Problem
A percent gap plus a count gap gives the total
A circle graph gives every language's percent, and the gap between two of them is given as a number of students. We need the count for a third language.
RatiosExpressions
Your answer
How to solve
Strategy Work Backwards — Every percent on the graph is measured against the same total, so the gap between two of them is a percent of that same total. That one sentence hands back the total, and everything else follows.
1STEP 1
Turn the gap into a percent
Both shares are measured against the same total, so their difference is a share of that total too.
30 - 20 = 10
The gap is 10% of the school.
6.SP.B.4Analyze The UnitsThe gap between two shares is itself a share of the same whole.
Why?
Both shares count hundredths of the same school, so taking one from the other leaves a number of those same hundredths.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Use the gap to find the total
That 10% is 8 students, so the whole school is ten times as many.
8 ÷ 10 × 100 = 80
There are 80 students altogether.
6.RP.A.3Work Backwards3STEP 3
Take the German share of it
German is 5% of the total just found.
80 × 0.05 = 4
4 students want German.
6.RP.A.3Organize Information In More WaysAnswer
4 students
Count the two languages directly: Chinese is 24 students and Japanese is 16, a gap of 8 exactly as stated.
Takeaway
A difference between two shares of the same whole is itself a share of that whole.
- Turn the gap into a percent
- Use the gap to find the total
- Take the German share of it