Farafonova Natalia Igorevna
After completing the topic “Actions with decimals» to practice counting skills and test mastery of the material, you can conduct individual work with students using cards. Each student must complete tasks for all activities without errors. There are many options for each action, this allows each student to solve the task for each action with decimals several times and achieve an error-free result or complete the task with minimum quantity errors. Since each student completes an individual task, the teacher has the opportunity, as completed tasks are presented to him, to discuss them personally with each student. If a student makes mistakes, the teacher corrects them and offers to do the task from a different option. So, until the student completes the entire task or most of it without errors. It is better to make cards on colored paper.
At the last stage of work, you can propose solving an example containing several actions.
For each error-free option, regardless of which attempt the task was completed correctly, students can be given an excellent mark, or an average grade can be given after completing all the work, at the discretion of the teacher.
Adding decimals.
1 option
7,468 + 2,85
9,6 + 0,837
38,64 + 8,4
3,9 + 26,117
Option 2
19,45 + 34,8
4,9 + 0,716
75,86 + 4,2
5,6 + 44,408
Option 3
24,38 + 7,9
6,5 + 0,952
48,59 + 1,8
35,906 + 2,8
Option 4
7,6 + 319,75
888,99 + 4,5
64,15 + 18,9
4,5 + 0,738
Option 5
7,62 + 8,9
25,38 + 0,09
12,842 + 8,6
412 + 78,83
Option 6
70,7 + 3,8645
3,65 + 0,89
61,22 + 31.719
12,842 + 8,6
Answers: Option 1: 10.318; 10.437; 47.04; 30.017;
Option 2: 54.25; 5.616; 80.06; 50.008;
Option 3: 32.28; 7.452; 50.19; 38.706;
Option 4: 327.35; 893.49; 83.05; 5.238;
Option 5: 16.52; 25.47; 21.442; 490.83;
Option 6: 74.5645; 4.54; 92.939; 21.442;
Subtracting decimals.
1 option
26,38 - 9,69
41,12 - 8,6
5,2 - 3,445
7 - 0,346
Option 2
47,62 - 8,78
54,06 - 9,1
7,1 - 6,346
3 - 1,551
Option 3
50,41 - 9,62
72,03 - 6,3
9,2 - 5,453
4 - 2,662
Option 4
60,01 - 8,364
123,61 - 69,8
8,7 - 4,915
10 - 3,817
Option 5
6,52 - 3,8
7,41 - 0,758
67,351 - 9,7
22 - 0,618
Option 6
4,5 - 0,496
61,3 - 20,3268
24,7 - 15,276
50 - 2,38
Answers: Option 1: 16.69; 32.52; 1.755; 6.654;
Option 2: 38.84; 44.96; 0.754; 1.449;
Option 3: 40.79; 65.73; 3.747; 1.338;
Option 4: 51.646; 53.81; 3.785; 6.183;
Option 5: 2.72; 6.652; 57.651; 21.382;
Option 6: 4.004; 40.9732; 9.424; 47.62;
Multiplying decimals.
1 option
7.4 3.5
20.2 3.04
0.68 0.65
2.5 840
Option 2
2.8 9.7
6.05 7.08
0.024 0.35
560 3.4
Option 3
6.8 5.9
6.06 8.05
0.65 0.014
720 4.6
Option 4
34.7 8.4
9.06 7.08
0.038 0.29
3.6 540
Option 5
62.4 2.5
0.038 9
1.8 0.009
4.125 0.16
Option 6
0.28 45
20.6 30.5
2.3 0.0024
0.0012 0.73
Option 7
68 0.15
0.08 0.012
1.4 1.04
0.32 2.125
Option 8
4.125 0.16
0.0012 0.73
1.4 1.04
720 4.6
Answers: Option 1: 25.9; 61.408; 0.442; 2100;
Option 2: 27.16; 42.834; 0.0084; 1904;
Option 3: 40.12; 48.783; 0.0091; 3312;
Option 4: 291.48; 64.1448; 0.01102; 1944;
Option 5: 156; 0.342; 0.0162; 0.66;
Option 6: 12.6; 628.3; 0.00552; 0.000876;
Option 7: 10.2; 0.00096; 1.456; 0.68;
Option 8: 0.66; 0.000876; 1.456; 3312;
Dividing a decimal fraction by natural number.
