Dividing Decimals 4.5


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1 4.5 Dividing Decimals 4.5 OBJECTIVES 1. Divide a decimal by a whole number 2. Divide a decimal by a decimal 3. Divide a decimal by a power of ten 4. Apply division to the solution of an application problem The division of decimals is very similar to our earlier work with dividing whole numbers. The only difference is in learning to place the decimal point in the quotient. Let s start with the case of dividing a decimal by a whole number. Here, placing the decimal point is easy. You can apply the following rule. Step by Step: To Divide a Decimal by a Whole Number Step 1 Step 2 Place the decimal point in the quotient directly above the decimal point of the dividend. Divide as you would with whole numbers. Example 1 Dividing a Decimal by a Whole Number NOTE Do the division just as if you were dealing with whole numbers. Just remember to place the decimal point in the quotient directly above the one in the dividend. Divide by The quotient is CHECK YOURSELF 1 Divide by 34. Let s look at another example of dividing a decimal by a whole number. NOTE Again place the decimal point of the quotient above that of the dividend. Example 2 Dividing a Decimal by a Whole Number Divide by
2 356 CHAPTER 4 DECIMALS We normally do not use a remainder when dealing with decimals. Add a 0 to the dividend and continue. NOTE Remember that adding a 0 does not change the value of the dividend. It simply allows us to complete the division process in this case Add a 0. So The quotient is CHECK YOURSELF 2 Divide by 68. Often you will be asked to give a quotient to a certain place value. In this case, continue the division process to one digit past the indicated place value. Then round the result back to the desired accuracy. When working with money, for instance, we normally give the quotient to the nearest hundredth of a dollar (the nearest cent). This means carrying the division out to the thousandths place and then rounding back. Example 3 Dividing a Decimal by a Whole Number and Rounding the Result NOTE Find the quotient to one place past the desired place, and then round the result. Find the quotient of to the nearest hundredth Add a 0 to carry the division to the thousandths place. So (to the nearest hundredth). CHECK YOURSELF 3 Find to the nearest hundredth. As we mentioned, problems similar to the one in Example 3 often occur when working with money. Example 4 is one of the many applications of this type of division.
3 DIVIDING DECIMALS SECTION Example 4 An Application Involving the Division of a Decimal by a Whole Number A carton of 144 items costs $ What is the price per item to the nearest cent? To find the price per item, divide the total price by 144. NOTE You might want to review the rules for rounding decimals in Section Carry the division to the thousandths place and then round back. The cost per item is rounded to $0.39, or 39. CHECK YOURSELF 4 An office paid $26.55 for 72 pens. What was the cost per pen to the nearest cent? We want now to look at division by decimals. Here is an example using a fractional form. Example 5 Rewriting a Problem That Requires Dividing by a Decimal Divide. NOTE It s always easier to rewrite a division problem so that you re dividing by a whole number. Dividing by a whole number makes it easy to place the decimal point in the quotient So CHECK YOURSELF 5 Write the division as a fraction. We multiply the numerator and denominator by 10 so the divisor is a whole number. This does not change the value of the fraction. Multiplying by 10, shift the decimal point in the numerator and denominator one place to the right. Our division problem is rewritten so that the divisor is a whole number. After we multiply the numerator and denominator by 10, we see that is the same as Rewrite the division problem so that the divisor is a whole number
4 358 CHAPTER 4 DECIMALS NOTE Of course, multiplying by any wholenumber power of 10 greater than 1 is just a matter of shifting the decimal point to the right. Do you see the rule suggested by this example? We multiplied the numerator and the denominator (the dividend and the divisor) by 10. We made the divisor a whole number without altering the actual digits involved. All we did was shift the decimal point in the divisor and dividend the same number of places. This leads us to the following rule. Step by Step: To Divide by a Decimal Step 1 Step 2 Step 3 Step 4 Move the decimal point in the divisor to the right, making the divisor a whole number. Move the decimal point in the dividend to the right the same number of places. Add zeros if necessary. Place the decimal point in the quotient directly above the decimal point of the dividend. Divide as you would with whole numbers. Let s look at an example of the use of our division rule. Example 6 Rounding the Result of Dividing by a Decimal Divide by 0.48 and give the quotient to the nearest tenth. Write ^ ^ Now divide: Shift the decimal points two places to the right to make the divisor a whole number. NOTE Once the division statement is rewritten, place the decimal point in the quotient above that in the dividend Note that we add a 0 to carry the division to the hundredths place. In this case, we want to find the 144 quotient to the nearest tenth Round 3.27 to 3.3. So (to the nearest tenth) CHECK YOURSELF 6 Divide, rounding the quotient to the nearest tenth Let s look at some applications of our work in dividing by decimals.
