Ordinary and decimal fractions and operations on them. Rules for arithmetic operations on ordinary fractions Fractional examples with decimal operations
The decimal is used when you need to perform operations with non-integer numbers. This may seem irrational. But this type of numbers greatly simplifies the mathematical operations that need to be performed with them. This understanding comes over time, when writing them becomes familiar, and reading them does not cause difficulties, and the rules of decimal fractions have been mastered. Moreover, all actions repeat already known ones, which have been learned with natural numbers. You just need to remember some features.
Decimal definition
A decimal is a special representation of a non-integer number with a denominator that is divisible by 10, giving the answer as one and possibly zeros. In other words, if the denominator is 10, 100, 1000, and so on, then it is more convenient to rewrite the number using a comma. Then the whole part will be located before it, and then the fractional part. Moreover, the recording of the second half of the number will depend on the denominator. The number of digits that are in the fractional part must be equal to the digit of the denominator.
The above can be illustrated with these numbers:
9/10=0,9; 178/10000=0,0178; 3,05; 56 003,7006.
Reasons for using decimals
Mathematicians needed decimals for several reasons:
Simplifying recording. Such a fraction is located along one line without a dash between the denominator and numerator, while clarity does not suffer.
Simplicity in comparison. It is enough to simply correlate numbers that are in the same positions, while with ordinary fractions you would have to reduce them to a common denominator.
Simplify calculations.
Calculators are not designed to accept fractions; they use decimal notation for all operations.
How to read such numbers correctly?
The answer is simple: just like an ordinary mixed number with a denominator that is a multiple of 10. The only exception is fractions without an integer value, then when reading you need to pronounce “zero integers.”
For example, 45/1000 should be pronounced as forty-five thousandths, at the same time 0.045 will sound like zero point forty five thousandths.
A mixed number with an integer part of 7 and a fraction of 17/100, which would be written as 7.17, would in both cases be read as seven point seventeen.
The role of digits in writing fractions
Correctly marking the rank is what mathematics requires. Decimals and their meaning can change significantly if you write the digit in the wrong place. However, this was true before.
To read the digits of the whole part of a decimal fraction, you simply need to use the rules known for natural numbers. And on the right side they are mirrored and read differently. If the whole part sounded “tens”, then after the decimal point it will be “tenths”.
This can be clearly seen in this table.
| Class | thousands | units | , | fraction | |||||||
| discharge | cell | dec. | units | cell | dec. | units | tenth | hundredth | thousandth | ten-thousandth | |
How to correctly write a mixed number as a decimal?
If the denominator contains a number equal to 10 or 100, and others, then the question of how to convert a fraction to a decimal is not difficult. To do this, it is enough to rewrite all its components differently. The following points will help with this:
write the numerator of the fraction a little to the side, at this moment the decimal point is located on the right, after the last digit;
move the comma to the left, the most important thing here is to count the numbers correctly - you need to move it by as many positions as there are zeros in the denominator;
if there are not enough of them, then there should be zeros in the empty positions;
the zeros that were at the end of the numerator are now not needed and can be crossed out;
Before the comma, add the whole part; if it was not there, then there will also be zero here.
Attention. You cannot cross out zeros that are surrounded by other numbers.
You can read below about what to do in a situation where the denominator has a number not only consisting of ones and zeros, and how to convert a fraction to a decimal. This is important information that you should definitely read.
How to convert a fraction to a decimal if the denominator is an arbitrary number?
There are two options here:
When the denominator can be represented as a number that is equal to ten to any power.
If such an operation cannot be performed.
How can I check this? You need to factor the denominator. If only 2 and 5 are present in the product, then everything is fine, and the fraction is easily converted to a final decimal. Otherwise, if 3, 7 and other prime numbers appear, the result will be infinite. It is customary to round such a decimal fraction for ease of use in mathematical operations. This will be discussed a little below.
Explores how decimals are made, 5th grade. Examples here will be very helpful.
Let the denominators contain the numbers: 40, 24 and 75. The decomposition into prime factors for them will be as follows:
- 40=2·2·2·5;
- 24=2·2·2·3;
- 75=5·5·3.
In these examples, only the first fraction can be represented as the final fraction.
Algorithm for converting a common fraction to a final decimal
Check the factorization of the denominator into prime factors and make sure that it will consist of 2 and 5.
Add as many 2s and 5s to these numbers so that there are an equal number of them. They will give the value of the additional multiplier.
Multiply the denominator and numerator by this number. The result will be an ordinary fraction, under the line of which there is 10 to some extent.
If in the problem these actions are performed with a mixed number, then it must first be represented as an improper fraction. And only then act according to the described scenario.
Representing a fraction as a rounded decimal
This method of converting a fraction to a decimal may seem even easier to some. Because it doesn't have a lot of action. You just need to divide the numerator by the denominator.
Any number with a decimal part to the right of the decimal point can be assigned an infinite number of zeros. This property is what you need to take advantage of.
First, write down the whole part and put a comma after it. If the fraction is correct, write zero.
Then you need to divide the numerator by the denominator. So that they have the same number of digits. That is, add the required number of zeros to the right of the numerator.
Perform long division until the required number of digits is reached. For example, if you need to round to hundredths, then the answer should be 3. In general, there should be one more number than you need to get in the end.
Write down the intermediate answer after the decimal point and round according to the rules. If the last digit is from 0 to 4, then you just need to discard it. And when it is equal to 5-9, then the one in front of it needs to be increased by one, discarding the last one.
Return from decimal to common fraction
In mathematics, there are problems when it is more convenient to represent decimal fractions in the form of ordinary fractions, in which there is a numerator with a denominator. You can breathe a sigh of relief: this operation is always possible.
For this procedure you need to do the following:
write down the whole part, if it is equal to zero, then there is no need to write anything;
draw a fraction line;
above it, write down the numbers from the right side, if the zeros come first, then they need to be crossed out;
under the line write one with as many zeros as there are digits after the decimal point in the original fraction.
