Download Solving Percent Problems 5.6 and more Exercises Reasoning in PDF only on Docsity!
224 Chapter 5 Ratios and Rates
5.6 Solving Percent Problems
How can you use mental math to find the percent of a number?
Work with a partner. a. How did Newton know that 1% of 80 is 0.8? b. How do you move the decimal point to find 1% of a number?
22 ACTIVITY: Finding 1% of a Number
Work with a partner. a. How did Newton know that 10% of 80 is 8?
Write 10% as a fraction. 10% = — = 1 —
Method 1: Use a model. 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
0 80
Method 2: Use multiplication.
10% of 80 = — 10 of 80 = — 10
× = —
10
b. How do you move the decimal point to find 10% of a number? Move the decimal point one place to the. 10% of 80. =
per
cent
11 ACTIVITY: Finding 10% of a Number
“I have a secret way for finding 21% of 80.”
“10% is 8, and 1% is 0.8.” “So, 21% is 8 + 8 + 0.8 = 16.8.”
COMMON
CORE
Percents In this lesson, you will ● (^) find percents of numbers. ● (^) find the whole given the part and the percent. Learning Standard 6.RP.3c
Section 5.6 Solving Percent Problems 225
Work with a partner. Use mental math to find each percent of a number. a. 12% of 40 b. 19% of 50 Think: 12% = 10% + 1% + 1% Think: 19% = 10% + 10% − 1%
33 ACTIVITY: Using Mental Math
10% of 40 = 1% of 40 = 10% of 50 = 1% of 50 =
Work with a partner. Use mental math to find each percent of a number.
a. 20% tip for a $30 meal
b. 18% tip for a $30 meal
c. 6% sales tax on a $20 shirt
d. 9% sales tax on a $20 shirt
e. 6% service charge for a $200 boxing ticket
f. 2% delivery fee for a $200 boxing ticket
g. 21% bonus on a total of 40,000 points
h. 38% bonus on a total of 80,000 points
44 ACTIVITY: Using Mental Math
5. IN YOUR OWN WORDS How can you use mental math to find the percent of a number? 6. Describe two real-life examples of fi nding a percent of a number. 7. How can you use 10% of a number to fi nd 20% of the number? 30%? Explain your reasoning.
Use what you learned about fi nding the percent of a number to complete Exercises 3 –10 on page 229.
g.
e
f.
Evaluate Results Does your answer seem reasonable? How can you check your answer?
Math Practice
Section 5.6 Solving Percent Problems 227
Finding the Whole Write the percent as a fraction. Then divide the part by the fraction. Words The part divided by the percent equals the whole. Numbers 20%^ of^60 is^ 12.
1 — 5
× 60 = 12 12 ÷ —^1
5
Multiplication equation Related division equation
EXAMPLE 33 Finding the Whole
75% of what number is 48?
48 ÷ 75% = 48 ÷ 3 — 4
Write the percent as a fraction and divide.
4 — 3
Multiply by the reciprocal.
= 64 Simplify.
So, 75% of 64 is 48. 0%^ 25%^ 50%^ 75%^ 100%
0 16 32 48 64
EXAMPLE 44 Finding the Whole Using a Ratio Table
120% of what number is 72? Use a ratio table to fi nd the whole. Find an equivalent ratio with 72 as the part.
Part 120 6 72 Whole 100 5 60
So, 120% of 60 is 72.
Find the whole. Explain your method. Exercises 27–36 5. 5% of what number is 10? 6. 62% of what number is 31?
You can use a related division equation to find the whole given the part and the percent.
The fi rst column represents the percent. part — whole
= —^120
100
÷ 20 × 12
÷ 20 × 12
228 Chapter 5 Ratios and Rates
15 5 f 5555 ftff ttttttt
Time
EXAMPLE 55 Real-Life Application
The width of a rectangular room is 80% of its length. What is the area of the room? Find 80% of 15 feet.
80% of 15 = 4 — 5
× 15 0%^ 20%^ 40%^ 60%^ 80%^ 100%
0 3 6 9 12 15
4 × 15 — 5
= 12 The width is 12 feet. Use the formula for the area A of a rectangle. A = 15 × 12 = 180
So, the area of the room is 180 square feet.
