Equivalent Ratios
Discover how to scale ratios up and down, build ratio tables, and determine whether two ratios are equivalent.
Every time you double a recipe, scale a map distance, or adjust a medication dose for a different body weight, you're working with equivalent ratios. A nurse who doesn't understand equivalent ratios may give a patient a dose calibrated for twice their weight. A baker who can't scale a ratio ends up with flat cookies. This skill is the backbone of every scaling decision you'll ever make.
- Multiply or divide both terms of a ratio by the same factor to produce an equivalent ratio
- Build and read a ratio table to scale quantities systematically
- Use the cross-product test to determine whether two ratios are equivalent
Two ratios are equivalent when they reduce to the same simplest form. You create equivalent ratios by multiplying or dividing both terms by the same non-zero number — the scale factor.
\[ 2 : 3 \;\xrightarrow{\times 4}\; 8 : 12 \qquad 8 : 12 \;\xrightarrow{\div 4}\; 2 : 3 \]The cross-product test: \(\dfrac{a}{b} = \dfrac{c}{d}\) when \(a \times d = b \times c\). This is the fastest way to check equivalence without simplifying both ratios.
Ratio Table — Scaling a Recipe
A muffin recipe uses flour and sugar in a 3 : 1 ratio. Here's the ratio table for multiple batch sizes:
Ratio Table — Flour to Sugar (3 : 1 base)
| Scale Factor | Flour (cups) | Sugar (cups) | Ratio |
|---|---|---|---|
| ×1 | 3 | 1 | 3 : 1 |
| ×2 | 6 | 2 | 6 : 2 |
| ×3 | 9 | 3 | 9 : 3 |
| ×5 | 15 | 5 | 15 : 5 |
Every row is an equivalent ratio — all simplify back to 3 : 1.
Are \(6 : 10\) and \(15 : 25\) equivalent?
Method 1 — Simplify both:
\[ 6 : 10 \;\xrightarrow{\div 2}\; 3 : 5 \qquad 15 : 25 \;\xrightarrow{\div 5}\; 3 : 5 \]Method 2 — Cross-product: \(6 \times 25 = 150\) and \(10 \times 15 = 150\). Equal ✓
Yes, they are equivalent. Both represent the same ratio 3 : 5.
Find the missing value: \(5 : 8 = ? : 40\).
- What scale factor takes 8 to 40? \(40 \div 8 = 5\). So the scale factor is 5.
- Apply the same factor to the first term: \(5 \times 5 = 25\).
Check: cross-products \(5 \times 40 = 200\) and \(8 \times 25 = 200\). ✓
A car travels 150 miles on 5 gallons. Using an equivalent ratio, find how many gallons are needed for 360 miles.
- Base ratio: 150 mi : 5 gal = \(150 : 5\). Simplify: \(30 : 1\) (30 mpg).
- Set up: \(30 : 1 = 360 : ?\). Scale factor: \(360 \div 30 = 12\).
- Gallons needed: \(1 \times 12 = 12\) gallons.
The 360-mile trip requires 12 gallons.
Test yourself before moving on:
- Are \(4 : 10\) and \(6 : 15\) equivalent? Show your work.
- Find the missing number: \(3 : 7 = 18 : ?\)
- A ratio table has the row \(? : 20\) where the base ratio is \(1 : 4\). What is the missing term?
▶ Show Answers
- \(4:10 \div 2 = 2:5\). \(6:15 \div 3 = 2:5\). Yes, equivalent. (Cross-check: \(4\times15=60\), \(10\times6=60\) ✓)
- Scale factor: \(18 \div 3 = 6\). Missing term: \(7 \times 6 =\) 42. So \(3:7 = 18:42\).
- Scale factor: \(20 \div 4 = 5\). Missing term: \(1 \times 5 =\) 5. Row is \(5 : 20\).
- Adding instead of multiplying to scale: Starting from 2 : 3, adding 4 to both gives 6 : 7 — not equivalent. You must multiply both terms by the same factor.
- Applying different factors to each term: Multiplying the first term by 3 and the second by 5 destroys the ratio. The scale factor must be identical for both.
- Mixing up the cross-product: For a/b = c/d, the cross-products are a×d and b×c — not a×b and c×d. Keep the diagonals straight.
- Equivalent ratios are created by multiplying or dividing both terms by the same non-zero number.
- A ratio table lists a family of equivalent ratios — every row is the same relationship at a different scale.
- Cross-product test: \(a : b = c : d\) when \(a \times d = b \times c\).
- Finding a missing term: identify the scale factor from the known terms, then apply it to the unknown side.
Nurses use equivalent ratios every shift when calculating medication doses. If a standard adult dose is 500 mg per 70 kg of body weight, a nurse must scale that ratio to match each patient's actual weight. A 56 kg patient receives a proportionally smaller dose — calculated using equivalent ratios. In pediatric care, where doses are tiny fractions of adult amounts, this precision is literally life-saving.
Calculator Connection
Use the Ratio Solver to quickly generate equivalent ratios or find a missing term. The Proportion Solver lets you set up \(a : b = c : d\) and solve for any unknown.
Try it with the Calculator
Apply what you've learned with these tools.
Equivalent Ratios — Quiz
5 questions per attempt · Beginner · Pass at 70%
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