That's all you need to do to convert a decimal to a fraction.
What can you do with decimals?
In mathematics, these will be certain operations with decimals that were previously performed for other numbers.
They are:
comparison;
addition and subtraction;
multiplication and division.
The first action, comparison, is similar to how it was done for natural numbers. To determine which is greater, you need to compare the digits of the whole part. If they turn out to be equal, then they move on to the fractional and also compare them by digits. The number with the largest digit in the most significant digit will be the answer.
Adding and subtracting decimals
These are perhaps the simplest steps. Because they are carried out according to the rules for natural numbers.
So, in order to add decimal fractions, they need to be written one below the other, placing commas in a column. With this notation, whole parts appear to the left of the commas, and fractional parts to the right. And now you need to add the numbers bit by bit, as is done with natural numbers, moving the comma down. You need to start adding from the smallest digit of the fractional part of the number. If there are not enough numbers in the right half, then zeros are added.
The same applies to subtraction. And here there is a rule that describes the possibility of taking a unit from the highest rank. If the fraction being reduced has fewer digits after the decimal point than the fraction being subtracted, then zeros are simply added to it.
The situation is a little more complicated with tasks where you need to multiply and divide decimal fractions.
How to multiply a decimal fraction in different examples?
The rule for multiplying decimal fractions by a natural number is:
write them down in a column, ignoring the comma;
multiply as if they were naturals;
Separate with a comma as many digits as there were in the fractional part of the original number.
A special case is the example in which a natural number is equal to 10 to any power. Then to get the answer you just need to move the decimal point to the right by as many positions as there are zeros in the other factor. In other words, when multiplied by 10, the decimal point moves by one digit, by 100 - there will already be two of them, and so on. If there are not enough numbers in the fractional part, then you need to write zeros in the empty positions.
The rule that is used when a task requires multiplying decimal fractions by another same number:
write them down one after another, not paying attention to commas;
multiply as if they were natural;
Separate with a comma as many digits as there were in the fractional parts of both original fractions together.
A special case are examples in which one of the multipliers is equal to 0.1 or 0.01 and so on. In them you need to move the decimal point to the left by the number of digits in the presented factors. That is, if it is multiplied by 0.1, then the decimal point is shifted by one position.
How to divide a decimal fraction in different tasks?
Dividing decimal fractions by a natural number is performed according to the following rule:
write them down for division in a column as if they were natural ones;
divide according to the usual rule until the whole part is over;
put a comma in the answer;
continue dividing the fractional component until the remainder is zero;
if necessary, you can add the required number of zeros.
If the integer part is equal to zero, then it will not be in the answer either.
Separately, there is division into numbers equal to ten, hundred, and so on. In such problems, you need to move the decimal point to the left by the number of zeros in the divisor. It happens that there are not enough numbers in a whole part, then zeros are used instead. You can see that this operation is similar to multiplying by 0.1 and similar numbers.
To divide decimals, you need to use this rule:
turn the divisor into a natural number, and to do this, move the comma in it to the right to the end;
move the decimal point in the dividend by the same number of digits;
act according to the previous scenario.
The division by 0.1 is highlighted; 0.01 and other similar numbers. In such examples, the decimal point is shifted to the right by the number of digits in the fractional part. If they run out, then you need to add the missing number of zeros. It is worth noting that this action repeats division by 10 and similar numbers.
Conclusion: It's all about practice
Nothing in learning comes easy or without effort. Reliably mastering new material takes time and practice. Mathematics is no exception.
To ensure that the topic about decimal fractions does not cause difficulties, you need to solve as many examples with them as possible. After all, there was a time when adding natural numbers was a dead end. And now everything is fine.
Therefore, to paraphrase a well-known phrase: decide, decide and decide again. Then tasks with such numbers will be completed easily and naturally, like another puzzle.
By the way, puzzles are difficult to solve at first, and then you need to do the usual movements. It’s the same in mathematical examples: having walked along the same path several times, then you will no longer think about where to turn.
Mathematical simulator on the topic
"Joint actions with decimals"
Compiled by a math teacher
Tolmacheva Nadezhda Alekseevna
MBOU Secondary School No. 69, Nizhny Tagil
Explanatory note
The mathematics simulator is intended for 5th-6th grade students; it can be used in working with any teaching materials in mathematics, as well as in preparing 9th grade students for taking the OGE.
The simulator is designed both for use in the classroom and for independent work at home.
The simulator provides the opportunity to develop a conscious application of all the rules for working with decimal fractions.
The simulator can be used as a primary control of knowledge, as well as in correctional work. The simulator tasks allow the student to perform a larger volume of calculations in a short time. In this way, not only computational skills are honed, but also attention is trained, and the student’s working memory is developed.
The simulator tasks can be offered for both individual and group work in the classroom.
