Problem
Scale a simple ratio up to bigger numbers
Among all the ratios with a given value, we need the one whose two terms add to a given number.
Ratios
Your answer
How to solve
Strategy Convert to Algebra — Every ratio with this value is 6 and 7 multiplied by the same number, so the sum is 13 of that number. One division finds it.
1STEP 1
Write the ratio with a multiplier
A value of 6/7 means the simplest ratio is 6 : 7, and every other one multiplies both terms by the same number.
(6 × ■) : (7 × ■)
One unknown decides the whole ratio.
6.RP.A.1Convert To Algebra2STEP 2
Add the two terms
Six multipliers plus seven multipliers is thirteen of them.
6 + 7 = 13
The sum is 13 multipliers.
6.RP.A.3Identify SubproblemsThe two terms together are thirteen multipliers.
Why?
Every ratio with this value is and of the same piece, because multiplying both terms by the same number leaves the value alone.
🧱Multiplicative identity (one)One group of a number is just that number — multiplying by one changes nothing.
Why?
Six pieces and seven pieces put together are thirteen pieces, with none counted twice.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Find the multiplier and build the ratio
Divide the given sum by 13, then multiply both terms.
52 ÷ 13 = 4, 6 × 4 = 24, 7 × 4 = 28
The ratio is 24 : 28.
6.RP.A.3Work BackwardsAnswer
24 : 28
Check both conditions: 24 + 28 = 52, and 24 ÷ 28 reduces to 6/7.
Takeaway
Every ratio with a given value is the simplest one scaled up -- find the scale factor.
- Write the ratio with a multiplier
- Add the two terms
- Find the multiplier and build the ratio