Solve problems involving percent, ratio, and proportion - TACHS Math

SECTION 1

Where Did Percents, Ratios, and Proportions Come From?

People have been comparing amounts for thousands of years. Ancient farmers needed to split harvests fairly. Merchants needed to figure out fair trades. Over time, math gave us three powerful tools for comparing: ratios, proportions, and percents.

~1800 BCE • Babylonian Ratios

Ancient Babylonians used clay tablets to record ratios for mixing building materials and splitting land.

~300 BCE • Greek Proportions

The Greek mathematician Euclid wrote about proportions in his famous book Elements. He showed how equal ratios could solve geometry problems.

~100 CE • Roman "Per Centum"

Roman tax collectors charged fees "per centum" (per hundred). This Latin phrase is where we get the word "percent."

1500s • The % Symbol Appears

Italian merchants began using a short-hand symbol that slowly evolved into the % sign we use today.

Today, percents, ratios, and proportions show up everywhere: sales tax, recipe adjustments, sports stats, and test scores. The big question this lesson answers is: How do you set up and solve problems that compare quantities?

SECTION 2

Core Definitions You Need to Know

Before you solve any problem, you need to understand three key ideas. They are all related, but each one has its own job.

Ratio

A ratio compares two quantities. You can write it as 3 to 5, 3:5, or 3/5. Order matters! The ratio of cats to dogs is different from dogs to cats.

Proportion

A proportion is an equation that says two ratios are equal. For example, 2/3 = 4/6. You use cross-multiplication to solve for a missing value.

Percent

A percent means "out of 100." So 40% means 40 out of 100. It is really a special ratio where the second number is always 100.

Cross-Multiplication

When you have a proportion like a/b = c/d, you can cross-multiply: a × d = b × c. This is the main tool for solving proportions.

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KEY TAKEAWAY

Think of a ratio like a recipe. If a smoothie uses 2 bananas for every 3 cups of milk, the ratio is 2:3. A proportion is like scaling that recipe up. If you want to make a bigger batch, you keep the same ratio: 4 bananas to 6 cups of milk. A percent is like grading your smoothie out of 100 taste points!

SECTION 3

Seeing the Connection

The diagram below shows how ratios, proportions, and percents connect. A ratio compares two amounts. When you set two equal ratios side by side, you get a proportion. A percent is just a ratio with 100 on the bottom.

This diagram shows how a single comparison (3 out of 5) can be written as a ratio, used in a proportion, or turned into a percent.

Notice how "3 out of 5" can be written three ways. As a ratio it's 3:5. As a proportion you can scale it up to 6/10. As a percent it's 60%. All three say the same thing, just in different ways.

SECTION 4

The Key Formulas

There are a few formulas you should memorize. They are your toolkit for solving problems on the TACHS and in everyday life.

PERCENT FORMULA

Part = Percent × Whole

Part = the piece you're looking at. Percent = the rate written as a decimal (divide % by 100). Whole = the total amount.

FINDING THE PERCENT

Percent = (Part ÷ Whole) × 100

Divide the part by the whole, then multiply by 100 to get the percent.

CROSS-MULTIPLICATION

If a/b = c/d, then a × d = b × c

Multiply across the diagonals. This lets you find any one missing value when you know the other three.

PERCENT CHANGE

Percent Change = ((New − Original) ÷ Original) × 100

Subtract the original from the new value. Divide by the original. Multiply by 100. A positive answer means increase; a negative answer means decrease.

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Quick Tip

When you see the word "of" in a percent problem, it usually means multiply. For example, "20% of 50" means 0.20 × 50 = 10.

SECTION 5

Types of Problems You'll See

On the TACHS, you'll run into several problem types. The bar model below shows six common types and how often they appear on state assessments.

The six most common problem types for percent, ratio, and proportion, with a mini-example of each at the bottom.

Problem Type	What You Know	What You Find
Find the Part	Percent and Whole	Part (multiply)
Find the Whole	Part and Percent	Whole (divide)
Find the Percent	Part and Whole	Percent (divide, then × 100)
Solve a Proportion	Three of four values	Missing value (cross-multiply)
Percent Change	Original and New values	Percent increase or decrease
Ratio Word Problem	Ratio and a total or one part	The other part or total

Summary of what you know vs. what you find for each type

SECTION 6

Step-by-Step Worked Examples

Example 1: Proportion Problem

A car travels 150 miles on 5 gallons of gas. How many miles can it travel on 8 gallons?

Proportion — Miles per Gallon

1
Step 1 — Write the known ratio

The car goes 150 miles on 5 gallons. Write this as a fraction: 150/5.

150/5
2
Step 2 — Set up the proportion

You want to find how many miles (call it x) for 8 gallons. Write: 150/5 = x/8.

150/5 = x/8
3
Step 3 — Cross-multiply

Multiply across: 5 × x = 150 × 8. This gives you 5x = 1,200.

5x = 1,200
4
Step 4 — Solve for x

Divide both sides by 5: x = 1,200 ÷ 5 = 240.

x = 240 miles

Example 2: Percent Problem

A jacket costs $80 and is on sale for 25% off. What is the sale price?

