Let us see a few examples. Example 18: a) Find the sum of 173.809 and 23.617. b) Subtract 216.735 from 563.726. Solution: a) b) 12 16 11 5 2/ 6/ 12 17 3 . 8 0 9 5 6/ 3/ . 7/ 2/ 6 + 23 . 617 –2 1 6 . 7 3 5 197 . 426 346 . 991 Example 19: Solve: a) 294.631 + 306.524 b) 11.904 – 6.207 Solution: a) 1 1 1 b) 11 8 9 14 294 . 631 1/ 1/ . 9/ 0/ 4/ +3 0 6 . 5 2 4 – 6 . 20 7 601 . 155 5 . 69 7 15 1/7/2019 3:24:19 PM NR_BGM_9789386663511 MAPLE G05 INTEGRATED TEXTBOOK TERM 3_Text.pdf 56

Chapter Decimals - II 12 Let Us Learn About • multiplying and dividing decimals by 1-digit and 2-digit numbers. • m ultiplying decimals by 10, 100 and 1000. • multiplying and dividing a decimal number by another decimal number. • the relationship between percentages, decimals and fractions. Concept 12.1: Multiply and Divide Decimals Think Pooja bought six different types of toys for ` 236.95 each. She calculated the total cost and paid the amount to the shopkeeper. Pooja then went to a sweet shop where 410.750 kg of a sweet was prepared. She wanted to know the number of 250 g packs that can be made from it. Do you know how to find the total cost of the toys? Can you calculate how many packs of sweets can be made? Recall Multiplication and division of decimal numbers are similar to that of usual numbers. Let us recall multiplication and division of numbers by answering the following. Solve: a) 267 × 14 b) 3218 × 34 c) 7424 × 14 d) 576 ÷ 12 e) 265 ÷ 5 f) 384 ÷ 4 19 NR_BGM_9789386663511 MAPLE G05 INTEGRATED TEXTBOOK TERM 3_Text.pdf 60 1/7/2019 3:24:19 PM

& Remembering and Understanding Multiplication of decimals is similar to multiplication of numbers. When two decimal numbers are multiplied, a) count the total number of digits after decimal point in both the numbers. Say it is ‘n’. b) multiply the two decimal numbers as usual and place the decimal point in the product after ‘n’ digits from the right. Multiply decimals by 1-digit and 2-digit numbers Let us understand the multiplication of decimals through a few examples. Example 1: Solve: a) 25.146 × 23 b) 276.32 × 6 Solution: a) 25.146 × 23 T Th Th H TO 1 1 1 46 11 23 25 1 38 20 × 58 1 + 75 4 502 9 5 7 8.3 Therefore, 25.146 × 23 = 578.358 b) 276.32 × 6 Step 1: To multiply the given numbers, follow the steps outlined here. Multiply the numbers without considering the decimal point. T Th Th H T O 4 3 11 2 7 6 32 ×6 1 6 5 7 92 Decimals - II 20 NR_BGM_9789386663511 MAPLE G05 INTEGRATED TEXTBOOK TERM 3_Text.pdf 61 1/7/2019 3:24:19 PM

Step 2: Count the number of decimal places in the given number. The number of decimal places in 276.32 is two. Step 3: Count from the right, the number of digits in the product as the number of decimal places in the given number. Then place the decimal point. Therefore, 276.32 × 6 = 1657.92. Multiply decimals by 10,100 and 1000 Example 2: Solve: a) 3.4567 × 10 b) 3.4567 × 100 c) 3.4567 × 1000 Solution: To multiply a decimal number by 10, 100 and 1000, follow the steps. Step 1: Write the decimal number as it is. Step 2: Shift the decimal point to the right by as many digits as the number of zeros in the multiplier. Therefore, a) 3 .4567 × 10 = 34.567 (The decimal point is shifted to the right by one digit as the multiplier is 10.) b) 3.4567 × 100 = 345.67 (The decimal point is shifted to the right by two digits as the multiplier is 100.) c) 3 .4567 × 1000 = 3456.7 (The decimal point is shifted to the right by three digits as the multiplier is 1000.) Multiply a decimal number by another decimal number Multiplication of a decimal number by another decimal number is similar to multiplication of a decimal number by a number. Let us understand this through an example. Example 3: Solve: 7.12 × 3.7 Solution: Step1: Multiply the numbers without considering the decimal point. 1 7 12 × 37 11 4 9 84 +2 1 3 6 0 26 3 44 21 1/7/2019 3:24:19 PM NR_BGM_9789386663511 MAPLE G05 INTEGRATED TEXTBOOK TERM 3_Text.pdf 62

