Solve ratio, percent, and proportional problems.

SECTION 1

Why Ratios and Percents Matter

Long before algebra existed, people needed ways to compare quantities fairly. Ancient traders asked questions like, "If 3 bags of grain cost 12 coins, how much do 7 bags cost?" That question is a proportion problem, and it's the same kind of reasoning you use today when you figure out a sale price, adjust a recipe, or calculate a tip. Ratios, percents, and proportions are among the most practical math skills you can build—and they make up a significant portion of the GED Mathematical Reasoning test.

~1800 BCE

Babylonian Clay Tablets

Ancient Babylonian scribes solved proportion problems on clay tablets to divide land, measure grain, and calculate wages.

~300 BCE

Euclid Formalizes Ratios

The Greek mathematician Euclid defined ratios rigorously in Book V of the Elements, establishing rules still used today.

1500s CE

Percent Symbol Emerges

Italian merchants began writing 'per cento' (per hundred) as 'pc°' in ledgers. Over time, this evolved into the modern '%' symbol.

Today

Everyday Applications

From sales tax to nutritional labels to interest rates, ratio and percent reasoning is woven into nearly every financial and scientific decision we make.

The central question these tools answer is simple but powerful: How do two quantities relate to each other, and how can I use that relationship to find an unknown value? Mastering this reasoning will help you on the GED and in real life.

SECTION 2

Core Principles & Definitions

Before solving problems, you need a clear understanding of three interconnected ideas: ratios, percents, and proportions. These are not three separate topics—they are three different lenses for viewing the same underlying concept of multiplicative comparison.

Ratio

A comparison of two quantities by division. Written as a:b, a/b, or "a to b." Example: 3 cups of flour to 2 cups of sugar = 3:2.

Rate

A special ratio that compares quantities with different units. Example: 150 miles in 3 hours = 50 miles per hour.

Percent

A ratio with a denominator of 100. 'Per cent' literally means 'per hundred.' 25% = 25/100 = 0.25.

Proportion

An equation stating that two ratios are equal. Example: 3/4 = 6/8. Solving proportions is the main technique for finding unknown values.

Cross-Multiplication

The primary method for solving a proportion a/b = c/d: multiply across the equals sign diagonally, giving a × d = b × c.

✦ KEY TAKEAWAY

Think of a ratio like a recipe. If you know the recipe for a small batch (say 2 cups of flour per 1 cup of sugar), you can scale it up or down to any batch size. A proportion is just the math that proves the big batch and the small batch have the same flavor—the same relationship between ingredients. A percent is what happens when you always scale the batch to exactly 100.

SECTION 3

Visualizing Ratios and Proportions

The diagram below shows how a ratio of 3:2 can be represented visually and how a proportion connects two equivalent ratios. Notice that in each bar, the relationship between the blue and pink sections stays the same—even though the total size changes.

Both bars maintain the same 3:2 ratio. The proportion 3/5 = 6/10 proves the relationship holds when the batch is doubled. Cross-multiplying both sides gives 30 = 30, confirming the ratios are equal.

The key insight is that equivalent ratios produce the same fraction when reduced. Whether you write 3:5, 6:10, or 60:100, the decimal value is always 0.6, and the percent is always 60%. This connection among ratios, fractions, decimals, and percents is at the heart of every problem in this lesson.

SECTION 4

Mathematical Framework

The GED formula sheet won't list these for you because they are considered foundational knowledge. However, every ratio, percent, and proportion problem boils down to one of the following equations. Learn to recognize which one applies to a given situation.

PROPORTION (CROSS-MULTIPLICATION)

a / b = c / d → a × d = b × c

If two ratios are equal, the cross products are equal. Use this to solve for any one unknown value when the other three are known.

PERCENT FORMULA

Part = Percent × Whole (or Part / Whole = Percent / 100)

This formula has three versions depending on which value is unknown. To find the part, multiply. To find the percent, divide part by whole and multiply by 100. To find the whole, divide the part by the percent (as a decimal).

