Problem
A rate is the ratio written as a number
A ratio has a known rate, and its two quantities differ by a known amount. We need the ratio itself.
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
The shape of the ratio is 3 : 5.
6.RP.A.1Convert To Algebra2STEP 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 × ■)
One unknown decides the size.
6.RP.A.3Guess And Check3STEP 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
The multiplier is 5.
6.EE.B.7Identify SubproblemsThe difference between the two quantities is two multipliers.
Why?
Both terms are built from the same multiplier, five of it and three of it, so taking the smaller from the larger leaves two of it.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Splitting off three copies from five leaves exactly two, with nothing unaccounted for.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
4STEP 4
Build the ratio
Multiply both 3 and 5 by that number.
3 × 5 = 15, 5 × 5 = 25
The ratio is 15 : 25.
6.RP.A.3Identify SubproblemsAnswer
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