Problem
Population density is people over area
Two villages have known populations, and one village's people-per-area rate is given. The other village is a known amount larger in area. We need its rate.
Ratios
Your answer
How to solve
Strategy Analyze the Units — The rate we want needs an area we do not have. But the other village's rate hands back its area, and the two areas are linked -- so go the long way round rather than looking for a shortcut.
1STEP 1
Get Hyuntae's area from his rate
The rate is population divided by area, so area is population divided by the rate.
160 ÷ 80 = 2
Hyuntae's village is 2 km².
6.RP.A.2Work BackwardsHyuntae's village covers .
Why?
A rate of means the population is eighty people for each square kilometre, so the population is that many equal groups of eighty.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Finding how many such groups make is the division that undoes that multiplying.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
2STEP 2
Step across to Eunwoo's area
Eunwoo's village is 1 km² larger.
2 + 1 = 3
Eunwoo's village is 3 km².
6.RP.A.3Identify Subproblems3STEP 3
Work out Eunwoo's rate
Divide his village's population by the area just found.
195 ÷ 3 = 65
Eunwoo's rate is 65.
6.RP.A.2Analyze The UnitsAnswer
65
Check it forwards: 65 × 3 = 195 people, and 80 × 2 = 160 people -- both villages come back out right.
Takeaway
When the rate you want needs a missing quantity, look for another rate that hands it back.
- Get Hyuntae's area from his rate
- Step across to Eunwoo's area
- Work out Eunwoo's rate