GED Mathematics · Chapter 3 · pp. 117–165
Ratio, Proportion, and Percent
Ratios, proportions and percents all compare one amount with another. They turn up all over the GED® Mathematical Reasoning test: in sale prices, recipes, maps, pay raises and loans. This chapter shows you how to set each one up, solve it by hand and on the TI-30XS MultiView™, and check that the answer makes sense.
Learning objectives
By the end of this chapter you can:
- Write a ratio three ways, simplify it, and keep its terms in the right order.
- Solve a proportion by cross multiplying, and find a rate such as miles per gallon.
- Change percents to fractions and decimals, and name the base, part and rate in a problem.
- Use the percent formula, part = base × rate, to find any one of the three.
- Solve percent problems on the TI-30XS with the % key.
- Find simple interest with I = prt, and the percent of increase or decrease.
In this chapter: Lesson 1 Ratio and proportion · Lesson 2 Percents · Lesson 3 Using the percent formula · Lesson 4 Percent calculator skills · Lesson 5 Simple interest · Lesson 6 Percent of change
🎯 About the test
Percent questions are some of the most common on the test, and the wrong answer choices are usually built from the most common mistakes: terms in the wrong order, multiplying instead of dividing, or dividing by the wrong number. The simple interest formula is on the formula sheet you get on the test; the others in this chapter you need to know yourself.
✏️ How to use this page
Work through the lessons in order. Each one has worked examples and a short Try it. Lessons 3, 5 and 6 have a calculator that shows its working, so you can check your own. The book's practice sets are in this chapter's printable assignment, and there is a quiz at the end.
Lesson 1 · Ratio and proportion
🔑 Key ideas
- A ratio has two terms. Write it with the word to, with a colon, or as a fraction.
- Simplify a ratio by reducing it to lowest terms, just like a fraction.
- The two terms of a ratio can have different labels, such as miles and gallons.
- A proportion says two ratios are equal. Solve it by cross multiplying.
Writing a ratio
A ratio compares two numbers. A softball pitcher who strikes out 4 batters for every 1 she walks has a ratio of strikeouts to walks of:
4 to 1 · 4 : 1 ·
All three mean the same thing. The order matters: write the terms in the same order as the question. The ratio of walks to strikeouts is 1 : 4, not 4 : 1.
📘 Example 1 · Credit card sales to cash sales
In two hours, Frank's drugstore makes 25 cash sales and 15 credit card sales. What is the ratio of credit card sales to cash sales?
- Write the terms in the order asked: credit card first, then cash.
- Reduce: both divide by 5. =
Answer: 3 to 5
Ratios follow most fraction rules, with two exceptions:
- Don't change a ratio like into a mixed number. It stays 8 : 3.
- Don't turn into the whole number 4. A ratio always keeps both terms.
The terms of a fraction share one label: of a pie is 5 slices out of 6 slices. A ratio's terms can have different labels. “6 cans for $1” is a 6 : 1 ratio of cans to dollars.
📘 Example 2 · Find a missing term first
A football team won 12 games and lost 8, with no ties. What is the ratio of games won to games played?
- The question doesn't give the games played. Add: 12 won + 8 lost = 20 played.
- Won to played: = .
Answer: 3 : 5. The team won 3 games for every 5 it played.
💡 Test writers like to put the reversed ratio (here 5 : 3) among the choices. Check the order before you pick.
Proportion
A proportion is an equation that says two ratios are equal. In a true proportion the cross products are equal: multiply diagonally across the equals sign and you get the same number both ways.
📘 Example 3 · Is the drink mixed correctly?
A drink mix says 3 cups of water for every 2 scoops. Matt uses 12 cups of water and 8 scoops. Is that right?
- Write both ratios in the same order, cups to scoops: and .
- Cross multiply: 3 × 8 = 24 and 2 × 12 = 24.
Answer: the cross products are equal, so the ratios are equal. Matt made it correctly.
Most proportion problems ask for a missing term. Call it x.
- Write a proportion with both ratios in the same form (for example inches to miles, both times).
- Multiply the two numbers that sit diagonally across from each other.
- Divide by the number that is left.
📘 Example 4 · A map scale
A map scale says 2 inches = 150 miles. What distance does 5 inches on the map show?
- Inches to miles, both sides: =
- The 150 and the 5 are diagonal: 150 × 5 = 750.
- Divide by the number left over, 2: 750 ÷ 2 = 375 miles.
A rate compares an amount to 1. In fraction form its bottom number is always 1, and in words it often uses per: miles per gallon, dollars per hour.
📘 Example 5 · Miles per gallon
Connie drove 276 miles on 12 gallons. How many miles per gallon did she get?
