Addition and subtraction of decimal fractions examples. Subtraction of decimal fractions: rules, examples, solutions

• First you need to equalize the number of decimal places.
• Next, you need to write decimal fractions under each other in such a way that the commas were under each other. This is the most important part!
• Next, subtract decimal fractions, excluding commas, according to the rules of subtraction in column of natural numbers.
• And finally, put a comma under the commas in the answer.

Second option subtracting decimal fractions:

If you are well versed in decimal fractions, what are tenths, hundredths, etc., then you will bethis option is interesting.

Rules for subtracting decimal fractions per line:

• Subtract decimals from right to left. That is, starting from the rightmost number after the decimal point.
• Subtract bit by bit. Wholes from whole, tenths from tenths, hundredths from hundredths, thousandths from thousandths and so on.
• When subtracting a larger number from a smaller one, we take ten from the neighbor on the left of the smaller number.

For example:

The rightmost digit in the given fractions is the hundredth digit. 1 - 1 = 0 ... We get zero, that is, in the categoryhundredths of the difference we write0 .

Subtract tenths from tenths. 2 - in decreasing, 3 - deductible. Because from 2 (smaller) cannot be subtracted3 (greater), then you need to take ten from the left digit for2. Here it is 5. 2 + 10 = 12. Thus, 3 subtract not from 2 , and from 12 .

12 - 3 = 9

We write down 9 in difference. Since we are from 5 deducted 1 a dozen, in the reduced there is no 15 , but 14 so that itdo not forget we put over5 empty circle or period, whichever is more convenient.

Subtract 8 from 14:

14 - 8 = 6

Note! Tenths can only be subtracted from tenths, hundredths from hundredths, thousandths from thousandths andetc. If in one of the fractions, there is no digit of the corresponding digit, instead of it write down 0 .

In the second number, the rightmost digit is two (hundredth place), and in the first number, hundredths are not visible.Hence, to the first number to the right of9 add 0 and then subtract based onBasic Rules.

The third option subtracting decimal fractions:

Arithmetic computational operations such as addition and subtraction of decimal fractions, are necessary in order to obtain the desired result by operating with fractional numbers. The particular importance of these operations is that in many spheres of human activity, the measures of many entities are represented precisely decimal fractions... Therefore, to carry out certain actions with many objects material world required fold or subtract exactly decimals... It should be noted that in practice these operations are used almost everywhere.

Procedures addition and subtraction of decimal fractions in its mathematical essence, it is carried out in almost the same way as analogous operations for integers. In its implementation, the value of each digit of one number must be written under the value of a similar digit of another number.

Submits to the following rules:

First, you need to make the equalization of the number of those signs that are located after the decimal point;

Then you need to write decimal fractions under each other in such a way that the commas contained in them are located strictly one under the other;

Carry out the procedure subtracting decimal fractions in full accordance with the rules that apply to the subtraction of integers. You don't need to pay any attention to the commas;

After receiving the answer, the comma in it must be placed strictly under those that are in the original numbers.

Operation adding decimal fractions is carried out in accordance with the same rules and algorithm as described above for the subtraction procedure.

Example of adding decimal fractions

Two point two tenths plus one hundredth plus fourteen point ninety five hundredths equals seventeen point sixteen hundredths.

2,2 + 0,01 + 14,95 = 17,16

Examples of adding and subtracting decimal fractions

Mathematical operations additions and subtracting decimal fractions in practice, they are used extremely widely, and they often relate to many objects of the material world around us. Below are some examples of such calculations.

Example 1

According to the design and estimate documentation, for the construction of a small production facility ten point five tenths of a cubic meter of concrete is required. Using modern technologies erection of buildings, to contractors without prejudice to quality characteristics the structures were used to carry out all the work, only nine point nine tenths of cubic meters of concrete. The amount of savings is:

Ten point five tenths minus nine point nine tenths equals zero point six tenths of a cubic meter of concrete.

