SECTION 4.2: Multiplying and Dividing Decimals

4.2 Multiplying and Dividing Decimals

1. Multiply decimals. SWBAT

2. Divide decimals.

3. Multiply and divide powers of ten. 4. Multiply and divide signed decimals. 5. Multiply and divide using scientific notation.

In grade school we learned to move the decimal point when we calculated products and quotients. Using the calculator, we will no longer need to multiply and divide each of the digits, but we will still be the one responsible for checking the place value of our answers. To be accurate and efficient, we need to work our calculators wisely when we multiply and divide. Keying in long strings of zeros is not a wise use of a calculator; in fact, it is a sure way to make mistakes! With scientific notation and powers of ten, we do not have to key these zeros into our calculator; we can calculate the place value of our products and quotients using mental math.

In this section we study multiplying and dividing signed decimals using powers of ten and scientific notation.

Multiplying Decimals When most of us think about multiplying decimals, we think about money. When we multiply with money, we can still round to the leading digit to make estimates.

Example 1: Working weekend nights at the emergency room, Bernie receives differential pay of 1.75 times his daytime wages. If the daytime wage is $325.89 per shift, what will Bernie earn for working a weekend night?

Think it through: Understand: We need to multiply the weekend night differential pay by the daytime wages. Plan: We need to multiply. Our estimate is . Solve: Write the expression. ≈ Estimate by rounding to leading digit. Use a calculator and notice that this fits our estimate. Round to the nearest cent since Bernie’s wages are paid in dollars and cents. ANSWER: Bernie will be paid $570.30 or $570.31 for the weekend night shift.

When working with money, our factors may have long decimal fractions. We do not round these until we are finished with all of our calculations.

Example 2: Josephine County has 19 tax districts. For one Grants Pass district, the total property tax rate was $13.9592 per thousand dollars of assessed property value for the 2011-20012 tax year. What are the taxes John would pay on a house assessed for tax purposes at in this district?

Think it through: Understand: We need to find the taxes on this property. The taxes are not charged per dollar, “but per thousand dollars.” That means we will think of $218,345 as “$218.345 thousand.” We can then multiply the tax rate by the thousands of dollars of assessed value to find the total tax amount. Plan: Estimate by rounding both the rate and the value of the home. Use a calculator to multiply the more exact answer. Round the final answer so that it is useful. Solve: Estimate and this will be low.

Check: Our calculated answer of is reasonably close to ANSWER: John would pay $3,047.92 property taxes for a house in this district. While our calculations gave us fractions of a cent, we round these results to a reasonable number of decimal places for applications. For the last application, we are only going to pay our taxes to the cent.

Check Point 1

Caleb’s Medford home was assessed for tax purposes at $218,000. His total property tax rate is $14.4672 per thousand dollars of assessed property value. How much will he pay in property taxes? Answer using reasonable digits.

Dividing Decimals To divide we will again use our calculators. First, we make reasonable estimates so that we can quickly check our results. We round our results to numbers that make sense.

Example 3: Johannes' gross pay (before deductions) is. He makes an hour, but he is not sure how many hours are on this pay check. He decides to divide to find out.

Think it through: Hours is the missing factor, so divide gross pay ($249.10) by the rate of pay ($9.40).

and Round and estimate. Estimate 25 hours. Use a calculator. The calculated result fits the estimate, so Johannes accepts it. ANSWER: Johannes was paid for 26.5 hours on this paycheck.

To estimate when we divide, we look for a close multiplication fact, then we find the missing factor. For instance so . Example 4: Chanterelle borrowed $5,500 for college from her family. They agreed to give her the money if she paid them back $6,500 within the first three years after she finishes college. If Chanterelle decides to pay back the loan by making equal payments for the first 36 months she is out of college, what will she pay each month?

Think it through: Understand: The payment amount is the missing factor, so divide. Plan: Round the money owed and months to create a division fact that we can calculate mentally. Then use a calculator for the exact answer. Solve: Write the expression. Estimate by rounding to a division fact. Use a calculator. Check: The calculated result fits the estimate, so Chanterelle accepts it. ANSWER: Chanterelle will round up and pay for $180.56 each month for 36 months.