Math simulator
| Option 1 |
|
| 15,3 * 5,4 - 4,2* (5,12 – 4,912) + 16,0036 |
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| 9,84 - 16,32 * (8 – 7,45) + 2,186 |
|
| (2,12 + 1,07) * (2,12 – 1,07) |
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| 86,4 * (17,01: 4,2) : 6,4 |
|
| 42,26 – 34,68: (33,32: 9,8) |
|
| 40 – (7,12 + 11,043: 2,7) |
|
| 12,6: (2,04 + 4,26) – 0,564 |
|
| 7,371: (5 – 3,18) + 2,05 *(17,82 – 7) |
|
| (5,2: 26 + 26: 5,2) *6,1 + 5,25: 5 |
|
| 27,5967: (8 – 1,186) + 3,02 |
|
| (20 – 13,7) * 7,4 + 18: 0,6 |
|
| (4,694 - 3,998) : 4,35 + (4,5 * 5,4 – 0,06) |
|
| (4,6 * 3,5 + 15,32) : 31,42 + (7,26 – 5,78) : 0,148 |
|
| (101,96 – 6,8 * 7,2) : 4,24 – 3,4 * (10 – 6,35) |
|
| 7,72 * 2,25 – 4,06: (0,824 + 1,176) – 12,423 |
|
| 51,328: 6, 4 + 3,2 * (10 – 4,7) * 2,05 |
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| (42,12 * 0,12 + 112,016* 0,1) : 1,6 – 9,424 |
|
| ((4,2 *0,81 – 6,8*0,05) : 0,5)) : 200 |
|
| 2,6* (4,4312 + 15,5688) – 6,66: (8,2 – 6,72) |
|
| (0,624: 4,16 + 6,867: 2,18) *2,08 – 4,664 |
|
| 4260 + 42,6: (62,06 + 37,94) – 42,6: (52,44 - 52,43) |
|
| 5: 0,25 + 0,6 *(9,275 – 4,275) : 0,1 |
|
| 3,1: 100 + (6 – 0,3: 100) *10 |
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| 0,415 +(2,85: 0,6*3,2 – 2,72: 8) + 5,134: 0,17 |
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| 0,1: 0,002 – 0,5*(7,91: 0,565 – 11,1:1,48) |
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| 0,2: 0,004 + (7,91: 0,565 – 44,4: 5,92) *0,5 |
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| 4,735: 0,5 + 14,95: 1,3 + 2,121: 0,7 |
|
| (0,1955 + 0,187) : 0,085 |
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| (86,9 + 667,6) : (37,1 + 13,2) |
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| (0,008 + 0,992) * (5 *0,6 – 1,4) |
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Math simulator
Operations with decimals
| Option 2 |
|
| (130,2 – 30,8) : 2,8 - 21,84 |
|
| 3,712: (7 – 3,8) + 1,3* (2,74 + 0,66) |
|
| (3,4: 1,7 + 0,57: 1,9)* 4,9 + 0,0825: 2,75 |
|
| 10,79: 8,3*0,7 - 0,46 * 3,15: 6,9 |
|
| (21,2544: 0,9 + 1,02 * 3,2) : 5,6 |
|
| 4,36: (3,15 + 2,3) + (0,792 – 0,78) * 350 |
|
| (3,91: 2,3 * 5,4 – 4,03) * 2,4 |
|
| 6,93: (0,028 + 0,36 * 4,2) - 3,5 |
|
| 42,165 – 22,165: (0,61 + 3,42) |
|
| ((4: 0,128 + 14628,25) : 1,011* 0,00008 + 6,84) : 12,5 |
|
| 687,8 + (88,0802 – 85,3712) : 0,045 |
|
| (3,1 * 5,3 – 14,39) : 1,7 + 0,8 |
|
| (3,8 * 1,75: 0,95 – 1,02) : 2,3 + 0,4 |
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| ((23,79: 7,8 – 6,8: 17) * 3,04 – 2,04) * 0,85 |
|
| 0,15: 0,01 + (6 + 9,728: 3,2) * 2,5 – 1,4 |
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| 1,44: 3,6 + 0,8 + 3,6: 1,44* (0,1 - 0,02) |
|
| 3,45 * (11,2 + 75,6) – 0,93 * 1,26 |
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| 4,25: 0,25 – 0,06 * 82 + 0,4 |
|
| (0,237 + 45,6) * 12,01 - 11,1* (237,1 – 229,9) |
|
| 5,8 – 0,27 * 3,6 + 5,172 |
|
| 12 – 5,3: (19,6: 0,35 - 0,06 * 50) |
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| (0,6 + 0,25 – 0,125) * 3,2 + 4,5: 100 |
|
| (15,5: 0,25 – 0,08 * 200) : 2,3 – 1,3 |
|
| (87,05 * 2,7 – 55,68:32) * 0,8: 0,02 |
|
| 522,348: 87 + 2,7 * (0,84 – 0,128: 0,16) |
|
| 6400 * 0,0145 – (1272,6: 0,42 – 3000) |
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| (0,7: 1,4 – 0,02) : 0,012 + 1,6 * (0,548 – 0,023) |
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| (1,184: 3,2 + 0,832: 0,4) : 0,5 + 1,5 |
|
| 4,96 ; 10 + 35,8: 100 - 0,0042 |
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| (0,04 + 3,59) * (7,35 + 2,65) : 300 |
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Math simulator
Operations with decimals
| Option 3 |
|
| 2,5 + 0,56* 28 + 0,125*15 – 0,12*7 |
|
| 12,8: 4 + 76,8: 12 – 42,6: 6 – 2,4 |
|
| 4,01 + 43,6: 10 – 73,2: 30 + 15,4: 100 |
|
| 176,4: 100 – 0,041*40 + 13,5:50 +0,3 |
