GED Math: Ratio, Proportion, and Percent
Percent is often used to show change. The amount of change is the difference between the new number and the original number. The percent of change compares that amount to the original number (the number before the change): percent of change = (amount of change ÷ original) × 100%. If the new number is larger it is a percent of increase, and if it is smaller it is a percent of decrease. An increase can be greater than 100%. For example, a number that doubles has increased by 100% and is also 200% of the original. Think carefully about which number came first, because dividing by the wrong number is the most common error. You can work backward to check: after a 20% decrease the new number should be 80% of the original.
1. Ratios: Writing and Simplifying
A ratio compares two numbers. You can write it with the word to (5 to 3), with a colon (5:3), or as a fraction (5/3). Like a fraction, a ratio is simplified by dividing both terms by their greatest common factor. Ratios are not exactly the same as fractions, though: write the terms in the same order as the words in the question (“made to missed” puts made first), do not turn an improper ratio into a mixed number, and if the second term is 1, leave it as a fraction such as 4/1 instead of writing 4. The terms of a ratio can also have different labels, such as 6 cans for $1, a 6:1 ratio. Sometimes you must do a quick calculation to find one of the terms, for example adding wins and losses to get games played.
Worked Example: A player makes 15 free throws and misses 9. Find (a) the ratio of made to missed and (b) the ratio of made to attempted.
(a) Made to missed is 15 to 9. Dividing both terms by 3 gives 5 to 3.
(b) The number attempted is not given, so find it: 15 + 9 = 24. Made to attempted is 15 to 24, which reduces to 5 to 8.
1. A printer prints 90 pages in 6 minutes. What is the ratio of pages to minutes in lowest terms?
Explanation: The order is pages first, then minutes: 90 to 6. Divide both terms by 6 to get 15 to 1, which you can leave as 15/1. (1 to 15 reverses the order.)
2. A math class has 12 men and 18 women. What is the ratio of women to the total number of students?
Explanation: The total is 12 + 18 = 30. Women to total is 18 to 30, which reduces to 3 to 5.
3. A team played 40 games and won 28 of them, with no ties. What is the ratio of wins to losses?
Explanation: Losses = 40 − 28 = 12. Wins to losses is 28:12, which reduces to 7:3.
4. In a survey of 500 voters, 200 chose A, 175 chose B, and 125 chose C. What is the ratio of voters who chose B to the total surveyed?
Explanation: B to total is 175:500. Divide both terms by 25 to get 7:20. (7:13 compares B with the voters who did not choose B.)
5. Soan made a $600 down payment on an item that costs $2400. What is the ratio of the amount he has paid to the amount he still owes?
Explanation: He still owes 2400 − 600 = $1800. Paid to owed is 600 to 1800, which reduces to 1 to 3.
2. Proportions and Rates
A proportion is an equation that says two ratios are equal. In a true proportion the cross products are equal: multiply diagonally across the equal sign and the two products match. To solve for a missing term, first write both ratios in the same form (for example inches over miles on both sides). Then find the known term that is diagonal from the unknown x, multiply the two known numbers that are diagonal to each other, and divide by the remaining known number. A rate compares a quantity to 1 of something, such as miles per gallon, and in fraction form its denominator is 1. You can solve proportions in one series of calculator keys: multiply the diagonal pair, then divide.
Worked Example: (a) A map scale says 4 inches = 120 miles. How many miles do 7 inches represent? (b) A car drives 312 miles on 13 gallons. What is its mileage in miles per gallon?
(a) Keep inches on top and miles on the bottom: 4/120 = 7/x. Cross multiply 120 × 7 = 840 and divide by 4: x = 210 miles.
(b) A rate compares miles to 1 gallon: 312/13 = x/1, so x = 312 ÷ 13 = 24 miles per gallon.
1. Solve for x: 23 = x18
Explanation: Cross multiply: 2 × 18 = 36, then divide by 3: x = 12.
2. Solve for x: 59 = 20x
Explanation: Cross multiply 9 × 20 = 180, then divide by 5: x = 36.
3. Three pens cost $1.35. At the same price, how much do 8 pens cost?
Explanation: 3/1.35 = 8/x, so x = 1.35 × 8 ÷ 3 = $3.60. ($0.45 is the price of one pen.)
4. A team wins 3 out of every 5 games it plays. At the same rate, how many games will it have to play to win 45 games?
Explanation: 3/5 = 45/x, so x = 5 × 45 ÷ 3 = 75 games.
