DWG NO. 07.1 — Lesson 1 of 5

Ratios, Proportions, and Scale

Unit 7: Similarity · ~15–30 min

Objective

Write and solve proportions, and use a scale factor to convert between a model or map and real-world measurements.

Ratios and proportions

A ratio compares two quantities by division. The ratio of a to b (with b ≠ 0) can be written a : b or as the fraction a/b. Ratios only make sense when the two quantities are measured in the same units — if they aren't, convert first.

A proportion is an equation stating that two ratios are equal: a/b = c/d. Proportions are the main tool of this unit, because similarity is really just "same shape" restated as "corresponding measurements form equal ratios."

Cross Products Property: If a/b = c/d, then a · d = b · c. This turns any proportion into a simple equation you can solve.
Reciprocal Property: If a/b = c/d, then b/a = d/c — flipping both ratios keeps the proportion true.

Scale and scale factor

A scale compares a length in a drawing, map, or model to the actual length it represents — for example, "1 inch = 50 miles" or "1 cm : 20 m." The scale factor is that same relationship written as a single reduced ratio, such as 1 : 2,000,000. To find a real-world length, set up a proportion between the scale and the measured length, then solve using cross products.

A scale of 1 cm : 20 m maps a small drawing onto a much larger real distance.

Worked Example 1 · Solving a proportion

ProblemSolve for x: x/6 = 15/9.

1.Cross multiply: 9x = 6(15) = 90.

2.Divide both sides by 9: x = 10.

x = 10

Worked Example 2 · Reading a map scale

ProblemA map has a scale of 1 in : 25 mi. Two cities are 4.5 in apart on the map. Find the actual distance.

1.Set up a proportion: 1/25 = 4.5/x.

2.Cross multiply: 1 · x = 25 · 4.5, so x = 112.5.

The cities are 112.5 miles apart.

Guided practice

1.Solve for x: x/8 = 21/24.
Show answer
Cross multiply: 24x = 8(21) = 168, so x = 7.
2.Write the ratio of 15 minutes to 2 hours in simplest form (same units).
Show answer
2 hours = 120 minutes, so the ratio is 15:120, which simplifies to 1:8.
3.A model car is built at a scale of 1 in : 4 ft. If the model is 11 in long, how long is the real car?
Show answer
1/4 = 11/x, so x = 44. The real car is 44 feet long — if that seems huge, remember the point is just practicing the setup.
4.Solve for x: (x + 2)/5 = 9/15.
Show answer
Cross multiply: 15(x + 2) = 45, so 15x + 30 = 45, 15x = 15, x = 1.
5.On a blueprint with scale 1 cm : 3 m, a wall measures 2.4 cm. How long is the actual wall, in meters?
Show answer
1/3 = 2.4/x, so x = 7.2. The actual wall is 7.2 meters long.

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