Solve proportional relationships. - ISEE Middle Level Quantitative Reasoning

SECTION 1

Where Do Proportions Come From?

People have been using proportions (two equal ratios) for thousands of years. Ancient builders needed them to scale up small drawings into huge structures. Traders needed them to exchange different currencies fairly.

Whenever you compare two quantities that grow or shrink together at the same rate, you are working with a proportional relationship. This idea is so useful that it appears in cooking, map reading, science, art, and of course, on the ISEE.

~1800 BCE • Babylonian Clay Tablets

Ancient Babylonians recorded ratio tables on clay tablets to help merchants convert weights and measures during trade.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid wrote formal rules for ratios and proportions in his famous book, laying the groundwork for geometry.

~1500 CE • Renaissance Art & Architecture

Artists like Leonardo da Vinci used proportions to create realistic drawings and scale models of buildings.

Today • Everyday & Test Applications

Proportions appear in recipe scaling, map distances, unit conversions, and standardized tests like the ISEE.

The central question is simple: if you know three of the four numbers in a proportion, how do you find the missing one? That is exactly what this lesson will teach you.

SECTION 2

Core Principles of Proportional Relationships

Before you can solve proportions on the ISEE, you need to understand a few key building blocks. These ideas will make every proportion problem feel familiar.

Ratio

A ratio compares two quantities. You can write it as a fraction, like 3/4, or with a colon, like 3 : 4.

Proportion

A proportion is an equation that says two ratios are equal. For example, 3/4 = 6/8 is a proportion.

Cross-Multiplication

Cross-multiplication is the main tool for solving proportions. Multiply diagonally across the equal sign, then solve for the unknown.

Equivalent Ratios

Two ratios are equivalent if you can multiply or divide both parts by the same number to get from one to the other.

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KEY TAKEAWAY

Think of a proportion like a balanced seesaw. Both sides must be equal. If you change one side, you need to adjust the other side by the same factor to keep it balanced. Cross-multiplication is your shortcut to finding the exact adjustment!

SECTION 3

Seeing Proportions in Action

A picture can make proportions click instantly. The diagram below shows how cross-multiplication works. Notice how the two diagonal arrows create two products that must be equal.

The green arrow shows the product of a × d, and the orange arrow shows b × c. In a true proportion, these two products are always equal.

This is the heart of solving proportions. When you see a/b = c/d, just multiply the numbers that sit diagonally from each other. The two products must be equal. If one number is missing, cross-multiply and solve for it!

SECTION 4

The Mathematical Framework

Let's look at the formulas you'll use on the ISEE. Don't worry — there are only a couple of key equations, and they're easier than they look.

PROPORTION EQUATION

a / b = c / d

This says that two ratios are equal. The numbers a and d are called the extremes. The numbers b and c are called the means.

CROSS-MULTIPLICATION RULE

a × d = b × c

Multiply the extremes together (a × d) and the means together (b × c). Set the two products equal. Then divide to solve for the unknown.

SOLVING FOR AN UNKNOWN

x / 5 = 12 / 20 → x × 20 = 5 × 12 → 20x = 60 → x = 3

Cross-multiply, simplify, then divide both sides by the number attached to x. Always check your answer by plugging it back in: 3/5 = 12/20? Yes, both simplify to 3/5. ✓

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ISEE Strategy Tip

On the ISEE, there is no penalty for wrong answers. Always answer every question! If you get stuck on a proportion problem, try plugging each answer choice back into the proportion. Whichever one makes both sides equal is the answer.

SECTION 5

Three Ways to Solve Proportions

Cross-multiplication is the most popular method, but the ISEE sometimes rewards students who can spot shortcuts. Here are three methods you should know, along with when each one works best.

Three methods compared side-by-side. Cross-multiplication always works. The scale factor method is fastest when you spot a clean multiplier. Plugging in choices is a great safety net on multiple-choice tests.

On the ISEE, time matters. Start by looking for a scale factor — it's the fastest approach. If you don't see one quickly, switch to cross-multiplication. And remember: since the ISEE is multiple choice, you can always test the answer choices!

SECTION 6

Step-by-Step Worked Example

Let's walk through a complete ISEE-style problem from start to finish. Follow each step carefully — this is exactly the process you'll use on test day.

A recipe calls for 3 cups of flour to make 24 cookies. How many cups of flour do you need to make 40 cookies?

1
Step 1 — Set Up the Proportion

Put the same type of quantity on top of each fraction and the other type on the bottom. We'll put flour on top and cookies on the bottom.

3 / 24 = x / 40
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Step 2 — Cross-Multiply

Multiply diagonally: 3 × 40 and 24 × x. Set these products equal to each other.

3 × 40 = 24 × x → 120 = 24x
3
Step 3 — Solve for x

Divide both sides of the equation by 24 to isolate x.

x = 120 ÷ 24 = 5
4
Step 4 — Check Your Answer

Plug 5 back in: 3/24 = 5/40. Simplify both: 3/24 = 1/8 and 5/40 = 1/8. They match!

You need 5 cups of flour.

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Pro Tip: Keep Units Consistent

Always label what's on top and what's on bottom. If flour is on top in the first fraction, flour must be on top in the second fraction too. Mixing them up is the #1 mistake students make!

SECTION 7

Common Mistakes & How to Avoid Them

Knowing the right method is important, but knowing what can go wrong is just as valuable. Here are the most common proportion mistakes that students make on the ISEE, and how to avoid each one.