|
| (16,4 + 13,2)*3 – (10,6 + 4,8) *2 – 23,2 |
|
| (40,65 - 32,6) : 5 + (4,72 _ 2,24)*3 |
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| 4,735: 0,5 + 14,95: 1,3 + 2,121: 0,7 – 21,6 |
|
| 0,01105 + 0,05 - 0,3417: 34 -_ 0,875: 125 |
|
| (5,72 – 3,21)*5 + (86,9 + 667,6) : (37,1 + 13,2) |
|
| (0,1955 + 0,187) : 0,085 – (4,72 – 4,72)*0,157 |
|
| 4,9 – (0,008 + 0,992) * (5 *0,6 – 1,4) |
|
| (50000 – 1397,3) : (20,4 + 33,603) – 856 |
|
| 3,7 *0,18 + 35,9 *0,26 – 0,109 *91 |
|
| 34,98: 6,6 + 5,141: 0,53 – 0,8379: 0,057 |
|
| 0,131 *470 + 26,97: 2,9 - 50,4 *1,4 |
|
| 0,439 *97 – 182,75: 4,3 + 31,9 *0,43 |
|
| (20,4 – 18,23)* 4,3 + (0,40713 + 0,44176) : 0,67 |
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| (0,357 + 7,043)*0,85 + (52 – 1,928) : 5,69 |
|
| (1,5 - 0,4732)* 35 – (0,6092 + 0,0718) : 0,75 |
|
| (139,4 + 16,6)* 0,039 - (20 – 17,54) : 2,5 |
|
| 4,1819 + 0,73 *(5,375 + 2,595) |
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| 5,0143 – 65,9*(0,0612 + 0,0058) |
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| (0,83 *3,7 + 9,741:51 – 0,012) : 0,325 |
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| (67,21: 0,143 – 0,546*850 + 2,1) : 1,25 |
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| (79* 0,63 – 9,558: 5,4 – 26,94) : 0,324 |
|
| (11,328: 16 + 7,752: 7,6) : 0,16 |
|
| 13,7 – (0,53 *6,7 + 1,77*3,1 + 0,004) : 0,66 |
|
| 5,3: (2,87* 0,53 – 0,043 *7,7 – 0,19) |
|
| (3,06 – 2,97) * (5,6*0,93 – 0,84*6,2) |
|
| (5,4*0,77 – 0,008) : (2,747: 0,67+ 0,05) |
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Math simulator
Operations with decimals
| Option 4 |
|
| 589,72:16 – 18,305:7 + 5,67: 4 |
|
| (86,9 + 667,6) : (37,1 +13,2) |
|
| (0,93 + 0,07) : (0,93 – 0,805) |
|
| 1,35: 2,7 + 6,02 – 5,9 + 0,4: 2,5 *(4,2 – 1,075) |
|
| ((14,068 + 15,78) : (1,875 + 0,175)) : (0,325+ 0,195) |
|
| (0,578 + 0,172)* (0,823 + 0,117) – 1,711: (4,418 + 1,382) |
|
| (39,3 + 116,7) *0,39 – (19,01 -16,56) : 2,5 |
|
| (2,747: 0,67 + 0,05) : (0,54* 7,7 – 0,008) |
|
| 5,76*4,76: 6,12 + 81,9: 58,5*2,05 |
|
| 25,6: (38,07 + 1,93) + 0,037 *10 |
|
| (3,7011: 0,73 – 9,27: 4,5 – 1,41) :1,6 |
|
| 40,86: 4,5 – 0,6039: 5,49 + 0.338: 0,13 |
|
| (85,9 +667,1) : ((37 +13,2) + (11,44 – 6,42)*10 |
|
| 1,224: (7 – 2,92) + 1,06*(13,5 – 3) |
|
| (7,5* 48 – 8,2* 9,5 + 141,4) : (254,1:4,2) |
|
| 0,63*69 – 10,048: 6,4 – 19,44: 32,4 *0,8 |
|
| (3,8: 19 + 1,9: 3,8) *5,2 + 7,28: 7 |
|
| (4,9 + 1,06 – 0,98) : (0,83*0,6) : 2,4 |
|
| (28,7 *0,15) : (0,25 *0,21) + 22,5:1,25 |
|
| 0,1: 0,002 + (7,91: 0,565 - 11,1: 1,48) |
|
| (0,2028:0,24 – 0,32 *1,5) *(4,05 – 13,1625: 4,05) |
|
| (97,44: 0,48 + 128,64: 3,2) *0,25 – 17,89 |
|
| 5,4 + ((4,7 – 2,85)*1,8 + 0,0156: 0,13) |
|
| (1,2 *0,15 + 12:100 – 1,4: 10) : 0,1 |
|
| 0,545: 0,5 +2,75 *0,4 – 0,45 *3,8 |
|
| 0,6 * (7,24: 0,8 – 0,968: 0,16) + 2,25 *0,04 |
|
| (6,4 *0,025 + 7,07: 3,5 – 3,68: 4) : 0,9 |
|
| 2,5 *(3: 6 – 0,2: 5 + 1,2 *0,15) |
|
| (5,508: 0,27 – 10,2 *1,3) : 0,7 + 1,3: 0,1 |
|
| 1,5 + 0,5*(4,214: 0,14 – 5,436: 1,8) * 0,1 |
|
Answers
Math simulator
Operations with decimals
| Option 1 | Option 2 | Option 3 | Option 4 |
|
Chapter 2 FRACTIONAL NUMBERS AND ACTIONS WITH THEM
§ 45. Problems and examples for all operations with natural numbers and decimal fractions
First level
1620. Find (orally):
1) 1,8 + 3,1; 2) 0,05 + 0,18; 3) 4,2 - 1,2;
4) 100 ∙ 0,15; 5) 57 ∙ 0,1; 6) 0,73: 0,1.
1621. Find (orally):
1) 7,8 + 4,9; 2) 3,7 + 2,51; 3) 1 - 0,6;
4) 2 - 0,17; 5) 0,001 ∙ 29; 6) 4,2: 0,7.
1622. Count (orally):
1) 0,57 + 1,43; 2) 4,27 - 2,07; 3) 4,1 - 2,01;
4) 8 ∙ 1,5; 5) 60: 0,2; 6) 739: 100.
1623. Count (orally):
1) 8,32 ∙ 10; 2) 117,3 ∙ 100; 3) 1,85 ∙ 1000;
4) 3,71 ∙ 0,1; 5) 4,92 ∙ 0,01; 6) 125,3 ∙ 0,001.
1624. Count (orally):
1) 32,7: 10; 2) 45,13: 100; 3) 2792: 1000;
4) 8,3: 0,1; 5) 37,3: 0,01; 6) 13,24: 0,001.
