Percentage copy - Lecture notes 1-2 PDF

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Three Types of Percent Problems A percent word problem has three values: the base (B) or whole, the rate (R) or percent, and the part (P). For example, if you got 40 correct answers on a 50-point exam, the base is 50, the part is 40, and your grade (rate) would be 40/50 = 0.80 = 80%.

Every percent problem contains three variables. They are the base (B), the rate (R) or percent, and the part (P). When you solve a percent problem, you are given the values of two of the three variables and you are asked to find the value of the third variable. The relationship of the three variables can be pictured in the circle shown in Figure

TYPE 1: FINDING THE PART Type 1 problems can be stated as follows: Find 20% of 60. What is 20% of 60? 20% of 60 is what number? In Type 1 problems, you are given the base and the rate and are asked to find the part. Use the formula P = R × B and multiply the rate by the base. Be sure to change the percent to a decimal or fraction before multiplying EXAMPLE: Find 60% of 90. SOLUTION: Change the percent to a decimal and multiply: 0.60 × 90 = 54. EXAMPLE: Find 45% of 80. SOLUTION: Use the formula P = R × B. Change 45% to a decimal and multiply: 0.45 × 80 = 36. PRACTICE: 1. Find 70% of 45. 2. Find 84% of 15. 3. What is 33% of 66? 4. 62.5% of 64 is what number? 5. Find 18% of 630. SOLUTIONS: 1. 0.70 × 45 = 31.5 2. 0.84 × 15 = 12.6 3. 0.33 × 66 = 21.78 4. 0.625 × 64 = 40 5. 0.18 × 630 = 113.4

TYPE 2: FINDING THE RATE Type 2 problems can be stated as follows: What percent of 16 is 10? 10 is what percent of 16? In Type 2 problems, you are given the base and the part and are asked to find the rate or percent. The formula is R = P/ B . In this case, divide the part by the base and then change the answer to a percent. EXAMPLE: What percent of 8 is 6? SOLUTION: Use the formula R = P/B . Divide 6/8 = 6 ÷ 8 = 0.75. Change the decimal to a percent: 0.75 = 75%. EXAMPLE: 18 is what percent of 90? SOLUTION: Use the formula R = P/B, and then divide 18 /90= 18 ÷ 90 = 0.20. Change the decimal to a percent: 0.20 = 20%.

PRACTICE: 1. What percent of 18 is 3? 2. 30 is what percent of 240? 3. 5 is what percent of 60? 4. What percent of 20 is 18?

5. What percent of 110 is 60? SOLUTIONS: 1. 3 ÷ 18 = 0.166 = 16.6% 2. 30 ÷ 240 = 0.125 = 12.5% 3. 5 ÷ 60 = 0.083 = 8.3% 4. 18 ÷ 20 = 0.9 = 90% 5. 60 ÷ 110 = 0.5454 = 54.54%

TYPE 3: FINDING THE BASE Type 3 problems can be stated as follows: 16 is 20% of what number? 20% of what number is 16? In Type 3 problems, you are given the rate and the part, and you are asked to find the base. Use the formula B = P/R . EXAMPLE: 52% of what number is 416? SOLUTION: Use the formula B = P/R . Change 52% to 0.52 and divide: 416 ÷ 0.52 = 800. EXAMPLE: 45 is 30% of what number? SOLUTION: Use the formula B = P/R . Change 30% to 0.30 and divide: 45 ÷ 0.30 = 150.

PRACTICE: 1. 6% of what number is 90? 2. 250 is 20% of what number? 3. 35 is 70% of what number? 4. 40% of what number is 200? 5. 19.2% of what number is 115.2? SOLUTION: 1. 90 ÷ 0.06 = 1500 2. 250 ÷ 0.20 = 1250 3. 35 ÷ 0.70 = 50 4. 200 ÷ 0.40 = 500 5. 115.2 ÷ 0.192 = 600

Percent word problems can be solved by identifying what you need to find and selecting the correct formula. In order to solve a percent problem 1. Read the problem. 2. Identify the base, rate (%), and part. One of these will be unknown. 3. Select the correct formula. 4. Substitute the values in the formula and evaluate

EXAMPLE: On a test consisting of 60 questions, a student received a grade of 85%. How many problems did the student answer correctly SOLUTION: The base is 60 and the rate is 85%. The number of correct answers is the part. Since you need to find the part, use the formula P = R × B. Change 85% to 0.85 and multiply: 0.85 × 60 = 51. Hence, the student got 51 problems correct.

EXAMPLE: A basketball team won 12 of its 20 games. What percent of the games played did the team win? SOLUTION: The base or total is 20 and the part is 12. The percent is the rate. Since you need to find the rate, use the formula R = P/B . Divide 12 ÷ 20 = 0.60 = 60%. Hence, the team won 60% of its games

1) The sales tax rate in a certain state is 6%. If the sales tax on an automobile was $1350, find the price of the automobile. . 2)The cost of a suit that was originally $300 was reduced to $180. What was the percent of the reduction? 3) A home was sold for $80,000. If the salesperson’s commission was 7%, find the amount of the person’s commission. 4) If a merchant purchased a clock for $30 and sold it for $50, find the rate of the markup based on the price that the merchant paid for the clock. 5). If the regular price of a picture frame is $25 and the price tag is marked 30% off, find the sale price. 6) On a 60-question examination, a student answered 45 questions correctly. What percent did she get correct? 7). The sales tax on a television set is $30.10. Find the cost of the television set if the tax rate is 7%. 8) A person saves $100 a month. If her annual income is $24,000, what percent of her income is she saving? 9). There are 40 students enrolled in Business Math 101. If 15% of the students were absent on a certain day, how many were

absent? 10). Last August, the Martin family paid $75 for electricity. In February, they paid $54. What is the percent of decrease? 11). A railroad inspector inspects 360 railcars. If 95% passed, how many cars passed the inspection? 12). An instructor announced that 25% of his students received an A on the last test. If 8 students received an A, how many students took the test?...