Free Pre-Algebra Lesson 41! page 1 Lesson 41 Solving Percent Equations A percent is really a ratio, usually of part to whole. In percent problems, the numerator of the ratio (the part) is called the, and the denominator (the whole) is called the base. The quotient, or ratio, is the percent. You can write the relationship as a division or as a multiplication. base = percent = percent base or Since the percent is a ratio, it really doesn t stand alone as a number, but always refers to some base. The word of following the percent points us to the base. Since the percent is a part of one hundred, you can picture the base as a set of one hundred, like the square containing one hundred little squares shown with each example. 30% of the class 90% of the points 65% of the televisions The class is the base, represented by the whole square. 30% is shaded. (All) the points is the base, represented by the whole square. 90% is shaded. The base is (all) the televisions, represented by the whole square. 65% is shaded. Translating Percent Sentences to Equations Percent sentences in English are often structured like the multiplication version of the percent equations above. 30% of the class is sick. The word of in the sentence is the multiplication in the equation, and the word is (or some form of it) is the equals sign. The square is the whole class. 30% of the square is shaded, representing the 30% of the class that is sick. Written as a ratio, it says, base = percent sick people in the class all people in the class = 30 100 = 30% There may not be 100 people in the class, but the ratio of sick people to all people in the class is equal to the ratio of 30 to 100. Look for the percent in the sentence first, since it is easily identifiable. Ask yourself what it is a percent of and look for the base following the word of. The is separated from the percent and base by some form of is.
Free Pre-Algebra Lesson 41! page 2 Example: Identify the percent, base, and, and write the percent sentence as a ratio. 65% of all televisions sold were HD TVs. base = % HD TVs all TVs sold = 65% The entire square represents all the TVs sold (the base). The shaded part is 65% of the base, representing the part of all the TVs that were HD TVs. The sentence may have the first or last. Example: Identify the percent, base, and, and write the percent sentence as a ratio. That piece is 75% of the pizza. = percent base base = % size of piece size of pizza = 75% 75% is 75/100 = 3/4 The piece of pizza is 3/4 of the whole pizza. 75% of the whole square is shaded. The shaded part is the. The whole square is the base. Example: Identify the percent, base, and, and write the percent sentence as a ratio. Her donation is 40% of the cost of the equipment. = percent base her donation cost of equipment = 40% The donation is being considered as part of the cost of equipment. Her donation is the shaded part of the entire square. The square represents the full cost of the mysterious equipment. Everyday sentences in English are not always structured so precisely. But when dealing with percent, always ask yourself percent of what? to find the base.
Free Pre-Algebra Lesson 41! page 3 Finding the Percent Percent is a ratio of the to the base. Once you have determined which quantities are the and base, you can find their ratio as a decimal, and then convert to a percent. In most cases, the is some part of the base. base = percent Example: Find the ratio of distance education students to all students in 2008, and round to the nearest hundredth. Write the ratio as a percent. 9#,.!:0(-*!&-1033;#-*!)-!<)$*,-"#!&.(",* =,33!4#1;$!7AAB!*0!7AAC =,33!4#1;$ <)$*,-"# &.(",*)0- :033#># &-1033;#-* 7AAE 6'EG7 77'6EH 7AA8 6'GG6 76'667 7AAF 7'BAI 7A'HA8 7AAG E'BG7 7A'66H 7AAH 8'BF6 7A'FGG 7AAI F'H6F 76'GIF distance ed students in 2008 all students in 2008 5,715 21,685! 0.26 0.26 = 26% = 5,715 21,685 Distance ed students were about 26 % of all students in 2008. Notice that the sentence that states the answer to the above example as a percent is in the form showing the and base: Distance ed students were 26% of all students. = percent base If the whole square represents all the students at the college, the 26% participating in distance education courses are the shaded part. Example: The shipment contained 2,000 microprocessors, but 40 were defective. What percent of the microprocessors were defective? Only part of the shipment of microprocessors were defective, so it makes sense that the 40 defective microprocessors are the and the entire shipment of 2000 is the base. The phrasing of the question What percent of the microprocessors were defective? reveals the and base: What percent of the microprocessors were defective? To find the percent, we find the ratio of to base: defective microprocessors all microprocessors Re-write the decimal as a percent, and state the result in a sentence: 0.02 = 2% = 40 2000 = 0.02 2% of the microprocessors were defective.