Check Point 2 Show your rounding and estimate first, and then divide. Stan’s entire annual bill for Rogue Community College tuition and fees is $5,268.90. He wants to divide this up into 9 monthly payments. How much is each payment?

Estimate: ______

Exact answer: ______

Multiplying and Dividing Powers of Ten Because exponents tell us how many times a base is multiplied by itself, when we multiply by powers of ten we will end up with a different power of ten. See if you can find the pattern in these next two examples before you read the rules. The first example shows what happens when we multiply powers of ten.

Example 5: Multiply these: a. b. c. d.

Think it through: The bases are the same and we are multiplying, add the exponents.

a. b. c. d.

RULE: To multiply powers of ten, add the exponents and keep ten as the base.

The next example shows what happens when we divide powers of ten. Again see if you can find the rule before it is given.

Example 6: Divide these: a. b. c. d.

Think it through: The bases are the same and we are dividing, subtract the exponents.

a. b. c. d.

RULE: To divide powers of ten, subtract the exponents and keep ten as the base.

Check Point 3 Multiply. a. b.

Divide. c. d.

Multiplying and Dividing Signed Decimals Decimals are fractions. When we multiply and divide, it may be easier to estimate and calculate using their fractional equivalents or it may be easier using scientific notation. When we multiply and divide decimals, the rules of signs apply.

Example 7: Multiply.

a. Think it through: Use fractions to multiply in order to see the power of ten.

Estimate using leading digits and decimal fractions. Regroup using the properties. Multiply the numerators. Rewrite as a decimal. The 3 is in the thousandths place. ANSWER:

b. Think it through: Use scientific notation to multiply to make the calculation easier.

Write the expression showing that the product is negative. Rewrite using scientific notation. Use the commutative property to regroup. Estimate by rounding to leading digit.

Use the calculator to multiply . Add the exponents for the power of ten. Move the decimal point 6 powers of ten to the left (smaller). ANSWER:

Check Point 4

Multiply. a. b.

When working with application problems, we round our final answer to a reasonable place value.

Example 8: The children’s dosage for medications is sometimes based on weight in kilograms. To change pounds to kilograms, we can use the formula where is the kilogram measure and is the weight in pounds. How many kilograms does a 62-pound child weigh?

Think it through: Understand: We need to convert the weight to pounds using a formula. Plan: Use the formula Solve: Write the formula. Estimate using a close fraction. ≈ Use a calculator and notice that this fits our estimate. Because the pounds were in whole numbers, round to whole kilograms.

Check: Because 0.4536 is about ½, the answer should be a bit less than 31 pounds. Since 28 is close to 31, we accept this answer. ANSWER: A child who weighs 62 lbs. weighs about 28 kg.

Check Point 5

Roger is machining a fine jewelry fitting part that has a length of 0.25” and a width of 0.125.” What is its area?

When we divide using decimal fractions like, we change the division problem to one that gives us an equivalent quotient while dividing by a whole number. We used this process for long division with paper and pencil.

For we wrote:

We moved the decimal point:

Only then did we fill in a zero and divide:

We can also use decimal fractions to see why we have equivalent quotients when we move the decimal point in a division problem.

We can also look at this problem another way. If we write our division , then we can multiply by a fraction equal to one that gives whole numbers in both the numerator and denominator. For this fraction, is a great choice. So we have . Next we can divide using whole numbers. These expressions, , , , and are all equivalent expressions. They simplify to the same number.

RULE: Every decimal division problem can be rewritten as an equivalent fraction with integers in both numerator and denominator.

Example 9: Divide.

a. Think it through: The quotient will be negative. Rewrite to divide by an integer.

Multiply each number by 1,000 to divide by an integer. Round to a convenient division fact and estimate. Divide. The result fits the estimate.

ANSWER:

b. Think it through: The quotient will be positive. Rewrite to divide by an integer.

Multiply each number by 10 to divide by an integer. Round to a convenient division fact and estimate. Use the calculator to divide.

ANSWER: . Again this answer is noteworthy, because we do not have an exact answer. If we round to three decimal places, the result is .  Check Point 6

Divide. a. b.

