary fractions; developing oral counting skills and the ability to apply rules when solving more difficult examples. Developing: development of skills of educational and cognitive activity; development of a culture of oral and writing; development of skills of self-control and self-assessment of the achieved knowledge and skills. Educational: education of attentiveness, activity, independence, responsibility.

3 What mathematicians, drummers and even hunters cannot do without?

4 What month is it? What season? What do you like about winter?

5 Today in the lesson we will sculpt a snowman, but not from snow, but from our knowledge

6 Evaluation paper(First name of the student) "Snowdrifts" "1 com" "2 com" "3 com" "Attributes" Total Grade

7 1. To compare (add, subtract) fractions with different ones, you must: 1) bring these fractions to; 2) compare (add, subtract) the resulting fractions. 2. To add (subtract) mixed numbers, you must: 1) bring the fractional parts to; 2) separately perform addition (subtraction) of parts and fractional parts. 3. To multiply a fraction by a natural number, you need to multiply it by this number, and leave it unchanged. denominators NOZ (least common denominator) NOZ integers numerator denominator 4. To multiply a fraction by a fraction, you need to find the product and the product. 5. In order to multiply mixed numbers, you need to write them as fractions, and then use the rule of fractions. 6. To divide one fraction by another, you need to multiply by the number, the divisor. numerators denominators of incorrect multiplication divisible inverse "SUGROBS" For each correct rule - 1 point

8 "1 kom" For each correct answer - 1 point

10 I Option 635(a) II Option 635(b) "2 com" For each correct action - 1 point

12 The grass is small, small. Trees are tall. The wind shakes the trees. It tilts to the right, then to the left. Up, then back. That bends down. The birds are flying away. The students sit quietly at their desks. Fizminutka

13 Problem The tourists went on a hike. On the first day they walked a kilometer, which is more than on the second day. And on the third day they walked 2 times less than on the first. How many kilometers did the tourists walk during these three days? "3 room"

14 1) find how many tourists traveled on the second day, for this we subtract 2) find how many tourists traveled on the third day, for this we divide by 2 3) add the result of 1 action and the result of the second action and find how much they traveled for these three days. Answer: Solution plan For each correct action - 1 point + 1 point for the correct answer

16 Test "Attributes" For each correct answer 1 point

18 27-30 points - "5" points - "4" points - "3" 0-14 points - "2"

19 Homework: 635 (d), 643 Prepare a report on the topic: the origin of ordinary fractions

20 Lesson summary I liked everything! Difficult but interesting! Tired!

21 The great Russian writer L.N. Tolstoy believed that a person is like a fraction, the denominator of which is what he thinks about himself, and the numerator is what they think about him. I wish you that the numerator in your life was greater than the denominator.

Students are introduced to fractions in 5th grade. Before people who knew how to perform actions with fractions were considered very smart. The first fraction was 1/2, that is, half, then 1/3 appeared, and so on. For several centuries, the examples were considered too complex. Now detailed rules have been developed for converting fractions, addition, multiplication and other actions. It is enough to understand the material a little, and the solution will be given easily.

An ordinary fraction, which is called a simple fraction, is written as a division of two numbers: m and n.

M is the dividend, that is, the numerator of the fraction, and the divisor n is called the denominator.

Select proper fractions (m< n) а также неправильные (m >n).

A proper fraction is less than one (for example, 5/6 - this means that 5 parts are taken from one; 2/8 - 2 parts are taken from one). An improper fraction is equal to or greater than 1 (8/7 - the unit will be 7/7 and one more part is taken as a plus).

So, a unit is when the numerator and denominator matched (3/3, 12/12, 100/100 and others).

Actions with ordinary fractions Grade 6

With simple fractions, you can do the following:

Expand fraction. If we multiply the top and lower part fractions for any the same number(only not by zero), then the value of the fraction will not change (3/5 = 6/10 (just multiplied by 2).
Reducing fractions is similar to expanding, but here they are divided by a number.
Compare. If two fractions have the same numerator, then the fraction with the smaller denominator will be larger. If the denominators are the same, then the fraction with the largest numerator will be larger.
Perform addition and subtraction. At same denominators this is easy to do (we sum the upper parts, and the lower part does not change). For different ones, you will have to find a common denominator and additional factors.
Multiply and divide fractions.

Examples of operations with fractions are considered below.

Reduced fractions Grade 6

To reduce means to divide the top and bottom of a fraction by some equal number.

The figure shows simple examples of reduction. In the first option, you can immediately guess that the numerator and denominator are divisible by 2.