Free Pre-Algebra Lesson 41! page 4 Finding the Amount or Base The percent equation, is the easiest way to find the or base when you know the percent. Example: The factory had grown to expect that 2% of the microprocessors they received would be defective. If the current shipment contains 850 microprocessors, how many are expected to be defective? Step 1 Determine which quantity is the and which the base, in words. You know the percent is 2%. Ask yourself 2% of what? 2% of the microprocessors they received. 2% of the microprocessors they received would be defective Step 2 Using the formula, fill in the values you know. The percent must be written as a decimal. Use a variable for the value you don t know. 2% of the microprocessors they received would be defective 0.02 850 = a Step 3 Solve the equation you ve written. a = 0.02 850 = 17 Step 4 Interpret the answer in the context of the problem and write it in a sentence, including units if applicable. They expected 17 microprocessors to be defective. Example: The 1215 signatures on the petition represented 45% of the students at the school. Find the number of students at the school. 1255 signatures on the petition represented 45% of the students at the school. = percent base 1215 = 0.45 b The base follows the percent, signified by the word of. Part of the students signed the petition, so that is the. 1215 = 0.45 b 1215 / 0.45 = 0.45 b / 0.45 2700 = b There were 2,700 students at the school. Identifying the base of a percent is the key to writing and solving percent equations.!
Free Pre-Algebra Lesson 41! page 5 Lesson 41: Solving Percent Equations Worksheet Name 1. Fill in the sentence chart to identify the percent, base and. Find the ratio of the to the base with a decimal. Round to the nearest hundredth and write it as a percent. base = percent a. There were 212 participants at the conference. Afterwards, 84 participants rated the food unsatisfactory. What percent of the participants rated the food unsatisfactory? What percent of the participants (were the ones who) rated the food unsatisfactory? b. The wedding cost $5,000 and Shani contributed $1,500. What percent of the cost was Shani s contribution? What percent of the cost (of the wedding) was Shani s contribution? 2. Find the missing value for the or base using the percent equation. a. 25% of the area of the back yard was devoted to a vegetable garden. The area of the vegetable garden was 288 square feet. What is the area of the back yard? 25% of the area of the back yard was devoted to a vegetable garden.
Free Pre-Algebra Lesson 41! page 6 2. (continued) Find the missing value for the or base using the percent equation. b. Albert s income was $3,588 per month, but he spent 45% of his income on rent. How much was Albert s rent? 45% of his income was spent on rent 3. Fill in the structure sentence or the values you know to identify the percent, base, and, then solve as before. a. The team won 78 of the 96 games that season. What percent of the games had the team won? b. One of the scientists had identified 18 new species of beetle. This was 60% of the new species identified on the expedition. How many new species were identified on the expedition? c. 6% of the 21,400 students missed their registration date. How many students missed their registration dates?
Free Pre-Algebra Lesson 41! page 7 Lesson 41: Solving Percent Equations Homework 41A 1. The bullet traveled a distance of 1653 feet in 1.5 seconds. What was the speed of the bullet? Name 2. Superman traveled at a rate of 1,800 feet per second. If he flew 162,000 feet, how long did it take? 3. There were two sizes available, the large with 30 rolls, priced at $44.99, or the medium size with 12 rolls, on sale for $16.20. Find the price per roll for each size package. 4. The density of gold is 19.3 grams per cubic centimeter. If a gold object weighs 39 grams, what is its volume? 5. The medication instructions say to give 50 mg per kg of your cat s weight. If your cat weighs 4.3 kg, how many milligrams of medication should you give? 6. The legend on the map shows 1 inch on the map is equivalent to 250 miles in the real world. If two islands are 2 3 / 8 inches apart on the map, what is the real distance between them, to the nearest mile? 7. If the two triangles are similar, what is the length of side b of the smaller triangle? Large triangle: Side a = 6.72 meters, side b = 5.6 meters Small triangle: Side a = 2.4 meters 8. The shadow of a tower is 108 feet at the same time the shadow of a 3-foot yardstick is 1.9 feet. What is the height of the tower?
Free Pre-Algebra Lesson 41! page 8 9. Write the numbers in scientific notation. a. 87,000,000,000 10. Write the numbers in standard form. a. 8.91 x 10 7 b. 4.5 million b. 7.0 x 10 16 c. 7,780,000,000,000 c. 36.2 billion 11. Fill in the blanks: FRACTION DECIMAL PERCENT 1/8 0.5 75% 3% 0.22 7/5 12. Carlos earned 62 of the 70 points possible on the assignment. What percent of the points did Carlos get? Round to the nearest whole percent. 13. 80% of the crowd of 5000 wore the team colors. How many people wore the team colors?. 14. Stuart had 480 car-themed songs on his ipod, which was only 16% of all his songs. How many songs did he have on his ipod?