1 option
62,5: 25
0,5: 25
9,6: 12
1,08: 8
Option 2
0,28: 7
0,2: 4
16,9: 13
22,5: 15
Option 3
0,75: 15
0,7: 35
1,6: 8
0,72: 6
Option 4
2,4: 6
1,5: 75
0,12: 4
1,69: 13
Option 5
3,5: 175
1,8: 24
10,125: 9
0,48: 16
Option 6
0,35: 7
1,2: 3
0,2: 5
7,2: 144
Option 7
151,2: 63
4,8: 32
0,7: 25
2,3: 40
Option 8
397,8: 78
5,2: 65
0,9: 750
3,4: 80
Option 9
478,8: 84
7,3: 4
0,6: 750
5,7: 80
Option 10
699,2: 92
1,8: 144
0,7: 875
6,3: 24
Answers: Option 1: 2.5; 0.02; 0.8; 0.135;
Option 2: 0.04; 0.05; 1.3; 1.5;
Option 3: 0.05; 0.02; 0.2; 0.12;
Option 4: 0.4; 0.02; 0.03; 0.13;
Option 5: 0.02; 0.075; 1.125; 0.03;
Option 6: 0.05; 0.4; 0.04; 0.05;
Option 7: 2.4; 0.15; 0.28; 0.0575;
Option 8: 5.1; 0.08; 0.0012; 0.0425;
Option 9: 5.7; 1.825; 0.0008; 0.07125;
Option 10: 7.6; 0.0125; 0.0008; 0.2625;
Division by decimal fraction.
1 option
32: 1,25
54: 12,5
6: 125
Option 2
50,02: 6,1
34,2: 9,5
67,6: 6,5
Option 3
2,8036: 0,4
3,1: 0,025
0,0008: 0,16
Option 4
4: 32
303: 75
687,4: 10
1,59: 100
Option 5
5: 16
336: 35
412,5: 10
24,3: 100
Option 6
41,82: 6,8
73,44: 3,6
7,2: 0,045
32,89: 4,6
Answers: Option 1: 25.6; 4.32; 0.048;
Option 2: 8.2; 3.6; 10.4;
Option 3: 7.009; 124; 0.005;
Option 4: 0.125; 4.04; 68.74; 0.0159;
Option 5: 0.3125; 9.6; 41.25; 0.243;
Option 6: 6.15; 20.4; 160; 7.15;
Joint operations with decimals.
824,72 - 475: (0,071 + 0,929) + 13,8
(7.351 + 12.649) 105 - 95.48 - 4.52
(3.82 - 1.084 + 12.264) (4.27 + 1.083 - 3.353) + 83
278 - 16,7 - (15,75 + 24,328 + 39,2)
57.18 42 - 74.1: 13 + 21.35: 7
(18.8: 16 + 9.86 3) 40 - 12.73
(2 - 0.25 0.8) : (0.16: 0.5 - 0.02)
(3,625 + 0,25 + 2,75) : (28,75 + 92,25 - 15) : 0,0625
Answers: 1) 363.52; 2) 2000; 3) 113; 4) 182.022; 5) 2398.91; 6) 1217.47; 7) 6; 8) 1.
Division by a decimal fraction is reduced to division by a natural number.
The rule for dividing a number by a decimal fraction
To divide a number by a decimal fraction, you need to move the decimal point in both the dividend and the divisor by as many digits to the right as there are in the divisor after the decimal point. After this, divide by a natural number.
Examples.
Divide by decimal fraction:
To divide by a decimal, you need to move the decimal point in both the dividend and the divisor by as many digits to the right as there are after the decimal point in the divisor, that is, by one digit. We get: 35.1: 1.8 = 351: 18. Now we perform the division with a corner. As a result, we get: 35.1: 1.8 = 19.5.
2) 14,76: 3,6
To divide decimal fractions, in both the dividend and the divisor we move the decimal point to the right one place: 14.76: 3.6 = 147.6: 36. Now we perform a natural number. Result: 14.76: 3.6 = 4.1.
To divide a natural number by a decimal fraction, you need to move both the dividend and the divisor to the right as many places as there are in the divisor after the decimal point. Since a comma is not written in the divisor in this case, we fill in the missing number of characters with zeros: 70: 1.75 = 7000: 175. Divide the resulting natural numbers with a corner: 70: 1.75 = 7000: 175 = 40.
4) 0,1218: 0,058
To divide one decimal fraction by another, we move the decimal point to the right in both the dividend and the divisor by as many digits as there are in the divisor after the decimal point, that is, by three decimal places. Thus, 0.1218: 0.058 = 121.8: 58. Division by a decimal fraction was replaced by division by a natural number. We share a corner. We have: 0.1218: 0.058 = 121.8: 58 = 2.1.