10.5 - 9.9 = 0.6 m 3

Example 2

The engine installed on the old car model consumes eight point two liters of fuel per hundred kilometers in the urban cycle. For a new power unit, this figure is seven point five liters. The amount of savings is:

Eight point two liters minus seven point five liters is equal to zero point seven liters per hundred kilometers in urban driving.

8.2 - 7.5 = 0.7L

The operations of addition and subtraction of decimal fractions are used extremely widely, and their implementation does not present any problems. In modern mathematics, these procedures are worked out almost perfectly, and almost everyone is fluent in them since school.

Chapter 2 FRACTIONAL NUMBERS AND ACTIONS WITH THEM

§ 37. Addition and subtraction of decimal fractions

Decimal fractions are written according to the same principle as natural numbers. Therefore, addition and subtraction are performed according to the corresponding schemes for natural numbers.

During addition and subtraction, decimal fractions are written in a "column" - one under the other so that the digits of the same name are one under the other. Thus, the comma will be below the comma. Next, we perform the action as with natural numbers, ignoring the commas. In the sum (or difference), we put a comma under the commas of the addends (or commas of the decrement and the subtractor).

Example 1.37.982 + 4.473.

Explanation. 2 thousandths plus 3 thousandths equals 5 thousandths. 8 acres plus 7 acres is equal to 15 acres, or 1 tenth and 5 acres. We write down 5 acres, and remember 1 tenth, etc.

Example 2. 42.8 - 37.515.

Explanation. Since the decreasing and subtracted have a different number of decimal places, it can be assigned in a decreasing required amount zeros. Figure out on your own how the example is done.

Note that when adding and subtracting zero, you can not add, but mentally represent them in those places where there are no bit units.

When adding decimal fractions, the previously studied permutable and connecting properties of addition come true:

First level

1228. Calculate (orally):

1) 8 + 0,7; 2) 5 + 0,32;

3) 0,39 + 1; 4) 0,3 + 0,2;

5) 0,12 + 0,37; 6) 0,1 + 0,01;

7) 0,02 + 0,003; 8) 0,26 + 0,7;

9) 0,12 + 0,004.

1229. Numbers:

1230. Calculate (orally):

1) 4,72 - 2; 2) 13,892 - 10; 3) 0,8 - 0,6;

4) 6,7 - 0,3; 5) 2,3 - 1,2; 6) 0,05 - 0,02;

7) 0,19 - 0,07; 8) 0,47 - 0,32; 9) 42,4 - 42.

1231. Numbers:

1232. Numbers:

1233. One car had 2.7 tons of sand, and the other - 3.2 tons. How much sand was there on two cars?

1234. Perform addition:

1) 6,9 + 2,6; 2) 9,3 + 0,8; 3) 8,9 + 5;

4) 15 + 7,2; 5) 4,7 + 5,29; 6) 1,42 + 24,5;

7) 10,9 + 0,309; 8) 0,592 + 0,83; 9) 1,723 + 8,9.

1235. Find the amount:

1) 3,8 + 1,9; 2) 5,6 + 0,5; 3) 9 + 3,6;

4) 5,7 + 1,6; 5) 3,58 + 1,4; 6) 7,2 + 15,68;

7) 0,906 + 12,8; 8) 0,47 + 0,741; 9) 8,492 + 0,7.

1236. Subtract:

1) 5,7 - 3,8; 2) 6,1 - 4,7; 3) 12,1 - 8,7;

4) 44,6 - 13; 5) 4 - 3,4; 6) 17 - 0,42;

7) 7,5 - 4,83; 8) 0,12 - 0,0856; 9) 9,378 - 8,45.

1237. Find the Difference:

1) 7,5 - 2,7; 2) 4,3 - 3,5; 3) 12,2 - 9,6;

4) 32,7 - 5; 5) 41 - 3,53; 6) 7 - 0,61;

7) 8,31 - 4,568; 8) 0,16 - 0,0913; 9) 37,819 - 8,9.

1238. The flying carpet flew 17.4 km in 2 hours, and in the first hour it flew 8.3 km. How many flying carpet flew in the second hour?

