m81295 Decimal Operations By the end of this section, you will be able to: Add and subtract decimals Multiply decimals Divide decimals Use decimals in money applications 47846d7d-94ab-47d7-965d-e95f964f2c1c Before you get started, take this readiness quiz. Simplify 70100. If you missed this problem, review . 710 Multiply 310·910. If you missed this problem, review . 27100 Divide −36÷(−9). If you missed this problem, review . 4

Let’s take one more look at the lunch order from the start of Decimals, this time noticing how the numbers were added together. All three items (sandwich, water, tax) were priced in dollars and cents, so we lined up the dollars under the dollars and the cents under the cents, with the decimal points lined up between them. Then we just added each column, as if we were adding whole numbers. By lining up decimals this way, we can add or subtract the corresponding place values just as we did with whole numbers. Write the numbers vertically so the decimal points line up. Use zeros as place holders, as needed. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers. Add: 3.7+12.4. 3.7+12.4 Write the numbers vertically so the decimal points line up. 3.7 +12.4_____ Place holders are not needed since both numbers have the same number of decimal places. Add the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers. 31.7 +12.4_____ 16.1 Add: 5.7+11.9. 17.6 Add: 18.32+14.79. 33.11 Add: 23.5+41.38. 23.5+41.38 Write the numbers vertically so the decimal points line up. Place 0 as a place holder after the 5 in 23.5, so that both numbers have two decimal places. Add the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers. Add: 4.8+11.69. 16.49 Add: 5.123+18.47. 23.593 How much change would you get if you handed the cashier a $20 bill for a $14.65 purchase? We will show the steps to calculate this in the next example. Subtract: 20−14.65. 20−14.65 Write the numbers vertically so the decimal points line up. Remember 20 is a whole number, so place the decimal point after the 0. Place two zeros after the decimal point in 20, as place holders so that both numbers have two decimal places. Subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers. Subtract: 10−9.58. 0.42 Subtract: 50−37.42. 12.58 Subtract: 2.51−7.4. If we subtract 7.4 from 2.51, the answer will be negative since 7.4>2.51. To subtract easily, we can subtract 2.51 from 7.4. Then we will place the negative sign in the result. 2.51−7.4 Write the numbers vertically so the decimal points line up. Place zero after the 4 in 7.4 as a place holder, so that both numbers have two decimal places. Subtract and place the decimal in the answer. Remember that we are really subtracting 2.51−7.4 so the answer is negative. 2.51−7.4=−4.89 Subtract: 4.77−6.3. −1.53 Subtract: 8.12−11.7. −3.58

Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first review multiplying fractions. Do you remember how to multiply fractions? To multiply fractions, you multiply the numerators and then multiply the denominators. So let’s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side in . Look for a pattern. A B (0.3)(0.7) (0.2)(0.46) Convert to fractions. (310)(710) (210)(46100) Multiply. 21100 921000 Convert back to decimals. 0.21 0.092 There is a pattern that we can use. In A, we multiplied two numbers that each had one decimal place, and the product had two decimal places. In B, we multiplied a number with one decimal place by a number with two decimal places, and the product had three decimal places. How many decimal places would you expect for the product of (0.01)(0.004)? If you said “five”, you recognized the pattern. When we multiply two numbers with decimals, we count all the decimal places in the factors—in this case two plus three—to get the number of decimal places in the product—in this case five. Once we know how to determine the number of digits after the decimal point, we can multiply decimal numbers without converting them to fractions first. The number of decimal places in the product is the sum of the number of decimal places in the factors. The rules for multiplying positive and negative numbers apply to decimals, too, of course. When multiplying two numbers, if their signs are the same, the product is positive. if their signs are different, the product is negative. When you multiply signed decimals, first determine the sign of the product and then multiply as if the numbers were both positive. Finally, write the product with the appropriate sign. Determine the sign of the product. Write the numbers in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders. Write the product with the appropriate sign. Multiply: (3.9)(4.075). (3.9)(4.075) Determine the sign of the product. The signs are the same. The product will be positive. Write the numbers in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points. Place the decimal point. Add the number of decimal places in the factors (1+3). Place the decimal point 4 places from the right. The product is positive. (3.9)(4.075)=15.8925 Multiply: 4.5(6.107). 27.4815 Multiply: 10.79(8.12). 87.6148 Multiply: (−8.2)(5.19). (−8.2)(5.19) The signs are different. The product will be negative. Write in vertical format, lining up the numbers on the right. 5.19 ×8.2_____ Multiply. 5.19 ×8.2_____ 1038 4152_____ 42558 5.19 ×8.2_____ 1038 4152_____ 42.558 The product is negative. (−8.2)(5.19)=−42.558 Multiply: (4.63)(−2.9). −13.427 Multiply: (−7.78)(4.9). −38.122 In the next example, we’ll need to add several placeholder zeros to properly place the decimal point. Multiply: (0.03)(0.045). (0.03)(0.045) The product is positive. Write in vertical format, lining up the numbers on the right. Multiply. Add zeros as needed to get the 5 places. The product is positive. (0.03)(0.045)=0.00135 Multiply: (0.04)(0.087). 0.00348 Multiply: (0.09)(0.067). 0.00603