7. The width of a rectangular stage is 55% of its length. The stage is 120 feet long. What is the area?
3
1
You win an online auction for concert tickets. Your winning bid is 60% of your maximum bid. How much more were you willing to pay for the tickets than you actually paid?
A $72 B $80 C $120 D $ Your maximum bid is the whole , and your winning bid is the part. Find your maximum bid by dividing the part by the percent.
120 ÷ 60% = 120 ÷ (^) —^3 5
Divide the part by the percent.
= 120 ⋅ —^5
3
Multiply by the reciprocal.
= 200 Simplify.
Your maximum bid is $200, and your winning bid is $120. So, you were willing to pay 200 − 120 = $80 more for the tickets.
The correct answer is B.
8. WHAT IF? Your winning bid is 96% of your maximum bid. How much more were you willing to pay for the tickets than you actually paid?
EXAMPLE 66 Real-Life Application
230 Chapter 5 Ratios and Rates
Find the whole. Explain your method.
27. 10% of what number is 14? 28. 20% of what number is 18? 29. 25% of what number is 21? 30. 75% of what number is 27? 31. 15% of what number is 12? 32. 85% of what number is 17? 33. 140% of what number is 35? 34. 160% of what number is 32? 35. 125% of what number is 25? 36. 175% of what number is 42? 37. ERROR ANALYSIS Describe and correct (^) 20% of what number is 5?
5 ÷ 20% =
5 — 20 = 1 — 4
the error in fi nding the whole.
38. COUPON You have a coupon for a restaurant. You save $3 on a meal. What was the original cost of the meal? 39. SURVEY The results of a survey are shown at the right. In the survey, 12 students said that they would like to learn French. a. How many students were surveyed? b. How many of the students surveyed would like to learn Spanish? 40. WEIGHT A sixth grader weighs 90 pounds, which is 120% of what he weighed in fourth grade. How much did he weigh in fourth grade? 41. PARKING LOT In a parking lot, 16% of the cars are blue. There are 4 blue cars in the parking lot. How many cars in the parking lot are not blue? 42. LOTION A bottle contains 20 fluid ounces of lotion and sells for $5.80. The 20-fl uid-ounce bottle contains 125% of the lotion in the next smallest size, which sells for $5.12. Which is the better buy? Explain.
OFF
dine-in, take-out, or delivery
Grand Opening
SPECIAL
Section 5.6 Solving Percent Problems 231
Multiply. (Section 2.5)
54. 0.6 × 8 55. 3.3 × 5 56. 0.74 × 9 57. 2.19 × 12 58. MULTIPLE CHOICE What is the quotient of 75 and 2.4? (Section 2.6) A 0.032 B 0.3125 C 3.2 D 31.
Copy and complete the statement using <, >, or =.
43. 80% of 60 60% of 80 44. 20% of 30 30% of 40 45. 120% of 5 0.8% of 250 46. 85% of 40 25% of 136 47. TIME How many minutes is 40% of 2 hours? 48. LENGTH How many inches is 78% of 3 feet? 49. GEOMETRY The width of the rectangle is 75% of its length.
24 in.
a. What is the area of the rectangle? b. The length of the rectangle is doubled. What percent of the length is the width now? Explain your reasoning.
50. BASKETBALL To pass inspection, a new basketball should bounce back to between 68% and 75% of the starting height. A new ball is dropped from 6 feet and bounces back 4 feet 1 inch. Does the ball pass inspection? Explain. 51. REASONING You know that 15% of a number n is 12. How can you use this to fi nd 30% of n? 45% of n? Explain. 52. SURFBOARD You have a coupon for 10% off the sale price of a surfboard. Which is the better buy? Explain your reasoning. ● 40% off the regular price ● 30% off the regular price and then 10% off the sale price 53. On three 150-point geography tests, you earned grades of 88%, 94%, and 90%. The fi nal test is worth 250 points. What percent do you need on the fi nal to earn 93% of the total points on all tests?
