4.3 Addition and Subtraction of Decimals

Transcription

1 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:

2 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:

3 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

4 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

5 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:

6 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:

7 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

8 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:

9 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

10 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

11 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 $ $ $ $

12 20. Add: (- 1.5) 21. Subtract: (- 3.17) 22. Add: (-57.8) 23. Add: Subtract: Subtract: (- 4) 294

13 4.3 Addition and Subtraction of Decimals Exercises Answers mm in cm km $ $9.25 = $

14 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 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 Show how to set up the subtraction problem to calculate by hand. Do not subtract

15 CCBC Math 081 Mid-Chapter Review 4 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. $ $