5. A map scale reads 3 cm = 80 km. How many kilometers does a map distance of 7.5 cm represent?
Explanation: 3/80 = 7.5/x, so x = 80 × 7.5 ÷ 3 = 200 km.
6. A truck drives 180 miles in 4 hours. Which expression gives how many miles it can drive in 9 hours at the same rate?
Explanation: Set up miles/hours: 180/4 = x/9. Multiply the diagonal pair 180 × 9 and divide by 4. The answer is 405 miles.
7. Carla drove 378 miles on 14 gallons of gasoline. How many miles per gallon did she get?
Explanation: A rate compares to 1 gallon: 378 ÷ 14 = 27 miles per gallon.
3. What Percent Means: Base, Part, and Rate
Percent means “per hundred” or “out of 100.” To change a percent to a fraction, write it over 100 and reduce. To change a percent to a decimal, drop the % sign and move the decimal point two places to the left. Every percent problem has three elements: the base is the whole amount the problem is about, the part (or percentage) is a portion of the base, and the rate is the number with the % sign. The word of usually comes right before the base (“75% of 250” has base 250). Because the part is to the base as the rate is to 100, you can write a proportion: part/base = rate/100. A percent can be greater than 100%, and then the part is larger than the base.
Worked Example: About 3% of the 600 fixtures made at a plant each day are defective. How many fixtures are defective?
The base is 600, the rate is 3%, and the part x is unknown. Write part/base = rate/100: x/600 = 3/100.
Cross multiply: 600 × 3 = 1800, then divide by 100: x = 18 defective fixtures.
1. Change 8% to a decimal.
Explanation: Drop the % sign and move the decimal point two places left: 8% = 0.08.
2. Write 45% as a fraction in lowest terms.
Explanation: 45% = 45/100. Divide both terms by 5 to get 9/20.
3. Victor owes his uncle $1000 and recently paid him $200, which is 20% of the money he owes. What is the base?
Explanation: The base is the whole amount the problem is about: the $1000 he owes. $200 is the part and 20% is the rate.
4. About 4% of 750 fixtures are expected to be defective. Which proportion, using x for the defective fixtures, solves this problem?
Explanation: The part is to the base as the rate is to 100: x/750 = 4/100. Cross multiplying gives x = 30.
5. Which proportion finds what percent 18 is of 72?
Explanation: The part is 18 and the base is 72, so part/base = rate/100 gives 18/72 = x/100. The answer is 25%.
4. The Percent Formula: Solving for the Part
Besides proportions, you can use the percent formula: Base × Rate = Part. The triangle below helps you remember how to rearrange it. To find the part, change the rate to a decimal and multiply it by the base. This is the same as asking “what is 35% of 240?” Because the part is a piece of the base, the answer should be smaller than the base when the rate is less than 100% and larger than the base when the rate is more than 100%. Use this reasonableness check on every answer. Some problems have two steps, such as finding a raise and then adding it to the old amount.
Worked Example: A company has 240 workers and 35% chose Plan A. How many chose Plan A? Then, if a worker earning $2,250 a month gets a 6% raise, what is the new monthly pay?
Plan A: the base is 240 and the rate is 35% = 0.35, so the part is 240 × 0.35 = 84 workers.
Raise: 2250 × 0.06 = $135. The new pay is the old pay plus the raise: 2250 + 135 = $2,385.
1. What is 20% of $35?
Explanation: 20% = 0.20, and 35 × 0.20 = $7.
2. What is 125% of 48?
Explanation: 125% = 1.25, and 48 × 1.25 = 60. The rate is more than 100%, so the part is bigger than the base.
3. Find 33 1/3% of 51. (Hint: 33 1/3% = 1/3)
Explanation: One third of 51 is 51 ÷ 3 = 17.
4. Douglas gets a 6% raise. His old monthly salary was $2,250. What is his new monthly salary?
Explanation: The raise is 2250 × 0.06 = $135, so the new salary is 2250 + 135 = $2,385. ($135 is only the raise.)
5. Levy’s restaurant bill is $46. How much should he leave for a 15% tip?
Explanation: 46 × 0.15 = $6.90.
6. Find 12 1/2% of $64. (Hint: 12 1/2% = 1/8)
Explanation: 12.5% = 1/8, and 64 ÷ 8 = $8.
