Ratios · Proportions · 3.2

Cross- products.

Two ratios a:b and c:d are equal exactly when their cross-products match: a×d = b×c. For 2:3 and 6:9, 2×9 = 18 and 3×6 = 18 — equal, so it’s a proportion.

01 / THE IDEA

See it work.

Watch the method

Cross-products

2:3 = 6:9 because 2×9 = 3×6 = 18.

1 / 1
2:32 : 3
1

Simplify

Reduce the ratio.

2:3=2:32:3 = 2:3
— / WHY IT WORKS

The idea underneath.

a×d = b×c

Cross-multiplying gives a quick test for equal ratios.

Why it works

2×9 = 3×6.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Is 4:3 equal to 12:8?

1

Answer

No

EX 2medium

Is 4:3 equal to 8:8?

1

Answer

No

EX 3hard

Is 6:2 equal to 24:8?

1

Answer

Yes

04 / COMMON TRAPS

Where students slip.

Trap

Multiplying straight across (a×c).

Fix

Cross-multiply: a×d and b×c.

Trap

Comparing only the numerators.

Fix

Use both cross-products.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Is 6:2 equal to 12:6?

Need a hint?
  • Multiply a×d and b×c.
  • Equal products mean a proportion.
C2

Is 6:2 equal to 18:6?

Need a hint?
  • Multiply a×d and b×c.
  • Equal products mean a proportion.
C3

Is 2:5 equal to 8:22?

Need a hint?
  • Multiply a×d and b×c.
  • Equal products mean a proportion.
C4

Is 5:4 equal to 10:9?

Need a hint?
  • Multiply a×d and b×c.
  • Equal products mean a proportion.
Apply

Apply (4)

A1

Is 4:5 equal to 12:16?

Need a hint?
  • Multiply a×d and b×c.
  • Equal products mean a proportion.
A2

Is 5:6 equal to 15:18?

Need a hint?
  • Multiply a×d and b×c.
  • Equal products mean a proportion.
A3

Is 2:4 equal to 8:17?

Need a hint?
  • Multiply a×d and b×c.
  • Equal products mean a proportion.
A4

Is 2:2 equal to 8:9?

Need a hint?
  • Multiply a×d and b×c.
  • Equal products mean a proportion.
Extension

Extension (4)

E1

Is 4:5 equal to 12:15?

Need a hint?
  • Multiply a×d and b×c.
  • Equal products mean a proportion.
E2

Is 4:5 equal to 8:10?

Need a hint?
  • Multiply a×d and b×c.
  • Equal products mean a proportion.
E3

Is 6:5 equal to 24:22?

Need a hint?
  • Multiply a×d and b×c.
  • Equal products mean a proportion.
E4

Is 2:4 equal to 6:12?

Need a hint?
  • Multiply a×d and b×c.
  • Equal products mean a proportion.