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PERCENTSPERCENTSPERCENTSPERCENTS
Percent means “per hundred.” Writing a number as a percent is a way of comparing the number with
- For example: 42% = (^) ⡩⡨⡨⡲⡰
Percents are really fractions (or ratios) with a denominator of 100. Any percent may be changed to an equivalent fraction by dropping the percent symbol and writing the number over 100. Usually it is best to put this fraction in simplest terms.
CHANGING PERCENTS TO DECIMALSCHANGING PERCENTS TO DECIMALSCHANGING PERCENTS TO DECIMALSCHANGING PERCENTS TO DECIMALS
RULE:RULE:RULE:RULE: To change a percent to a decimal, drop the % symbol and move the decimal point two places to the left.
Examples:Examples:Examples:Examples: 25% = 0.25 75% = 0.75 6.8% = 0.068 0.63% = 0.
CHANGING DECIMALS TO PERCENTSCHANGING DECIMALS TO PERCENTSCHANGING DECIMALS TO PERCENTSCHANGING DECIMALS TO PERCENTS
RULE:RULE:RULE:RULE: To change a decimal to a percent, move the decimal point two places to the right and use the % symbol.
Examples:Examples:Examples:Examples: 0.27= 27% 4.89 = 489% 0.2 = 20% 25 = 2500%
CHANGING PERCENTS TO FRACTIONSCHANGING PERCENTS TO FRACTIONSCHANGING PERCENTS TO FRACTIONSCHANGING PERCENTS TO FRACTIONS
RULE:RULE:RULE:RULE: To change a percent to a fraction, drop the % symbol and write the original number over 100. Simplify the fraction to lowest terms.
Examples:Examples:Examples:Examples: 62% = (^) ⡩⡨⡨⡴⡰ = ⡱⡩⡳⡨
4.5% = (^) ⡩⡨⡨⡲.⡳ = (^) ⡩⡨⡨ 㐀 ⡩⡨⡲.⡳ 㐀 ⡩⡨ = (^) ⡩⡨⡨⡨⡲⡳ = (^) ⡰⡨⡨⡷ To create a whole number in the numerator, multiply the numerator and denominator by 10. Simplify.
32 ⡩⡰ % = ⡱⡰
ㄗㄘ ⡩⡨⡨ =
ㄢㄡㄘ ⡩⡨⡨ =^
⡴⡳ ⡰ 㐀^
⡩ ⡩⡨⡨ =^
⡴⡳ ⡰⡨⡨ =^
⡩⡱ ⡲⡨ Writing 32 ⡩⡰ % over 100 produces a complex fraction, so we change 32 ⡩⡰ to an improper fraction and simplify.
CHANGING FRACTIONS TO PERCENTSCHANGING FRACTIONS TO PERCENTSCHANGING FRACTIONS TO PERCENTSCHANGING FRACTIONS TO PERCENTS
RULE:RULE:RULE:RULE: To change a fraction to a percent, change the fraction to a decimal and then change the decimal to a percent.
Examples:Examples:Examples:Examples: ⡵
⡩⡨ = 0.7 = 70% Change (^) ⡩⡨⡵ to a decimal by dividing 7 by 10. Then change the resulting decimal 0.7 to a percent by moving the decimal point two places to the right and use the % symbol.
⡱ ⡶ = 0.375 = 37.5% Change ⡱⡶ to a decimal by dividing 3 by 8. Then change the decimal to a percent by moving the decimal point tow places to the right and use the % symbol. Division equals 0.375 which becomes 37.5%.
BASIC PERCENT WORD PROBLEMSBASIC PERCENT WORD PROBLEMSBASIC PERCENT WORD PROBLEMSBASIC PERCENT WORD PROBLEMS
There are three types of word problems associated with percents: Type A: What number is 15% of 63? Type B: What percent of 42 is 21? Type C: 25 is 40% of what number? The method we use to solve all three types of problems involves translating the sentences into equations and then solving the equations.
The following translations are used to equations: EnglishEnglishEnglishEnglish MathematicsMathematicsMathematicsMathematics is = of x (multiply) a number n what percent n what number n The word is always translates to an = sign, the word of almost always means multiply, and the number we are looking for can be represented with a letter, such as n or x.
ExampleExampleExampleExample 1 (1 (1 (1 (Type AType AType AType A))):)::: What number is 15% of 63? We translate the sentence into an equation as follows: What number is 15% of 63?
n = 0.15 · 63 To do arithmetic with percents, we have to change percents to decimals. Solving the equation, we have: n = 0.15 · 63 n = 9.