= , so x = 276 × 1 ÷ 12 = 23 miles per gallon.
📘 Example 6 · All in one go on the calculator
Find x in = . You need 16 × 21, then ÷ 6.
clear 16 × 21 ÷ 6 enter → the display shows 56.
💡 Think “proportion” whenever a question compares two quantities, or gives you three numbers and asks for a fourth.
✏️ Try it
- Stan made 24 sales in 6 hours. Write the ratio of sales to hours as a fraction in lowest terms.
- A basketball player tried 32 free throws and made 20. What is the ratio of throws made to throws missed?
- Solve: =
- Soup is 4 cans for $0.98. To the nearest cent, what do 5 cans cost?
Answers
1. = · 2. 32 − 20 = 12 missed, so 20 : 12 = 5 : 3 · 3. 2 × 18 ÷ 3 = 12 · 4. 0.98 × 5 ÷ 4 = 1.225 → $1.23
Lesson 2 · Percents
🔑 Key ideas
- The percent sign (%) means “out of 100”.
- Every percent problem has three parts: the base, the part and the rate.
- You can solve a percent problem with a proportion whose second ratio is over 100.
Percent means “per hundred”. If you have $100 and spend $25, you have spent 25 out of 100, or 25%, of your money.
Percents, fractions and decimals
- Percent to fraction: write the percent over 100 and reduce. 25% = =
- Percent to decimal: drop the % sign and move the decimal point two places left. 25% = 0.25, 8% = 0.08, 150% = 1.5
| Percent | Fraction | Decimal |
|---|---|---|
| 10% | 0.1 | |
| 20% | 0.2 | |
| 25% | 0.25 | |
| 50% | 0.5 | |
| 75% | 0.75 | |
| 100% | (the whole) | 1 |
Base, part and rate
- The base is the whole amount the problem is about.
- The part (also called the percentage) is a piece of the base.
- The rate is the number with the % sign.
📘 Example 1 · A restaurant tip
Janice's bill is $20. She leaves a $3 tip, which is 15% of the bill. Name the base, part and rate.
The whole bill, $20, is the base. The $3 tip is the part. The rate is 15%.
💡 The word of often comes just before the base: in “75% of 250”, 250 is the base.
Percent as a proportion
The part is to the base as the rate is to 100:
=
In Example 1, = . Check by cross multiplying: 3 × 100 = 300 and 20 × 15 = 300.
📘 Example 2 · Defective fixtures
A factory expects about 2% of its lighting fixtures to have a defect. If it makes 900 in a day, about how many will be defective?
- 900 is the base and 2% is the rate. The part is unknown: =
- Cross multiply and divide: 900 × 2 = 1800, then 1800 ÷ 100 = 18.
Answer: about 18 defective fixtures.
✏️ Try it
- Victor owes his uncle $1000. He pays back $200, which is 20% of what he owes. Name the base, part and rate.
- Write 35% as a fraction in lowest terms and as a decimal.
- A $128 jacket is 25% off. Use a proportion to find how much the price drops.
Answers
1. Base $1000, part $200, rate 20% · 2. = , and 0.35 · 3. = , so x = 128 × 25 ÷ 100 = $32
Lesson 3 · Using the percent formula
🔑 Key ideas
- The percent formula is part = base × rate.
- Percent to decimal: drop the % and move the decimal point two places left (70% = 0.7).
- Decimal to percent: move the decimal point two places right and add % (0.4 = 40%).
part = base × rate
Cover the letter you want to find. What's left tells you what to do:
| To find | Cover | Calculate |
|---|---|---|
| the part | P | base × rate |
| the rate (percent) | R | part ÷ base |
| the base (whole) | B | part ÷ rate |
💡 On the test, sketch this triangle on your wipe-off board before you start a percent problem.
Solving for the part
📘 Example 1 · 70% of 320 workers
70% of a company's 320 workers chose Plan A. How many chose Plan A?
- The rate is 70% and the base is 320. You want the part, so multiply.
- 70% = 0.7, and 320 × 0.7 = 224 workers.
Solving for the rate
📘 Example 2 · A $640 discount
A computer is usually $1600. The manager cuts the price by $640. By what percent was it discounted?
- The base is $1600 and the part is $640. You want the rate, so divide the part by the base.
- 640 ÷ 1600 = 0.4
- Move the decimal point two places right: 0.4 = 40%.
Does it make sense? 50% of $1600 is $800. $640 is a bit less than $800, and 40% is a bit less than 50%. ✔
Solving for the base
The base is the whole item or group. Read carefully to work out which of the three is missing before you multiply or divide.
📘 Example 3 · How big is the class?
75% of a class got at least a B. If 18 students got at least a B, how many are in the class?
