CHAPTER 4: DECIMALS. Image from Microsoft Office Clip Art CHAPTER 4 CONTENTS

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1 CHAPTER 4: DECIMALS Image from Microsoft Office Clip Art CHAPTER 4 CONTENTS 4.1 Introduction to Decimals 4.2 Converting between Decimals and Fractions 4.3 Addition and Subtraction of Decimals 4.4 Multiplication and Division of Decimals 4.5 Metric Measurement 4.6 Applications 263

2 CCBC Math 081 Introduction to Decimals Section pages 4.1 Introduction to Decimals Our numbering system is based on powers of 10 and is therefore called the decimal number system, from the Latin root decem, meaning ten. In this chapter, we will study what decimal values represent and how they are connected to fractions. We will also learn how to round decimals and perform basic arithmetic operations on decimals. Finally, we will look at some applications of decimals to the metric system, geometry, finance, and statistics. We use decimals every time we talk about money. When we get a bill at a restaurant or buy something at the store or online, we rarely must pay an amount that is a whole number dollar amount. You might buy a large iced coffee for $2.75 or download a Coldplay album for $ In the previous chapter, we learned about fractions and mixed numbers that had whole number parts and fractional parts. The decimal 2.75 expresses a similar idea: you have two wholes (two whole dollars) and 75 cents, which is 75. So the 100 of another dollar. In fact, numbers to the right of the decimal point represent a part of a whole in the same way that the fractional part of a mixed number does. Of course, the use of decimals is not limited to money. We can calculate a baseball player s batting average to be or measure the weight of a bag of imported dehydrated lima beans to be pounds. When referring to decimals, we talk about place values just like we do with whole numbers. When we read the number 3,125 we say three thousand, one hundred twenty-five. That really means we have 3 thousands, plus 1 hundred, plus 2 tens, plus 5 ones. There are also place values for the numbers to the right of the decimal point as in the diagram below: thousandths hundredths tenths ones tens hundreds thousands Notice how the values behind (to the right of) the decimal point end in ths. Later, we will see why this is the case, but for now let s say that it pertains to the connection between decimals and fractions. Also notice that there is no oneths place; the first place to the right of the decimal point is the tenths place. Although they are not listed in the diagram above, after the thousandths place come the ten thousandths, then the hundred thousandths, then the millionths, and so on. 264

3 CCBC Math 081 Introduction to Decimals Section pages Example 1: Consider the number hundreds tens ones ten thousandths thousandths hundredths tenths a. What digit is in the hundreds place? Answer: The digit 3 is in the hundreds place. b. What digit is in the tenths place? Answer: The digit 4 is in the tenths place. c. What digit is in the thousandths place? Answer: The digit 5 is in the thousandths place. d. Name the place value of the digit 7. Answer: The digit 7 is in the hundredths place. Practice 1: Consider the number 9, a. Which digit is in the hundreds place? b. Which digit is in the hundredths place? c. Name the place value of the digit 5. d. Name the place value of the digit 7. Answers: a. 0 b. 4 c. tens d. ten thousandths Example 2: Consider the number 3, a. What digit is in the thousands place? Answer: The digit 3 is in the thousands place. b. What digit is in the hundreds place? Answer: The digit 1 is in the hundreds place. c. What digit is in the tenths place? Answer: The digit 0 is in the tenths place. d. Name the place value of the digit 7. Answer: The digit 7 is in the thousandths place. Practice 2: Consider the number 35, a. Which digit is in the ones place? b. Which digit is in the tenths place? c. Name the place value of the digit 5. d. Name the place value of the digit 0. Answers: a. 7 b. 2 c. thousands d. thousandths 265

4 CCBC Math 081 Introduction to Decimals Section pages Ordering Decimal Numbers Sometimes it is necessary to compare decimal numbers. Note that adding 0 after the rightmost digit does not change the value of the decimal number. As examples, notice: 2.75 = AND 0.6 = 0.60 = = =... and so on. We will see why this is true when we investigate the relationship between decimal numbers and fractions. COMPARING DECIMALS 1. Start from the left and compare digits in the corresponding place value positions. 2. Moving from left to right, the first number with a greater digit in the corresponding place position is the greater number. Example 3: Compare the decimal numbers by filling in the blank with <, >, or = The first place from the left where their digits differ is in the tenths place. Compare the digits: Since 6 is greater than 5, 0.6 is greater than 0.59 and we write 0.6 > Practice 3: Compare the decimal numbers by filling in the blank with <, >, or = Answer: > Example 4: Compare the decimal numbers by filling in the blank with <, >, or = Notice these two decimal numbers have the same value in the ones place, the tenths place, and the hundredths place. The first place from the left where their digits differ is in the thousandths place. Compare the digits there: Since 1 is less than 2, is less than and we write: < Practice 4: Compare the decimal numbers by filling in the blank with <, >, or = Answer: <

5 CCBC Math 081 Introduction to Decimals Section pages Example 5: Write the decimal numbers in order from least to greatest: Notice we can write 31 as Now compare digits in the three numbers place-by-place and rewrite from least to greatest as: Practice 5: Write the decimal numbers in order from least to greatest: Answer: 0.5, , Example 6: The birth weights (in pounds) of quintuplet babies were: Find the median birth weight. Start by listing the data values in order from least to greatest (by comparing their digits place-byplace): Because there are an odd number of data values (5), there is one middle value: Answer: The median weight is pounds. Practice 6: A grocery store sells bananas in bags with each bag containing approximately two pounds of bananas. Find the median weight for seven bags if their actual weights are: Answer: 2.01 pounds Estimating Numbers Sometimes, we don t need exact numbers; approximating or estimating is often good enough. If you are at the supermarket and notice that a box of Fructose Flakes cereal costs $4.79, you could estimate its cost to be about $5. Doing so would make it easier to calculate that four boxes will cost you roughly $20. Now, this is not an exact answer, but it s a pretty close estimate. We don t have to bother making a precise calculation if we just want to know approximately how much four boxes of this cereal would cost. So how would we estimate a number like 2,132? The easiest answer would be to say that it is approximately 2,000 although you could also say that it is approximately 2,100 or even 2,130. But 2,000 is the simplest (for use in computations) estimate of 2,132. In this course, we will estimate numbers in the following way: 267

6 CCBC Math 081 Introduction to Decimals Section pages HOW TO ESTIMATE 1. Determine the left-most non-zero digit. 2. Look at the digit immediately to its right. - If that digit is less than 5, then the left-most digit remains unchanged. - If that digit is 5 or more, then we add 1 (round up) to the left-most digit. 3. Hold the remaining digits in place with appropriate zeros. Example 7: Estimate The left-most non-zero digit is the 3 in the hundreds place. 2. The digit immediately to its right is 4, which is less than 5; thus, the 3 remains unchanged. 3. Replace both of the digits to the right of the 3 with a 0 to keep the 3 in the hundreds place: 300 Answer: The estimate for 348 is 300. Practice 7: Estimate 1,234 Answer: 1,000 Example 8: Estimate 4, The left-most non-zero digit is the 4 in the thousands place. 2. The digit immediately to its right is 8, which is more than 5; thus, we add 1 to ( bump up by 1) the 4, making Replace each of the three digits to the right of the 5 with a 0 to keep the 5 in the thousands place: 5000 Answer: The estimate for 4,897 is Practice 8: Estimate 37.5 Answer:

7 CCBC Math 081 Introduction to Decimals Section pages Example 9: Estimate 9, The left-most non-zero digit is the 9 in the thousands place. 2. The digit immediately to its right is 5; thus, we add 1 to ( bump up by 1) the 9, making Replace each of the three digits to the right of the 10 with a 0 to keep the number 10 in the thousands place: 10,000 Answer: The estimate for 9,502 is 10,000. Practice 9: Estimate 961 Answer: 1,000 Example 10: Estimate The left-most non-zero digit is the 2 in the thousandths place. 2. The digit immediately to its right is 3, which is less than 5; thus, the 2 remains unchanged. 3. The digit 2 is in the thousandths place. Therefore we do not need zeros to the right of the 2: Answer: The estimate for is Practice 10: Estimate Answer: Rounding Decimals If you buy a pizza that costs $12.38, you could say that it costs you about $12. You would then expect to receive about $8 in change from a $20 bill. In this case, we are not estimating; instead we are rounding $12.38 to the nearest whole dollar to get $ (The estimate for $12.38, according to the rules above, would be $10.00). To round to the nearest dollar (that is, to the ones place), we look at the 2 in the ones place and then look at the digit immediately to its right, the 3. Since 3 is less than 5, the 2 remains unchanged. Then, as before, each digit to the right of the 2 is replaced with a 0. Rounding $12.38 to the nearest whole dollar (to the ones place) gives us $12.00 = $12 Because $12 is less than $12.38, we say that we have rounded down. Suppose that we want to round $4.375 to the nearest penny. We would have to look at the 7 in the hundredths place (each hundredth of a dollar is a penny) and then look at the digit 5 immediately to its right. Since that 5 is 5 or more, we add 1 to ( bump up by 1) the 7, making an 8. We also replace each digit to the right of the 8 with a 0: $4.380 = $4.38 Rounding $4.375 to the nearest penny (to the nearest hundredth) gives us $4.38. Because $4.38 is more than $4.375, we say that we have rounded up. 269

8 CCBC Math 081 Introduction to Decimals Section pages IMPORTANT NOTE: Notice that we do not move the decimal point when we round. For example, rounding to two decimal places does not mean that we move the decimal point two places; it means that the answer will have two digits to the right of the decimal point. We can introduce some notation that is a helpful tool in the rounding process. ROUNDING DECIMALS 1. Identify the digit that you are rounding by circling it or pointing to it. 2. Place a vertical line immediately to the right of the digit. 3. Look at the digit immediately to the right of the vertical line to determine if you round up or leave unchanged. 4. If needed, write zeros to keep the remaining digits in their original place. Let s take another look at rounding to the nearest hundredth: 1. Identify the digit that you are rounding by circling it or pointing to it: Place a vertical line immediately to the right of that digit: The digit immediately to the right of the vertical line is what we use to determine whether we round up or leave unchanged. 4. We do not need to write zeros because the remaining digits are in their original place value: Answer: rounded to the nearest hundredth is Example 11: Round to the nearest hundredth. The digit 1 is in the hundredths place. Place a vertical line to right of the 1: The digit to the right of the vertical line is 8, which is greater than or equal to 5. So, we add 1 to ( bump up by 1) the 1, making 2. We do not need to write zeros because the remaining digits are in their original place value: Answer: rounded to the nearest hundredth is Practice 11: Round to the nearest tenth. Answer:

9 CCBC Math 081 Introduction to Decimals Section pages Example 12: Round 12,345 to the nearest hundred. The digit 3 is in the hundreds place. Place a vertical line to right of the 3: 12,3 45 The digit to the right of the vertical line is a 4, which is less than 5. Thus, the 3 remains unchanged. We also replace each digit to the right of the vertical line with a 0 to keep the remaining digits in their original place value. 12,3 00 Answer: 12,345 rounded to the nearest hundred is 12,300 Practice 12: Round 12,345 to the nearest ten. Answer: 12,350 Example 13: Round 7.96 to the nearest tenth. The digit 0 is in the tenths place. Place a vertical line to right of the 0: The digit to the right of the vertical line is a 6, which is greater than 5. So, we add 1 to ( bump up by 1) the 9. However, we cannot write a 10 where the 9 is, so we write a 0 in place of the 9 and carry a 1 (add 1 to) the place immediately to the left of the 9. = 8.0 We do not need to write zeros because the remaining digits are in their original place value: 8.0 Notice that we must keep the digit 0 to the left of the vertical line and not write 8.0 as 8. Because we were asked to round to the tenths place, there must be a digit appearing in the tenths place, even if that digit is 0. Answer: 7.96 rounded to the nearest tenth is 8.0 Practice 13: Round to the nearest hundredth. Answer:

10 CCBC Math 081 Introduction to Decimals Section pages Example 14: Round to three decimal places. The digit 9 is in the third decimal place. Place a vertical line to right of the 9: The digit to the right of the vertical line is a 5. So, we add 1 to ( bump up by 1) the 9. However, we cannot write a 10 where the 9 is, so we write a 0 in place of the 9 and carry a 1 (add 1 to) the place immediately to the left of the 9. We do not need to write zeros because the remaining digits are in their original place value: Note that we keep the 0 to the left of the vertical line, to fulfill the directions to round to 3 decimal places. Answer: rounded to 3 decimal places is Practice 14: Round to three decimal places. Answer: Example 15: Round to three decimal places. The digit 9 is in the third decimal place. Place a vertical line to right of the The digit to the right of the vertical line is a 7. So, we add 1 to ( bump up by 1) the 9. However, we cannot write a 10 where the 9 is, so we write a 0 in place of the 9 and carry a 1 (add 1 to) the place immediately to the left of the 9 which is a 9 itself, so we again write a 0 in that place as well and carry a 1 to the digit immediately to its left. We do not need to write zeros because the remaining digits are in their original place value: Note that we keep the zeros to the left of the vertical line, to fulfill the directions to round to 3 decimal places. Answer: rounded to 3 decimal places is Practice 15: Round to two decimal places. Answer: Watch All: 272

11 CCBC Math 081 Introduction to Decimals Section pages 4.1 Introduction to Decimals Exercises 1. In the number , the digit in the a. tenths place is: b. hundredths place is: c. thousandths place is: d. ones place is: e. tens place is: f. hundreds place is: 2. Consider the number a. Name the place value of 2: b. Name the place value of 4: c. Name the place value of 7: d. Name the place value of 1: e. Name the place value of 9: f. Name the place value of 5: 3. In any decimal, name the place value. a. 2 places to the right of the decimal point: b. 2 places to the left of the decimal point: c. 3 places to the right of the decimal point: d. 1 place to the left of the decimal point: e. 1 place to the right of the decimal point: 273

12 CCBC Math 081 Introduction to Decimals Section pages 4. Compare the decimal numbers by filling in the blank with <, >, or = Compare the decimal numbers by filling in the blank with <, >, or = Compare the decimal numbers by filling in the blank with <, >, or = Write the decimal numbers in order from least to greatest: According to Wikipedia, Ty Cobb s batting averages for seven years were: Find his median batting average for those years. 9. Estimate 289: 10. Estimate 14,567: 11. Estimate 963,146: 12. Estimate : 13. Estimate : 14. Estimate 0.65: 15. Estimate : 16. Round to the nearest tenth: 17. Round to the nearest hundredth: 18. Round to 3 decimal places: 19. Round 1, to the nearest hundredth: 20. Round to the nearest ten-thousandth: 274

13 CCBC Math 081 Introduction to Decimals Section pages 4.1 Introduction to Decimals Exercises Answers 1. a. 3 b. 5 c. 9 d. 2 e. 4 f a. tenths b. tens c. ones d. hundredths e. thousandths f. hundreds 3. a. hundredths b. tens c. thousandths d. ones e. tenths < > = , 4.16, 4.162, , ,000, ,