5. Solving for the Rate
When the base and the part are known and you need the percent, rearrange the formula to Rate = Part ÷ Base. Divide the part by the base, then convert the decimal answer to a percent by moving the decimal point two places to the right and adding the % sign. Be careful to put the right number on the bottom: the base is the whole, or the original amount, and the part is the piece that is compared with it. For a discount, the reduction is the part and the regular price is the base. If the part is larger than the base, the rate is over 100%.
Worked Example: A jacket regularly costs $80 and is reduced by $28. What is the percent of the discount?
The base is the regular price $80 and the part is the $28 reduction. Divide: 28 ÷ 80 = 0.35. Move the decimal point two places right: 35%.
Check for reasonableness: 35% is a little more than 1/3, and 1/3 of $80 is about $27, close to $28.
1. What percent is 19 out of 20?
Explanation: 19 ÷ 20 = 0.95 = 95%.
2. 42 is what percent of 168?
Explanation: 42 ÷ 168 = 0.25 = 25%.
3. Pat called 150 customers about a software upgrade, and 90 bought it. What percent bought the upgrade?
Explanation: The base is the 150 customers called. 90 ÷ 150 = 0.6 = 60%.
4. A package is on sale for $45.50 and was originally $65. What percent of the original price is the sale price?
Explanation: The base is the original price: 45.50 ÷ 65 = 0.70 = 70%. (The discount itself is 30%.)
5. $15.60 is what percent of $156.00?
Explanation: 15.60 ÷ 156 = 0.10 = 10%.
6. 240 is what percent of 120?
Explanation: 240 ÷ 120 = 2.0 = 200%. The part is larger than the base, so the rate is over 100%.
6. Solving for the Base
When you know the part and the rate and need the whole amount, rearrange the formula to Base = Part ÷ Rate. Convert the rate to a decimal first, then divide the part by it. Read the situation carefully to find which of the three elements is missing, and do not begin calculating until you do. The words of and is help: the number after “of” is usually the base, and the number next to “is” is usually the part. If the rate is over 100%, the base is smaller than the part. You can also solve base problems with the proportion part/base = rate/100, and both methods give the same answer.
Worked Example: In a class, 18 students earned at least a B. That is 60% of the class. How many students are in the class?
The part is 18 and the rate is 60% = 0.60, so divide: 18 ÷ 0.60 = 30 students.
Proportion check: 18/x = 60/100, and 18 × 100 ÷ 60 = 30. Both methods agree.
Rate over 100%: a company has 180 employees this year, which is 240% of last year. Base = 180 ÷ 2.40 = 75 employees last year.
1. $35 is 20% of what amount?
Explanation: 35 ÷ 0.20 = $175. Check: 20% of 175 is 35.
2. 5% of a number is 14. What is the number?
Explanation: 14 ÷ 0.05 = 280.
3. $105 is 125% of what amount?
Explanation: 105 ÷ 1.25 = $84. The rate is over 100%, so the base is less than the part.
4. 9 is 1% of what number?
Explanation: 9 ÷ 0.01 = 900.
5. Jack earns a 5% commission on each sale and was paid $160. What was the amount of the sale?
Explanation: The part is the $160 commission and the rate is 5%. Base = 160 ÷ 0.05 = $3,200.
6. Kevin’s payroll deductions are 30% of his earnings. If his deductions are $369 for a two-week period, how much were his earnings?
Explanation: 369 ÷ 0.30 = $1,230. Check: 30% of 1,230 is 369.
7. Percent Calculator Skills
There are two ways to solve percent problems on the TI-30XS MultiView calculator. First, you can change the percent to a decimal (or a fraction with the fraction key) and multiply or divide as usual. Second, you can use the percent 2nd function, found in green above the left parenthesis key: type the base, press the multiplication sign, type the rate, press 2nd and then the percent key, and press enter. To find a base, type the part, press the division sign, type the rate, and use the percent function. If you make a mistake, the delete key erases the latest entry and the clear key erases the whole entry.
| To find | Keys to press | Display |
|---|---|---|
| 46% of $130 | 130 [×] 46 [2nd] [%] [enter] | 59.8 |
| Base: $54 is 15% of what? | 54 [÷] 15 [2nd] [%] [enter] | 360 |
| 25% of 120 (decimal) | 0.25 [×] 120 [enter] | 30 |
| 25% of 120 (fraction) | 25/100 [×] 120 [enter] | 30 |
Worked Example: (a) What is 18% of $450? (b) $54 is 15% of what amount?