1625. Calculate:
1) 5,18 + 25,37; 2) 0,805 + 7,105;
3) 5,97 + 0,032; 4) 8,91 - 1,328;
5) 71,5 - 16,07; 6) 42 - 7,18.
1626. Calculate:
1) 4,27 + 37,42; 2) 0,913 + 8,39;
3) 4,13 + 0,9027; 4) 4,17 - 0,127;
5) 42,7 - 17,08; 6) 78 - 14,53.
1627. Calculate:
1) 42 ∙ 0,13; 2) 3,6 ∙ 2,5; 3) 7,05 ∙ 800;
4) 15: 4; 5) 72: 2,25; 6) 15,3: 17.
1628. Calculate:
1) 38 ∙ 0,25; 2) 4,8 ∙ 3,5; 3) 4,07 ∙ 900;
4) 18,3: 2; 5) 53,55: 4,25; 6) 406,6: 19.
1629. Write as a decimal:
1630. Write as a common fraction or mixed number:
1) 2,3; 2) 4,07; 3) 0,23; 4) 10,073.
1631. Compare:
1) 4.897 and 4.879; 2) 7.520 and 7.52;
3) 42.57 and 42.572; 4) 9.759 and 9.758.
1632. Compare:
1) 7.896 and 7.869; 2) 8.01 and 8.1;
3) 47.53 and 47.530; 4) 4.571 and 4.578.
Average level
1633. Calculate 2.5 x + 0.37 if:
1) x = 1.6; 2) x = 3.4.
1634. Find the arithmetic mean of the numbers:
1) 0,573; 1,96; 35,24;
2) 4,82; 89,59; 0,462; 9,368.
1635. Find the arithmetic mean of the numbers 20.76; 80.43; 90.24.
1636. In 2.5 hours the train traveled 195 km. How many kilometers will the train travel in 3.6 hours if it moves at the same speed?
1637. Car during t I drove for hours at a speed of 85 km/h. Write an expression to find the distance traveled by the car and calculate it if t is 0.5; 0.8; 1.4; 3.
1638. Calculate the value of the expression 27.3 - a: b if:
1) a = 33.5; b = 2.5; 2) a = 32.16; b = 13.4.
1639. Solve the equations:
1) 12.5 + x = 37.4; 2) in + 13.72 = 18.1;
3) in - 137.8 = 27.41; 4) 17 - x = 12.42.
1640. Solve the equations:
1) 13.7 + a = 18.4; 2) x + 13.42 = 18.9;
3) b - 142.3 = 15.73; 4) 14 - y = 12.142.
1641. Compare the values:
1) 0.4 m and 4 dm; 2) 0.2 dm and 20 cm;
3) 0.07 m and 7 cm; 4) 0.03 km and 300 m
1642. Compare the values:
1) 0.2 t and 2 c; 2) 0.3 c and 31 kg;
3) 0.8 t and 785 kg; 4) 0.08 kg and 80 g.
1643. The speed of a motor ship in still water is 25.4 km/h, and the speed of the river flow is 1.8 km/h. How many kilometers does the ship travel?
1) in 1.5 hours along the river;
2) in 2.4 hours against the flow of the river?
1644. The boat moved first for 1.6 hours along the lake at a speed of 25.5 km/h, and then for 0.8 hours along the river against the current. The current speed is 1.7 km/h. How far did the boat travel?
1645. Find the meaning of the expression:
1) 15 ∙ (2,7 + 4,2);
2) (5,7 - 2,3) : 4;
3) (5,47 - 4,25) ∙ 10;
4) (4,47 + 2,7) : 10;
5) (13,42 - 4,15) ∙ (12,3 - 0,3);
6) (2,17 + 4,45) : (12,6 - 12,5).
1646. Find the meaning of the expression:
1) (2,43 + 4,15) ∙ 1,7;
2) (12,49 - 3,57) : 0,4;
3) (4,17 - 3,8) ∙ (10,1 - 8,1);
4) (15,7 + 14,9) : (2,91 - 1,21).
1647. Solve the equations:
1) 12.5 x = 45; 2) in ∙ 4.8 = 60.6;
3) x: 4.7 = 12.3; 4) 12.7: b = 0.01.
1648. Development of equations:
1) 3.7 y = 7.77; 2) x ∙ 3.48 = 8.7;
3) in: 5.4 = 13.5; 4) 52.54: x = 3.7.
1649. Make up an expression: from the sum of the numbers a and 42.3, subtract the difference between the numbers 15.7 and b . Calculate the value of the expression if a = 3.7; b = 2.3.
1650. Of the 360 students at the school, 40% took part in cross-country. How many students participated in cross-country?
1651. Find the meaning of the expression:
1) (120,21 - 37,59) : 34 + 5,43 ∙ 19;
2) (8,57 + 9,585: 4,5) ∙ 3,8 - 42,7: 4.
1652. Find the meaning of the expression:
1) (5,02 - 3,89) ∙ 29 + 0,27: 18;
2) (32,526: 3,9 + 2,26) ∙ 5,4 - 47,2 ∙ 0,5.
1653. How much greater is the sum of the numbers 19.4 and 4.72 than the difference of these same numbers?
1654. Find the sum of 25.3 dm + 13.7 cm + 15 mm in centimeters.
1655. 32 students collected 152 kg of strawberries and 33.6 kg of raspberries. How many kilograms of berries did each student collect if they picked an equal amount of each type of berry?
1656. From a field of 420 hectares it was planned to collect 35 centners of grain per hectare, but 1785 tons of grain were collected. How many centners is the yield per hectare higher than planned?
1657. Find the surface area of a cube with an edge of 1.5 cm.
1658. Find the area and perimeter of a square with a side of 4.7 dm.
1659. Write the fractions in descending order: 0.27; 0.372; 0.423; 0.279; 0.51; 0.431; 0.307.
1660. Write the fractions in ascending order: 4.23; 4.32; 4.222; 43.2; 4.232; 4.323.
1661. A rope 15.3 m long was cut into three parts. One of them is ropes, second
longer than the first by 1.8 m. Find the length of each part.