PERCENT CHANGE

Percent Change = (New Value − Original Value) / Original Value × 100

A positive result means percent increase; a negative result means percent decrease. The denominator is always the original value.

CONVERTING BETWEEN FORMS

Fraction → Decimal: divide numerator by denominator Decimal → Percent: multiply by 100

Example: 3/8 = 0.375 = 37.5%. To reverse: move the decimal point two places left and write as a fraction over 100.

💡 GED Test Tip

Part 1 of the GED is no-calculator. Practice converting simple fractions to percents mentally: 1/4 = 25%, 1/5 = 20%, 1/3 ≈ 33.3%, 3/4 = 75%. Knowing these cold will save you precious time.

SECTION 5

Types of Problems You'll See

GED ratio and percent questions come in several common formats. The diagram below maps out the main problem types, and the table that follows gives you a strategy for each one.

This map shows the major categories of proportional reasoning problems on the GED. Ratio problems split into part-to-part and part-to-whole. Percent problems ask you to find one of three values: part, percent, or whole. Rate problems include unit rates, percent change, and scale conversions.

Strategy chart for ratio, percent, and proportion problem types

Problem Type	What You're Given	Strategy
Find the Part	Percent and Whole	Convert % to decimal, then multiply by Whole.
Find the Percent	Part and Whole	Divide Part by Whole, then multiply by 100.
Find the Whole	Part and Percent	Divide Part by the Percent (as a decimal).
Percent Change	Original and New values	Subtract Original from New, divide by Original, × 100.
Proportion / Scale	Two ratios, one unknown	Set up a/b = c/d, cross-multiply, solve for the unknown.

SECTION 6

Worked Example: Multi-Step Percent and Proportion

Let's work through a realistic GED-style problem step by step. Read the scenario carefully, then follow each step.

📋 Problem

A store is having a 30% off sale. Maria wants to buy a jacket originally priced at $85. The sales tax rate is 7%. What is the total amount Maria will pay, including tax, rounded to the nearest cent?

Solution

Step 1 — Find the Discount AmountConvert 30% to a decimal: 30% = 0.30. Multiply by the original price: 0.30 × $85 = $25.50.

Discount = $25.50

Step 2 — Find the Sale PriceSubtract the discount from the original price: $85.00 − $25.50 = $59.50.

Sale Price = $59.50

Step 3 — Calculate the Sales TaxTax is applied to the sale price, not the original price. Convert 7% to a decimal: 7% = 0.07. Multiply: 0.07 × $59.50 = $4.165, which rounds to $4.17.

Tax = $4.17

Step 4 — Find the TotalAdd the sale price and the tax: $59.50 + $4.17 = $63.67.

Total = $63.67

⚠ WATCH OUT

A common GED trap is applying tax to the original price instead of the discounted price. In real life (and on the test), tax is calculated on the amount you actually pay. Always discount first, then add tax.

SECTION 7

Strategies and Common Mistakes

Knowing the formulas is only half the battle. You also need to recognize which formula fits the situation and avoid the traps that test-makers build into wrong answer choices. The table below compares common mistakes with the correct approach.

Common mistakes vs. correct approaches for ratio and percent problems

Common Mistake	Why It's Wrong	Correct Approach
Confusing part-to-part with part-to-whole	A ratio of 3:2 means 3 and 2 parts, totaling 5. Using 3/2 instead of 3/5 gives the wrong answer.	Read carefully: does the problem ask for a comparison between groups or each group's share of the total?
Using the wrong denominator for percent change	Dividing by the new value instead of the original gives a different (incorrect) percentage.	Always divide by the ORIGINAL value: (New − Original) / Original × 100.
Forgetting to convert % to a decimal	Multiplying by 25 instead of 0.25 makes the answer 100 times too large.	Divide the percent by 100 (or move the decimal two places left) before multiplying.
Setting up the proportion with mismatched units	Putting miles on top on one side and hours on top on the other makes the cross-multiplication invalid.	Label each ratio. Make sure the same unit is in the numerator on both sides: miles/hours = miles/hours.