5) 0,0456: 3,8
The use of equations is widespread in our lives. They are used in many calculations, construction of structures and even sports. Man used equations in ancient times and since then their use has only increased. A linear equation with decimals is solved in the same way as many other equations, but you need to start solving them by shortening the equation and getting rid of the decimals.
Suppose we are given an equation of the following form:
This equation can be solved in two different ways.
Method No. 1:
We begin the solution by simplifying the equation by opening parentheses, and since we have a number in front of the brackets, we multiply this number by each term in brackets:
Now our equation has a linear form, thanks to which we carry out the transfer of unknowns in one direction, integer to another:
\[ - 7.2x + 5.2x = 1.7 - 14.4 - 4.3\]
Divide 2 parts by the number before \
\[ - 2x = - 17\]
Answer: \
Method number 2:
In this method, multiply the left and right sides by 10:
This linear equation, which is solved by analogy with method 1:
\[ - 72x + 52x = 17 - 144 - 43\]
\[ - 20x = - 170\]
Answer: \
Where can I solve decimal equations online?
You can solve the equation on our website https://site. Free online solver will allow you to solve online equations of any complexity in a matter of seconds. All you need to do is simply enter your data into the solver. You can also watch video instructions and learn how to solve the equation on our website. And if you still have questions, you can ask them in our VKontakte group http://vk.com/pocketteacher. Join our group, we are always happy to help you.
Of the many fractions found in arithmetic, those that have 10, 100, 1000 in the denominator - in general, any power of ten - deserve special attention. These fractions have a special name and notation.
A decimal is any number fraction whose denominator is a power of ten.
Examples of decimal fractions:
Why was it necessary to separate out such fractions at all? Why do they need their own recording form? There are at least three reasons for this:
- Decimals are much easier to compare. Remember: to compare ordinary fractions, you need to subtract them from each other and, in particular, reduce the fractions to common denominator. In decimals nothing like this is required;
- Reduce computation. Decimal fractions are added and multiplied by own rules, and after a little training you will work with them much faster than with regular ones;
- Ease of recording. Unlike ordinary fractions, decimals are written on one line without loss of clarity.
Most calculators also give answers in decimals. In some cases, a different recording format may cause problems. For example, what if you ask for change in the store in the amount of 2/3 of a ruble :)
Rules for writing decimal fractions
The main advantage of decimal fractions is convenient and visual notation. Namely:
Decimal notation is a form of writing decimal fractions where the integer part is separated from the fractional part by a regular period or comma. In this case, the separator itself (period or comma) is called a decimal point.
For example, 0.3 (read: “zero point, 3 tenths”); 7.25 (7 whole, 25 hundredths); 3.049 (3 whole, 49 thousandths). All examples are taken from the previous definition.
In writing, a comma is usually used as a decimal point. Here and further throughout the site, the comma will also be used.
To write an arbitrary decimal fraction in this form, you need to follow three simple steps:
- Write out the numerator separately;
- Shift the decimal point to the left by as many places as there are zeros in the denominator. Assume that initially the decimal point is to the right of all digits;
- If the decimal point has moved, and after it there are zeros at the end of the entry, they must be crossed out.
It happens that in the second step the numerator does not have enough digits to complete the shift. In this case, the missing positions are filled with zeros. And in general, to the left of any number you can assign any number of zeros without harm to your health. It's ugly, but sometimes useful.
At first glance, this algorithm may seem quite complicated. In fact, everything is very, very simple - you just need to practice a little. Take a look at the examples:
Task. For each fraction, indicate its decimal notation:
The numerator of the first fraction is: 73. We shift the decimal point by one place (since the denominator is 10) - we get 7.3.
Numerator of the second fraction: 9. We shift the decimal point by two places (since the denominator is 100) - we get 0.09. I had to add one zero after the decimal point and one more before it, so as not to leave a strange entry like “.09”.
The numerator of the third fraction is: 10029. We shift the decimal point by three places (since the denominator is 1000) - we get 10.029.
The numerator of the last fraction: 10,500. Again we shift the point by three places - we get 10,500. There are extra zeros at the end of the number. Cross them out and we get 10.5.
Pay attention to the last two examples: the numbers 10.029 and 10.5. According to the rules, the zeros on the right must be crossed out, as was done in the last example. However, you should never do this with zeros inside a number (which are surrounded by other numbers). That's why we got 10.029 and 10.5, and not 1.29 and 1.5.