14 CCBC Math 081 Converting between Decimals and Fractions Section pages 4.2 Converting between Decimals and Fractions In the introduction, we talked about how $2.75 represents two whole dollars plus 75 cents, which is a fractional part of another whole dollar. Since it takes 100 cents to make a dollar, 75 cents 75 can be written in fraction form as 100 which reduces to 3. So, 2.75 in decimal form is 4 3 equivalent to 2 in fraction form. 4 Let s look at some fractions using powers of 10 to see their connection to decimal place values. Dividing 1 into 10 equal parts gives us tenths; a tenth can be represented as tenth = 1 10 as a fraction = 0.1 as a decimal Dividing 1 into 100 equal parts gives us hundredths; a hundredth can be represented as 1 hundredth = as a fraction = 0.01 as a decimal Dividing 1 into 1000 equal parts gives us thousandths; a thousandth can be represented as 1 thousandth = as a fraction = as a decimal Notice that the number of 0s in the denominator of the fraction is equal to the number of places to the right of the decimal point where the digit 1 is located in the decimal number. Converting from a Decimal to a Fraction We will learn how to convert decimals into fractions using the place value of the digits in the decimal number. 276

15 CCBC Math 081 Converting between Decimals and Fractions Section pages Example 1: Represent the decimal 6.37 as a fraction. The digit 6 is in the ones place and represents six ones or six wholes. The digit 3 is in the tenths place so the 3 represents The digit 7 in the hundredths place so the 7 represents We can write 6.37 as 6. To add fractions, we need a common denominator, which will be 100: Answer: Rewrite the whole numbers over one Determine what to multiply by to get Multiply Add the numerators together Let s look at a little more closely. Since the rightmost place of 6.37 is the hundredths place, it makes sense that the denominator of the fraction is 100 as well. All we really need to do to convert a decimal to a fraction is determine the number of digits to the right of the decimal point. An easy way to do this is to see that there are two digits to the right of the decimal point in 6.37, corresponding to the two zeros in the denominator of Practice 1: Represent 2.7 as a fraction in simplest form. Answer:

16 CCBC Math 081 Converting between Decimals and Fractions Section pages Example 2: Represent the decimal as a fraction in simplest form. Since the rightmost decimal place is the thousandths place, the denominator will be 1,000. Similarly, we can reason that because there are three digits to the right of the decimal point, there should be three zeros (in the power of 10) in the denominator, giving us the denominator 1,000. Answer: or Note that this fraction is in simplest terms and cannot be reduced. It is okay to leave the conversion as an improper fraction as long as it is in simplest form. Practice 2: Represent as a fraction in simplest form. Answer: Example 3: Represent the decimal as a fraction in simplest form. Since the rightmost decimal place is the hundred thousandths place, the denominator will be 100,000. Similarly, we can reason that because there are five digits to the right of the decimal point, there should be five zeros (in the power of 10) in the denominator, giving us the denominator 100,000. So: which reduces to Answer: Practice 3: Represent as a fraction in simplest form. Answer: Converting from a Fraction to a Decimal 278

17 CCBC Math 081 Converting between Decimals and Fractions Section pages Note: In this section we will work only with fractions having denominators easily changed to a power of 10. Later in the chapter, you will learn more skills allowing you to convert any fraction to a decimal. We should be able to convert a fraction to decimal form using the same idea as above. Example 4: Represent the fraction as a decimal. Since the denominator is 100, the hundredths place must contain the rightmost digit of 27, which is 7. Thus, 0.27 is the decimal form. Notice we would read both the fraction and the decimal as 27 hundredths. We could also reason that since the power of 10 in the denominator 100 in this fraction has two zeroes, there should be two digits to the right of the decimal point. Answer: Practice 4: Represent the fraction as a decimal. Answer: Example 5: Represent the fraction 1043 as a decimal. 10 Since the denominator is 10, the tenths place must contain the rightmost digit of 1043, which is 3. Thus, is the decimal form. We could also reason that since the power of 10 in the denominator 10 in this fraction has one zero, there should be just one digit to the right of the decimal point. Thus, is the decimal form. Answer: Notice that the numerator of is greater than the denominator, which means that the fraction is greater than or equal to 1 in value. Thus, there will be at least one nonzero digit to the left of the decimal point in the decimal conversion of the fraction. Also, we could write Read both the fraction and the decimal as one hundred four and three tenths. 973 Practice 5: Represent the fraction as a decimal. Answer:

18 CCBC Math 081 Converting between Decimals and Fractions Section pages Example 6: Represent the fraction 7 20 as a decimal. We will have to begin this example a little differently than Examples 4 and 5, because the denominator is not a power of 10 (10 or 100 or 1000, etc.). Fortunately, 20 is a factor of 100; Multiply both the numerator and denominator by 5 to write an equivalent fraction with a denominator of 100. Then write 35 hundredths as 0.35: Answer: 7 20 = Practice 6: Represent the fraction 5 4 as a decimal. Answer: Calculator Note Converting from a fraction to a decimal by hand helps us understand the connection between fractions and decimals. We can also use our calculator to convert. Start by rewriting the fraction as a division problem. Then, use the calculator to perform that division, remembering to enter the numerator first. For example, to represent the fraction 7 20 on the calculator to get as a decimal, read 7 20 as 7 divided by 20 and enter Watch All: 280

19 CCBC Math 081 Converting between Decimals and Fractions Section pages 4.2 Converting Between Decimals and Fractions Exercises In Exercises 1 10, represent each decimal as a fraction in simplest form In Exercises 11 20, represent each fraction as a decimal

20 CCBC Math 081 Converting between Decimals and Fractions Section pages 4.2 Converting Between Decimals and Fractions Exercises Answers or or or or or or or

21 CCBC Math 081 Addition and Subtraction of Decimals Section pages 4.3 Addition and Subtraction of Decimals Now that we have a good understanding of decimal numbers and their relationship to fractions, we will cover the basic arithmetic operations on decimal numbers. In this section, we will consider the operations of addition and subtraction. While we could do all of these calculations on a calculator, it is important to understand how to do them by hand as well. Addition of Decimals To add decimal numbers, use a vertical display to line up the decimal points of each addend. If necessary, insert additional zeros as placeholders after the last digit to the right of the decimal point. Then add the numbers as you would when adding whole numbers. The decimal point of the sum (answer) is written directly below the decimal points of the addends. ADDITION OF DECIMALS 1. Set up the problem in a vertical display with the decimal points lined up. 2. If necessary, insert additional zeros as placeholders after the last digit to the right of the decimal point. 3. Add the numbers as you would whole numbers. 4. Place the decimal point of the sum (answer) directly below the decimal points of the addends. Example 1: Add Line up the decimal points and add the digits down, regrouping ( carrying ) as needed Answer: = Practice 1: Add Answer:

22 CCBC Math 081 Addition and Subtraction of Decimals Section pages Example 2: Add Where is the decimal point for the whole number 12? Think of 12 as 12 dollars or $12.00, and then you ll see the decimal point. Any whole number can be written as a decimal with the decimal point at the right end of the number (to the right of the ones digit) and with zeros to the right of the decimal point. Line up the decimal points and add the digits down Insert two zeros as placeholders Answer: = Practice 2: Add Answer: Example 3: Add Line up the decimal points and add the digits down Insert a 0 as a placeholder. Answer: = Practice 3: Add Answer:

23 CCBC Math 081 Addition and Subtraction of Decimals Section pages Example 4: Calculate the perimeter of the given trapezoid. 2.8 in 3.4 in 3.8 in 7.5 in Remember that perimeter is the distance around an object s boundary. So add the lengths of the four sides of the trapezoid: Perimeter: P = = 17.5 in. Practice 4: Calculate the perimeter of the given trapezoid. Answer: P = 11.3 cm 3.1 cm 2 cm 2.2 cm 4 cm 285