On a note! If the number is even, then it can be divisible by 2. Even numbers is 2, 4, 6…32 8 (ends in even), etc.

In the second case, when dividing 6 by 18, it is immediately clear that the numbers are divisible by 2. Dividing, we get 3/9. This fraction is also divisible by 3. Then the answer is 1/3. If you multiply both divisors: 2 by 3, then 6 will come out. It turns out that the fraction was divided by six. This gradual division is called successive reduction of the fraction by common divisors.

Someone will immediately divide by 6, someone will need division by parts. The main thing is that at the end there is a fraction that cannot be reduced in any way.

Note that if the number consists of digits, the addition of which will result in a number divisible by 3, then the original can also be reduced by 3. Example: the number 341. Add the numbers: 3 + 4 + 1 = 8 (8 is not divisible by 3, so the number 341 cannot be reduced by 3 without a remainder). Another example: 264. Add: 2 + 6 + 4 = 12 (divided by 3). We get: 264: 3 = 88. This will simplify the reduction of large numbers.

In addition to the method of successive reduction of a fraction by common divisors, there are other ways.

GCD is the largest divisor for a number. Having found the GCD for the denominator and numerator, you can immediately reduce the fraction by the desired number. The search is carried out by gradually dividing each number. Next, they look at which divisors match, if there are several of them (as in the picture below), then you need to multiply.

Mixed fractions grade 6

All improper fractions can be converted into mixed fractions by isolating the whole part in them. The integer is written on the left.

Often you have to make a mixed number from an improper fraction. The conversion process in the example below: 22/4 = 22 divided by 4, we get 5 integers (5 * 4 = 20). 22 - 20 = 2. We get 5 integers and 2/4 (the denominator does not change). Since the fraction can be reduced, we divide the upper and lower parts by 2.

It is easy to turn a mixed number into an improper fraction (this is necessary when dividing and multiplying fractions). To do this: multiply the whole number by the lower part of the fraction and add the numerator to this. Ready. The denominator does not change.

Calculations with fractions Grade 6

Mixed numbers can be added. If the denominators are the same, then this is easy to do: add up the integer parts and numerators, the denominator remains in place.

When adding numbers with different denominators, the process is more complicated. First, we bring the numbers to one itself small denominator(NOZ).

In the example below, for the numbers 9 and 6, the denominator will be 18. After that, additional factors are needed. To find them, you should divide 18 by 9, so an additional number is found - 2. We multiply it by the numerator 4, we get the fraction 8/18). The same is done with the second fraction. We already add the converted fractions (whole numbers and numerators separately, we do not change the denominator). In the example, the answer had to be converted to a proper fraction (initially, the numerator turned out to be greater than the denominator).

Please note that with the difference of fractions, the algorithm of actions is the same.

When multiplying fractions, it is important to place both under the same line. If the number is mixed, then we turn it into simple fraction. Next, multiply the top and bottom parts and write down the answer. If it is clear that fractions can be reduced, then we reduce immediately.

In this example, we didn’t have to cut anything, we just wrote down the answer and highlighted the whole part.

In this example, I had to reduce the numbers under one line. Though it is possible to reduce also the ready answer.

When dividing, the algorithm is almost the same. First, we turn the mixed fraction into an improper one, then we write the numbers under one line, replacing the division with multiplication. Do not forget to swap the upper and lower parts of the second fraction (this is the rule for dividing fractions).

If necessary, we reduce the numbers (in the example below, they reduced it by five and two). We transform the improper fraction by highlighting the integer part.

Basic tasks for fractions Grade 6

The video shows a few more tasks. For clarity, we used graphic images solutions to help visualize fractions.

Examples of fraction multiplication Grade 6 with explanations

Multiplying fractions are written under one line. After that, they are reduced by dividing by the same numbers (for example, 15 in the denominator and 5 in the numerator can be divided by five).

Comparison of fractions Grade 6

To compare fractions, you need to remember two simple rules.

Rule 1. If the denominators are different

Rule 2. When the denominators are the same

For example, let's compare the fractions 7/12 and 2/3.

We look at the denominators, they do not match. So you need to find a common one.
For fractions, the common denominator is 12.
We divide 12 first by the lower part of the first fraction: 12: 12 = 1 (this is an additional factor for the 1st fraction).
Now we divide 12 by 3, we get 4 - add. multiplier of the 2nd fraction.
We multiply the resulting numbers by numerators to convert fractions: 1 x 7 \u003d 7 (first fraction: 7/12); 4 x 2 = 8 (second fraction: 8/12).
Now we can compare: 7/12 and 8/12. Turned out: 7/12< 8/12.

