Scaling in Both Directions to a Non-Unit Intermediate Ratio
Sometimes scaling a ratio to a target value requires first scaling backward to a simpler non-unit intermediate ratio, then scaling forward to the target. For a ratio of 12:18 where you need the first quantity to equal 10: divide both by 6 to get the intermediate ratio 2:3, then multiply both by 5 to reach 10:15. This two-step scaling strategy from Reveal Math, Course 1, Module 1 helps 6th graders solve ratio problems where direct scaling would produce difficult fractions or decimals.
Key Concepts
To scale in both directions using a non unit intermediate ratio, divide both quantities by a common factor $c$ to find a simpler equivalent ratio, then multiply by a new factor $d$ to reach the target ratio: $$a : b \xrightarrow{\div c} \frac{a}{c} : \frac{b}{c} \xrightarrow{\times d} \left(\frac{a}{c} \times d\right) : \left(\frac{b}{c} \times d\right)$$ where the intermediate ratio $\frac{a}{c} : \frac{b}{c}$ does not contain a $1$.
Common Questions
What is scaling in both directions for ratio problems?
It means first scaling a ratio backward (dividing) to a simpler intermediate ratio, then scaling forward (multiplying) to reach the target value. The intermediate ratio is not necessarily a unit rate.
Find an equivalent ratio to 12:18 where the first quantity is 10.
Divide both by 6 to get the intermediate ratio 2:3. Multiply both by 5 to get 10:15. The equivalent ratio is 10:15.
Find an equivalent ratio to 20:15 where the second quantity is 21.
Divide both by 5 to get the intermediate ratio 4:3. Multiply both by 7 to get 28:21. The equivalent ratio is 28:21.
When do I need to use a non-unit intermediate ratio?
Use this method when you cannot scale directly from the original to the target using a single whole number, especially when the target value is not a multiple of the original denominator.
How is this different from finding a unit rate?
Finding a unit rate scales all the way down to a denominator of 1. This method stops at a simpler but non-unit intermediate ratio before scaling up to the target.
When do 6th graders learn this two-step scaling strategy?
Module 1 of Reveal Math, Course 1 covers scaling in both directions in the Ratios and Rates unit.