Depth · one concept from first principles · step 2 of 4

A rate is the ratio written as a number

Find the ratio that satisfies both conditions.

The rate is $0.6$ .
The base and the compared quantity differ by $10$ .

RatiosExpressions

Your answer

How to solve

Strategy Convert to Algebra — A rate alone leaves infinitely many ratios, all the same shape. Write the simplest one, carry an unknown multiplier, and let the difference pin that multiplier down.

1STEP 1

Write the rate as the simplest ratio

A rate of 0.6 is 6/10, which reduces to 3/5, so the compared quantity is 3 when the base is 5.

0.6 = 6/10 = 3/5

2STEP 2

Carry an unknown multiplier

Every ratio with this rate is 3 and 5 both multiplied by the same number, so write them that way.

(3 × ■) : (5 × ■)

3STEP 3

Use the difference to find the multiplier

The two quantities differ by 5 multipliers minus 3 multipliers, which is 2 of them, and that difference is 10.

2 × ■ = 10 → ■ = 5

4STEP 4

Build the ratio

Multiply both 3 and 5 by that number.

3 × 5 = 15, 5 × 5 = 25

Answer

15 : 25

Check both conditions: 15 ÷ 25 = 0.6, and 25 - 15 = 10.

Takeaway

A rate fixes a ratio's shape but not its size -- one more condition sets the size.

Write the rate as the simplest ratio
Carry an unknown multiplier
Use the difference to find the multiplier
Build the ratio

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