Finding a Discount

1
Step 1 — Find the discount amount

"25% of $80" means 0.25 × 80. Remember, change the percent to a decimal by dividing by 100.

0.25 × 80 = $20
2
Step 2 — Subtract from original

The discount is $20. Subtract that from the original price: $80 − $20.

Sale price = $60

SECTION 7

Common Mistakes and How to Avoid Them

Even strong math students make mistakes with percents, ratios, and proportions. Here are the most common errors and how to fix them.

Common Mistake	Why It Happens	How to Fix It
Forgetting to convert percent to a decimal	Students multiply by 25 instead of 0.25	Always divide the percent by 100 before multiplying
Mixing up the ratio order	Writing dogs:cats when the question asks cats:dogs	Read the question carefully. Underline what goes first
Setting up the proportion wrong	Putting miles over gallons on one side but gallons over miles on the other	Keep the same units on top for both fractions
Using the wrong base for percent change	Dividing by the new value instead of the original	Always divide by the ORIGINAL value
Not checking if the answer makes sense	Rushing through without estimating	Ask: is this answer reasonable? 25% of 80 should be less than 80

Five common errors and their fixes

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KEY TAKEAWAY

Think of setting up a proportion like lining up puzzle pieces. The labels on top and bottom must match on both sides. If the left fraction is miles/gallons, the right fraction must also be miles/gallons — never gallons/miles.

SECTION 8

Connecting to Bigger Ideas

The skills you're building now are the foundation for topics you'll see in high school and beyond. Here's how they connect.

What You Learn Now	Where It Leads
Ratios (3:5)	Rates, unit rates, slope in algebra
Proportions (a/b = c/d)	Similar triangles in geometry, scale drawings, linear equations
Percent (Part/Whole × 100)	Interest rates, tax calculations, statistics
Percent change	Growth and decay in science, profit/loss in business
Cross-multiplication	Solving rational equations in Algebra 2

How today's skills connect to future math topics

In algebra, you'll discover that a proportion is really just a special kind of equation. The cross-multiplication you practice now will become second nature. Every time you calculate a tip at a restaurant, figure out a batting average, or resize a photo, you are using the same ideas from this lesson.

SECTION 9

Practice Problems

Try these five problems. They start easy and get harder. Work each one step by step, and then check the answer.

Question 1: CONCEPTUAL

A classroom has 12 boys and 18 girls. What is the ratio of boys to girls in simplest form?

Show answer

The ratio of boys to girls is 12:18. To simplify, find the greatest common factor (GCF) of 12 and 18, which is 6. Divide both numbers by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3. The simplest form is 2:3. This means for every 2 boys, there are 3 girls.

Question 2: BASIC CALCULATION

What is 35% of 200?

Show answer

Convert 35% to a decimal: 35 ÷ 100 = 0.35. Then multiply: 0.35 × 200 = 70. So 35% of 200 is 70. You can check by thinking: 10% of 200 is 20, so 30% is 60, and 5% is 10. That gives 60 + 10 = 70. ✓

Question 3: INTERMEDIATE

A map uses a scale of 2 cm = 15 miles. If two cities are 9 cm apart on the map, how far apart are they in real life?

Show answer

Set up the proportion: 2/15 = 9/x. Cross-multiply: 2 × x = 15 × 9, so 2x = 135. Divide both sides by 2: x = 67.5. The cities are 67.5 miles apart. Notice that 9 cm is 4.5 times bigger than 2 cm, and 67.5 is 4.5 times bigger than 15. The proportion keeps the relationship consistent.

Question 4: APPLIED

A store raised the price of a pair of sneakers from $60 to $75. What is the percent increase?

Show answer

Use the percent change formula: (New − Original) ÷ Original × 100. Substitute: (75 − 60) ÷ 60 × 100 = 15 ÷ 60 × 100 = 0.25 × 100 = 25. The price increased by 25%. Remember to always divide by the original price ($60), not the new price ($75). This is one of the most common mistakes students make.

Question 5: CRITICAL THINKING

Maria scored 18 out of 24 on Test 1 and 22 out of 30 on Test 2. On which test did she do better, and by how much (in percent)?

Show answer

First, convert each score to a percent. Test 1: (18 ÷ 24) × 100 = 0.75 × 100 = 75%. Test 2: (22 ÷ 30) × 100 ≈ 0.7333 × 100 ≈ 73.3%. Maria did better on Test 1 by about 1.7 percentage points (75% − 73.3% ≈ 1.7%). Even though she got more questions right on Test 2 (22 vs. 18), the total number of questions was also bigger. This is why converting to percents gives a fair comparison.

SUMMARY

Lesson Summary

Summary

A ratio compares two quantities (like 3:5). A proportion sets two ratios equal and lets you find a missing value using cross-multiplication (a × d = b × c). A percent is a ratio out of 100. To find the part, multiply the percent (as a decimal) by the whole. To find the percent, divide the part by the whole and multiply by 100.

For percent change, subtract the original from the new value, divide by the original, and multiply by 100. Always keep units consistent in proportions — same labels on top, same labels on bottom. These skills show up on the TACHS in word problems about shopping, recipes, maps, and scores. Practice converting between ratios, fractions, decimals, and percents so you can pick the fastest path to the answer.