5 DIVIDING DECIMALS SECTION Example 7 Solving an Application Involving the Division of Decimals Andrea worked 41.5 hours in a week and earned $ What was her hourly rate of pay? To find the hourly rate of pay we must use division. We divide the number of hours worked into the total pay. NOTE Notice that we must add a zero to the dividend to complete the division process ^ ^ Andrea s hourly rate of pay was $5.78. CHECK YOURSELF 7 A developer wants to subdivide a 12.6acre piece of land into 0.45acre lots. How many lots are possible? Example 8 Solving an Application Involving the Division of Decimals At the start of a trip the odometer read 34,563. At the end of the trip, it read 36,235. If 86.7 gallons (gal) of gas were used, find the number of miles per gallon (to the nearest tenth). First, find the number of miles traveled by subtracting the initial reading from the final reading. 36,235 Final reading 34,563 Initial reading 1672 Miles covered Next, divide the miles traveled by the number of gallons used. This will give us the miles per gallon ^ ^ Round to 19.3 mi/gal.
6 360 CHAPTER 4 DECIMALS CHECK YOURSELF 8 John starts his trip with an odometer reading of 15,436 and ends with a reading of 16,238. If he used 45.9 gallons (gal) of gas, find the number of miles per gallon (mi/gal) (to the nearest tenth). Recall that you can multiply decimals by powers of 10 by simply shifting the decimal point to the right. A similar approach will work for division by powers of 10. Example 9 Dividing a Decimal by a Power of 10 (a) Divide (b) Divide The dividend is The quotient is The decimal point has been shifted one place to the left. Note also that the divisor, 10, has one zero. Here the dividend is 378.5, whereas the quotient is The decimal point is now shifted two places to the left. In this case the divisor, 100, has two zeros. CHECK YOURSELF 9 Perform each of the following divisions. (a) (b) Example 9 suggests the following rule. Rules and Properties: To Divide a Decimal by a Power of 10 Move the decimal point to the left the same number of places as there are zeros in the power of 10.