24 CCBC Math 081 Addition and Subtraction of Decimals Section pages Example 5: Calculate the perimeter of the given triangle. 1.7 cm 3.2 cm 0.7 cm 4.7 cm Add the lengths of the three sides of the triangle: Perimeter: P = = 9.6 cm Practice 5: Calculate the perimeter of the given triangle. Answer: in 2 in 4.1 in 1.3 in 5.27 in Subtraction of Decimals To subtract decimal numbers, use a vertical display to line up the decimal points. If necessary, insert additional zeros as placeholders after the last digit to the right of the decimal point. Then subtract the numbers as you would when subtracting whole numbers. The decimal point of the difference (answer) is written directly below the decimal points of the subtrahend. 286

25 CCBC Math 081 Addition and Subtraction of Decimals Section pages SUBTRACTION OF DECIMALS 1. Set up the problem in a vertical display with the decimal points lined up. 2. If necessary, insert additional zeros as placeholders after the last digit to the right of the decimal point. 3. Subtract the numbers as you would whole numbers. 4. Place the decimal point of the difference (answer) directly below the decimal points of the subtrahend. Example 6: Subtract Line up the decimal points: Insert a 0 as a placeholder Answer: = Practice 6: Subtract Answer:

26 CCBC Math 081 Addition and Subtraction of Decimals Section pages Example 7: Subtract Line up the decimal points: Insert a 0 as a placeholder We need to regroup ( borrow ) in order to finish the subtraction Answer: = Practice 7: Subtract Answer: Example 8: Subtract Write the whole number 5 as Line up the decimal points and subtract: Answer: = Practice 8: Subtract Answer:

27 CCBC Math 081 Addition and Subtraction of Decimals Section pages Example 9: (Source: Interactive online study via Cengage Learning) The following data represent the consumption of energy (in quadrillion BTUs) in Note: one quadrillion = 1,000,000,000,000,000 Source 1999 Petroleum Natural Gas Coal Nuclear 7.73 Renewable Energy (hydroelectric, geothermal, etc.) 3.75 a. In 1999, how much more energy (in quadrillion BTUs) was consumed from petroleum than coal? Reading from the table, quadrillion BTUs were consumed from petroleum and quadrillion BTUS were consumed from coal. Find the difference in energy consumption by subtracting: quadrillion BTUs b. In 1999, how much total energy (in quadrillion BTUs) was consumed from Nuclear and Renewable Energy? Reading from the table, 7.73 quadrillion BTUs were consumed from Nuclear Energy and 3.75 quadrillion BTUs were consumed from Renewable Energy. Find the total in energy consumption by adding: quadrillion BTUs 289

28 CCBC Math 081 Addition and Subtraction of Decimals Section pages Practice 9: The table below shows the postal rates for priority mail in Postal Rate for Priority Mail 2009 Weight Zones Not Over Local, (pounds) 1 & $ 4.95 $ 4.95 $ 4.75 $ 4.95 $ 4.95 $ $ 4.95 $ 5.20 $ 5.75 $ 7.10 $ 7.60 $ $ 5.50 $ 6.25 $ 7.10 $ 9.05 $ 9.90 $ $ 6.10 $ 7.10 $ 8.15 $ $ $ $ 6.85 $ 8.15 $ 9.45 $ $ $ $ 7.55 $ 9.25 $ $ $ $ a. How much more does it cost to send a 2.7 pound package to Zone 6 than to Zone 5? b. A customer sends one 1.5 pound package to Zone 3 and another 1.5 pound package to Zone 7. How much will it cost to mail both packages? Answers: a. $ 0.85 b. $ Now we will practice addition and subtraction involving signed decimal numbers. The rules for adding and subtracting signed numbers are the same whether we are working with integers, fractions, or decimals. Example 10: Add (- 4.12) We will use the Triple-S method since we are adding two numbers that have the same sign. Compute the sum of the absolute values of each of the two numbers. Add 0.65 and 4.12 by lining up the decimal points: The final answer takes the same sign as the sign of the two original numbers. Since and are both negative numbers, the sum is a negative number. Answer: (- 4.12) = Practice 10: Add ( ) Answer:

29 CCBC Math 081 Addition and Subtraction of Decimals Section pages Example 11: Subtract 3.1 (-1.23) First, recognize that we are subtracting a negative number. Using the rules for operations on integers, 3.1 (-1.23) is equivalent to We have rewritten the subtraction problem as an equivalent addition problem. Now add by lining up the decimal points: 3.10 Insert a 0 as a placeholder Answer: 3.1 (-1.23) = 4.33 Practice 11: Subtract Answer: Example 12: Add We will use the Triple-D method since we are adding two numbers that have different signs. Compute the difference between the absolute value of each of the two numbers. We subtract by lining up the decimal points: The final answer takes the sign of the dominant number. Notice that has a larger absolute value than 27.2 so the final answer is a negative number. Answer: = Practice 12: Add Answer: Watch All: 291

30 CCBC Math 081 Addition and Subtraction of Decimals Section pages 4.3 Addition and Subtraction of Decimals Exercises 1. Without actually finding the sum, show how you would set up the addition problem when calculating by hand: Without actually finding the sum, show how you would set up the addition problem when calculating by hand: Add: Add: Add: Add: Add: Calculate the perimeter of the given trapezoid. 5.7 mm 3.4 mm 4.2 mm 6.7 mm 9. Calculate the perimeter of the given trapezoid. 5.3 in 2.8 in 3.5 in 8.7 in 10. Calculate the perimeter of the given triangle. 2.8 cm 5 cm 1.3 cm 6.9 cm 292

31 CCBC Math 081 Addition and Subtraction of Decimals Section pages 11. Calculate the perimeter of the given triangle km 12 km 12.8 km km 12. Without actually finding the difference, show how you would set up the subtraction problem when calculating by hand: Without actually finding the difference, show how you would set up the subtraction problem when calculating by hand: Without actually finding the difference, show how you would set up the subtraction problem when calculating by hand: Subtract: Subtract: Subtract: Subtract: The table below shows the postal rates for priority mail in How much more does it cost to send a 5.25 pound package to Zone 7 than to Zone 3? Postal Rate for Priority Mail 2009 Weight Zones Not Over Local, (pounds) 1 & $ 4.95 $ 4.95 $ 4.75 $ 4.95 $ 4.95 $ $ 4.95 $ 5.20 $ 5.75 $ 7.10 $ 7.60 $ $ 5.50 $ 6.25 $ 7.10 $ 9.05 $ 9.90 $ $ 6.10 $ 7.10 $ 8.15 $ $ $ $ 6.85 $ 8.15 $ 9.45 $ $ $ $ 7.55 $ 9.25 $ $ $ $

32 CCBC Math 081 Addition and Subtraction of Decimals Section pages 20. Add: (- 1.5) 21. Subtract: (- 3.17) 22. Add: (-57.8) 23. Add: Subtract: Subtract: (- 4) 294

33 CCBC Math 081 Addition and Subtraction of Decimals Section pages 4.3 Addition and Subtraction of Decimals Exercises Answers mm in cm km $ $9.25 = $