1239.1) Multiply 7.2831 by 2.423.

2) Decrease the number 5.372 by 4.47.

Average level

1240. Solve the equations:

1) 7.2 + x = 10.31; 2) 5.3 - x = 2.4;

3) x - 2.8 = 1.72; 4) x + 3.71 = 10.5.

1241. Solve the equations:

1) x - 4.2 = 5.9; 2) 2.9 + x = 3.5;

3) 4.13 - x = 3.2; 4) x + 5.72 = 14.6.

1242. How is it more convenient to add? Why?

4.2 + 8.93 + 0.8 = (4.2 + 8.93) + 0.8 or

4,2 + 8,93 + 0,8 = (4,2 + 0,8) + 8,93.

1243. Count (orally) in a convenient way:

1) 7 + 2,8 + 1,2; 2) 12,4 + 17,3 + 0,6;

3) 3,42 + 4,9 + 5,1; 4) 12,11 + 7,89 + 13,5.

1244. Find the meaning of the expression:

1) 200,01 + 0,052 + 1,05;

2) 42 + 4,038 + 17,25;

3) 2,546 + 0,597 + 82,04;

4) 48,086 + 115,92 + 111,037.

1245. Find the meaning of the expression:

1) 82 + 4,042 + 17,37;

2) 47,82 + 0,382 + 17,3;

3) 15,397 + 9,42 + 114;

4) 152,73 + 137,8 + 0,4953.

1246. From metal pipe with a length of 7.92 m, first cut 1.17 m, and then another 3.42 m. What is the length of the remaining pipe?

1247. Apples together with the box weigh 25.6 kg. How many kilograms do apples weigh if an empty box weighs 1.13 kg?

1248. Find the Length of the Polyline ABC if AB = 4.7 cm, and BC is 2.3 cm less than AB.

1249. One can contains 10.7 liters of milk, while the other contains 1.25 liters less. How much milk is in two cans?

1250. Numbers:

1) 147,85 - 34 - 5,986;

2) 137,52 - (113,21 + 5,4);

3) (157,42 - 114,381) - 5,91;

4) 1142,3 - (157,8 - 3,71).

1251. Calculate:

1) 137,42 - 15 - 9,127;

2) 1147,58 - (142,37 + 8,13);

3) (159,52 - 142,78) + 11,189;

4) 4297,52 - (113,43 + 1298,3).

1252. Find the value of the expression a - 5.2 - b if a = 8.91, b = 0.13.

1253. The speed of the boat in still water is 17.2 km / h, and the speed of the current is 2.7 km / h. Find the speed of the boat upstream and upstream.

1254. Fill in the table:

Own speed, km / h	Speed currents, km / h	Downstream speed, km / h	Speed ​​upstream, km / h
13,1
17,2		18,5
12,35			10,85
		13,5
	1,65		12,95

1255. Find the missing numbers in the chain:

1256. Measure in centimeters the sides of the quadrangle shown in Figure 257 and find its perimeter.

1257. Draw an arbitrary triangle, measure its sides in centimeters and find the perimeter of the triangle.

1258. Point B was designated on the segment AC (Fig. 258).

1) Find AC if AB = 3.2 cm, BC = 2.1 cm;

2) find BC if AC = 12.7 dm, AB = 8.3 dm.

Rice. 257

Rice. 258

Rice. 259

1259. How many centimeters is the segment AB is longer than the segment CD (Fig. 259)?

1260. One side of the rectangle is 2.7 cm, and the other is 1.3 cm shorter. Find the perimeter of the rectangle.

1261. The base of an isosceles triangle is 8.2 cm, and the lateral side is 2.1 cm less than the base. Find the perimeter of the triangle.

1262. The first side of the triangle is 13.6 cm, the second is 1.3 cm shorter than the first. Find the third side of the triangle if its perimeter is 43.1 cm.