To represent fractions better, you can use drawings for clarity, where an object is divided into parts (for example, a cake). If you want to compare 4/7 and 2/3, then in the first case, the cake is divided into 7 parts and 4 of them are chosen. In the second, they divide into 3 parts and take 2. With the naked eye, it will be clear that 2/3 will be more than 4/7.

Examples with fractions grade 6 for training

As an exercise, you can perform the following tasks.

Compare fractions

do the multiplication

Tip: if it is difficult to find the lowest common denominator of fractions (especially if their values are small), then you can multiply the denominator of the first and second fractions. Example: 2/8 and 5/9. Finding their denominator is simple: multiply 8 by 9, you get 72.

Solving equations with fractions Grade 6

In solving equations, you need to remember the actions with fractions: multiplication, division, subtraction and addition. If one of the factors is unknown, then the product (total) is divided by the known factor, that is, the fractions are multiplied (the second is turned over).

If the dividend is unknown, then the denominator is multiplied by the divisor, and to find the divisor, you need to divide the dividend by the quotient.

Imagine simple examples solving equations:

Here it is only required to produce the difference of fractions, without leading to a common denominator.

The division by 1/2 was replaced by multiplication by 2 (the fraction was reversed).

Adding 1/2 and 3/4, we came to a common denominator of 4. At the same time, an additional factor of 2 was needed for the first fraction, 2/4 came out of 1/2.

Added 2/4 and 3/4 - got 5/4.

We did not forget about multiplying 5/4 by 2. By reducing 2 and 4 we got 5/2.

The answer is an improper fraction. It can be converted to 1 whole and 3/5.

In the second method, the numerator and denominator were multiplied by 4 to shorten the bottom rather than flip the denominator.

496. To find X, if:

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

2) If you subtract 10 1/2 from 7/10 of an unknown number, you get 15 2/5. Find an 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 an unknown number is increased by 2/3 of it, you get 60. What is this number?

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

501. 1) The yield of potatoes with a square-nest planting is on average 150 centners per 1 ha, and with a normal planting 3/5 of this amount. How many more potatoes can be harvested from an area of 15 hectares if potatoes are planted in a square-nest way?

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

502. 1) Pioneers assembled within three days 56 kg of different seeds. 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 wheat, it turned out: flour 4/5 of the total amount of wheat, semolina - 40 times less than flour, and the rest is bran. How much flour, semolina and bran separately did you get when grinding 3 tons of wheat?

503. 1) Three garages fit 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 in the third garage there are 1 1/2 times as many cars as in the first. How many cars fit in each garage?

2) The plant, which has three workshops, employs 6,000 workers. The number of workers in the second workshop is 1 1/2 times less 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 in each shop?

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

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

505. 1) The icebreaker made its way through the ice field for three days. On the first day he covered 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 distance traveled by the icebreaker in three days.

2) Three detachments of schoolchildren planted trees for landscaping the village. The first detachment planted 7/20 of all the 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 total 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 plot?

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

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

2) The city consists of four districts. And in the first district live 4/13 of all the inhabitants of the city, in the second 5/6 of the inhabitants of the first district, in the third 4/11 of the inhabitants of the first; two districts combined, and the fourth district is home to 18,000 people. 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 path, on the second 9/10 of what he walked on the first day, and on the third the rest of the path, 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 traveled all the way 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 day, 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 took away the land workers of three plants for garden plots. The first plant was assigned 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 rest of the plots. How many plots were allotted to the workers of three factories if the first plant was given 50 fewer plots than the third?

2) The plane delivered a change of winterers to polar station from Moscow in three days. On the first day he flew 2/5 of the entire path, on the second - 5/6 of the path he traveled 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 factory workers; 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 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 the families of 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 fewer than in the second. How many families are on the collective farm?

511. 1) The Artel spent in the first week 1/3 of its stock of raw materials, and in the second 1/3 of the remainder. 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 for heating the house in the first month, 1/6 of it was spent, and in the second month - 3/8 of the remainder. How much coal is left for heating the house if 1 3/4 more was used in the second month than in the first month?

512. 3 / 5 of the entire land of the collective farm is allocated for sowing grain, 13 / 36 of the rest is occupied by vegetable gardens and meadows, the rest of the land is forested, and the sown area of the collective farm is 217 hectares more area forests, 1 / 3 of the land allotted for grain crops is sown with rye, and the rest with 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. During this route, the tram makes 18 stops, spending on average up to 1 1/6 minutes per stop. The average tram speed along the entire route is 12 1/2 km per hour. How long does it take for a tram to make one trip?