In many fields, especially in the sciences, it is common to multiply decimals by powers of 10. Let’s see what happens when we multiply 1.9436 by some powers of 10. Look at the results without the final zeros. Do you notice a pattern? 1.9436(10)=19.4361.9436(100)=194.361.9436(1000)=1943.6 The number of places that the decimal point moved is the same as the number of zeros in the power of ten. summarizes the results. Multiply by Number of zeros Number of places decimal point moves 10 1 1 place to the right 100 2 2 places to the right 1,000 3 3 places to the right 10,000 4 4 places to the right We can use this pattern as a shortcut to multiply by powers of ten instead of multiplying using the vertical format. We can count the zeros in the power of 10 and then move the decimal point that same of places to the right. So, for example, to multiply 45.86 by 100, move the decimal point 2 places to the right. Sometimes when we need to move the decimal point, there are not enough decimal places. In that case, we use zeros as placeholders. For example, let’s multiply 2.4 by 100. We need to move the decimal point 2 places to the right. Since there is only one digit to the right of the decimal point, we must write a 0 in the hundredths place. Move the decimal point to the right the same number of places as the number of zeros in the power of 10. Write zeros at the end of the number as placeholders if needed. Multiply 5.63 by factors of ⓐ 10 ⓑ 100ⓒ 1000. By looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right. ⓐ 56.3(10) There is 1 zero in 10, so move the decimal point 1 place to the right. 56.3 ⓑ 5.63(100) There are 2 zeros in 100, so move the decimal point 2 places to the right. 563 ⓒ 5.63(1000) There are 3 zeros in 1000, so move the decimal point 3 places to the right. A zero must be added at the end. 5,630 Multiply 2.58 by factors of ⓐ 10 ⓑ 100 ⓒ 1000. ⓐ 25.8 ⓑ 258 ⓒ 2,580 Multiply 14.2 by factors of ⓐ 10 ⓑ 100 ⓒ 1000. ⓐ 142 ⓑ 1,420 ⓒ 14,200

Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed. To understand decimal division, let’s consider the multiplication problem (0.2)(4)=0.8 Remember, a multiplication problem can be rephrased as a division problem. So we can write 0.8÷4=0.2 We can think of this as “If we divide 8 tenths into four groups, how many are in each group?” shows that there are four groups of two-tenths in eight-tenths. So 0.8÷4=0.2.