Enough level

1263. Write down a sequence of five numbers if:

1) the first number is 7.2, and each next number is 0.25 more than the previous one;

2) the first number is 10.18, and each next one is 0.34 less than the previous one.

1264. The first box contained 12.7 kg of apples, which is 3.9 kg more than the second. The third box contained 5.13 kg less apples than the first and second combined. How many kilograms of apples were in three boxes together?

1265. On the first day, tourists covered 8.3 km, which is 1.8 km more than on the second day, and 2.7 km less than on the third. How many kilometers did the tourists walk in three days?

1266. Perform addition, choosing a convenient calculation order:

1) 0,571 + (2,87 + 1,429);

2) 6,335 + 2,896 + 1,104;

3) 4,52 + 3,1 + 17,48 + 13,9.

1267. Perform addition, choosing a convenient calculation order:

1) 0,571 + (2,87 + 1,429);

2) 7,335 + 3,896 + 1,104;

3) 15,2 + 3,71 + 7,8 + 4,29.

1268. Put numbers instead of asterisks:

1269. Put such numbers in the cells so that correctly executed examples are formed:

1270. Simplify the expression:

1) 2.71 + x - 1.38; 2) 3.71 + c + 2.98.

1271. Simplify the expression:

1) 8.42 + 3.17 - x; 2) 3.47 + y - 1.72.

1272. Find the regularity and write down the three numbers of the sequence:

1) 2; 2,7; 3,4 ... 2) 15; 13,5; 12 ...

1273. Solve the equations:

1) 13.1 - (x + 5.8) = 1.7;

2) (x - 4.7) - 2.8 = 5.9;

3) (y - 4.42) + 7.18 = 24.3;

4) 5.42 - (c - 9.37) = 1.18.

1274. Solve the equations:

1) (3.9 + x) - 2.5 = 5.7;

2) 14.2 - (6.7 + x) = 5.9;

3) (c - 8.42) + 3.14 = 5.9;

4) 4.42 + (y - 1.17) = 5.47.

1275. Find the value of the expression in a convenient way, using the properties of subtraction:

1) (14,548 + 12,835) - 4,548;

2) 9,37 - 2,59 - 2,37;

3) 7,132 - (1,132 + 5,13);

4) 12,7 - 3,8 - 6,2.

1276. Find the value of the expression in a convenient way using the properties of subtraction:

1) (27,527 + 7,983) - 7,527;

2) 14,49 - 3,1 - 5,49;

3) 14,1 - 3,58 - 4,42;

4) 4,142 - (2,142 + 1,9).

1277. Calculate, writing these values ​​in decimeters:

1) 8.72 dm - 13 cm;

2) 15.3 dm + 5 cm + 2 mm;

3) 427 cm + 15.3 dm;

4) 5 m 3 dm 2 cm 4 m 7 dm 2 cm.

1278. The perimeter of an isosceles triangle is

17.1 cm and the side is 6.3 cm. Find the length of the base.

1279. The speed of a freight train is 52.4 km / h, and that of a passenger train is 69.5 km / h. Determine whether these trains are moving away or approaching and by how many kilometers per hour, if they left at the same time:

1) from two points, the distance between which is 600 km, towards each other;

2) from two points, the distance between which is 300 km, and the passenger one overtakes the freight one;

1280. The speed of the first cyclist is 18.2 km / h, and the speed of the second is 16.7 km / h. Determine whether the cyclists are moving away or approaching and how many kilometers per hour, if they left at the same time:

1) from two points, the distance between which is 100 km, towards each other;

2) from two points, the distance between which is 30 km, and the first one overtakes the second;

3) from one point in opposite directions;

4) from one point in one direction.

1281. Calculate, the answer is rounded to the nearest hundredth:

1) 1,5972 + 7,8219 - 4,3712;

2) 2,3917 - 0,4214 + 3,4515.

1282. Calculate by writing these values ​​in centners:

1) 8 q - 319 kg;

2) 9 q 15 kg + 312 kg;

3) 3 t 2 c - 2 c 3 kg;

4) 5 t 2 q 13 kg + 7 t 3 q 7 kg.