ExampleExampleExampleExample 2 (2 (2 (2 (TypeTypeTypeType B)B)B)B):::: What percent of 42 is 21? We translate the sentence into an equation as follows: What percent of 42 is 21?
n · 42 = 21 We solve for n by dividing both sides by 42. ⤤ · ⡲⡰ ⡲⡰ =^
⡰⡩ ⡲⡰ ᡦ = ⡰⡩⡲⡰ ᡦ = 0. Since the original problem asked for a percent, we change 0.50 to a percent. ᡦ = 0.50 = 50%
15% of 63 is 9.
21 is 50% of 42
APPLICATIONS OF PERCENTSAPPLICATIONS OF PERCENTSAPPLICATIONS OF PERCENTSAPPLICATIONS OF PERCENTS
Example 1Example 1Example 1Example 1 On a 120-question test, a student got 96 correct answers. What percent of the problems did the student work correctly?
The problem states that we have 96 correct answers out of a possible 120. The problem can be restated as: 96 is what percent of 120?96 is what percent of 120?96 is what percent of 120?96 is what percent of 120? 96 = n · 120 ⡷⡴ ⡩⡰⡨ =^
ぁ ·⡩⡰⡨ ⡩⡰⡨ n = 0. n = 80% The test score was 80%.
Example 2Example 2Example 2Example 2 How much HCI (hydrochloric acid) is in a 60-milliliter bottle that is marked 80% HCI?
If the bottle is marked 80% HCI, that means that 80% of the solution is HCI and the rest is water. Since the bottle contains 60 milliliters, we can restate the question as: What is 80% of 60?What is 80% of 60?What is 80% of 60?What is 80% of 60? n= 0.80 · 60 n = 48 There are 48 milliliters of HCI in 60 milliliters of 80% HCI solution.
Example 3Example 3Example 3Example 3 If 48% of the students in a certain college are female and there are 2,400 female students, what is the total number of students in the college?
We restate the problem as:2,400 is 48% of what number?2,400 is 48% of what number? 2,400 is 48% of what number?2,400 is 48% of what number? 2400 = 0.48 · n ⡰⡲⡨⡨ ⡨.⡲⡶ =^
⡨.⡲⡶ぁ ⡨.⡲⡶ n = 5, There are 5,000 students.
Example 4Example 4Example 4Example 4 If 25% of the students in elementary algebra courses receive a grade of A, and there are 300 students enrolled in elementary algebra this year, how many students will receive an A?
We restate the problem as:What number is 25% of 300?What number is 25% of 300? What number is 25% of 300?What number is 25% of 300? n = 0.25 · 300 n = 75 So, 75 students will receive A’s in elementary algebra.
PRACTICEPRACTICEPRACTICEPRACTICE
Write each percent as a fraction with a denominator of 100.
- 20% 2. 40% 3. 60% 4. 80%
- 24% 6. 48% 7. 65% 8. 35% Change each percent to a decimal.
- 23% 10. 34% 11. 92% 12. 87%
- 9% 14. 7% 15. 3.4% 16. 5.8%
- 6.34% 18. 7.25% 19. 0.9% 20. 0.6% Change each decimal to a percent.
- 0.23 22. 0.34 23. 0.92 24. 0.
- 0.45 26. 0.54 27. 0.03 28. 0.
- 0.6 30. 0.9 31. 0.8 32. 0. Change each percent to a fraction in lowest terms.
- 4% 34. 2% 35. 26.5% 36. 34.2%
- 71.87% 38. 63.6% 39. 0.75% 40. 0.45%
- 6 ⡩⡲% 42. 5⡩⡲% 43. 33 ⡩⡱% 44. 66 ⡰⡱%
Change each fraction or mixed number to a percent.
- ⡩⡰ 46. ⡩⡲ 47. ⡱⡲ 48. ⡰⡱
- ⡵⡶ 50. ⡩⡶ 51. (^) ⡳⡨⡵ 52. (^) ⡰⡳⡷
- 3 ⡩⡲ 54. 2 ⡩⡶ 55. 1 ⡩⡰ 56. 1 ⡱⡲
- What number is 25% of 32?
- What number is 10% of 80?
- What number is 20% of 120?
- What number is 15% of 75?
- What number is 54% of 38?
- What number is 72% of 200?
- What number is 11% of 67?
- What percent of 24 is 12?
- What percent of 80 is 20?
- What percent of 50 is 5?
- What percent of 20 is 4?
- What percent of 36 is 9?
ANSWERSANSWERSANSWERSANSWERS
- 400 78. 70% 79. 90% correctly, 10% incorrectly
- 84% 81. 25% 82. 45 mi
- 4 liters acetic acid, 1 liter water 84. 18.2 acres
- 126 are first-year, 294 are not 86. 3,000 students 87. 400 students
- 500 parts 89. 1,664 women 90. 225 students