Using long division notation, we would write Notice that the decimal point in the quotient is directly above the decimal point in the dividend. To divide a decimal by a whole number, we place the decimal point in the quotient above the decimal point in the dividend and then divide as usual. Sometimes we need to use extra zeros at the end of the dividend to keep dividing until there is no remainder. Write as long division, placing the decimal point in the quotient above the decimal point in the dividend. Divide as usual. Divide: 0.12÷3. 0.12÷3 Write as long division, placing the decimal point in the quotient above the decimal point in the dividend. Divide as usual. Since 3 does not go into 0 or 1 we use zeros as placeholders. 0.12÷3=0.04 Divide: 0.28÷4. 0.07 Divide: 0.56÷7. 0.08 In everyday life, we divide whole numbers into decimals—money—to find the price of one item. For example, suppose a case of 24 water bottles cost $3.99. To find the price per water bottle, we would divide $3.99 by 24, and round the answer to the nearest cent (hundredth). Divide: $3.99÷24. $3.99÷24 Place the decimal point in the quotient above the decimal point in the dividend. Divide as usual. When do we stop? Since this division involves money, we round it to the nearest cent (hundredth). To do this, we must carry the division to the thousandths place. Round to the nearest cent. $0.166≈$0.17 $3.99÷24≈$0.17 This means the price per bottle is 17 cents. Divide: $6.99÷36. $0.19 Divide: $4.99÷12. $0.42

So far, we have divided a decimal by a whole number. What happens when we divide a decimal by another decimal? Let’s look at the same multiplication problem we looked at earlier, but in a different way. (0.2)(4)=0.8 Remember, again, that a multiplication problem can be rephrased as a division problem. This time we ask, “How many times does 0.2 go into 0.8?” Because (0.2)(4)=0.8, we can say that 0.2 goes into 0.8 four times. This means that 0.8 divided by 0.2 is 4. 0.8÷0.2=4 We would get the same answer, 4, if we divide 8 by 2, both whole numbers. Why is this so? Let’s think about the division problem as a fraction. 0.80.2(0.8)10(0.2)10824 We multiplied the numerator and denominator by 10 and ended up just dividing 8 by 2. To divide decimals, we multiply both the numerator and denominator by the same power of 10 to make the denominator a whole number. Because of the Equivalent Fractions Property, we haven’t changed the value of the fraction. The effect is to move the decimal points in the numerator and denominator the same number of places to the right. We use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.It may help to review the vocabulary for division: Determine the sign of the quotient. Make the divisor a whole number by moving the decimal point all the way to the right. Move the decimal point in the dividend the same number of places to the right, writing zeros as needed. Divide. Place the decimal point in the quotient above the decimal point in the dividend. Write the quotient with the appropriate sign. Divide: −2.89÷(3.4). Determine the sign of the quotient. The quotient will be negative. Make the divisor the whole number by 'moving' the decimal point all the way to the right. 'Move' the decimal point in the dividend the same number of places to the right. Divide. Place the decimal point in the quotient above the decimal point in the dividend. Add zeros as needed until the remainder is zero. Write the quotient with the appropriate sign. −2.89÷(3.4)=−0.85 Divide: −1.989÷5.1. −0.39 Divide: −2.04÷5.1. −0.4 Divide: −25.65÷(−0.06). −25.65÷(−0.06) The signs are the same. The quotient is positive. Make the divisor a whole number by 'moving' the decimal point all the way to the right. 'Move' the decimal point in the dividend the same number of places. Divide. Place the decimal point in the quotient above the decimal point in the dividend. Write the quotient with the appropriate sign. −25.65÷(−0.06)=427.5 Divide: −23.492÷(−0.04). 587.3 Divide: −4.11÷(−0.12). 34.25 Now we will divide a whole number by a decimal number. Divide: 4÷0.05. 4÷0.05 The signs are the same. The quotient is positive. Make the divisor a whole number by 'moving' the decimal point all the way to the right. Move the decimal point in the dividend the same number of places, adding zeros as needed. Divide. Place the decimal point in the quotient above the decimal point in the dividend. Write the quotient with the appropriate sign. 4÷0.05=80 We can relate this example to money. How many nickels are there in four dollars? Because 4÷0.05=80, there are 80 nickels in $4. Divide: 6÷0.03. 200 Divide: 7÷0.02. 350