1283. Calculate by writing these values ​​in meters:

1) 7.2 m - 25 dm;

2) 2.7 m + 3 dm 5 cm;

3) 432 dm + 3 m 5 dm + 27 cm;

4) 37 dm - 15 cm.

1284. The perimeter of an isosceles triangle is

15.4 cm and the base 3.4 cm. Find the length of the side.

1285. The perimeter of the rectangle is 12.2 cm, and the length of one of the sides is 3.1 cm. Find the length of the side that is not equal to the given one.

1286. There are 109.6 kg of tomatoes in three boxes. In the first and second boxes together 69.9 kg, and in the second and third 72.1 kg. How many kilograms of tomatoes are in each box?

1287. Find the numbers a, b, c, d in the chain:

1288. Find the numbers a and b in the chain:

High level

1289. Replace the asterisks with the signs "+" and "-" so that the equality is fulfilled:

1) 8,1 * 3,7 * 2,7 * 5,1 = 2;

2) 4,5 * 0,18 * 1,18 * 5,5 = 0.

1290. Chip had 5.2 UAH. After Dale lent him UAH 1.7, Dale got UAH 1.2. less than Chip's. How much money did Dale have first?

1291. Two brigades pave the highway and move towards each other. When the first brigade paved 5.92 km of the highway, and the second - 1.37 km less, then 0.85 km were left before their meeting. How long was the section of the highway that needed to be asphalted?

1292. How will the sum of two numbers change if:

1) increase one of the terms by 3.7, and the other by 8.2;

2) increase one of the terms by 18.2, and decrease the other by 3.1;

3) reduce one of the terms by 7.4, and the other by 8.15;

4) increase one of the terms by 1.25 and decrease the other by 1.25;

5) increase one of the terms by 7.2, and decrease the other by 8.9?

1293. How will the difference change if:

1) decrease the decreasing by 7.1;

2) increase the decreasing by 8.3;

3) increase the deductible by 4.7;

4) decrease the deductible by 4.19?

1294. The difference between the two numbers is 8.325. What is the new difference if the decreasing one is increased by 13.2 and the deductible is increased by 5.7?

1295. How will the difference change if:

1) increase the decreasing by 0.8, and the deductible by 0.5;

2) increase the decreasing by 1.7, and the deductible by 1.9;

3) increase the decreasing by 3.1, and decrease the deductible by 1.9;

4) decrease the decreasing by 4.2, and increase the deductible by 2.1?

Repetition exercises

1296. Compare the values ​​of expressions without performing any actions:

1) 125 + 382 and 382 + 127; 2) 473 ∙ 29 472 ∙ 29;

3) 592 - 11 and 592 - 37; 4) 925: 25 and 925: 37.

1297. The dining room has two types of first courses, 3 types of second and 2 types of third courses. How many ways can you choose a three-course meal in this dining room?

1298. The perimeter of the rectangle is 50 dm. The length of the rectangle is 5 dm longer than the width. Find the sides of the rectangle.

1299. Write down the largest decimal fraction:

1) with one decimal place, less than 10;

2) with two decimal places, less than 5.

1300. Write down the smallest decimal fraction:

1) with one decimal place, greater than 6;

2) with two decimal places, greater than 17.

Home independent work № 7

2. Which of the inequalities is correct:

A) 2.3> 2.31; B) 7.5< 7,49;

B ) 4.12> 4.13; D) 5.7< 5,78?

3. 4,08 - 1,3 =

A) 3.5; B) 2.78; B) 3.05; D) 3.95.

4. Write down the decimal fraction 4.0701 with a mixed number:

5. Which of the rounding to the nearest hundredth is correct:

A ) 2.729 ≈ 2.72; B) 3.545 ≈ 3.55;

B ) 4.729 ≈ 4.7; D) 4.365 ≈ 4.36?

6. Find the root of the equation x - 6.13 = 7.48.

A) 13.61; B) 1.35; B) 13.51; D) 12.61.