34 CCBC Math 081 Section CHAPTER 4 Mid-Chapter Review In the number , what digit is in the: 1. hundredths place 2. tens place In the number : 3. name the place value of the 9 4. name the place value of the 1 Fill in the blank with <, >, or = Write the numbers in order (least to greatest) , 1.374, 1.382, 1.309, 1.38 Estimate each number Round to the nearest tenth 12. to the nearest thousand 13. to the nearest thousandth Write each as a fraction in simplest form Write each as a decimal Show how to set up the addition problem to calculate by hand. Do not add Add Show how to set up the subtraction problem to calculate by hand. Do not subtract Subtract Perform the indicated operation The average rainfall (in inches) for five months was 4.23, 2.45, 1.46, 3.45, and Find the median rainfall. Calculate the perimeter of the trapezoid m 6.70 m 7.38 m m Use the table below to answer each question. 35. What would have been the cost for purchasing a loaf of bread, a gallon of milk, and a dozen eggs in 1990? 36. How much more was a pound of coffee in 2010 compared to 1980? Item Bread (1 lb) $0.52 $0.70 $0.99 $1.39 Coffee (1 lb) $2.82 $2.94 $3.21 $4.15 Eggs (1 dz) $0.87 $1.01 $0.91 $1.37 Milk (1 gal) $0.78 $1.32 $2.79 $3.32

35 CCBC Math 081 Section M i d - C h a p t e r 4 R e v i e w A n s w e r s Tenths 4. Thousands 5. > 6. < , 1.365, 1.374, 1.38, in m 35. $ $

36 CCBC Math 081 Multiplication and Division of Decimals Section pages 4.4 Multiplication and Division of Decimals In this section, we ll consider the operations of multiplication and division of decimal numbers. While we could do all of these calculations on a calculator, it is important to understand how to do them by hand as well. We will also review order of operations and a few applications. Multiplication of Decimals Multiplying decimals is very much like multiplying whole numbers. However, unlike adding and subtracting decimals, we do NOT need to line up the decimal points when multiplying decimal numbers. So, where does the decimal point go in the product (answer)? STEPS TO PLACING THE DECIMAL POINT IN THE PRODUCT 1. Count the number of digits to the RIGHT of the decimal point in the first number. 2. Count the number of digits to the RIGHT of the decimal point in the second number 3. Add these two counts. 4. In the answer, place the decimal point so that there are as many digits to the right of the decimal point as there are the sum total of the number of digits to the right of the decimal in the two factors. Example 1: Multiply digits to the right of the decimal point digits to the right of the decimal point digits to the right of the decimal point. Put the decimal point here so that there are 4 digits to the right of the decimal point. Answer: = Practice 1: Multiply Answer:

37 CCBC Math 081 Multiplication and Division of Decimals Section pages Example 2: Multiply digits to the right of the decimal point digits to the right of the decimal point digits to the right of the decimal point Put the decimal point here so that there are 3 digits to the right of the decimal point. Answer: = Practice 2: Multiply Answer: Example 3: Multiply The rules for multiplying signed numbers are the same whether we are multiplying integers, fractions, or decimals. In this problem, we are multiplying a negative number with a positive number; thus, the product (answer) is a negative number digits to the right of the decimal point digits to the right of the decimal point digits to the right of the decimal point. Put the decimal point here so there are 2 digits to the right of the decimal point. Answer: = Practice 3: Multiply 3.15 ( 6) Answer:

38 CCBC Math 081 Multiplication and Division of Decimals Section pages Example 4: Calculate the area of the given rectangle. 4.2 m 2.5 m 2.5 m 4.2 m Remember that the area of a rectangle can be found by multiplying its length L by its width W. Area = L W digits to the right of the decimal point digits to the right of the decimal point digits to the right of the decimal point m (Don't forget squared units for area.) Practice 4: Calculate the area of the given rectangle. Answer: A = 5.92 cm cm 1.6 cm 1.6 cm 3.7 cm 300

39 CCBC Math 081 Multiplication and Division of Decimals Section pages Division of Decimals Let s review terminology used in division problems: PARTS OF A DIVISION STATEMENT Dividend Divisor Quotient Dividend 8 Quotient 4 Quotient Divisor Divisor 28 Dividend 2 4 Divisor: The divisor is the number outside of the division sign. Dividend: The dividend is the number inside of the division sign. Quotient: The quotient is the answer. Dividing decimals is very much like dividing whole numbers. However, if the divisor has a decimal point, we will have to move it all the way to the right end of the number to make it a whole number. Then, we must move the decimal point in the dividend by the same number of places to the right. Once the decimal point is in the proper place in the dividend, we can put the decimal point in the proper location in the quotient (answer). The decimal point in the quotient will be directly above the decimal point in the dividend. DIVIDING DECIMALS 1. Move the decimal point in the divisor all the way to the right end of the number to make it a whole number. 2. Move the decimal point in the dividend by the same number of places to the right. 3. Put the decimal point in the quotient (answer) directly above the decimal point in the dividend. 4. Use long division method to divide. 301

40 CCBC Math 081 Multiplication and Division of Decimals Section pages Example 5: Divide by 3. The number is the dividend so it is placed inside of the division sign. The number 3 is the divisor so it is placed outside of the division sign Since there is no decimal place in the divisor, the decimal gets moved straight up into the quotient. Then divide as you would whole numbers Answer: = Practice 5: Divide Answer:

41 CCBC Math 081 Multiplication and Division of Decimals Section pages Example 6: Divide Since we have a decimal point in the divisor, we need to follow the steps below to move the decimal point to the correct place in the quotient: Write as a long division problem Place the decimal. (Remember 496 = 496.0) Move the decimal to the end of the divisor. Move the decimal the same number of spaces in the dividend Place the decimal of the quotient directly above the decimal in the dividend Divide using long division. Remember that every digit in the dividend must have an answer in the quotient (if possible) up to the decimal point. Answer: 80 Practice 6: Divide Answer:

42 CCBC Math 081 Multiplication and Division of Decimals Section pages Example 7: Divide by 7 and round off your answer to the nearest hundredth Write as a long division problem. Since there is no decimal place in the divisor, place the decimal in the answer directly above the decimal in the dividend To round to the hundredths place, we will need to know the digit of the thousandths place in the answer. Therefore we will need to place a 0 to the right of the divisor in the thousandths place Since we are rounding the answer to nearest hundredth, the 5 (which is 5 or greater) in the third decimal place is an indication to bump the 6 in the hundredths place to make it 7. Answer: Practice 7: Divide Note: Round your answer to the nearest hundredth. Answer:

43 CCBC Math 081 Multiplication and Division of Decimals Section pages Example 8: Convert the fraction 9 8 to a decimal. Round to one decimal place. We can think of the fraction 9 8 as 9 8. Set up the division problem as 89. Since we need to round the answer to one decimal place, we need to have 2 decimal places in the quotient. So, we write the division as Since we are rounding the answer to one decimal place, the 2 (which is less than 5) in the second decimal place is an indication to leave the 1 in the tenths place unchanged. Answer: 9 8 rounded to one decimal place is Practice 8: Convert the fraction to a decimal. Round your answer to one decimal place. 6 Answer:

44 CCBC Math 081 Multiplication and Division of Decimals Section pages Example 9: Convert the fraction 3 4 to a decimal. We can think of the fraction 3 4 as 3 4. Set up the division problem as 43. To begin the division, we need to put some zeros in the problem. We will begin with three zeros and add more if necessary. So, we write the division as Answer: is 0.75 Practice 9: Convert the fraction 4 5 to a decimal. Answer: Now that we have studied how to add, subtract, multiply, and divide decimal numbers, we can evaluate expressions using order of operations. Example 10: Evaluate Using Order of Operations, compute inside the parentheses = (0.4) Now square 0.4 by multiplying 0.4 by 0.4: digit to the right of the decimal = 0.16 Answer: digit to the right of the decimal digits to the right of the decimal 306

