Pre-Algebra Class 9: Proportional Reasoning — Solving Simple Proportions

Short answer up top. A proportion is an equation that says two ratios are equal, written a/b = c/d. Two ratios form a proportion when they simplify to the same fraction — or, equivalently, when their cross products are equal: a × d = b × c. To solve for a missing value, use the scale-factor method when one side scales cleanly (e.g., 5 → 20 by ×4) or cross-multiplication otherwise, which always works. Class 9 is the class that turns the ratios and unit rates from Class 8 into a real solving tool — and it is the exact machinery behind every scaling problem in Class 10 percent, every similar-triangle problem in Regents Geometry, and every unit-conversion or word problem on the SAT. Book a free 30-minute evaluation at /evaluation or call (646) 668-6151, and any 5th- or 6th-grader can join Young Fermats — Pre-Algebra on the next Monday at our 226 W 79th Street classroom on the Upper West Side.

What a proportion actually is

A ratio compares two quantities — 3 apples for every 5 oranges is a ratio, written 3:5 or 3/5. A proportion is a statement that two ratios are equal: 3/5 = 6/10. The equals sign is the whole point. A ratio stands alone; a proportion is an equation.

Two ratios are proportional exactly when they name the same amount — that is, when they are equivalent fractions. 3/5 = 6/10 because both simplify to 3/5. 3/5 = 9/15 because both simplify to 3/5. 3/5 ≠ 4/6 because 4/6 simplifies to 2/3, not 3/5.

Two tests: are these ratios proportional?

You will get two clean tools in this class. Both answer the same question — “is a/b = c/d a true equation?” — and each is faster in different situations.

Cross-product test. For any proportion a/b = c/d, multiplying diagonally gives a × d and b × c. If those two products are equal, the ratios are proportional. If they are not equal, the ratios are not proportional. That is it.

Example. Is 15/20 = 9/12? Cross products: 15 × 12 = 180 and 20 × 9 = 180. Yes — both ratios simplify to 3/4.

Example. Is 3/8 = 5/12? Cross products: 3 × 12 = 36 and 8 × 5 = 40. 36 ≠ 40, so no.

The two clean methods to solve for a missing value

The heart of Class 9. You are given a proportion with one unknown — usually x — and you have to find its value. Two methods, both bulletproof.

Method A — Scale factor (fastest when the numbers cooperate)

The rule is one sentence: look at the two known matching parts (the two numerators or the two denominators). Figure out what you multiply the known one by to get the other one — that is the scale factor. Multiply the third known number by the same scale factor to find the missing value.

Full example — 3/5 = x/20:

1. Look at the two denominators (both known): 5 and 20.
2. What number times 5 gives 20? 20 ÷ 5 = 4. The scale factor is 4.
3. Multiply the numerator on the known side by that same 4: x = 3 × 4 = 12.

Sanity check: 12/20 simplifies to 3/5. ✓

Full example — 21/x = 7/2:

1. Look at the two numerators: 21 and 7. What times 7 gives 21? 21 ÷ 7 = 3. Scale factor 3, going from the right side to the left side.
2. Multiply the denominator 2 by 3: x = 2 × 3 = 6.

Sanity check: 21/6 simplifies to 7/2. ✓

When to use scale factor. Use it whenever one pair of matching parts has a clean whole-number scale factor (2, 3, 4, 5, 10…). It is fast, mental, and it keeps proportional reasoning “visible” instead of hidden inside an algebra step. It is also the method that shows up most on the SHSAT and SAT no-calculator math.

Method B — Cross-multiplication (always works)

The rule is one sentence: for a proportion a/b = c/d, replace it with the equivalent equation a × d = b × c, then solve the resulting one-step equation.