7. Which of the proposed equalities is correct:

A) 7 cm = 0.7 m; B) 7 dm2 = 0.07 m2;

in) 7 mm = 0.07 m; D) 7 cm3 = 0.07 m3?

8. Names the greatest natural number that does not exceed 7.0809:

A) 6; B) 7; AT 8; D) 9.

9. How many digits are there that can be put instead of an asterisk in the approximate equality 2.3 * 7 * 2.4 so that rounding to decimal is correct?

A) 5; B) 0; AT 4; D) 6.

10.4 a 3 m2 =

A) 4.3 a; B) 4.003 a; B) 4.03 a; D) 43.

11. Which of the proposed numbers can be substituted for a so that the double inequality 3.7< а < 3,9 была правильной?

A) 3.08; B) 3.901; B) 3.699; D) 3.83.

12. How will the sum of three numbers change if the first term is increased by 0.8, the second is increased by 0.5, and the third is decreased by 0.4?

A ) will increase by 1.7; B) will increase by 0.9;

B ) will increase by 0.1; D) will decrease by 0.2.

Knowledge Test Tasks No. 7 (§34 - §37)

1. Compare decimal fractions:

1) 47.539 and 47.6; 2) 0.293 and 0.2928.

2. Perform the addition:

1) 7,97 + 36,461; 2) 42 + 7,001.

3. Subtract:

1) 46,63 - 7,718; 2) 37 - 3,045.

4. Round up to:

1) tenths: 4.597; 0.8342;

2) hundredths: 15,795; 14.134.

5. Express it in kilometers and write it down in decimal:

1) 7 km 113 m; 2) 219 m; 3) 17 m; 4) 3129 m.

6. The boat's own speed is 15.7 km / h, and the current speed is 1.9 km / h. Find the speed of the boat upstream and upstream.

7. On the first day, 7.3 tons of vegetables were brought to the warehouse, which is 2.6 tons more than on the second, and 1.7 tons less than on the third day. How many tons of vegetables were delivered to the warehouse in three days?

8. Find the meaning of the expression by choosing a convenient procedure:

1) (8,42 + 3,97) + 4,58; 2) (3,47 + 2,93) - 1,47.

9. Write down three numbers, each less than 5.7 but greater than 5.5.

10. Additional task. Write down all the numbers that can be put in place of * so that the inequality is approximated correctly:

1) 3,81*5 ≈3,82; 2) 7,4*6≈ 7,41.

11. Additional task. At what natural values n inequalities 0.7< n < 4,2 и 2,7 < n < 8,9 одновременно являются правильными?

In this article, we will focus on subtracting decimal fractions... Here we will look at the rules for subtracting end decimal fractions, dwell on subtraction of decimal fractions in a column, and also consider how the subtraction of infinite periodic and non-periodic decimal fractions is carried out. Finally, let's talk about subtracting decimal fractions from natural numbers, fractions and mixed numbers, and subtracting natural numbers, fractions and mixed numbers from decimals.

Let's say right away that here we will only consider the subtraction of a smaller decimal fraction from a larger decimal fraction, we will analyze other cases in the articles subtraction of rational numbers and subtraction of real numbers.

Page navigation.

General principles of subtracting decimal fractions

At its core subtraction of trailing decimal fractions and infinite periodic decimal fractions represents the subtraction of the corresponding fractions. Indeed, the indicated decimal fractions are the decimal notation of ordinary fractions, as described in the article on the conversion of ordinary fractions to decimal fractions and vice versa.

Let's consider examples of subtraction of decimal fractions, starting from the stated principle.

Example.

Subtract from the decimal 3.7 decimal 0.31.

Solution.

Since 3.7 = 37/10 and 0.31 = 31/100, then. So the subtraction of decimal fractions was reduced to the subtraction of ordinary fractions with different denominators:. We represent the resulting fraction as a decimal fraction: 339/100 = 3.39.

Answer:

3,7−0,31=3,39 .