45 CCBC Math 081 Multiplication and Division of Decimals Section pages Practice 10: Evaluate Answer: bch0 Example 11: Evaluate = Using Order of Operations, divide first: Now add, lining up at the decimal point (and inserting a 0): Answer: Practice 11: Evaluate Answer:

46 CCBC Math 081 Multiplication and Division of Decimals Section pages Now let s review some geometry applications but here, our data will consist of decimal values. Example 12: Calculate the perimeter of the given rectangle. 4.2 m 2.5 m 2.5 m 4.2 m Use the formula for perimeter of a rectangle: P = 2L + 2W. Use L = 2.5 and W = 4.2. Perimeter = 2 L + 2W m (Don't forget the units in the final answer.) Practice 12: Calculate the perimeter of the given rectangle. Answer: P = inches in 3.6 in 3.6 in in In previous sections, we learned about measures of central tendency: mean, median, and mode. Let s continue with these concepts where the data values are decimal numbers. Example 13: Find the mean of 14.70, 15.3, 12.9, 13.1, 14.7, To calculate the mean, start by adding the six data values: = 84.6 Now, divide that sum by 6 (the number of data values): Mean = Practice 13: Find the mean of Answer:

47 CCBC Math 081 Multiplication and Division of Decimals Section pages Example 14: Find the median of 14.70, 15.3, 12.9, 13.1, 14.7, Note: = 14.7 Start by listing the data values in order from least to greatest: Because there is an even number of data values (6), there are two middle values: 13.9 and Find the mean of these two values: Practice 14: Find the median of Answer: Example 15: Find the mode of 14.70, 15.3, 12.9, 13.1, 14.7, The mode is the data value that occurs most often. Order the data from smallest to largest: 12.9, 13.1, 13.9, 14.70, 14.7, 15.3, Because 14.7 appears twice and every other data value appears only once, the mode is (Remember = 14.7) Practice 15: Find the mode of Answer: Watch All: 309

48 CCBC Math 081 Multiplication and Division of Decimals Section pages 4.4 Multiplication and Division of Decimals Exercises 1. Without actually finding the product, determine how many places will show to the right of the decimal point in the product: Without actually finding the product, determine how many places will show to the right of the decimal point in the product: Without actually finding the product, determine how many places will show to the right of the decimal point in the product: Multiply: Multiply: Multiply: Multiply: Multiply: (-12) (-3.5) 9. Calculate the perimeter and area of the given rectangle. 4.2 m 2.5 m 2.5 m 4.2 m 10. Calculate the perimeter and area of the given rectangle in 3 in 3 in 1.25 in 310

49 CCBC Math 081 Multiplication and Division of Decimals Section pages 11. Identify the dividend and the divisor in the division problem. Then, without actually finding the quotient, show how you would set up the long division problem Identify the dividend and the divisor in the division problem. Then, without actually finding the quotient, show how you would set up the long division problem Without actually finding the quotient, show where the decimal point will be located in the quotient: Without actually finding the quotient, show where the decimal point will be located in the quotient: Without actually finding the quotient, show where the decimal point will be located in the quotient: Divide: Divide: Divide: Round to the nearest hundredth. 19. Divide: Round to one decimal place. 20. Divide: Divide:

50 CCBC Math 081 Multiplication and Division of Decimals Section pages 22. Convert the fraction 32 5 to a decimal. 23. Convert the fraction 5 9 to a decimal. Round to the nearest thousandth. 24. Convert the fraction 7 6 to a decimal. Round to two decimal places. 25. Evaluate: 2 ( ) 26. Evaluate: Evaluate: (12 1.5) Evaluate: 29. Evaluate: According to Weather.com, the average rainfall (in inches) for six months in Baltimore is: Using the data above, find the mean rainfall. 31. Using the data above, find the median rainfall. 32. Using the data above, find the mode(s). The following data represent the wait time (in minutes) in line for a hamburger at McWendy s Drive Thru Burger Joint Using the data above, find the mean wait time. Round the answer to three decimal places. 34. Using the data above, find the median wait time. 35. Using the data above, find the mode(s). 312

51 CCBC Math 081 Multiplication and Division of Decimals Section pages 4.4 Multiplication and Division of Decimals Exercises Answers Perimeter: 13.4 meters Area: 10.5 square meters 10. Perimeter: 8.5 inches Area: 3.75 square inches 11. Dividend: 5.6 Divisor: Dividend: Divisor: in in 32. no mode min min mi 313

52 CCBC Math 081 Metric Measurement Section pages 4.5 Metric Measurement Previously, we used U. S. units to measure length, weight, and volume. We will now look at a different system of measures known as the metric system. The metric system was first adopted by France in 1799 and is now the basic system of measure used in most of the world. It bases all measures on powers of 10. The metric system uses the following basic units for length, volume, and weight. Meter: The basic unit of length is the meter; the symbol for a meter is m. (A meter is about 3 inches longer than a yard. Michael Jordan is about 2 meters tall.) Liter: The basic unit of volume is the liter; the symbol for a liter is L. (Slightly less than 4 liters are equivalent to 1 gallon. It takes about 50 liters of gasoline to fill a Mini Cooper s gas tank.) Gram: The basic unit of weight is the gram; the symbol for a gram is g. (Slightly more than 28 grams make an ounce, and about 453 grams make a pound. The average weight of a healthy newborn baby is about 3,000 grams.) In the metric system, a prefix can be attached to the basic unit to produce a new unit. The new unit is smaller or larger than the basic unit by a power of 10. The table shows some of the more common prefixes and their meanings: Prefix KILO k- HECTO h- DEKA da- BASIC UNIT DECI d- CENTI c- MILLI m- Meaning Thousand Hundred Ten (m,l,g) Tenth Hundredth Thousandth 1, For example, a centimeter (cm) is smaller than a meter because one centimeter is equal to one hundredth of a meter. A kilogram (kg) is larger than a gram because one kilogram is equal to one thousand grams. When we convert from feet to inches, we multiply by 12; when we convert from inches to feet, we divide by 12. Metric system conversions can be completed more easily because the metric system, like our numbering system, is based on powers of 10. Conversions between metric system units will require us either to multiply or to divide by 10 or by 100 or by 1000, etc. As an example of a simple metric system conversion that we will encounter, multiply the number by 10: the answer is , a shift of the decimal point one place to the right. As 314

53 CCBC Math 081 Metric Measurement Section pages another example, multiply the same original number by 100: the answer is , a shift of the decimal point two places to the right. In short, multiplying by a power of 10 shifts the decimal point to the right. On the other hand, dividing by a power of ten shifts the decimal point to the left. As an example, divide the number by 10: the answer is , a shift of the decimal point one place to the left. As another example, divide the same original number by 100: the answer is , a shift of the decimal point two places to the left. Since metric system conversions involve multiplying or dividing by powers of 10, a quick way to make a metric-to-metric conversion is to move the decimal point. If you can remember the order of the prefixes, you can shift the decimal the correct number of places, either right or left. Use the saying King Henry s daughter makes delicious chocolate milk to help you keep the prefix order straight. KILO k- HECTO h- DEKA da- BASIC UNIT meter liter gram DECI d- CENTI c- MILLI m- King Henry s Daughter Makes Likes Gives Delicious Chocolate Milk Here are some lesser known alternative sayings to help you memorize the prefix order. Pick your favorite. Klingons hate Dr. McCoy s deadly cold medicine. Kmart has district managers distributing cash machines. Kansas hockey divisions meet during cold months. STEPS TO CONVERT USING THE PREFIXES IN THE TABLE ABOVE: 1. Locate the prefix originally given. 2. Count how many positions and in what direction you need to move to get to the prefix you want. 3. Move the decimal point in the original value by that same number of places and in the same direction. Example 1: Convert 3 kilometers (km) to meters (m). 315