1662. The yacht “Trouble” covered 234.9 km in 3 days of the regatta. During the first day the yacht coveredthis distance, and for the second - 8.3 km less than for the first. How many kilometers did the yacht "Trouble" travel every day?
1663. The car traveled 471 km. He drove the first 205 km at a speed of 82 km/h, and the rest at a speed of 76 km/h. How long did it take the car to cover the entire distance?
1664. The perimeter of an isosceles triangle is 15.4 cm. Find its base if the lateral side of the triangle is 5.3 cm.
1665. Find the perimeter of an isosceles triangle, the base of which is 4.2 in., and the side is 1.5 times larger than the base.
1666. Calculate:
1) (88,57 + 66,87) : 29 - 0,27 ∙ 18;
2) 20,8: (12 - 11,36) - 8: 12,5 + 4,7 ∙ 5,2.
1667. Calculate:
1) (1,37 + 4,86) ∙ 17 - 556,89: 19;
2) (3,81 + 59,427: 9,3) ∙ 7,6 - 10,2 ∙ 4,7.
1668. How much is the sum of the numbers 8.1 and 7.2 greater than their fraction?
1669. How much is the difference between the numbers 3.7 and 2.5 less than their product?
1670. Find the value of the expression a ∙ 2.5 - b if a = 3.6; b = 1.117.
1671. Between which adjacent natural numbers is the fraction placed:
1672. Rounded to:
1) units: 25.17; 37.89;
2) tenths: 37.893; 42.012;
3) hundredths: 108.112; 213.995.
1673. Rounded to:
1) units: 25.372; 37.51;
2) tenths: 13.185; 14.002;
3) hundredths: 15.894; 17,377.
1674. Draw a coordinate ray, taking 10 cells as a unit segment. Mark points A(0,7) on it, B (1.3), C (1), D (0.2), D (1.9).
1675. Draw a coordinate ray, taking 10 cells as a unit segment. Mark the points M(0,6) on it, N (1.4), K (0.3), L (2), P (1.8).
1676. A polar bear weighs 720 kg, and the mass of a brown bear is 40% of the mass of a polar bear. Calculate the mass of a brown bear.
1677. Simplify the expression 2.7 x - 0.05 x + 0.75 x and find its value if x = 2.7.
1678. The base of an isosceles triangle is 10.8 cm, and the length of the side isbase length. Find the perimeter of the triangle.
1679. Simplify the expression and calculate its meaning:
1) 2.7 a ∙ 2, if a = 3.5;
2) 3.2 x ∙ 5y, if x = 0.1; in = 1.7.
1680. Find the volume of a rectangular parallelepiped whose dimensions are equal to:
1) 1.2 cm, 5 cm, 1.8 cm; 2) 1.2 dm, 3 cm, 23 mm.
1681. Express in tons and write as a decimal:
1) 7314 kg; 2) 2 t 511 kg; 3) 3 c 12 kg; 4) 18 kg.
1682. Express in meters and write as a decimal fraction:
1) 527 cm; 2) 12 dm; 3) 3 m 5 dm; 4) 5 m 4 cm. 336
Enough level
1683. Perform division and round the resulting fraction:
1) 110: 57 to ones; 2) 18: 7 to tenths;
3) 15.2: 0.7 to hundredths; 4) 14: 5.1 to thousandths.
1684. Perform the division and round the resulting fraction:
1) 120: 37 to tenths; 2) 5.2: 0.17 to hundredths.
1685. The plant operated for 15 days and produced an average of 45.4 tons of mineral fertilizers daily. All fertilizers were loaded equally into 25 railway cars. How much fertilizer was loaded into each car?
1686. The sum of the two lengths of a triangle is 15 cm, and the length of the third side is 80% of this sum. Find the perimeter of the triangle.
1687. One of the sides of the rectangle is 14.4 cm, and the length of the second is 75% of the first. Find the area and perimeter of this rectangle.
1688. The perimeter of a triangle is 36 cm. The length of one of the sides isperimeter, and the length of the second is 40% of the perimeter. Find the sides of the triangle.
1689. The length of a rectangular parallelepiped is 16 dm, the width islength, and height - 70% of width. Find the volume of a rectangular parallelepiped.
1690. Find the sum of three numbers, the first of which is 4.27, and each next one is 10 times larger.
1691. The height of a rectangular parallelepiped is 16 cm, which islength and 40% width. Find the volume of a rectangular parallelepiped.
1692. One side of the rectangle is 8.5 cm, and the second is 60% of the first. Find the perimeter and area of the rectangle.
1693. One of the workers produced 96 parts in 6 hours, and the other made 45 parts in 2.5 hours. How many hours will it take them to produce 119 parts working together?
1694. What is more profitable to buy?
1695. What is more profitable to buy?
1696. Compose problems using diagrams and solve them.
1697. Compose problems using diagrams and solve them.
1698. How much will the volume of a cube increase if its edge is increased from 2.5 cm to 3.5 cm?
1699. Make up a numerical expression and find its value:
1) the difference between the sums of numbers 2.72 and 3.82 and
2) the product of the difference between the numbers 18.93 and 9.83 and the number 10.
1700. Two cyclists left village A for village B at the same time at speeds of 15.6 km/h and 18.4 km/h. After 3.5 hours, one of the cyclists arrived in village B. How many kilometers should the other cyclist travel?
1701. Two cars left the same city at the same time in opposite directions. The speed of one of them is 76 km/h, which is 95% of the speed of the other. After how many hours will the distance between the cars be 390 km?