(a) Press 450 × 18 2nd % enter. The display reads 81, so the answer is $81. Check with a decimal: 450 × 0.18 = 81.
(b) Press 54 ÷ 15 2nd % enter. The display reads 360, so the amount is $360.
1. Use a calculator to find 46% of $130.
Explanation: 130 × 0.46 = 59.8, or $59.80. ($282.61 results from dividing by 0.46 instead of multiplying.)
2. 14% of what amount is $13.44?
Explanation: The part is 13.44 and the rate is 14%, so base = 13.44 ÷ 0.14 = $96.
3. 12 is what percent of 400?
Explanation: 12 ÷ 400 = 0.03 = 3%.
4. A shop uses this shipping chart. Chanel places an $84 order. How much shipping and handling is she charged?
| Order total | Shipping and handling |
|---|---|
| $20 or less | 3% + $1.50 |
| $20.01 to $50 | 4% + $2.50 |
| $50.01 to $100 | 5% + $4.00 |
| $100.01 or more | 8% |
Explanation: $84 is in the $50.01 to $100 range, so the charge is 5% of 84 plus $4.00: 4.20 + 4.00 = $8.20.
5. In an election, 2,460 of 3,000 registered voters voted. What percent of registered voters voted?
Explanation: 2460 ÷ 3000 = 0.82 = 82%.
6. 90% of what number is 63?
Explanation: 63 ÷ 0.90 = 70.
8. Simple Interest
Interest is a fee paid for using someone else’s money. The amount invested or borrowed is the principal. Simple interest is found with the formula I = prt, which says Interest = principal × rate × time. Letters written side by side are multiplied. The formula is on the GED formula sheet, so you do not need to memorize it, but you do need to use it correctly. The rate is a yearly rate, so write the rate as a decimal and the time in years. A number of months becomes a fraction of a year by putting it over 12, so 8 months = 8/12 = 2/3 year. To find the amount paid back or the total in the account, add the principal and the interest.
Worked Example: (a) Find the interest on $4000 borrowed for 2 years at 5% simple interest and the total paid back. (b) How much interest does $1500 earn in 8 months at 9%?
(a) Rate: 5% = 0.05. I = prt = 4000 × 0.05 × 2 = $400. Amount paid back = 4000 + 400 = $4,400.
(b) Rate: 9% = 0.09. Time: 8/12 = 2/3 year. I = 1500 × 0.09 × 2/3 = $90.
1. Leah borrows $1500 for 2 years at a 12% simple interest rate. How much interest does she pay?
Explanation: I = prt = 1500 × 0.12 × 2 = $360. ($1,860 would be the total paid back.)
2. How much interest would you pay on a loan of $800 for 6 months at 14% simple interest?
Explanation: 6 months = 1/2 year. I = 800 × 0.14 × 1/2 = $56.
3. How much interest is earned on an investment of $600 for 8 years at 10% simple interest?
Explanation: I = 600 × 0.10 × 8 = $480.
4. Todd invests $6500 for 3 1/2 years at 5% simple interest. How much interest will he be paid?
Explanation: I = 6500 × 0.05 × 3.5 = $1,137.50. ($7,637.50 is the total in the account.)
5. Ricardo borrows $1850 for 8 months at 12% simple interest. What total amount will he pay back?
Explanation: Time = 8/12 = 2/3 year. I = 1850 × 0.12 × 2/3 = $148. Total = 1850 + 148 = $1,998.
6. Jean borrows $1300 for 1 year 6 months at 9% simple interest. Which expression gives the amount she will pay back?
Explanation: 1 year 6 months = 1.5 years and 9% = 0.09. Interest is $1300 × 0.09 × 1.5, and the amount paid back is the principal plus that interest.
7. Charlotte needs to borrow $3000. Option A is 2 1/2 years at 12% simple interest and Option C is 4 years at 9%. How much more interest does she pay with Option C?
Explanation: Option A: 3000 × 0.12 × 2.5 = $900. Option C: 3000 × 0.09 × 4 = $1,080. The difference is 1080 − 900 = $180.
9. Percent of Change: Increase and Decrease
Percent is often used to show change. The amount of change is the difference between the new number and the original number. The percent of change compares that amount to the original number (the number before the change): percent of change = (amount of change ÷ original) × 100%. If the new number is larger it is a percent of increase, and if it is smaller it is a percent of decrease. An increase can be greater than 100%. For example, a number that doubles has increased by 100% and is also 200% of the original. Think carefully about which number came first, because dividing by the wrong number is the most common error. You can work backward to check: after a 20% decrease the new number should be 80% of the original.