- The 18 students are part of the class, and they are 75% of it. So 18 is the part, 75% the rate, and the base is missing.
- Cover B: divide the part by the rate. 75% = 0.75, and 18 ÷ 0.75 = 24 students.
A percent under 100% gives a part smaller than the base. A percent over 100% gives a part bigger than the base.
📘 Example 4 · A percent over 100%
Eastland Inc. has 225% as many employees this year as last year. It has 135 now. How many did it have last year?
- This year's number is a percent of last year's, so last year's is the base. The part is 135 and the rate is 225%.
- 225% = 2.25, and 135 ÷ 2.25 = 60 employees.
The proportion method gives the same answer: = , so x = 135 × 100 ÷ 225 = 60.
🧪 Try it: percent solver
Choose what to find, then type the other two.
✏️ Try it
- What is 35% of 400?
- 42 is what percent of 168?
- $35 is 20% of what amount?
- Douglas gets a 6% raise on a monthly salary of $2250. What is his new salary?
Answers
1. 400 × 0.35 = 140 · 2. 42 ÷ 168 = 0.25 = 25% · 3. 35 ÷ 0.2 = $175 · 4. 2250 × 0.06 = $135 raise, so $2250 + $135 = $2385
Lesson 4 · Percent calculator skills
🔑 Key ideas
- One way is to change the percent to a decimal or a fraction first, then multiply or divide.
- On the TI-30XS MultiView™, the % function finds the part or the base without any converting.
- % is the green 2nd function above the left bracket: press 2nd, let go, then press (.
Method 1 · Change the percent first
📘 Example 1 · What is 25% of 120?
| Change 25% to… | Keys | Display shows |
|---|---|---|
| the decimal 0.25 | clear .25 × 120 enter | 30 |
| the fraction | clear n/d 25 ↓ 100 → × 120 enter | 30 |
Either way, 25% of 120 is 30. The n/d key builds a fraction: type the top, press the down arrow, type the bottom, then press the right arrow to leave the fraction.
Method 2 · The % key
Type the percent as an ordinary number and press 2nd ( after it. The display shows it with a % sign, and the calculator treats it as a percent for you.
| Task | Keys | Display shows |
|---|---|---|
| Find the part: what is 65% of $360? | clear 360 × 65 2nd ( enter | 360*65% → 234 |
| Find the base: $150 is 20% of the original price. What was it? | clear 150 ÷ 20 2nd ( enter | 150÷20% → 750 |
| Find the rate: 60 is what percent of 240? | clear 60 ÷ 240 enter | 0.25, which is 25% |
- To find the part: base × percent 2nd (. So $234 is 65% of $360.
- To find the base: part ÷ percent 2nd (. Ned's stereo first cost $750.
- To find the rate, just divide the part by the base and move the decimal point two places right. (Chapter 1 showed a shortcut: 2nd ) before enter shows the answer as a percent.)
💡 Made a mistake? delete erases your last entry; clear erases the whole line.
Choosing the method
Easy percents such as 10%, 25% and 50% are often quicker in your head or as a fraction: 25% of 120 is a quarter of 120. Awkward ones such as 14% or 8.5% are where the % key saves time. Either way, the hard part is deciding whether you need the part, the base or the rate. The calculator can't do that for you.
📘 Example 2 · A two-step percent
Shipping on an $84 order is 5% of the order plus $4.00. What is the shipping charge?
- 5% of $84: clear 84 × 5 2nd ( enter → 4.2
- Add the $4.00 handling: $4.20 + $4.00 = $8.20.
✏️ Try it on your calculator
- Find 46% of $130.
- 14% of what amount is $13.44?
- 36 is what percent of 144?
- In an election, 3190 of 3625 registered voters voted against a tax increase. What percent is that?
Answers
1. 130 × 46% = $59.80 · 2. 13.44 ÷ 14% = $96 · 3. 36 ÷ 144 = 0.25 = 25% · 4. 3190 ÷ 3625 = 0.88 = 88%
Lesson 5 · Simple interest
🔑 Key ideas
- Simple interest = principal × rate × time: I = prt.
- Time must be in years. Write months as a fraction of a year (months over 12).
- The amount paid back is the principal plus the interest.
Interest is a fee for using someone else's money. Put money in a savings account and the bank pays you interest; borrow money and you pay it. Either way, the amount you invest or borrow is the principal.
I = p × r × t
I is the interest, p the principal, r the rate per year (as a decimal) and t the time in years. Letters written side by side, prt, are multiplied. This formula is on the GED® formula sheet, near the bottom.
📘 Example 1 · Interest on a loan
Asher borrows $2500 from his uncle for 3 years at 6% simple interest. How much interest will he pay?