We often apply decimals in real life, and most of the applications involving money. The Strategy for Applications we used in The Language of Algebra gives us a plan to follow to help find the answer. Take a moment to review that strategy now. Identify what you are asked to find. Write a phrase that gives the information to find it. Translate the phrase to an expression. Simplify the expression. Answer the question with a complete sentence. Paul received $50 for his birthday. He spent $31.64 on a video game. How much of Paul’s birthday money was left? What are you asked to find? How much did Paul have left? Write a phrase. $50 less $31.64 Translate. 50−31.64 Simplify. 18.36 Write a sentence. Paul has $18.36 left. Nicole earned $35 for babysitting her cousins, then went to the bookstore and spent $18.48 on books and coffee. How much of her babysitting money was left? $16.52 Amber bought a pair of shoes for $24.75 and a purse for $36.90. The sales tax was $4.32. How much did Amber spend? $65.97 Jessie put 8 gallons of gas in her car. One gallon of gas costs $3.529. How much does Jessie owe for the gas? (Round the answer to the nearest cent.) What are you asked to find? How much did Jessie owe for all the gas? Write a phrase. 8 times the cost of one gallon of gas Translate. 8($3.529) Simplify. $28.232 Round to the nearest cent. $28.23 Write a sentence. Jessie owes $28.23 for her gas purchase. Hector put 13 gallons of gas into his car. One gallon of gas costs $3.175. How much did Hector owe for the gas? Round to the nearest cent. $41.28 Christopher bought 5 pizzas for the team. Each pizza cost $9.75. How much did all the pizzas cost? $48.75 Four friends went out for dinner. They shared a large pizza and a pitcher of soda. The total cost of their dinner was $31.76. If they divide the cost equally, how much should each friend pay? What are you asked to find? How much should each friend pay? Write a phrase. $31.76 divided equally among the four friends. Translate to an expression. $31.76÷4 Simplify. $7.94 Write a sentence. Each friend should pay $7.94 for his share of the dinner. Six friends went out for dinner. The total cost of their dinner was $92.82. If they divide the bill equally, how much should each friend pay? $15.47 Chad worked 40 hours last week and his paycheck was $570. How much does he earn per hour? $14.25 Be careful to follow the order of operations in the next example. Remember to multiply before you add. Marla buys 6 bananas that cost $0.22 each and 4 oranges that cost $0.49 each. How much is the total cost of the fruit? What are you asked to find? How much is the total cost of the fruit? Write a phrase. 6 times the cost of each banana plus 4 times the cost of each orange Translate to an expression. 6($0.22)+4($0.49) Simplify. $1.32+$1.96 Add. $3.28 Write a sentence. Marla's total cost for the fruit is $3.28. Suzanne buys 3 cans of beans that cost $0.75 each and 6 cans of corn that cost $0.62 each. How much is the total cost of these groceries? $5.97 Lydia bought movie tickets for the family. She bought two adult tickets for $9.50 each and four children’s tickets for $6.00 each. How much did the tickets cost Lydia in all? $43.00 The Links to Literacy activity "Alexander Who Used to be Rich Last Sunday" will provide you with another view of the topics covered in this section. Adding and Subtracting Decimals Multiplying Decimals Multiplying by Powers of Ten Dividing Decimals Dividing by Powers of Ten

Add or subtract decimals. Write the numbers vertically so the decimal points line up. Use zeros as place holders, as needed. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers. Multiply decimal numbers. Determine the sign of the product. Write the numbers in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders. Write the product with the appropriate sign. Multiply a decimal by a power of 10. Move the decimal point to the right the same number of places as the number of zeros in the power of 10. Write zeros at the end of the number as placeholders if needed. Divide a decimal by a whole number. Write as long division, placing the decimal point in the quotient above the decimal point in the dividend. Divide as usual. Divide decimal numbers. Determine the sign of the quotient. Make the divisor a whole number by moving the decimal point all the way to the right. Move the decimal point in the dividend the same number of places to the right, writing zeros as needed. Divide. Place the decimal point in the quotient above the decimal point in the dividend. Write the quotient with the appropriate sign. Strategy for Applications Identify what you are asked to find. Write a phrase that gives the information to find it. Translate the phrase to an expression. Simplify the expression. Answer the question with a complete sentence.