Computers, calculators and all computing technologies have a limit to the number of characters that can be displayed on a screen. If your calculator’s limit is reached, the result displayed is an estimate, not an exact value.

 Check Point 7

So that we can see many of the ways calculators display results when they reach their limits for decimal notation, please take part in collecting data for our math class. Use your calculator to multiply the following problems and bring your answer to class.

a. The 2007 average income per person in the U.S. was about $34,000 according to the U.S. Census Bureau. In 2007 the average population of the U.S. was about 301,000,000 people.1 Multiply these two numbers to find the total income in the U.S. Write your calculator result: ______

b. The rectangular surface area of a nanofiber captures the sun’s energy in a solar cell. This nanofiber has a length of 0.000 000 5 meters and a width of 0.000 000 05 meters. What is the area of the fiber that is exposed to the sun? Write your calculator result: ______

1http://www.census.gov/compendia/statab/brief.html Even if we have not yet reached the calculator’s limit, we eliminate mistakes when we keep track of place value rather than writing long strings of zeros. The fewer key strokes we use on our calculator, the fewer our key stroke errors. To use a calculator to multiply and divide long decimals, we often must enter these numbers in our calculator using powers of ten because we have reached the limit of the place value display of our calculator. Even when we have a choice of methods, we use scientific notation with our calculators to avoid leaving out (or adding extra) zeros as we punch keys.

Multiplying and Dividing Using Scientific Notation

Example 10: Given the information about the 2011 population and the average income per person in Check Point 7a, calculate the total personal income in the U.S.A. for 2011 using scientific notation with a calculator.

Think it through: Understand: We need to multiply to find our solution.

Plan: Use scientific notation to multiply decimals in our calculator and apply powers of ten using mental math. Solve: Write the multiplication problem using scientific notation. Use the commutative property to regroup. Estimate Use the calculator for Add the exponents mentally. Multiply. ANSWER:  Our answer fits our estimate. The total income in the U.S. for 2011 is about $13,062,000,000,000. This is 13.062 trillion dollars!

When we divide large numbers, we can write this division as a fraction and eliminate factors of 10. For instance, if we want to divide the approximate U.S. national debt by the number of people in the U.S., we can use fractions: or we can write this using powers of 10 and division,

Using a fraction we divide factors of 10: .

Using scientific notation we regroup and subtract powers of ten:

We can use powers of ten to simplify our calculations so that when we need to use a calculator, we use the fewest key strokes possible. Fewer key strokes mean less chance of error. If we can think a problem through and eliminate the calculator altogether, this is even better.

Example 11: Planners are already working to replace petroleum when we run out. Their planning is based on the estimates for the number of barrels of crude oil left on the planet. These estimates run between 1.2 trillion and 2 trillion barrels. The high estimate was made in 2011 by Chevron CTO Don Paul, and this is the one we will use.2 The world population was estimated at 7,000,000,000 people in 2012. Using these figures, what was the average number of barrels of crude oil per person left on Earth in 2012?

Think it through: Understand: We need to divide the barrels of oil left by the people on the planet.

Plan: We will use the highest estimate for the barrels of oil remaining in the ground. 2 trillion is since a trillion is . A billion is so the world population

2 http://news.cnet.com/8301-10784_3-9803819-7.html can be written as . We divide these to find the unit rate, the barrels per person. Write the problem using powers of ten. Use the properties to regroup. Divide 9 factors of 10 away from the powers of ten. Use the calculator to divide the significant figures. Use powers of ten to move the decimal point 3 to the right. Round up to the barrel. Solve:

Check: and , and Because , we accept our answer.

ANSWER: If we divide the oil evenly, each person on the planet has about 285 barrels of crude oil left.

(You may be interested to know that each barrel of oil yields between 19 and 20 gallons of gasoline3. According to the information used and our results, we have less than 5700 gallons of gasoline per person left on Earth).

Check Point 8

NASA has already sent robotic missions to Mars and has other Martian missions planned. Mars is 1.52 Astronomical Units from Earth. An astronomical unit is the distance from Earth to our sun. Astronomical unit is abbreviated AU and is equal to 92,955,800 miles. How many miles is it from Earth to Mars? (Round your answer to the nearest million).