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

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

2) A steamboat travels downstream between two cities in 3 days. and back the same distance in 4 days. How many days will the rafts float 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 h 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) Playground rectangular shape has a length of 45 1/2 m, and its width is 5/13 of the length. This area is bordered by a path 4/5 m wide. Find the area of the path.

516. Find the mean arithmetic numbers:

517. 1) The arithmetic mean of two numbers 6 1 / 6 . One of the numbers 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 walked 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 ha, on the second day 15 3/4 ha, and on the third day 14 1/2 ha. How many hectares of land did a tractor driver plow on average per day?

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

520. 1) Three families live in the house. The first family for lighting the apartment has 3 light bulbs, the second 4 and the third 5 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) The polisher rubbed the floors in the apartment where three families lived. The first family had a living area of 36 1/2 sq. m, the second in 24 1/2 sq. m, and the third - in 43 sq. m. For all the work was paid 2 rubles. 08 kop. How much did each family pay?

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

2) A field-growing team on an area of 300 ha received a harvest of 20 1/2 centners of winter wheat per 1 ha, from 80 hectares 24 centners per 1 ha, and from 20 hectares - 28 1/2 centners per 1 ha. What is the average yield in a brigade from 1 hectare?

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

2) If you add up the numbers expressing the width of the Tatarsky and the width 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) Amount three numbers 35 2 / 3 . The first number is 5 1/3 greater than the second and 3 5/6 greater than the third. 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 more area 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 sq. m more than the area of the third. What is the area of the second room?

2) The cyclist during the three-day competition on the first day traveled 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 4 3/4 kg of cereals are poured from one box into another, then in both boxes there will be equal amounts of cereals. How many cereals are 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 to the second, then the first will still be more than the second by 2 17/30. Find both numbers.

2) Two boxes contain 24 1/4 kg of apples. If 3 1/2 kg are transferred 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 was moving along the river at a speed of 15 1/2 km per hour, and against the current 8 1/4 km per hour. What is the speed of the river?

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) Living space apartment, consisting of two rooms, is equal to 47 1 / 2 sq. 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 more than the first. Find these numbers.

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

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

2) Father older than son for 24 years. The number of the son's years is 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 more than its numerator. What is a fraction equal to 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 greater is the second number 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 picked 100 mushrooms together. 3/8 number of mushrooms, collected first boy, 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 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 greater than another. Find these numbers if 2/5 of one number is equal to 2/3 of another.

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 brigade can complete some work in 36 days, and the second in 45 days. How many days will it take both teams working together to complete this task?

2) A passenger train travels the distance between two cities in 10 hours, and a freight train travels 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 travels the distance between two cities in 6 1/4 hours, and a passenger train in 7 1/2 hours. In how many hours 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 two cities at the same time 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 the departure will the motorcyclists meet? (Round answer to the nearest 1 hour.)

544. 1) Three cars of different carrying capacity can carry 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 move the same cargo by 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) There are two taps connected to the bath. 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 bath if both taps are opened at once?

2) Two typists must retype the manuscript. The first woman can do this job in 3 1/3 days, and the second one 1 1/2 times faster. In how many days will both typists complete the work if they work at the same time?

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?

Instruction. 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 to plow this field, working alone?

547 *. Two trains leave two stations at the same time towards each other and meet after 18 hours. after its release. How long does it take the second train to travel the distance between stations if the first train travels this distance in 1 day and 21 hours?

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

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

2) From the collective farm to the city 24 km. A truck has left the collective farm and travels 1 km in 2 1/2 minutes. After 15 min. after the departure of this car from the city, a cyclist left the collective farm, at a speed half that of a truck. How long after his departure will the cyclist meet the truck?

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

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

551. 1) From two collective farms, through which the road to the district center passes, two collective farmers left at the same time to the district on horseback. The first of them traveled 8 3/4 km per hour, and the second 1 1/7 times the first. The second collective farmer overtook the first in 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 is 60 km per hour, the TU-104 aircraft took off in the same direction, at a speed 14 1/6 times the speed of the train. How many hours after the flight will the plane overtake the train?

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

2) Powerboat traveled 207 km downstream in 13 1/2 hours, spending 1/9 of that 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 on the reservoir covered a distance of 52 km without stopping in 3 hours and 15 minutes. Further, going along the river against the current, the speed of which is 1 3 / 4 km per hour, this boat traveled 28 1 / 2 km in 2 1 / 4 hours, making 3 equal stops in the process. How many minutes did the boat stop at each stop?