Add and Subtract Decimals In the following exercises, add or subtract. 16.92+7.56 24.48 18.37+9.36 256.37−85.49 170.88 248.25−91.29 21.76−30.99 −9.23 15.35−20.88 37.5+12.23 49.73 38.6+13.67 −16.53−24.38 −40.91 −19.47−32.58 −38.69+31.47 −7.22 −29.83+19.76 −4.2+(−9.3) −13.5 −8.6+(−8.6) 100−64.2 35.8 100−65.83 72.5−100 −27.5 86.2−100 15+0.73 15.73 27+0.87 2.51+40 42.51 9.38+60 91.75−(−10.462) 102.212 94.69−(−12.678) 55.01−3.7 51.31 59.08−4.6 2.51−7.4 −4.89 3.84−6.1 Multiply Decimals In the following exercises, multiply. (0.3)(0.4) 0.12 (0.6)(0.7) (0.24)(0.6) 0.144 (0.81)(0.3) (5.9)(7.12) 42.008 (2.3)(9.41) (8.52)(3.14) 26.7528 (5.32)(4.86) (−4.3)(2.71) −11.653 (−8.5)(1.69) (−5.18)(−65.23) 337.8914 (−9.16)(−68.34) (0.09)(24.78) 2.2302 (0.04)(36.89) (0.06)(21.75) 1.305 (0.08)(52.45) (9.24)(10) 92.4 (6.531)(10) (55.2)(1,000) 55,200 (99.4)(1,000) Divide Decimals In the following exercises, divide. 0.15÷5 0.03 0.27÷3 4.75÷25 0.19 12.04÷43 $8.49÷12 $0.71 $16.99÷9 $117.25÷48 $2.44 $109.24÷36 0.6÷0.2 3 0.8÷0.4 1.44÷(−0.3) −4.8 1.25÷(−0.5) −1.75÷(−0.05) 35 −1.15÷(−0.05) 5.2÷2.5 2.08 6.5÷3.25 12÷0.08 150 5÷0.04 11÷0.55 20 14÷0.35 Mixed Practice In the following exercises, simplify. 6(12.4−9.2) 19.2 3(15.7−8.6) 24(0.5)+(0.3)2 12.09 35(0.2)+(0.9)2 1.15(26.83+1.61) 32.706 1.18(46.22+3.71) $45+0.08($45) $48.60 $63+0.18($63) 18÷(0.75+0.15) 20 27÷(0.55+0.35) (1.43+0.27)÷(0.9−0.05) 2 (1.5−0.06)÷(0.12+0.24) [$75.42+0.18($75.42)]÷5 $17.80 [$56.31+0.22($56.31)]÷4 Use Decimals in Money Applications In the following exercises, use the strategy for applications to solve. Spending money Brenda got $40 from the ATM. She spent $15.11 on a pair of earrings. How much money did she have left? $24.89 Spending money Marissa found $20 in her pocket. She spent $4.82 on a smoothie. How much of the $20 did she have left? Shopping Adam bought a t-shirt for $18.49 and a book for $8.92 The sales tax was $1.65. How much did Adam spend? $29.06 Restaurant Roberto’s restaurant bill was $20.45 for the entrée and $3.15 for the drink. He left a $4.40 tip. How much did Roberto spend? Coupon Emily bought a box of cereal that cost $4.29. She had a coupon for $0.55 off, and the store doubled the coupon. How much did she pay for the box of cereal? $3.19 Coupon Diana bought a can of coffee that cost $7.99. She had a coupon for $0.75 off, and the store doubled the coupon. How much did she pay for the can of coffee? Diet Leo took part in a diet program. He weighed 190 pounds at the start of the program. During the first week, he lost 4.3 pounds. During the second week, he had lost 2.8 pounds. The third week, he gained 0.7 pounds. The fourth week, he lost 1.9 pounds. What did Leo weigh at the end of the fourth week? 181.7 pounds Snowpack On April 1, the snowpack at the ski resort was 4 meters deep, but the next few days were very warm. By April 5, the snow depth was 1.6 meters less. On April 8, it snowed and added 2.1 meters of snow. What was the total depth of the snow? Coffee Noriko bought 4 coffees for herself and her co-workers. Each coffee was $3.75. How much did she pay for