1702. Solve the equations:
1) 1.17 x + 0.32 x = 3.725;
2) 4.7 x - 1.2 x = 4.34;
3) 2.47 x - 1.32 x + 1.3 = 4.221;
4) 1.4 x + 2.7 x - 8.113 = 2.342.
1703. Solve the equations:
1) 4.13 x - 0.17 x = 9.9;
2) 5.3 x + 4.8 x - 5.13 = 43.35.
1704. The unfolded angle was divided by rays into cocked hats. The first isexpanded, and the second -first. Find the degree measures of the three formed corners
1705. Compose problems using diagrams and solve them:
1706. Compose problems using diagrams and solve them:
1707. Solve the equations:
1) 2.7(x - 4.7) = 9.45; 2) (4.7 + x): 3.8 = 10.5;
3) 2.4 + (x: 3 - 5) = 0.8; 4) 2.45: (2 x - 1.4) = 3.5.
1708. Solve the equations:
1) 21: (4 x + 1.6) = 2.5;
2) 3.7 - (x: 2 + 1.5) = 0.8.
1709. A ball was made with 2.5 g of copper wire, the mass of 1 m of which is 1.2 kg, and a piece of brass wire, the length of which is 8 times that of copper, and the mass of 1 m is 0.2 kg. How much alloy will remain if the bullet mass is 6.4 kg?
1710. Bought 2.5 kg of cookies at a price of 13.6 UAH. per kilogram and 1.6 kg of sweets, the price per kilogram is 1.5 times more than the price of one kilogram of cookies. What change should you get from 100 UAH?
1711. Fill in the cells with numbers to form the correct examples:
1712. Fill in the cells with such numbers to form the correct examples:
1713. The number 5.2 is the arithmetic mean of the numbers 2.1; 3.2 and x. Find x.
1714. Find the arithmetic mean of four numbers, the first of which is 3.6, and each subsequent one is 0.2 more than the previous one.
1715. Two motorcyclists set off simultaneously from one city to another in the same direction at a speed of 72.4 km/h and 67.8 km/h. After what time will the distance between motorcyclists be 11.5 km?
1716. The price of some goods is 120 UAH. How much will this product cost if the price is:
1) increase by 15%;
2) reduce by 10%;
3) first increase by 5%, and then reduce the new price by 20%?
1717. Find the numbers that are missing in the chain of calculations:
1718. The car traveled 170.4 km in the first two hours, and 0.45 of this distance in the next. Find the average speed of the car.
1719. The train traveled 210.5 km in the first three hours, and 0.6 of this distance in the next two hours. Find the average speed of the train.
1720. The side of an equilateral triangle is 11.2 cm. Find the side of a square whose perimeter is equal to the perimeter of the triangle.Determine the area of this square.
1721. Find the shaded part of the circle:
1722. Find the sum of three numbers, the first of which is 37.6, the second isfrom the first, and the third is the arithmetic mean of the first two.
1723. The boat covered 231 km against the river flow in 6 hours. How far will he travel along the river in 4 hours if the current speed is 1.4 km/h?
1724. Two pedestrians simultaneously left two points, the distance between which is 8.5 km, in opposite directions, moving away from each other. The speed of one of them is 4.2 km/h, which isspeed of the second. What will be the distance between pedestrians after 2.5 hours?
1725. The car moved for 4 hours at a speed of 82.5 km/h and 6 hours at a speed of 83.7 km/h. Find the average speed of the car along the entire route.
High level
1726. Carlson and the Kid together ate 3.6 kg of jam, and Carlson ate 3 times more than the Kid. How much jam did Carlson eat and how much did Baby eat?
1727. A load weighing 4.8 tons was placed on two trucks, and the first one was loaded with 0.6 tons more than the second. How many tons of cargo are in each car?
1728. Three workers, working together, produced 1001 parts in 7 hours. And the first one madeall the details, and the second -all the details. How many parts did the third worker produce per hour?
1729. Subtract 10% from a certain number and get 48.6. Find this number.
1730. We added 20% to a certain number and got 74.4. Find this number.
1731. Find two numbers if their sum is 4.7 and their difference is 3.1.
1732. The sum of two numbers is 27.2. Find these numbers if one of them is three times larger than the other.
1733. A rope 10.6 m long was cut into three parts. Find their lengths if the third part is 0.4 m longer than both the first and second.
1734. The boat's own speed is 13 times the speed of the current. Moving with the current for 2.5 hours, the boat covered 63 km. Find the boat's own speed and the speed of the current.
1735. From two stations, the distance between which is 385 km, two trains departed simultaneously towards each other and met after 2.5 hours. Find the speeds of the trains if it is known that the speed of one of them is 1.2 times the speed of the other.
1736. The sum of the length and width of a rectangle is 9.6 cm, with the width being 60% of the length. Find the area and perimeter of the rectangle.
1737. The length of one side of the triangle isperimeter, and the length of the other side isperimeter. Find the lengths of these sides if the third side is 10.4 cm.
1738. The student first read 0.25 of the entire book, and then another 0.4 of the rest, after which it turned out that the student had read 30 pages more than he had left to read. How many pages are there in the book?
1739. Find the meaning of the letters g, h, m, n, k, l, if:
g: n = 1.8; n ∙ k = 1.71; h + m = 2.13;
k + l = 10.44; m ∙ 0.9 = 1.17; g - h = 0.79.
1740. IS Three boxes together contain 62.88 kg of goods. The first box contains 1.4 times more goods than the second, and the third contains as much goods as there are in the first and second combined. How many kilograms of goods are in each box?
Exercises to repeat
1741. 1) Follow these steps:
2) Follow these steps:
3) Compare the numbers indicated by the figures:
1742. 1) Follow these steps:
2) Follow these steps:
2. Find the arithmetic mean of the numbers 1.8 and 2.6.
A) 1.8; B) 2; B) 2.6; D) 2.2.
3. Write the mixed number as a decimal fraction
A) 3.13; B) 13.3; B) 13.003; D) 13.03.
