Problem
Write the rate as a fraction to read off both terms
A proportion is to be filled in so that its value and the product of its outer terms are both as given.
Ratios
Your answer
How to solve
Strategy Convert to Algebra — Only one box has a partner already filled in, so start there -- the value settles it in one step. That first term is an outer term, so the product of the outer terms then gives the last one, and the value gives the one left.
1STEP 1
Fill the box whose partner is known
The first ratio is something to 20, and its value is 1.5, so the first term is 1.5 times 20.
20 × 1.5 = 30
The first term is 30.
7.RP.A.2Convert To Algebra2STEP 2
Use the product of the outer terms
The outer terms are the first and the last, and they multiply to 360.
360 ÷ 30 = 12
The last term is 12.
6.RP.A.3Work BackwardsThe first and last terms multiply to .
Why?
In a proportion the two ratios have the same value, so scaling the first pair by the second's bottom and the second pair by the first's bottom gives the same amount twice.
🧱Multiplicative identity (one)One group of a number is just that number — multiplying by one changes nothing.
Why?
That is exactly the statement that the outer two multiply to the same as the inner two, and the problem gives that product as .
🧱Transitivity of equalityIf the first equals the second and the second equals the third, the first equals the third.
3STEP 3
Fill the last box from the value
The second ratio is something to 12 with the same value 1.5.
12 × 1.5 = 18
The third term is 18.
7.RP.A.2Eliminate PossibilitiesAnswer
30 : 20 = 18 : 12
Check all three conditions: 30 ÷ 20 = 1.5 and 18 ÷ 12 = 1.5; the outer terms give 30 × 12 = 360; and the inner terms give 20 × 18 = 360 too, as a proportion demands.
Takeaway
Start where only one thing is unknown -- then each answer unlocks the next.
- Fill the box whose partner is known
- Use the product of the outer terms
- Fill the last box from the value