Worked Example: (a) A wage goes from $12.50 to $13.50 per hour. What is the percent of increase? (b) A coat drops from $150 to $120. What is the percent of decrease? (c) A business grows from 8 employees to 40. What is the percent of increase?
(a) Change = 13.50 − 12.50 = $1.00. Divide by the original: 1.00 ÷ 12.50 = 0.08 = 8% increase.
(b) Change = 150 − 120 = $30. 30 ÷ 150 = 0.20 = 20% decrease. Check: 80% of 150 is 120.
(c) Change = 40 − 8 = 32. 32 ÷ 8 = 4.0 = 400% increase.
1. Find the percent of increase from 2000 to 3000.
Explanation: Change = 1000. 1000 ÷ 2000 (the original) = 0.5 = 50%.
2. What is the percent of decrease from 500 to 340?
Explanation: Change = 160. 160 ÷ 500 = 0.32 = 32%. (68% is what the new number is as a percent of the original.)
3. What is the percent of increase from 30 to 90?
Explanation: Change = 60. 60 ÷ 30 = 2.0 = 200%. (The new number, 90, is 300% of the original.)
4. A worker’s weekly pay goes from $250 to $400. By what percent did his pay increase?
Explanation: Change = 150. 150 ÷ 250 = 0.60 = 60%. (37.5% divides by the new pay.)
5. A game was $75 and is now on sale for $60. What is the percent of decrease?
Explanation: Change = 15. 15 ÷ 75 (the original) = 0.20 = 20%.
6. Rent rises from $750 to $795 per month. By what percent did it increase?
Explanation: Change = 45. 45 ÷ 750 = 0.06 = 6%.
7. A printer has a wholesale price of $80 and a retail price of $164. What is the percent of increase from wholesale to retail?
Explanation: Change = 164 − 80 = 84. 84 ÷ 80 = 1.05 = 105%.
8. Find the percent of increase from 65 to 338.
Explanation: Change = 338 − 65 = 273. 273 ÷ 65 = 4.2 = 420%.
Common Mistakes to Avoid
- Writing the terms of a ratio in the wrong order. “Women to total” puts women first; an answer choice with the numbers reversed is a common trap.
- Using a part-to-part ratio when the question asks for part-to-whole. Add the parts to find the total first when the total is not given.
- Setting up a proportion with the labels in different positions (inches over miles on one side and miles over inches on the other).
- Forgetting to change the percent to a decimal before multiplying or dividing. 25% is 0.25, not 25.
- Mixing up the base and the part. The base is the whole (usually right after the word “of”), and the part is the piece being compared.
- Dividing by the wrong number in a percent of change problem. Always divide by the original amount, not the new one.
- Entering the time in months or the rate as a whole number in I = prt. Convert months to a fraction of a year and the rate to a decimal.
- Stopping at the interest when the question asks for the total amount paid back (principal plus interest), or stopping at the raise when it asks for the new salary.
- Not checking whether the answer is reasonable. A part of a base at less than 100% must be smaller than the base.
Frequently Asked Questions
What is the difference between a ratio and a proportion?
A ratio compares two numbers, like 3 to 5. A proportion is an equation that says two ratios are equal, like 3/5 = 6/10. You use proportions to find a missing term.
How do I solve a proportion with a missing number?
Write both ratios with matching labels, cross multiply the two known numbers that are diagonal from each other, and divide by the remaining known number. For 2/3 = x/18, multiply 2 × 18 = 36 and divide by 3 to get x = 12.
How do I find the base, part, and rate in a percent problem?
The base is the whole amount and often follows the word “of.” The rate has the % sign. The part is the piece of the base. Whichever one is missing tells you which formula to use: part = base × rate, rate = part ÷ base, or base = part ÷ rate.
Can a percent be greater than 100%?
Yes. When the part is larger than the base, the rate is over 100%. For example, 135 is 225% of 60, and a number that triples has increased by 200%.
How do I find a percent increase or decrease?
Subtract to find the amount of change, divide by the original number, and change the decimal to a percent. If the new number is smaller it is a decrease; if it is larger it is an increase.
What is the simple interest formula and is it on the GED formula sheet?
Interest = principal × rate × time, or I = prt. It is on the on-screen formula sheet, but remember that the rate must be a decimal and the time must be in years.
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.