554. From Leningrad to Kronstadt at 12 noon. the next day a steamboat set out and covered the entire distance between these cities in 1 1/2 hours. On the way, he met another steamer that left Kronstadt for Leningrad at 12:18. and walking at a speed 1 1/4 times greater than 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 delayed for 1 hour and 10 minutes. At what speed must he continue his journey in order to arrive at his destination without delay?

556. At 4 o'clock 20 min. in the morning a freight train left Kyiv for Odessa average speed 31 1/5 km per hour. After some time, a mail train left Odessa to meet it, the speed of which is 1 17/39 times the speed of the freight train, and met with a freight train 6 1/2 hours after its departure. At what time did the postal train leave Odessa if the distance between Kyiv and Odessa is 663 km?

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

558. 1) The factory has three workshops. The number of workers in the first workshop is 9/20 of all the 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 workers less than in the second. How many workers are 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 students less than in the second. How many students are in the three schools?

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

2) Two brothers bought a camera. One had 5/8, and the second had 4/7 of the cost of the camera, and the first had 2 rubles. 25 kop. more than the second. Each paid half the cost of the apparatus. How much money does each have?

560. 1) From city A to city B, the distance between them is 215 km, a car left 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 car travel before meeting the truck if the speed of the truck per hour was 18/25 of the speed of the car?

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

561. The collective farm harvested wheat and rye. Wheat was sown 20 hectares more than rye. General fee rye amounted to 5/6 of the total wheat harvest with a yield of 20 centners 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 meet its needs. How many trips did the two-ton trucks need to make to take out the grain sold to the state?

562. Rye and wheat flour was brought to the bakery. The weight of wheat flour was 3/5 of the weight of rye flour, and rye flour was brought 4 tons more than wheat. How much wheat and how much rye bread will be baked by the bakery from this flour, if the baked goods are 2/5 of all flour?

563. Within three days, a team of workers completed 3/4 of the entire work to repair the highway between the two collective farms. On the first day, 2 2/5 km of this highway was 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 vacancies 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 plot.

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 plot.

566. 1) The perimeter of the rectangle is 6 1/2 dm, its base is 1/4 dm more height. Find the area of this rectangle.

2) The perimeter of a 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, dividing them into rectangles and finding the dimensions of the rectangle by measuring.

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

2) How many boards 4 1/2 l long and 1/4 m wide will be required 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 crop was harvested from a rectangular field at 25 centners per 1 ha. How much wheat was harvested from the whole field if the field is 800 m long and 3/8 of its length wide?

570 . 1) A rectangular plot of land, having a length of 78 3/4 m and a width of 56 4/5 m, 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 proposes to plant a garden. How many trees will be planted in this garden if, on average, an area of 36 square meters is required for each tree?

571. 1) For normal daylight illumination of the room, it is necessary that the area of \u200b\u200ball windows be at least 1/5 of the floor area. Determine if there is enough light in a room that is 5 1/2 m long and 4 m wide. Does the room have one window measuring 1 1/2 m x 2 m?

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

572. 1) The barn measures 5 1/2 m x 4 1/2 m x 2 1/2 m. m of hay weighs 82 kg?

2) The woodpile is shaped cuboid, whose dimensions are 2 1/2 m x 3 1/2 m x 1 1/2 m. What is the weight of the woodpile if 1 cu. m of firewood weighs 600 kg?

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

2) The pool, having the shape of a rectangular parallelepiped, has a length of 6 1/2 m, a width of 4 m and a height of 2 m. 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 is to be built around a rectangular piece of land 75 m long and 45 m wide. How many cubic meters of boards should go to his device 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 of the minute and hour hand at 13 o'clock? at 15 o'clock? at 17 o'clock? at 21 o'clock? at 23:30?

2) By how many degrees will the hour hand turn 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 circle? 5 / 24 circles?

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

2) Measure the angles of the figure with a protractor and find the sum of all the angles of each figure (Fig. 31).

577. Run actions:

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

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

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

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

579. 1) The chart "Literacy of the population in the USSR" (Fig. 32) shows the number of literate per hundred people of the population. According to the diagram and its scale, determine the number of literate men and women for each of the indicated years.

Record the results in a table:

2) Using the data of the diagram "Soviet envoys to space" (Fig. 33), make up tasks.

580. 1) According to the sector diagram "Daily routine for a student of grade V" (Fig. 34), fill in the table and answer the questions: what part of the day is devoted to sleep? for homework? to school?

2) Build a pie chart about the mode of your day.