✦ KEY TAKEAWAY

Think of setting up a proportion like balancing a seesaw. Each side must be structured the same way—same units on top, same units on the bottom. If one side has apples/oranges, the other side must also have apples/oranges. A seesaw balances only when both sides match.

SECTION 8

Connections to Advanced Topics

Ratio and percent reasoning is not just a standalone skill—it's a foundation that connects to more advanced topics on the GED and beyond. Once you're comfortable with proportions, you're already doing early algebra. The table below shows how today's concepts extend into other areas you may encounter.

How ratio and percent skills connect to advanced GED topics

This Lesson's Concept	Advanced Connection	Where You'll See It
Proportion: a/b = c/d	Linear equations: y = kx (direct variation)	GED algebra questions, graphing lines
Percent change	Simple and compound interest	GED word problems, personal finance
Unit rate	Slope of a line (rise over run)	GED coordinate geometry
Scale factor (maps, models)	Similar figures, area and volume scaling	GED geometry questions

The big idea is that proportional reasoning is the bridge between arithmetic and algebra. When you set up a proportion and solve for an unknown, you're already using the same logic as solving a one-variable equation. Building confidence here makes algebra feel much more natural.

SECTION 9

Practice Problems

Work through these five problems in order. They increase in difficulty, just like sections of the GED. Try each problem on your own before reading the answer.

A classroom has 12 boys and 18 girls. A student says the ratio of boys to the total number of students is 12:18. Which of the following best explains why the student is incorrect? A. The ratio should be simplified to 2:3. B. The ratio 12:18 compares boys to girls, not boys to total students. C. The ratio should be written as a percent instead. D. The student should have subtracted 12 from 18 first.

A map uses a scale of 1 inch = 25 miles. Two cities are 3.5 inches apart on the map. How many miles apart are the two cities? A. 28.5 miles B. 75.0 miles C. 87.5 miles D. 100.0 miles

Last year, a gym had 240 members. This year, it has 312 members. What is the percent increase in membership from last year to this year? A. 23.1% B. 30.0% C. 72.0% D. 130.0%

A recipe for 4 servings of soup calls for 2.5 cups of broth and 1.5 cups of vegetables. Kevin wants to make 10 servings. He currently has 5 cups of broth and 4 cups of vegetables. Does he have enough of each ingredient? How many cups of broth and how many cups of vegetables does he need for 10 servings? A. He needs 6.25 cups of broth and 3.75 cups of vegetables; he has enough of both. B. He needs 6.25 cups of broth and 3.75 cups of vegetables; he has enough vegetables but not enough broth. C. He needs 5.0 cups of broth and 3.0 cups of vegetables; he has enough of both. D. He needs 25.0 cups of broth and 15.0 cups of vegetables; he doesn't have enough of either.

PROBLEM 5 — CRITICAL THINKING

A store advertises a television at 20% off, and then offers an additional 15% off the already-reduced price during a holiday sale. A customer believes this is the same as getting 35% off the original price. Determine the actual total percent discount off the original price. Show your work and explain why the customer's reasoning is incorrect.

SUMMARY

Lesson Summary

A ratio compares two quantities by division, and a percent is simply a ratio with a denominator of 100. A proportion is an equation stating two ratios are equal, and cross-multiplication (a × d = b × c) is the go-to method for solving them. The three core percent formulas—Part = Percent × Whole, find the percent, and find the whole—cover nearly every percent question on the GED. For percent change, always divide by the original value.

Watch out for common traps: confusing part-to-part with part-to-whole ratios, forgetting to convert a percent to a decimal before multiplying, and adding successive percent discounts instead of multiplying them. On the no-calculator section, memorize key fraction-to-percent conversions (1/4 = 25%, 1/5 = 20%, 1/3 ≈ 33.3%). These skills form the bridge between arithmetic and algebra, connecting directly to linear equations, slope, and interest calculations you'll encounter elsewhere on the GED.