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Stage 3 · Geometry

Scale Factors & Proportions

A proportion is two equal ratios. Set them up in the right order, cross multiply, and divide. That's the whole skill — used everywhere from similar figures to maps to recipes.

📖 What Is a Ratio?

A ratio compares two quantities. It can be written three ways — they all mean the same thing.

Fraction
23
Colon
2 : 3
Words
2 to 3

What Is a Proportion?

A proportion says two ratios are equal. If you know three of the four values, you can find the fourth.

Example — Equal Ratios
23 = 46   ← these are equal ratios (a proportion)
23 = x9   ← find x using cross multiplication

⚠️ Order Matters

The most common mistake on the GED is setting up a proportion in the wrong order. Both ratios must compare the same things in the same order.

✓ Correct — same order
smalllarge = smalllarge

410 = 6x
✗ Wrong — mixed order
smalllarge = largesmall

410 = x6

Solving with Cross Multiply & Divide

Cross multiply means multiplying diagonally across the equals sign — numerator of one side times denominator of the other. The two multiplications form an X shape, which is why some people call it the butterfly method.

3 4 x 12 = 3 × 12 = 36 4 × x = 4x
The Butterfly / Cross Multiply: The two arcs form a butterfly wing shape — or an X cross. Orange: 3 × 12 = 36. Purple: 4 × x = 4x. Set them equal: 36 = 4x. Divide by 4. x = 9.
Worked Example — Solve the Proportion

Solve: 34 = x12

34 = x12
→ Step 1: Cross multiply — multiply the orange pair and the purple pair
3 × 12 = 4 × x
36 = 4x
→ Step 2: Divide both sides by 4
x = 9
→ Check: 3/4 = 9/12 ✓
Worked Example — Similar Figures

A small triangle has sides 4 cm and 6 cm. A similar large triangle has a corresponding side of 10 cm. Find the missing side.

→ Set up: small/large = small/large
410 = 6x
→ Cross multiply: 4 × x = 10 × 6
4x = 60
→ Divide by 4
x = 15 cm
1
Identify corresponding values — what matches what?
2
Set up the proportion in the correct order — same units on the same side
3
Cross multiply — multiply diagonally
4
Divide to isolate x
5
Check — substitute back and verify both ratios are equal
Common mistakes:
• Mixing up the order — always keep the same units on the same side
• Cross multiplying the wrong way — multiply diagonally, not straight across
• Forgetting to divide after cross multiplying
• Matching wrong sides on similar figures — only corresponding sides form equal ratios
🎯 Test Strategy — Check the Answer Choices First
1.Estimate first. If a map scale is 1 cm = 50 km and you measure 3 cm, the answer must be around 150 km. Eliminate anything wildly different.
2.Check units. Your answer should make physical sense — a missing side of a small triangle shouldn't be 1000 cm.
3.Plug back in. Once you find x, substitute it back into your proportion. If both sides are equal, you're right.
4.Eliminate unreasonable answers. If the large shape should be bigger than the small shape, cross off any answer that's smaller.
Reference
GED Formula Sheet
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📋 Open Formula Sheet

✏️ Practice Questions

Bank 1 — Build the Skill & Error Analysis
Mix of multiple choice and fill-in-the-blank
You've answered 5 questions. Keep going or check your score.
Bank 2 — GED-Level Applications
Similar figures, maps, scale drawings, and word problems
You've answered 5 questions. Keep going or check your score.
Up Next in Stage 3
Perimeter & Area
Next Skill →