Note that it is convenient to subtract final decimal fractions in a column, we will talk about this method in.

Now let's look at an example of subtracting periodic decimal fractions.

Example.

Subtract 0, (4) from the periodic decimal fraction 0.41 (6).

Solution.

Answer:

0,(4)−0,41(6)=0,02(7) .

It remains to voice principle of subtraction of infinite non-periodic fractions.

Subtracting infinite non-periodic fractions is reduced to subtracting trailing decimal fractions. To do this, subtracted infinite decimal fractions are rounded to a certain digit, usually to the lowest possible (see rounding numbers).

Example.

Subtract the final decimal 0.52 from the infinite non-periodic decimal 2.77369….

Solution.

Let's round the infinite non-periodic decimal fraction to 4 decimal places, we have 2.77369… ≈2.7737. Thus, 2,77369…−0,52≈2,7737−0,52 ... Calculating the difference between the final decimal fractions, we get 2.2537.

Answer:

2,77369…−0,52≈2,2537 .

Column subtraction of decimal fractions

A very convenient way to subtract trailing decimals is column subtraction. Column subtraction of decimal fractions is very similar to column subtraction of natural numbers.

To execute column subtraction of decimal fractions, necessary:

• equalize the number of decimal places in the notation of decimal fractions (if it is, of course, different) by adding a number of zeros to one of the fractions on the right;
• write the subtracted under the decreasing so that the digits of the corresponding digits are under each other, and the comma is under the comma;
• perform subtraction in a column, ignoring the commas;
• in the resulting difference, put a comma so that it is located under the commas of the reduced and subtracted.

Consider an example of column subtraction of decimal fractions.

Example.

Subtract the decimal 10.30501 from the decimal 4 452.294.

Solution.

Obviously, the number of decimal places in fractions is different. Let's equalize it by adding two zeros to the right in the fraction 4 452.294, and we get the decimal fraction equal to it 4 452.29400.

Now let's write the subtracted under the decrement, as the method of subtracting decimal fractions in a column suggests:

We carry out the subtraction, ignoring the commas:

It remains only to put a decimal point in the resulting difference:

At this stage, the record has taken on a finished form, and the subtraction of decimal fractions in a column is completed. The result is the following.

Answer:

4 452,294−10,30501=4 441,98899 .

Subtracting a decimal from a natural number and vice versa

Subtracting a final decimal from a natural number It is most convenient to do it in a column, writing down the reduced natural number in the form of a decimal fraction with zeros in the fractional part. Let's deal with this when solving an example.

Example.

Subtract the decimal fraction 7.32 from the natural number 15.

Solution.

We represent the natural number 15 as a decimal fraction, adding two digits 0 after the decimal point (since the subtracted decimal fraction has two digits in the fractional part), we have 15.00.

Now we will subtract decimal fractions in a column:

As a result, we get 15-7.32 = 7.68.

Answer:

15−7,32=7,68 .

Subtract an infinite periodic decimal from a natural number can be reduced to subtracting an ordinary fraction from a natural number. To do this, it is enough to replace the periodic decimal fraction with the corresponding ordinary fraction.

Example.

Subtract from the natural number 1 the periodic decimal fraction 0, (6).

Solution.

Periodic decimal fraction 0, (6) corresponds to the ordinary fraction 2/3. Thus, 1−0, (6) = 1−2 / 3 = 1/3. Received common fraction can be written as a decimal fraction 0, (3).

Answer:

1−0,(6)=0,(3) .

Subtracting an infinite non-periodic decimal fraction from a natural number is reduced to subtracting the final decimal fraction. To do this, an infinite non-periodic decimal fraction must be rounded to a certain digit.

Example.

Subtract the infinite non-periodic decimal fraction 4.274… from the natural number 5.

Solution.

First, round off the infinite decimal fraction, we can round to hundredths, we have 4.274… ≈4.27. Then 5-4.274… ≈5-4.27.

