Scaling Ratios: Formula, Examples and Aptitude Problems
Scaling ratios means multiplying or dividing every term of a ratio by the same non-zero number. The ratio remains equivalent because the comparison between its terms does not change. This page covers the scaling ratios formula, scale factors, equivalent ratios, part-to-total calculations and solved scaling ratios problems for quantitative aptitude.
What Are Scaling Ratios?
For example, scaling 3:5 by 4 gives 12:20. Dividing 12:20 by 4 returns 3:5. The scale factor is the number used on every term. Scaling changes the numerical form of a ratio but not its value, provided the same factor is applied to all terms.
Scaling Ratios Video Lessons
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Ratio Basics: Simplify, Compare and Scale
Learn the fundamentals of ratios by simplifying ratio expressions, comparing quantities, and scaling ratios while preserving their relationships.
Scaling Ratios Formula & Tricks
Important Formulas
Multiply both terms of the ratio by the same non-zero number k.
Divide both terms by the same non-zero number k to obtain an equivalent ratio.
For corresponding values, divide the new value by the old value. The same factor must apply to every related quantity.
For a:b, the total number of parts is a+b. Each quantity equals its ratio part divided by the total parts, multiplied by the given total.
Quick Tricks
To change a ratio into a required equivalent form, divide the required first term by the original first term. Confirm that the same multiplier gives the required second term.
Add the ratio terms first, then divide the total by this sum to find the value of one part.
Two ratios a:b and c:d are equivalent when ad = bc. This verifies whether both terms were scaled by the same factor.
Scaling Ratios Concepts
Finding an Equivalent Ratio
For a:b, multiplying both terms by k gives ka:kb. The value of the ratio remains a/b because ka ÷ kb = a ÷ b. Applying a factor to only one term does not produce an equivalent ratio.
Finding the Scale Factor
If a:b changes to c:d, the ratios are equivalent only when c/a = d/b. This common value is the scale factor. A scale factor greater than 1 enlarges both terms, while a positive factor less than 1 reduces both terms.
Scaling Ratios in Proportional Quantities
If two quantities are x and y in the ratio a:b, write x = ak and y = bk. Use the given total, difference or one quantity to determine k. For a total T, k = T/(a+b). For a difference D, k = D/(b-a), assuming b>a.
Scaling Ratios with Units
Convert each quantity to a common unit, simplify the ratio, and then apply the required scale factor. A ratio of measurements with different units cannot be scaled correctly until the units match.
Ratio of Parts to the Whole
The whole contains a+b equal parts. Thus, the first quantity is a/(a+b) of the total and the second is b/(a+b) of the total. This method applies when the given total is distributed in the stated ratio.
Scaling Ratios Data Table
Use the structured data below while following the explanation and examples.
Worked Scaling Ratio Examples
Each row applies one common factor to all terms of the original ratio.
| Original ratio | Scale factor | Scaled ratio | Check or use |
|---|---|---|---|
| 3:5 | 4 | 12:20 | 12 ÷ 20 = 3 ÷ 5 |
| 8:13 | 2 | 16:26 | Both terms are doubled |
| 18:24 | 1/6 | 3:4 | Both terms are divided by 6 |
| 5:7 | Total = 1,440 | 600:840 | One part = 1,440 ÷ 12 = 120 |
Note: A scale factor may be greater than, equal to or less than 1, but it must be applied to every term.
Practice Scaling Ratios Questions
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Scaling Ratios Revision Points
Use these rules to solve scaling ratios questions and related aptitude problems.
- Multiply or divide every term by the same non-zero factor.
- For a:b scaled by k, write ka:kb.
- Scale factor = new corresponding quantity ÷ original corresponding quantity.
- For a total divided in a:b, use a/(a+b) and b/(a+b).
- For two quantities in a:b with difference D, the common multiplier is D/(b-a), when b>a.
- Convert all measurements to the same unit before forming or scaling a ratio.
- A ratio remains equivalent only when both terms receive the same operation.
Scaling Ratios FAQs
What is the scaling ratios formula?
If the original ratio is a:b and the scale factor is k, the scaled ratio is ka:kb, where k is non-zero.
How do you scale 9:14 to make the first term 45?
The scale factor is 45 ÷ 9 = 5. Therefore, 9:14 becomes 45:70.
Are 12:18 and 2:3 equivalent scaling ratios?
Yes. Dividing both terms of 12:18 by 6 gives 2:3. Equivalently, 12 × 3 = 18 × 2.
What happens if only one term of a ratio is multiplied?
The ratio generally changes and is not equivalent to the original ratio. For example, multiplying only the first term of 3:5 by 2 gives 6:5, not an equivalent ratio.
How do you divide 2,100 in the ratio 4:6?
The total parts are 4+6 = 10, so one part is 2,100 ÷ 10 = 210. The two shares are 840 and 1,260.
How do you find two numbers in the ratio 7:9 when their sum is 128?
The total parts are 16, so one part is 128 ÷ 16 = 8. The numbers are 7 × 8 = 56 and 9 × 8 = 72.