3 http://tonto.eia.doe.gov/ask/gasoline_faqs.asp Estimate: ______Answer: ______

Check Point 9

The U.S. health care system spent $2.2 trillion on health care in 20084. How much money was this for each of the 306 million people in the United States in 2008?

Estimate: ______Answer: ______

4 http://money.cnn.com/2009/08/10/news/economy/healthcare_money_wasters/ 4.2 Exercise Set Name ______

Skills Estimate and then use paper and pencil or mental math strategies to find the exact answer. For exampleuses the method of halving and doubling. Do not use a calculator.

1. 2. Est.______Est.______Ans.______Ans.______3. 4. Est.______Est.______Ans.______Ans.______5. 6. Est.______Est.______Ans.______Ans.______7. 8. Est.______Est.______Ans.______Ans.______9. 10. Est.______Est.______Ans.______Ans.______11. 12. Est.______Est.______Ans.______Ans.______13. 14. Est.______Est.______Ans.______Ans.______Estimate the quotient using multiplication facts and place value. Then divide by hand and show your work on a paper that you staple to this assignment. Round your hand-calculated answer to the nearest thousandth.

Estimate Answer 15. 16. 17. 18. 19. 20. 21. 22.

Round to the nearest multiplication fact, divide and complete your estimate using place value. Then divide using your calculator. Round your calculator answer to the nearest thousandth. Estimate Answer 23. 24. 25. 26. 27. 28. 29. 1.26 30. 31. 32. 33. 34.

For problems 35 - 46 multiply or divide by the given power of 10. No calculators are necessary.

35. ______36. ______37. ______38. ______39. ______40. ______41. ______42. ______43. ______44. ______45. ______46. ______

Applications UPS 47. Use the City of Grants Pass 2007 Tax table on page 286 for a and b. a. How much are the total property taxes for a GP home assessed at $249,000?

b. How many of these tax dollars would go to support Rogue Community College?

48. a. How much are the total property taxes for a GP home assessed at $152,000?

b. How many of these tax dollars would go to support School District 7?

49. Answer the questions below using the 2011 City and County of San Francisco Tax table on the right. Rates for $100 of Assessed Property Value a. How much are the total property SF Community College 0.074 taxes for a San Francisco home SF Unified School District 0.3237 assessed at $249,000? City and County of S. F. 0.565900123 Library Preservation Fund 0.025 SF Children's Fund 0.03 b. What are the total property taxes Open Space Acquisition 0.025 for a San Francisco home assessed Bond Interest & Redemption 0.1147 at $2.49 million? Superintendent of Schools 0.001 General Obligation Bonds 0.0063 Bay Area Air Quality District 0.00208539 Total Tax Assessment (per $100)

50. If a person owned a San Francisco home assessed at $2.3 million, how much of this property tax money goes to the Library Preservation Fund? How much of this property tax money goes to the Bay Area Air Quality District?

51. Alexis is getting a cost of living raise on her next paycheck. If her current salary is $1490.61 and the cost of living has increased by a factor of 0.015, how much will Alexis's raise be?

52. Oswald is getting a step raise in pay after working for his company for one year. His current monthly salary is $4,230, and his salary will increase by a factor of 0.045. What is his new salary? 53. In the checkpoints for this lesson you found the distance to Mars in astronomical units (AU). Find the distances from the sun to other planets in miles and complete this table: Number of Distance to sun in Distance to sun in Planet astronomical units miles in scientific miles (AU) notation Mercury 0.39 Venus 0.72 Earth 1 92,955,800 Mars 1.52 Jupiter 5.2 Saturn 9.54 Uranus 19.18 Neptune 30.06 * Pluto is no longer considered a planet.

54. The following table lists 2 estimates of the U.S. debt. The first is an estimate of the total indebtedness of all of us who live in the U.S. This first estimate includes personal, business and financial sector debt. The second is the U.S. national debt which is owed by us through our government. If the U.S. population is about 314 million in 2012, how much would each of us have to pay to get ourselves out of these total debts? Estimate of U.S. Debt Debt in Scientific Notation Amount Owed per Person $52,000,000,000,000 $15,980,000,000,000