Mistake	What It Looks Like	How to Fix It
Flipped units	Writing cups/cookies on the left but cookies/cups on the right	Label each fraction. Keep the same unit on top for both sides.
Multiplying straight across	Doing a × c and b × d instead of crossing diagonally	Draw an X through the equal sign to remind yourself to go diagonally.
Forgetting to divide	Writing 24x = 120 and stopping, or saying x = 120	After cross-multiplying, always divide by the number next to x.
Not checking	Selecting an answer without verifying it works	Plug your answer back in. Do both fractions simplify to the same value?

Common proportion mistakes on the ISEE

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REMEMBER

Think of setting up a proportion like putting matching socks together. You wouldn't put the left shoe with the right sock! Always pair the same type of quantity on the same level of each fraction.

SECTION 8

Connecting Proportions to Bigger Ideas

Proportions are the stepping stone to many powerful math topics. Understanding them now will give you a head start when you encounter these ideas in later courses.

What You Know Now	Where It Leads
Equivalent ratios: 2/3 = 4/6	Linear equations: y = kx, where k is the constant rate
Scale factors between shapes	Similar triangles and geometric proofs
Unit rates (miles per hour, cost per item)	Slope of a line on a graph
Percent problems using proportions	Probability and statistics calculations

How proportions connect to advanced math

On the ISEE specifically, proportions show up in many disguises. You might see them as rate problems, percent problems, map-distance problems, or even geometry questions about similar shapes. If you can recognize the proportion hiding inside these questions, you'll have a huge advantage.

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ISEE Quantitative Comparison Tip

In Quantitative Comparison questions, you may need to compare two quantities that involve ratios. Simplify each ratio to its simplest form, or convert both to decimals, before comparing Column A and Column B. If both columns contain only numbers (no variables), answer (D) — 'cannot be determined' — is never correct.

SECTION 9

Practice Problems

Now it's your turn! Try these five problems. They start easy and get harder, just like the real ISEE. Remember: there's no penalty for guessing, so always pick an answer.

Question 1: CONCEPTUAL

If 5/8 = 15/x, what is the value of x? (A) 20 (B) 24 (C) 30 (D) 40

Show answer

The correct answer is (B) 24. Cross-multiply: 5 × x = 8 × 15, so 5x = 120, and x = 24. You can also use the scale factor: 5 × 3 = 15, so 8 × 3 = 24. Choice (A) 20 is wrong because 5/8 ≠ 15/20 (15/20 simplifies to 3/4, not 5/8). Choice (C) 30 is wrong because 15/30 = 1/2, not 5/8. Choice (D) 40 is wrong because 15/40 = 3/8, not 5/8.

Question 2: BASIC CALCULATION

A car travels 150 miles in 3 hours at a constant speed. How far will it travel in 7 hours at the same speed? (A) 300 miles (B) 350 miles (C) 450 miles (D) 500 miles

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The correct answer is (B) 350 miles. Set up the proportion: 150/3 = x/7. Cross-multiply: 150 × 7 = 3 × x, so 1050 = 3x, and x = 350. Alternatively, the unit rate is 150 ÷ 3 = 50 miles per hour, so 50 × 7 = 350 miles. Choice (A) 300 would mean only 6 hours of travel. Choice (C) 450 would require a speed of about 64 mph. Choice (D) 500 would require about 71 mph.

Question 3: INTERMEDIATE

On a map, 2 centimeters represents 50 kilometers. Two cities are 7 centimeters apart on the map. What is the actual distance between the cities? (A) 100 km (B) 150 km (C) 175 km (D) 200 km

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The correct answer is (C) 175 km. Set up: 2/50 = 7/x. Cross-multiply: 2x = 50 × 7 = 350. Divide: x = 175. You can check with the unit rate: 50 ÷ 2 = 25 km per cm, and 25 × 7 = 175 km. Choice (A) 100 km would be correct for 4 cm. Choice (B) 150 km would be correct for 6 cm. Choice (D) 200 km would be correct for 8 cm.

Question 4: APPLIED

This is a Quantitative Comparison question. A store sells 4 notebooks for $6. Column A: The cost of 10 notebooks at this rate Column B: $16 (A) Column A is greater (B) Column B is greater (C) The two quantities are equal (D) Cannot be determined

Show answer

The correct answer is (B) Column B is greater. Set up a proportion for Column A: 4/$6 = 10/x. Cross-multiply: 4x = 60, so x = $15. Column A = $15 and Column B = $16. Since $15 < $16, Column B is greater. Both columns are pure numbers with no variables, so answer (D) cannot apply here. Choice (A) is wrong because $15 is not greater than $16. Choice (C) is wrong because $15 ≠ $16.

Question 5: CRITICAL THINKING

This is a Quantitative Comparison question. x is a positive number. Column A: The value of x when x/12 = 3/4 Column B: The value of x when 6/x = 2/5 (A) Column A is greater. (B) Column B is greater. (C) The two quantities are equal. (D) Cannot be determined.

Show answer

The correct answer is (B) Column B is greater. For Column A: cross-multiply x/12 = 3/4 to get 4x = 36, so x = 9. For Column B: cross-multiply 6/x = 2/5 to get 2x = 30, so x = 15. Since 9 < 15, Column B is greater. Both columns produce a single specific value (no ambiguity), so (D) does not apply. Choice (A) is wrong because 9 < 15. Choice (C) is wrong because 9 ≠ 15.

SUMMARY

Lesson Summary

Summary

A proportion is an equation that says two ratios are equal. To solve for an unknown, use cross-multiplication: multiply diagonally, set the products equal, then divide. Always keep your units consistent — the same type of quantity should be on top in both fractions.

You have three solving methods: cross-multiplication (works every time), the scale factor shortcut (fastest when numbers divide evenly), and plugging in answer choices (a great backup on multiple-choice tests). On the ISEE, always answer every question — there's no penalty for guessing. Check your work by plugging your answer back into the original proportion.