7 DIVIDING DECIMALS SECTION Example 10 Dividing a Decimal by a Power of 10 Divide. (a) ^ Shift one place to the left (b) ^ Shift two places to the left NOTE As you can see, we may have to add zeros to correctly place the decimal point. (c) ^ Shift three places to the left. (d) ^ The decimal after the 85 is implied. REMEMBER: 10 4 is a 1 followed by four zeros. (e) ^ Shift four places to the left CHECK YOURSELF 10 Divide. (a) (b) Let s look at an application of our work in dividing by powers of 10. Example 11 Solving an Application Involving a Power of 10 To convert from millimeters (mm) to meters (m), we divide by How many meters does 3450 mm equal? 3450 mm m Shift three places to the left to divide by ^ m
8 362 CHAPTER 4 DECIMALS CHECK YOURSELF 11 A shipment of 1000 notebooks cost a stationery store $658. What was the cost per notebook to the nearest cent? Recall that the order of operations is always used to simplify a mathematical expression with several operations. You should recall the order of operations as the following. Rules and Properties: The Order of Operations 1. Perform any operations enclosed in parentheses. 2. Apply any exponents. 3. Do any multiplication and division, moving from left to right 4. Do any addition and subtraction, moving from left to right. Example 12 Applying the Order of Operations Simplify each expression. (a) 4.6 ( ) (2.2) (b) 16.5 ( ) (3) parentheses exponent add (left of the subtraction) subtract parentheses exponent multiply subtraction (left of the addition) add CHECK YOURSELF 12 Simplify each expression. (a) 6.35 ( ) (b) ( ) CHECK YOURSELF ANSWERS $0.37, or lots mi/gal 9. (a) 5.26; (b) (a) 0.384; (b) (a) 5.54; (b) 9.62
9 Name 4.5 Exercises Section Date Divide ANSWERS
10 ANSWERS Divide by moving the decimal point Divide and round the quotient to the indicated decimal place tenth hundredth hundredth thousandth tenth thousandth hundredth tenth tenth hundredth hundredth thousandth Solve the following applications. 51. Cost of CDs. Marv paid $40.41 for three CDs on sale. What was the cost per CD? 52. Contributions. Seven employees of an office donated $ during a charity drive. What was the average donation? 53. Book purchases. A shipment of 72 paperback books cost a store $ What was the average cost per book to the nearest cent?
11 ANSWERS 54. Cost. A restaurant bought 50 glasses at a cost of $ What was the cost per glass to the nearest cent? 55. Cost. The cost of a case of 48 items is $ What is the cost of an individual item to the nearest cent? 56. Office supplies. An office bought 18 handheld calculators for $284. What was the cost per calculator to the nearest cent? 57. Monthly payments. Al purchased a new refrigerator that cost $ with interest included. He paid $100 as a down payment and agreed to pay the remainder in 18 monthly payments. What amount will he be paying per month? 58. Monthly payments. The cost of a television set with interest is $ If you make a down payment of $50 and agree to pay the balance in 12 monthly payments, what will be the amount of each monthly payment? Mileage. In five readings, Lucia s gas mileage was 32.3, 31.6, 29.5, 27.3, and 33.4 miles per gallon (mi/gal). What was her average gas mileage to the nearest tenth of a mile per gallon? 60. Pollution. Pollution index readings were 53.3, 47.8, 41.9, 55.8, 43.7, 41.7, and 52.3 for a 7day period. What was the average reading (to the nearest tenth) for the 7 days? 61. Label making. We have inches (in.) of plastic labeling tape and wish to make labels that are 1.25 in. long. How many labels can be made? 62. Wages. Alberto worked 32.5 hours (h), earning $ How much did he make per hour? 63. Cost per pound. A roast weighing 5.3 pounds (lb) sold for $ Find the cost per pound to the nearest cent. 64. Weight. One nail weighs ounce (oz). How many nails are there in 1 lb? (1 lb is 16 oz.) 65. Mileage. A family drove 1390 miles (mi), stopping for gas three times. If they purchased 15.5, 16.2, and 10.8 gallons (gal) of gas, find the number of miles per gallon (the mileage) to the nearest tenth. 66. Mileage. On a trip an odometer changed from 36,213 to 38,319. If 136 gal of gas were used, find the number of miles per gallon (to the nearest tenth). 365