54 CCBC Math 081 Metric Measurement Section pages Locate the given unit (km) in the prefix table. To get to the unit you want (m), move three places to the right. So, move the decimal point in 3 = by three places to the right. 3 km km m Answer: 3 km = 3000 m Practice 1: Convert 3.52 kilograms (kg) to grams (g). Answer: 3520 g Example 2: Convert 88 centigrams (cg) to grams (g). Locate the given unit (cg) in the prefix table. To get to the unit we want (g), move two places to the left. So move the decimal point in 88 = 88. by two places to the left. 88 cg 8 8. cg g Answer: 88 cg = 0.88 g. Practice 2: Convert 5.6 centimeters (cm) to meters (m). Answer: m Example 3: Convert 41 liters (L) to kiloliters (kl). Locate the given unit (L) in the prefix table. To get to the unit we want (kl), move three places to the left. So move the decimal point in 41 = 41. by three places to the left. Notice we need to insert a 0 as a placeholder to the left of the digit 4 so we can move the decimal point by three places. 41 L L kl Answer: 41 L = kl Practice 3: Convert 2 meters (m) to kilometers (km). Answer: km 316

55 CCBC Math 081 Metric Measurement Section pages Example 4: Convert 423 milligrams (mg) to grams (g). Locate the given unit (mg) in the prefix table. To get to the unit we want (g), move three places to the left. So move the decimal point in 423 = 423. by three places to the left. 423 mg mg g Answer: 423 mg = g Practice 4: Convert milliliters (ml) to liters (L). Answer: L Example 5: Convert 2.57 liters (L) to milliliters (ml). Locate the given unit (L) in the prefix table. To get to the unit we want (ml), move three places to the right. So move the decimal point in 2.57 by three places to the right. Notice we need to insert a 0 as a placeholder to the right of the digit 7 so we can move the decimal point by three places L L ml 2570 ml Answer: 2.57 L = 2570 ml. Practice 5: Convert 52.9 grams (g) to milligrams (mg). Answer: 52,900 mg Example 6: Convert 59 dekagrams (dag) to decigrams (dg). Locate the given unit (dag) in the prefix table. To get to the unit we want (dg), move two places to the right. So move the decimal point in 59 = 59. by two places to the right. 59 dag dag dg 5900 dg Answer: 59 dag = 5900 dg Practice 6: Convert 5000 centimeters (cm) to kilometers (km). Answer: 0.05 km Because the metric system and our decimal numbering system are both based on powers of 10, we were able to convert within the metric system by moving decimal points as above. We can also convert units by using conversion factors as we did in a previous section. 317

56 CCBC Math 081 Metric Measurement Section pages Let s write the information in the prefix table using conversion factors instead. The conversion factors below are shown using meters (the metric unit for length). Notice that meter could be replaced with liter (the metric unit for volume) or gram (the metric unit for weight). 1 kilometer (km) = 1000 meters (m) 1 hectometer (hm) = 100 meters (m) 1 dekameter (dam) = 10 meters (m) 10 decimeters (dm) = 1 meters (m) 100 centimeters (cm) = 1 meter (m) 1000 millimeter (mm) = 1 meter (m) If we write a conversion factor as a fraction, its value is 1. For example, we can write the conversion factor relating meters and centimeters as both: 1m cm and 100 cm 1 1m USING CONVERSION FACTORS TO CONVERT: Multiply the given value by the conversion factor written as a fraction. Form the fraction with the unit we want in the numerator and the unit we are given in the denominator. Example 7: Convert 3 kilometers (km) to meters (m). The given unit is kilometers and the wanted unit is meters. Write the conversion factor 1 kilometer (km) = 1000 meters (m) as 1000 m 1km. Multiply: 3 km 1000 m m 3000 m 1 1km Answer: 3 km = 3000 m. Practice 7: Convert 3.52 kilograms (kg) to grams (g). Answer: 3520 g Example 8: Convert 88 centigrams (cg) to grams (g). 318

57 CCBC Math 081 Metric Measurement Section pages The given unit is centimeters and the wanted unit is meters. Write the conversion factor 100 centigrams (cg) = 1 gram (g) as 1g 100 cg. Multiply: 88 cg 1g 88 g 0.88 g cg 100 Answer: 88 cg = 0.88 g. Practice 8: Convert 5.6 centimeters (cm) to meters (m). Answer: m Example 9: Convert 567 milliliters (ml) to kiloliters (kl). The given unit is milliliters and the wanted unit is kiloliters. We will use two conversion factors: one to convert from milliliters to liters and another to convert from liters to kiloliters. Multiply: 567 ml 1L 1kL 567 kl kl ml 1000 L 1,000,000 Answer: 567 ml = kl. Practice 9: Convert 5000 centimeters (cm) to kilometers (km). Answer: 0.05 km Watch All: 319

58 CCBC Math 081 Metric Measurement Section pages 4.5 Metric Measurement Exercises 1. Convert 6804 m to km. 2. Convert 4.2 kg to g. 3. Convert 1970 mg to g. 4. Convert 18,000 L to hl. 5. Convert 450 mm to m. 6. Convert 73.9 cm to m. 7. Convert 357 ml to L. 8. Convert hm to m. 9. Convert 345 g to mg. 10. Convert 8.75 m to cm. 11. Convert 8 dal to L. 12. Convert 7.5 cg to kg. 13. Convert 14 dm to m. 14. Convert 6 hg to cg. 15. Convert 553 dl to hl. 16. Convert 2.8 g to dag. 17. Convert 65 cm to dm. 18. Convert km to cm. 19. Convert 50 mg to hg. 20. Convert 0.35 L to dl. 320

59 CCBC Math 081 Metric Measurement Section pages 4.5 Metric Measurement Exercises Answers km 2. 4,200 g g or 1.97 g hl m or 0.45 m m L 8. 4,389 m ,000 mg cm L kg m ,000 cg hl dag dm 18. 7,800 cm hg dl 321

60 CCBC Math 081 Applications Section pages 4.6 Applications We studied geometry in earlier sections of this book. Now, we will revisit some geometry applications to use decimal numbers. 1 Recall that the area of a triangle can be written as A bh where b is the length of the base 2 and h is the height. In this chapter, since our data values are decimal numbers, instead of using the fraction 1, we will use its decimal equivalent AREA OF A TRIANGLE: Area of a triangle is 0.5 A b h h b = base Example 1: Calculate the area of the given triangle. 1.7 cm 3.2 cm 0.7 cm 4.7 cm Notice the base b = 4.7 cm and the height h = 0.7 cm. So calculate: Area 0.5b h square centimeters (Don't forget squared units for area.) Answer: Area = cm 2 Practice 1: Calculate the area of the given triangle. Answer: A = 4.25 in in 4.5 in 1.7 in 5 in 322

61 CCBC Math 081 Applications Section pages Now recall the formulas for circumference and area of a circle: CIRCUMFERENCE AND AREA OF A CIRCLE: Circumference isc r r Area of a Circle is A r When we studied fractions, we used the fraction approximation of : 7 since our data values are decimal numbers, we will use the decimal approximation for instead: In this chapter, Example 2: Determine the circumference and area of the given circle, using mm Notice the radius r of the circle is 2.3 mm. Circumference: C 2 r C = mm Area: A r A = = mm 2 Practice 2: Determine the circumference and area of the circle. Answer: C 4.396ft A ft ft 323