Step 2: Count the number of decimal places in both the multiplicand and the multiplier and add them. The number of decimal places in 7.12 is two The number of decimal places in 3.7 is three Total number of decimal places = 2 + 1 = 3 Step 3: Count as many digits in the product from the right as the total number of decimal places. Then place the decimal point. Therefore, 7.12 × 3.7 = 26.344. Divide decimal numbers by 1-digit and 2-digit numbers Division of decimal numbers is similar to the division of usual numbers. Let us understand this through a few examples. Example 4: Divide: a) 147.9 ÷ 3 b) 64.2 ÷ 6 Solution: Step 1: Follow the steps to divide. Divide the decimal number (dividend) by the 1-digit number (divisor) as usual. Step 2: Place the decimal point in the quotient exactly above the decimal point in the dividend. a) 49.3 b) 10.7 )3 147.9 )6 64.2 −12 −6 27 042 − 27 − 42 09 00 − 09 Example 5: 00 b) 56.96 ÷ 32 Divide: a) 20.475 ÷ 25 Solution : a) 0.819 b) 1.78 )25 20.475 )32 56.96 − 200 − 32 47 249 − 25 − 224 225 256 − 225 − 256 000 000 Decimals - II 22 NR_BGM_9789386663511 MAPLE G05 INTEGRATED TEXTBOOK TERM 3_Text.pdf 63 1/7/2019 3:24:19 PM

Divide decimals by 10,100 and 1000 Example 6: Solve: a) 3.4567 ÷ 10 b) 3.4567 ÷ 100 c) 3.4567 ÷ 1000 Solution: To divide a decimal number by 10, 100 and 1000, follow these steps: Step 1: Write the decimal number as it is. Step 2: Shift the decimal point to the left by as many digits as the number of zeros in the divisor. Therefore, a) 3.4567 ÷ 10 = 0.34567 (The decimal point is shifted to the left by one digit as the divisor is 10.) b) 3.4567 ÷ 100 = 0.034567 (The decimal point is shifted to the left by two digits as the divisor is 100.) c) 3.4567 ÷ 1000 = 0.0034567 (The decimal point is shifted to the left by three digits as the divisor is 1000.) Use decimals to continue division of numbers resulting in remainders Recall that sometimes we get remainders in the division of numbers. We can use the decimal point to divide the remainder up to the desired number of decimal places. Let us understand this through a few examples. Example 7: Solve: 54487 ÷ 46 Solution: To divide the given numbers, follow the steps given here. Step 1: Divide as usual, till you get a remainder. 1184 )46 54487 − 46 84 − 46 388 − 368 207 − 184 23 Step 2: Place a point to the right of the quotient. Add a zero to the right of the remainder and continue the division. 23 1/7/2019 3:24:19 PM NR_BGM_9789386663511 MAPLE G05 INTEGRATED TEXTBOOK TERM 3_Text.pdf 64

1184.5 )46 54487 − 46 84 − 46 388 − 368 207 − 184 230 − 230 000 In this case, the division is stopped after one decimal place as the remainder is zero. In some cases, the division continues for more than three decimal places. But usually, we divide up to three decimal places. We then round off the quotient to two decimal places. Example 8: Divide the following up to two decimal places. a) 91158 ÷ 28 b) 78323 ÷ 15 Solution: a) 3255.642 b) 5221.533 )28 91158 )15 78323 − 84 − 75 71 33 − 56 − 30 155 32 − 140 − 30 158 23 − 140 − 15 180 80 − 168 − 75 120 50 − 112 − 45 80 50 − 56 − 45 4 5 Therefore, 91158 ÷ 28 = 3255.64 and 78323 ÷ 15 = 5221.53 after rounding off to two decimal places. Decimals - II 24 NR_BGM_9789386663511 MAPLE G05 INTEGRATED TEXTBOOK TERM 3_Text.pdf 65 1/7/2019 3:24:19 PM

Divide a decimal number by another decimal number Let us understand the division of a decimal number by another through an example. Example 9: Solve: 3.0525 ÷ 5.5 Solution: To divide a decimal number by another, follow these steps. Step 1: Convert the decimals into fractions. Step 2: Step 3: 3.0525 = 30525 and 5.5 = 55 Step 4: 10000 10 Find the reciprocal of the divisor. 55 10 Reciprocal of is . 10 55 Multiply dividend by the reciprocal of divisor. 30525 10 30525 555 × = = 10000 55 55´1000 1000 Convert the fraction to a decimal number. 555 = 0.555 1000 Therefore, 3.0525 ÷ 5.5 = 0.555 25 1/7/2019 3:24:19 PM NR_BGM_9789386663511 MAPLE G05 INTEGRATED TEXTBOOK TERM 3_Text.pdf 66

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