12 ANSWERS Conversion. To convert from millimeters (mm) to inches, we can divide by If film is 35 mm wide, find the width to the nearest hundredth of an inch. 68. Conversion. To convert from centimeters (cm) to inches, we can divide by The rainfall in Paris was 11.8 cm during 1 week. What was that rainfall to the nearest hundredth of an inch? 69. Construction. A roadpaving project will cost $23,500. If the cost is to be shared by 100 families, how much will each family pay? Conversion. To convert from milligrams (mg) to grams (g), we divide by A tablet is 250 mg. What is its weight in grams? Conversion. To convert from milliliters (ml) to liters (L), we divide by If a bottle of wine holds 750 ml, what is its volume in liters? 72. Unit cost. A shipment of 100 calculators cost a store $ Find the cost per calculator (to the nearest cent). 73. The blood alcohol content (BAC) of a person who has been drinking is determined by the formula BAC oz of alcohol % of alcohol of body wt. (hours of drinking 0.015) A 125lb person is driving and is stopped by a policewoman on suspicion of driving under the influence (DUI). The driver claims that in the past 2 hours he consumed only six 12oz bottles of 3.9% beer. If he undergoes a breathalyzer test, what will his BAC be? Will this amount be under the legal limit for your state? 74. Four brands of soap are available in a local store. Brand Ounces Total Price Unit Price Squeaky Clean 5.5 $0.36 Smell Fresh Feel Nice Look Bright Compute the unit price, and decide which brand is the best buy. 366
13 ANSWERS 75. Sophie is a quality control expert. She inspects boxes of #2 pencils. Each pencil weighs 4.4 grams (g). The contents of a box of pencils weigh 66.6 g. If a box is labeled CONTENTS: 16 PENCILS, should Sophie approve the box as meeting specifications? Explain your answer Write a plan to determine the number of miles per gallon (mpg) your car (or your family car) gets. Use this plan to determine your car s actual mpg. 77. Express the width and length of a $1 bill in centimeters (cm). Then express this same length in millimeters (mm) If the perimeter of a square is 19.2 cm, how long is each side? P 19.2 cm If the perimeter of an equilateral triangle (all sides have equal length) is 16.8 cm, how long is each side? P 16.8 cm 80. If the perimeter of a regular (all sides have equal length) pentagon is 23.5 in., how long is each side? P 23.5 in. Simplify each expression ( ) ( ) ( ) 2 ( ) ( ) 2 367
14 Answers $ $ $0.59, or $ mi/gal labels 63. $ mi/gal in. 69. $ L cm
15 Using Your Calculator to Divide Decimals It would be most surprising if you had reached this point without using your calculator to divide decimals. It is a good way to check your work, and a reasonable way to solve applications. Let s first use the calculator for a straightforward problem. Example 1 Dividing Decimals Use your calculator to find the quotient Enter the problem in the calculator to find that the answer is CHECK YOURSELF 1 Use your calculator to find the quotient Now that you re convinced that it is easy to divide decimals on your calculator, let s introduce a twist. Example 2 An Application Involving the Division of Decimals Omar drove miles on a tank of gas. When he filled up the tank, it took 9.1 gallons. What was his gas mileage? Here s where students get into trouble when they use a calculator. Entering these values, you may be tempted to answer miles per gallon. The difficulty is that there is no way you can compute gas mileage to the nearest hundredmillionth mile. How do you decide where to round off the answer that the calculator gives you? A good rule of thumb is to never report more digits than the least number of digits in any of the numbers that you are given in the problem. In this case, you were given a number with four digits and another with two digits.your answer should not have more than two digits. Instead of , the answer could be 28 miles per gallon. Think about the question. If you were asked for gas mileage, how precise an answer would you give? The best answer to this question would be to give the nearest whole number of miles per gallon: 28 miles per gallon. CHECK YOURSELF 2 Emmet gained a total of 857 yards (yd) in 209 times that he carried the football. How many yards did he average for each time he carried the ball? CHECK YOURSELF ANSWERS yd 369
16 Name Section ANSWERS Date Divide and check. Calculator Exercises Divide and round to the indicated place hundredth tenth thousandth hundredth two decimal places three decimal places Solve the following applications. 13. Salary. In 1 week, Tom earned $ by working hours (h). What was his hourly rate of pay to the nearest cent? 14. Area. An 80.5acre piece of land is being subdivided into 0.35acre lots. How many lots are possible in the subdivision? If the circumference of a circle is known, the diameter can be determined by dividing the circumference by p. In each of the following, determine the diameter for the given circumference. Use 3.14 for p cm in cm mm Answers $ cm cm 370
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