Let's represent the natural number 5 as 5.00, and perform the subtraction of decimal fractions in a column:

Answer:

5−4,274…≈0,73 .

It remains to voice rule for subtracting a natural number from a decimal fraction: to subtract a natural number from a decimal fraction, subtract this natural number from the whole part of the decimal fraction to be reduced, and leave the fractional part unchanged. This rule applies to both finite and infinite decimal fractions. Let's consider the solution of an example.

Example.

Subtract the natural number 17 from the decimal fraction 37.505.

Solution.

Whole part decimal 37.505 is 37. Subtract the natural number 17 from it, we have 37−17 = 20. Then 37.505−17 = 20.505.

Answer:

37,505−17=20,505 .

Subtracting a decimal from a fraction or mixed number and vice versa

Subtract a finite decimal or infinite periodic decimal from a fraction can be reduced to subtraction of ordinary fractions. To do this, it is enough to convert the subtracted decimal fraction into an ordinary fraction.

Example.

Subtract 0.25 from the decimal 4/5.

Solution.

Since 0.25 = 25/100 = 1/4, the difference between the ordinary fraction 4/5 and the decimal fraction 0.25 is equal to the difference between the ordinary fractions 4/5 and 1/4. So, 4/5−0,25=4/5−1/4=16/20−5/20=11/20 ... In decimal notation, the resulting ordinary fraction looks like 0.55.

Answer:

4/5−0,25=11/20=0,55 .

Likewise subtracting a final decimal or a periodic decimal from a mixed number is reduced to subtracting an ordinary fraction from a mixed number.

Example.

Subtract the decimal 0, (18) from the mixed number.

Solution.

First, let's translate the periodic decimal fraction 0, (18) into an ordinary fraction:. Thus, . Received mixed number in decimal notation has the form 8, (18).

Date: 25.02.16 I approve:

Topic: Subtraction of Decimals

Goals:

Provide students with knowledge of decimal subtraction

Develop students' intelligence and cognitive interest

Carry out labor education

Equipment: textbook, chalkboard

Lesson type : combined

Method: work with laggards

During the classes :

Greetings

Checking for absent

Examination homework

Frontal poll

Explanation of the new material:

As well as addition, subtraction of decimal fractions is performed according to the rules natural numbers.

Basic rules for subtracting decimal fractions.

Equalize the number of decimal places.

We write decimal fractions under each other so that the commas are under each other.

We perform subtraction of decimal fractions, ignoring the commas, according to the rules of subtraction into a column of natural numbers.

We put a comma under the commas in the answer.

If you feel confident in decimal fractions and understand well what are called tenths, hundredths, etc., we suggest that you try another way of subtracting (adding) decimal fractions without writing them in a column. Another waysubtracting decimal fractions , like addition, is based on three basic rules.

Subtract decimalsfrom right to left ... That is, starting from the rightmost digit after the decimal point.

When subtracting a larger number from a smaller one, we take ten from the neighbor on the left of the smaller number.

As usual, consider an example:

Subtract from right to left from the rightmost digit. We have the rightmost number in both fractions - hundredths. 1 - in the first number, 1 - in the second. So we subtract them. 1 - 1 = 0. It turned out 0, which means that in place of the hundredths of the new number we write zero.

Subtract tenths from tenths. 2 - in the first number, 3 - in the second number. Since we cannot subtract 3 (more) from 2 (less), we borrow ten from the neighbor on the left for 2. We have 5. Now we do not subtract 3 from 2, but subtract 3 from 12.
12 − 3 = 9.
In place of the tenths of the new number, we write 9. Do not forget that after taking ten out of 5, we must subtract one from 5. In order not to forget this, we put an empty circle over 5.

Finally, subtract whole parts. 14 - in the first number (do not forget that we subtracted 1 from 5), 8 - in the second number. 14 - 8 = 6

Remember!

In the second number, the rightmost digit is 2 (hundredths), and in the first number, hundredths are not explicitly. Therefore, add zero to the first number to the right of 9 and subtract it according to the basic rules.