- Write the rate as a decimal: 6% = 0.06.
- Substitute and multiply: I = 2500 × 0.06 × 3 = $450.
📘 Example 2 · The amount paid back
At the end of the 3 years, Asher pays back the interest and the money he borrowed: $2500 + $450 = $2950.
📘 Example 3 · Time in months
Eva invests $3000 for 9 months at 8% simple interest. How much interest will she earn?
- Rate as a decimal: 8% = 0.08.
- Time as a fraction of a year: = = 0.75 years.
- Multiply: I = 3000 × 0.08 × 0.75 = $180.
💡 1 year 6 months is 1.5 years, and 2 years 3 months is 2.25 years. Putting 18 or 27 in for t is the classic mistake.
🧪 Try it: simple interest calculator
✏️ Try it
- Leah borrows $1500 for 2 years at 12% interest. Find the interest.
- How much interest would you pay on $800 for 6 months at 14%?
- Ricardo borrows $1850 for 8 months at 12%. What is the total he pays back?
Answers
1. 1500 × 0.12 × 2 = $360 · 2. 800 × 0.14 × 0.5 = $56 · 3. 1850 × 0.12 × = $148, so $1850 + $148 = $1998
Lesson 6 · Percent of change
🔑 Key ideas
- The amount of change is the difference between the new number and the original number.
- Percent of change = amount of change ÷ original number.
- A percent of increase can be more than 100%.
Percents are often used to describe a change. Say Michelle earned $1000 last month and $2000 this month. All of these are true:
- Her earnings doubled.
- Her earnings increased by 100%. (The change, $1000, is 100% of the original $1000.)
- This month's earnings are 200% of last month's.
percent of change =
(For a decrease, take the smaller number from the bigger one, so the change is positive.)
📘 Example 1 · A pay raise
Lisa earned $10.50 an hour. Now she earns $11.34. What percent raise did she get?
- Amount of change: $11.34 − $10.50 = $0.84.
- Divide by her wage before the change: 0.84 ÷ 10.50 = 0.08 = 8%.
Think carefully about which number is the original: it's the one from before the change, whichever is bigger.
📘 Example 2 · A sale price
A jacket is on sale for $90. Three days ago it was $120. By what percent was the price cut?
- Amount of change: $120 − $90 = $30.
- $120 came first, so it is the original. 30 ÷ 120 = 0.25 = 25% decrease.
Work backward to check: after a 25% cut the price should be 100% − 25% = 75% of the original. 75% of $120 = $90. ✔
📘 Example 3 · More than 100%
Calvin started his business with 10 employees. Now he has 60. By what percent has his workforce grown?
- Amount of change: 60 − 10 = 50.
- Divide by the original: 50 ÷ 10 = 5 = 500% increase.
Notice the trap: 60 is 600% of 10, but the increase is 500%.
🧪 Try it: percent of change calculator
✏️ Try it
- Find the percent of increase from 2000 to 3000.
- Find the percent of decrease from $2.00 to $1.25.
- The Utleys' rent rose from $600 to $636 a month. By what percent did it rise?
- Justin's weekly pay went from $280 to $448. By what percent did it increase?
Answers
1. 1000 ÷ 2000 = 50% · 2. 0.75 ÷ 2.00 = 37.5%, about 38% · 3. 36 ÷ 600 = 6% · 4. 168 ÷ 280 = 60%
Chapter summary
- A ratio compares two numbers: 3 to 5, 3 : 5 or . Keep the terms in the order the question gives, reduce to lowest terms, and never turn a ratio into a mixed or whole number.
- A proportion sets two ratios equal. Write both in the same form, multiply the diagonal pair, and divide by the number left over. A rate compares an amount to 1 (“per”).
- Percent means out of 100. Every percent problem has a base (the whole), a part and a rate; = .
- Part = base × rate. Rate = part ÷ base. Base = part ÷ rate. Work out which one is missing before you calculate.
- On the TI-30XS, % is 2nd (. Base × percent % finds the part; part ÷ percent % finds the base.
- I = prt, with t in years (months ÷ 12). The amount paid back is principal + interest.
- Percent of change = amount of change ÷ the original number. Increases can be more than 100%.
📄 Practice questions
The book's own practice sets for Lessons 1–6 and its chapter practice questions are in the printable assignment for this chapter.
Mini-quiz
Source: GED® Test Prep Mathematics, Chapter 3 (Ratio, Proportion, and Percent). Explanations, the Try it questions' answers, the tools and the quiz are Guru Jazzy's own. GED® is a registered trademark of the American Council on Education; this page is not made or endorsed by GED Testing Service.