So, we figured out the definition and form of writing decimal fractions. Now let's find out how to convert ordinary fractions to decimals - and vice versa.
Conversion from fractions to decimals
Let's consider a simple numerical fraction of the form a /b. You can use the basic property of a fraction and multiply the numerator and denominator by such a number that the bottom turns out to be a power of ten. But before you do, read the following:
There are denominators that cannot be reduced to powers of ten. Learn to recognize such fractions, because they cannot be worked with using the algorithm described below.
That's it. Well, how do you understand whether the denominator is reduced to a power of ten or not?
The answer is simple: factor the denominator into prime factors. If the expansion contains only factors 2 and 5, this number can be reduced to a power of ten. If there are other numbers (3, 7, 11 - whatever), you can forget about the power of ten.
Task. Check whether the indicated fractions can be represented as decimals:
Let us write out and factor the denominators of these fractions:
20 = 4 · 5 = 2 2 · 5 - only the numbers 2 and 5 are present. Therefore, the fraction can be represented as a decimal.
12 = 4 · 3 = 2 2 · 3 - there is a “forbidden” factor 3. The fraction cannot be represented as a decimal.
640 = 8 · 8 · 10 = 2 3 · 2 3 · 2 · 5 = 2 7 · 5. Everything is in order: there is nothing except the numbers 2 and 5. A fraction can be represented as a decimal.
48 = 6 · 8 = 2 · 3 · 2 3 = 2 4 · 3. The factor 3 “surfaced” again. It cannot be represented as a decimal fraction.
So, we’ve sorted out the denominator - now let’s look at the entire algorithm for moving to decimal fractions:
- Factor the denominator of the original fraction and make sure that it is generally representable as a decimal. Those. check that only factors 2 and 5 are present in the expansion. Otherwise, the algorithm does not work;
- Count how many twos and fives are present in the expansion (there will be no other numbers there, remember?). Choose an additional factor such that the number of twos and fives is equal.
- Actually, multiply the numerator and denominator of the original fraction by this factor - we get the desired representation, i.e. the denominator will be a power of ten.
Of course, the additional factor will also be decomposed only into twos and fives. At the same time, in order not to complicate your life, you should choose the smallest multiplier of all possible ones.
And one more thing: if the original fraction contains an integer part, be sure to convert this fraction to an improper fraction - and only then apply the described algorithm.
Task. Convert these numerical fractions to decimals:
Let's factorize the denominator of the first fraction: 4 = 2 · 2 = 2 2 . Therefore, the fraction can be represented as a decimal. The expansion contains two twos and not a single five, so the additional factor is 5 2 = 25. With it, the number of twos and fives will be equal. We have:
Now let's look at the second fraction. To do this, note that 24 = 3 8 = 3 2 3 - there is a triple in the expansion, so the fraction cannot be represented as a decimal.
The last two fractions have denominators 5 (prime number) and 20 = 4 · 5 = 2 2 · 5 respectively - only twos and fives are present everywhere. Moreover, in the first case, “for complete happiness” a factor of 2 is not enough, and in the second - 5. We get:
Transition from decimals to common fractions
The reverse conversion - from decimal to regular notation - is much simpler. There are no restrictions or special checks here, so you can always convert a decimal fraction to the classic “two-story” fraction.
The translation algorithm is as follows:
- Cross out all the zeros on the left side of the decimal, as well as the decimal point. This will be the numerator of the desired fraction. The main thing is not to overdo it and do not cross out the inner zeros surrounded by other numbers;
- Count how many decimal places are in the original fraction after the decimal point. Take the number 1 and add as many zeros to the right as there are characters you count. This will be the denominator;
- Actually, write down the fraction whose numerator and denominator we just found. If possible, reduce it. If the original fraction contained an integer part, we now get improper fraction, which is very convenient for further calculations.
Task. Convert decimal fractions to ordinary fractions: 0.008; 3.107; 2.25; 7,2008.
Cross out the zeros on the left and the commas - we get the following numbers (these will be numerators): 8; 3107; 225; 72008.
In the first and second fractions there are 3 decimal places, in the second - 2, and in the third - as many as 4 decimal places. We get the denominators: 1000; 1000; 100; 10000.
Finally, let's combine the numerators and denominators into ordinary fractions:
As can be seen from the examples, the resulting fraction can very often be reduced. Let me note once again that any decimal fraction can be represented as an ordinary fraction. The reverse conversion may not always be possible.