4. After distillation of oil, 30% kerosene is obtained. How much kerosene is obtained from 18 tons of oil?
A) 6 t; B) 5.4 t; B) 54 t; D) 0.6 t.
5. Milk makes 9% of cheese. How much milk was taken if you received 36 kg of cheese?
A) 400 kg; B) 40 kg; B) 324 kg; D) 300 kg.
6. In a basketball team, two players are 19 years old, two are 21 years old, and one player is 26 years old. What is the average age of the players on this team?
A) 19 years old; B) 21 years old;
B ) 21.2 years; D) 21.4 years.
7. During drying, mushrooms lose 89% of their mass. How many dry mushrooms will we get from 60 kg of fresh ones?
A) 53.4 kg; B) 6.6 kg; B) 6 kg; D) 5.34 kg.
8. When the student had read 30% of the book, he noticed that he still had 105 pages left to read. How many pages are there in the book?
A) 350 sec.; B) 250 sec.; B) 150 sec.; D) 160s.
9. One of the computer typing operators typed 45 pages of text in 6 hours, and another typed 26 pages of text in 4 hours. How many hours will it take them working together to complete 35 pages?
A) 2 hours; B) 2.5 hours C) 3 hours; D) 3.5 hours.
10. A box contains white and black balls, with white ones making up 30% of all balls. How many balls are there in total if there are 32 more black balls than white balls?
A) 80; B) 70; B) 56; D) 180.
11. The arithmetic mean of two numbers, one of which is 4 times larger than the other, is 6. Find the smaller of these two numbers.
A) 1.5; B) 2.4; B) 2.5; D) 9.6.
12. The price of some goods is 150 UAH. How much will this product cost if the price of the product was initially increased by 10% and then the new price was decreased by 15%?
A) 142.5 UAH; B) 157.5 UAH;
V) 155 UAH; D) 140.25 UAH.
Knowledge testing tasks No. 9 (§42 - §45)
1. Write as a decimal:
1) 15 %; 2) 3 %.
2. Write the decimal fraction as a percentage:
1) 0,45; 2) 1,37.
3. Follow these steps:
1) 3,7 + 13,42; 2) 15,8 - 13,12;
3) 4,2 ∙ 2,05; 4) 8,64: 2,4.
4. Of the 1200 students studying at the school, 65% took part in the sports competition. How many students took part in the sports competition?
5. Sergei bought a book for 8 UAH, which is 40% of the money he had. How many hryvnia did Sergei have?
6. Find the arithmetic mean of the numbers 48.5; 58.2; 46.8; 42.2.
7. The worker produced 320 parts. In the first hour - 35% of all parts, the second - 40%, and in the third - the rest. How many parts did the worker produce in the third hour?
8. The car drove for 2 hours at a speed of 66.7 km/h and for 3 hours at a speed of 72.8 km/h. Find his average speed along the entire path.
9. The tourist walked 56 km in three days. On the first day, he covered 30% of the entire path, which is 80% of the distance covered by the tourist on the second day. How many kilometers did the tourist walk on the third day?
10. Additional task. The length of a rectangular parallelepiped is 8.5 cm, which is 2.5 times greater than the width and 5.1 cm greater than the height. Find the volume of this rectangular parallelepiped.
11. Additional task. The arithmetic mean of two numbers is 12.4, and the arithmetic mean of the other eight numbers is 10.7. Find the arithmetic mean of these ten numbers.
When adding decimal fractions, you need to write them one under the other so that the same digits are under each other, and the comma is under the comma, and add the fractions the same way you add natural numbers. Let's add, for example, the fractions 12.7 and 3.442. The first fraction contains one decimal place, and the second contains three. To perform addition, we transform the first fraction so that there are three digits after the decimal point: , then
The subtraction of decimal fractions is performed in the same way. Let's find the difference between the numbers 13.1 and 0.37:
When multiplying decimal fractions, it is enough to multiply the given numbers, not paying attention to commas (like natural numbers), and then, as a result, separate as many digits from the right with a comma as there are after the decimal point in both factors in total.
For example, let's multiply 2.7 by 1.3. We have. We use a comma to separate two digits on the right (the sum of the digits of the factors after the decimal point is two). As a result, we get 2.7 1.3 = 3.51.
If the product contains fewer digits than must be separated by a comma, then the missing zeros are written in front, for example:
Let's consider multiplying a decimal fraction by 10, 100, 1000, etc. Let's say we need to multiply the fraction 12.733 by 10. We have . Separating three digits to the right with a comma, we get But. Means,
12 733 10=127.33. Thus, multiplying a decimal fraction by 10 is reduced to moving the decimal point one digit to the right.
In general, to multiply a decimal fraction by 10, 100, 1000, you need to move the decimal point in this fraction 1, 2, 3 digits to the right, adding, if necessary, a certain number of zeros to the fraction on the right). For example,
Dividing a decimal fraction by a natural number is performed in the same way as dividing a natural number by a natural number, and the comma in the quotient is placed after the division of the integer part is completed. Let us divide 22.1 by 13:
If the integer part of the dividend is less than the divisor, then the answer is zero integers, for example:
Let us now consider dividing a decimal by a decimal. Let's say we need to divide 2.576 by 1.12. To do this, in both the dividend and the divisor, move the comma to the right by as many digits as there are after the decimal point in the divisor (in this example, two). In other words, if we multiply the dividend and the divisor by 100, the quotient will not change. Then you need to divide the fraction 257.6 by the natural number 112, i.e. the problem reduces to the case already considered:
To divide a decimal fraction by, you need to move the decimal point in this fraction to the left (and, if necessary, add the required number of zeros to the left). For example, .
Just as division is not always feasible for natural numbers, it is not always feasible for decimal fractions. For example, let's divide 2.8 by 0.09.
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