Free Pre-Algebra Lesson 41! page 9 Lesson 41: Solving Percent Equations Homework 41A Answers 1. The bullet traveled a distance of 1653 feet in 1.5 seconds. What was the speed of the bullet? 1653 feet rate = 1.5 seconds = 1102 feet per second 3. There were two sizes available, the large with 30 rolls, priced at $44.99, or the medium size with 12 rolls, on sale for $16.20. Find the price per roll for each size package. price roll price roll = $44.99! $1.50 per roll 30 rolls = $16.20! $1.35 per roll 12 rolls 5. The medication instructions say to give 50 mg per kg of your cat s weight. If your cat weighs 4.3 kg, how many milligrams of medication should you give? 50 mg 1 kg = x mg 4.3 kg 50 4.3 = x x = 215 You should give your cat 215 mg. 7. If the two triangles are similar, what is the length of side b of the smaller triangle? Large triangle: Side a = 6.72 meters, side b = 5.6 meters Small triangle: Side a = 2.4 meters a b = 6.72 5.6 = 2.4 b 6.72b = 13.44 b = 2 Side b is 2 meters long. 2. Superman traveled at a rate of 1,800 feet per second. If he flew 162,000 feet, how long did it take? d = rt 162,000 = 1800t 162,000 / 1800 = 1800t / 1800 t = 90 It took him 90 seconds. 4. The density of gold is 19.3 grams per cubic centimeter. If a gold object weighs 39 grams, what is its volume? 39 grams = 19.3 g x cm 3 1 cm 3 19.3x = 39 19.3x / 19.3 = 39 / 19.3 x = 2.020725389 The volume is about 2 cm 3 6. The legend on the map shows 1 inch on the map is equivalent to 250 miles in the real world. If two islands are 2 3 / 8 inches apart on the map, what is the real distance between them, to the nearest mile? 3 2 1 inch 250 miles = 8 inches x miles 250 2.375 = x x = 593.75 The islands are about 594 miles apart. 8. The shadow of a tower is 108 feet at the same time the shadow of a 3-foot yardstick is 1.9 feet. What is the height of the tower? height h shadow 108 = 3 1.9 1.9h = 324 1.9h / 1.9 = 324 / 1.9 h = 170.5263158... The height of the tower is about 170.5 feet.
Free Pre-Algebra Lesson 41! page 10 9. Write the numbers in scientific notation. a. 87,000,000,000 8.7 x 10 10 b. 4.5 million 4,500,000 = 4.5 x 10 6 c. 7,780,000,000,000 7.78 x 10 12 10. Write the numbers in standard form. a. 8.91 x 10 7 89,100,000 b. 7.0 x 10 16 70,000,000,000,000,000 c. 36.2 billion 36,200,000,000 11. Fill in the blanks: FRACTION DECIMAL PERCENT 1/8 0.125 12.5% 1/2 0.5 50% 3/4 0.75 75% 3/100 0.03 3% 11/50 0.22 22% 7/5 1.4 140% 12. Carlos earned 62 of the 70 points possible on the assignment. What percent of the points did Carlos get? Round to the nearest whole percent. base Carlo's points = points possible 62 70 =.885714... Carlos got 89% of the possible points. 13. 80% of the crowd of 5000 wore the team colors. How many people wore the team colors? 0.80 5000 = 4000 4000 people wore the team colors. 14. Stuart had 480 car-themed songs on his ipod, which was only 16% of all his songs. How many songs did he have on his ipod? 0.16b = 480 0.16b / 0.16 = 480 / 0.16 b = 3000 He had 3000 songs on his ipod.
Free Pre-Algebra Lesson 41! page 11 Lesson 41: Solving Percent Equations Homework 41B 1. The bullet traveled a distance of 3000 feet in 2.5 seconds. What was the speed of the bullet? Name 2. Superman traveled at a rate of 1,800 feet per second. If he flew 99,000 feet, how long did it take? 3. There were two sizes available. One package held 17 ounces and cost $3.57. The other held 22 oz and cost $4.51. Find the price per ounce for each package. 4. The density of gold is 19.3 grams per cubic centimeter. If a gold object weighs 965 grams, what is its volume? 5. The medication instructions say to give 50 mg per kg of your cat s weight. If your cat weighs 3.7 kg, how many milligrams of medication should you give? 6. The legend on the map shows 1 inch on the map is equivalent to 150 miles in the real world. If two islands are 4 3 / 4 inches apart on the map, what is the real distance between them, to the nearest mile? 7. If the two triangles are similar, what is the length of side b of the smaller triangle? Large triangle: Side a = 7.5 meters, side b = 13.5 meters Small triangle: Side a = 2.0 meters 8. The shadow of a tower is 58 feet at the same time the shadow of a 3-foot yardstick is 0.8 feet. What is the height of the tower?
Free Pre-Algebra Lesson 41! page 12 9. Write the numbers in scientific notation. a. 96,000,000 10. Write the numbers in standard form. a. 8.91 x 10 11 b. 14.5 million b. 1.2 x 10 5 c. 1,028,000,000,000,000 c. 9.4 trillion 11. Fill in the blanks: FRACTION DECIMAL PERCENT 1/4 0.6 13% 6% 0.35 5/4 12. Debbie earned 51 of the 60 points possible on the assignment. What percent of the points did Debbie get? Round to the nearest whole percent. 13. 90% of the crowd of 6000 wore the team colors. How many people wore the team colors? 14. Kasey had 630 car-themed songs on his ipod, which was 21% of all his songs. How many songs did he have on his ipod?