all the coffees? $15.00 Subway Fare Arianna spends $4.50 per day on subway fare. Last week she rode the subway 6 days. How much did she spend for the subway fares? Income Mayra earns $9.25 per hour. Last week she worked 32 hours. How much did she earn? $296.00 Income Peter earns $8.75 per hour. Last week he worked 19 hours. How much did he earn? Hourly Wage Alan got his first paycheck from his new job. He worked 30 hours and earned $382.50. How much does he earn per hour? $12.75 Hourly Wage Maria got her first paycheck from her new job. She worked 25 hours and earned $362.50. How much does she earn per hour? Restaurant Jeannette and her friends love to order mud pie at their favorite restaurant. They always share just one piece of pie among themselves. With tax and tip, the total cost is $6.00. How much does each girl pay if the total number sharing the mud pie is ⓐ 2?ⓑ 3?ⓒ 4?ⓓ 5?ⓔ 6? ⓐ $3 ⓑ $2 ⓒ $1.50 ⓓ $1.20 ⓔ $1 Pizza Alex and his friends go out for pizza and video games once a week. They share the cost of a $15.60 pizza equally. How much does each person pay if the total number sharing the pizza is ⓐ 2?ⓑ 3?ⓒ 4?ⓓ 5?ⓔ 6? Fast Food At their favorite fast food restaurant, the Carlson family orders 4 burgers that cost $3.29 each and 2 orders of fries at $2.74 each. What is the total cost of the order? $18.64 Home Goods Chelsea needs towels to take with her to college. She buys 2 bath towels that cost $9.99 each and 6 washcloths that cost $2.99 each. What is the total cost for the bath towels and washcloths? Zoo The Lewis and Chousmith families are planning to go to the zoo together. Adult tickets cost $29.95 and children’s tickets cost $19.95. What will the total cost be for 4 adults and 7 children? $259.45 Ice Skating Jasmine wants to have her birthday party at the local ice skating rink. It will cost $8.25 per child and $12.95 per adult. What will the total cost be for 12 children and 3 adults?

Paycheck Annie has two jobs. She gets paid $14.04 per hour for tutoring at City College and $8.75 per hour at a coffee shop. Last week she tutored for 8 hours and worked at the coffee shop for 15 hours. ⓐ How much did she earn?ⓑ If she had worked all 23 hours as a tutor instead of working both jobs, how much more would she have earned? ⓐ $243.57 ⓑ $79.35 Paycheck Jake has two jobs. He gets paid $7.95 per hour at the college cafeteria and $20.25 at the art gallery. Last week he worked 12 hours at the cafeteria and 5 hours at the art gallery. ⓐ How much did he earn? ⓑ If he had worked all 17 hours at the art gallery instead of working both jobs, how much more would he have earned?

At the 2010 winter Olympics, two skiers took the silver and bronze medals in the Men's Super-G ski event. Miller's time was 1 minute 30.62 seconds and Weibrecht's time was 1 minute 30.65 seconds. Find the difference in their times and then write the name of that decimal. The difference: 0.03 seconds. Three hundredths of a second. Find the quotient of 0.12÷0.04 and explain in words all the steps taken.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?