Full example — 3/5 = x/20:

1. Cross-multiply: 3 × 20 = 5 × x.
2. Simplify: 60 = 5x.
3. Divide both sides by 5: x = 12.

Full example — 4/9 = 12/x:

1. Cross-multiply: 4 × x = 9 × 12.
2. Simplify: 4x = 108.
3. Divide both sides by 4: x = 27.

Full example — 2.5/1 = x/6 (decimals):

1. Cross-multiply: 2.5 × 6 = 1 × x.
2. Simplify: 15 = x. So x = 15.

When to use cross-multiplication. When the scale factor is not a clean whole number, or when you have decimals or fractions in the ratios, cross-multiplication is more reliable. It always works. The trade-off is that it hides the proportional-reasoning intuition inside an algebra step, so use scale factor first when you can.

Setting up a proportion from a word problem

Almost every real-world proportion problem is the same recipe: identify the two quantities that are related, set up two ratios that compare them the same way, put the unknown in the right spot, and solve. The one rule that matters is consistency of units and order.

Example — recipe scaling. A recipe uses 2 cups of flour for every 3 cookies. How much flour for 12 cookies?

1. Set up matching units: flour/cookies = flour/cookies.
2. Fill in: 2/3 = x/12.
3. Scale factor: 3 → 12 is ×4. So x = 2 × 4 = 8 cups.

Example — unit rate as a proportion. A car drives 90 miles in 2 hours at a steady speed. How far in 5 hours?

1. Set up matching units: miles/hours = miles/hours.
2. Fill in: 90/2 = x/5.
3. Cross-multiply: 90 × 5 = 2 × x → 450 = 2x → x = 225 miles.

Sanity check: unit rate is 90 miles ÷ 2 hours = 45 mph. 45 × 5 = 225 miles. ✓ (The proportion is just the unit-rate equation in a different costume.)

Where proportions show up later

• Class 10 (percent). Every percent problem is a proportion: part/whole = percent/100. Tip, tax, discount, percent-of-a-number, and percent-change all live in this single template.
• Regents Geometry — similar figures. Corresponding side lengths of similar triangles are always in proportion. Every scale-factor and similar-triangle question is a Class-9 proportion in disguise.
• SAT and SHSAT. Unit conversion, rate problems, mixtures, maps and scale drawings, and geometry with similar figures all reduce to setting up and solving one proportion.
• Science. Density = mass/volume, speed = distance/time, and every dilution problem in chemistry are proportions.

Three mistakes students always make

Fast benchmarks and sanity checks

These are the sanity checks that catch most proportion mistakes. If your setup gives an answer that scales the wrong direction (bigger when it should be smaller), the units on your two sides are almost certainly flipped. Rewrite the ratio.

The 25 questions below march from checking whether ratios are proportional, through the scale-factor and cross-multiplication methods, and finish with the classic real-life word problems that put the tool to work.

25 practice questions with hidden answers

About the Young Fermats Pre-Algebra course

Class 9 is one of 48 rolling classes in our Young Fermats Pre-Algebra program for grade 5–6 students (ages 10–12). The full arc runs from integers on the number line through systems of linear equations and probability — the direct on-ramp to Algebra 1 in 7th or 8th grade. Classes are small-group (max 6 students), 120 minutes per week, and students can start any Monday because the syllabus is rolling.

Every new family starts with a free 30-minute evaluation in our 226 W 79th Street classroom plus a written diagnostic delivered within 48 hours. The diagnostic maps your child’s current level against the readiness signals for Pre-Algebra and Algebra 1 — and gives you a clear next step whether you enroll or not.

Related posts

• Pre-Algebra Class 7: Add, Subtract, Multiply & Divide Decimals — Including Dividing by a Decimal (25 Practice Questions)
• Pre-Algebra Class 6: Add, Subtract, Multiply & Divide Fractions and Mixed Numbers (25 Practice Questions)
• Pre-Algebra Class 5: Equivalent Fractions, Comparing & Converting to Decimals (25 Practice Questions)
• Pre-Algebra Class 12: Evaluating Expressions & Combining Like Terms (25 Practice Questions)