62 CCBC Math 081 Applications Section pages In previous sections, we learned how to perform operations on decimal numbers. Now let s explore some additional applications of when to use those operations. Example 3: If Jean has 1.08 pounds of butter and 0.93 pounds of margarine, find the sum of the weights. The sum is the answer to an addition problem. Line up the decimal points and add: Answer: 2.01 pounds Practice 3: At his restaurant job, Drew earned $43.90 in tips on Friday and $57.75 in tips on Saturday. How much did he earn in tips altogether? Answer: $ Example 4: Abby purchased a book costing $7.99 with a $20 bill. How much change will she receive? Subtract the cost of the book from the amount paid. Line up the decimal points to subtract: $ $ $ Answer: $12.01 Practice 4: Alex purchased a notebook costing $3.91 with a $5 bill. How much change will he receive? Answer: $

63 CCBC Math 081 Applications Section pages Example 5: If Joe bought a bicycle for $ and Ted bought a bicycle for $182.95, what is the difference in the price of the two bicycles? The difference is the answer to a subtraction problem. Subtract the cost of Ted s bicycle from the cost of Joe s bicycle. Line up the decimal points and subtract: Answer: $ $ $ $ Practice 5: Lisa bought a spool of ribbon containing 5 yards. She used 3.25 yards of the ribbon to make a bow. How many yards of ribbon are left on the spool? Answer: 1.75 yards Example 6: A computer CD costs $1.88. How many CDs can be purchased with $18.80? Divide the total amount by the cost of one item. So divide: $18.80 $1.88 division with the dividend as $18.80 and the divisor as Set up the long Answer: Move the decimal points in the divisor and dividend two places to the right Place decimal point in the answer above the CDs can be purchased. decimal point in the dividend. Note: In general, to determine how many items of the same price can be purchased with a particular amount of money, divide the amount of money by the cost per item. Practice 6: A pack of soda containing 24 cans costs $6.99. How much does each can of soda cost? Round the answer to the hundredths place. Answer: $

64 CCBC Math 081 Applications Section pages Example 7: You need to buy 8 packages of computer paper. Each package of paper costs $4.50. How much money do you need? Multiply the number of packages (8) by the cost of each package ($4.50): Answer: You need $ Note: In general, to determine how much money is needed to buy many items where each item costs the same amount, multiply the number of items by the cost per item. Practice 7: Bananas cost $0.59 per pound. How much will 4 pounds of bananas cost? Answer: $ Many of the examples above involved money because money is a practical application of decimal numbers. We will continue to address some of the mathematical skills needed to live a financially healthy life. Let s consider bank accounts. There are generally two types of accounts: savings accounts and checking accounts. Savings accounts are one way of putting money aside and earning interest on it. By saving small amounts of money, you can build wealth slowly but steadily over time. Money placed in these accounts is not intended for everyday expenses, like purchasing movie tickets or buying a new music CD. Instead, the purpose of a savings account is to provide the individual with a safe place to save money that can be used at a later date to make a major purchase such as a car, or to fund a large expense such as a college education or a house. Have you ever tried to save up for something that you really wanted, only to be unsuccessful because you were constantly taking small amounts of cash out of the money you were saving? While most of us have good intentions about saving money and understand that it takes some time and effort to save up for a major purchase, many of us don t have the willpower to keep our hands off the cash when we have access to it. A savings account can help with this. Some people find it helpful to think of a savings account like a pail of water. The amount of water in the pail represents the money that you have placed in the savings account. When you place the pail under the tap and turn on the tap, the amount of water in the pail increases. The water from the tap is a deposit. Let s assume that your pail is also fitted with a tap at the bottom. 326

65 CCBC Math 081 Applications Section pages Each time you open the bottom tap, the amount of water in the pail decreases. When you make a withdrawal from your savings account, you decrease its value. Just like keeping your pail full, the key to successful saving is making sure that you have more money going into the account than you do coming out of it. In order for the amount of water in the pail to increase, water must flow into the pail faster than it flows out of the tap at the bottom of the pail. Similarly, to make your savings grow, the amount that you deposit into the account should be greater than the amount that you withdraw from the account. You also need to remember that with a savings account, there is a little extra inflow into the account coming from the interest earnings that are paid to you by the bank each month. Checking accounts, on the other hand, are designed to make it easy for people to pay their bills or purchase things without having to go to the bank and withdraw cash. Traditional checking accounts grant check-writing privileges. The privileges allow the account holder to make payments with checks for items such as utilities, rent, mortgage payments, food, and a variety of other expenses. The bank will provide you with a check register to keep with your checks. In the check register, you can record the date and amount of deposits as well as the date, check number, payee (the person to whom the check is written) and amount of each check as it is written. It is important to keep your check register up-to-date after each transaction. While Electronic Funds Transfers (EFTs) are immediately debited from your account, paper checks take much longer to process sometimes days or weeks, depending on when the recipient of the check decides to submit the check for payment. The account holder could be charged a fee because there are not enough funds in the account to cover a check/debit. The fee is called a NSF (non-sufficient fund) fee. At the end of each month, the bank will send you a statement which includes a statement balance. In addition to the balance, the statement will list all of the debits and credits for the account made before the statement date. It is important to remember that the statement balance may be different from the actual balance in the account because additional transactions have been made and not all debits cleared since the statement was printed and mailed to you. At the end of each month, you should balance or reconcile your checkbook by finding your account balance. Use your checkbook register and compare it to the statement to verify its accuracy and to ensure that your account has sufficient funds to cover outstanding debits. The example below shows how a typical check register looks and how to balance the checkbook. 327

66 CCBC Math 081 Applications Section pages Example 9: Below is a list of transactions made to your checking account for the month of September Record each transaction in the check register below. As you record each one, calculate the current, updated balance in the account. a. On September 1, your account balance was $ b. On September 1, you used Check #100 at the supermarket to buy groceries costing $ c. On September 2, you used Check #101 at the gas station to pay for $40 worth of gas. d. On September 6, your paycheck in the amount of $810 was deposited directly into your checking account via an EFT. e. On September 6, you used Check #102 to pay a bill for $ for your rent. Consider how each of those transactions is entered into the check register below. After recording each entry, calculate the current balance in the account. a. Enter the beginning balance of $ in the first line of the register. b. Enter Check 100 on 9/1/2013 to the Supermarket for a check amount of $ Now calculate the current balance. Since this amount is a withdrawal from the account, subtract: $ $64.14 = $50.98 [Enter this amount into the Balance column.] c. Enter Check 101 on 9/2/2013 to the Gas Station for a check amount of $ To calculate the current balance after this withdrawal, subtract: $ $40.00 = $10.98 [Enter this amount into the Balance column.] d. Enter for 9/6/2013 a Payroll Deposit of $ This amount is a deposit so add its amount to the previous balance: $ $ = $ [Enter this amount into the Balance column.] e. Enter Check 102 on 9/6/2013 for Rent for a check amount of $ To calculate the current balance after this withdrawal, subtract: $ $ = $ [Enter this amount into the Balance column.] Check Register Check Number Date Transaction Description Check/Debit Amount Deposit/Credit Amount Balance Beginning Balance $ /1/2013 Supermarket $ $ /2/2013 Gas Station $ $ /6/2013 Payroll Deposit $ $ /6/2013 Rent $ $ Notice also if you were only interested in the account balance at the end of the month, you could use the following formula: Account balance: Account balance equals the starting balance plus the total amount deposits made during the month minus the total amount of checks written during the month. 328

67 CCBC Math 081 Applications Section pages Account balance = Starting Account Balance + Total Deposits Total of Amount of Checks As shown in the check register: Start of the month account balance = $ Total amount of deposits made = $ Total amount of withdrawals made = $ ( = $ $ $650.00) So the account balance at the end of the month is: $ $ $ = $ The picture below shows how Check #100 would be written: The picture below shows how Check #100 would be written: 9/1/