Problem
Recover the whole from a percent
Every sixth grader is to get one notebook. 60 % of them already have theirs. The 24 still waiting are everybody else. Find how many sixth graders there are.
Reasoning · Grade 6-1 Ratio and Rate (1)
Your answer
Watch out The 24 look like the 60 %, but they are the 40 % still waiting.
How to solve
Strategy Draw a Diagram — The one number I am handed, 24, is attached to the group the problem never names a percent for, so the first job is to see that 24 is the other 40 %. A bar drawn for the whole sixth grade and cut into ten equal 10 % strips makes that visible and, better, makes the whole thing countable: six strips are shaded, four strips together hold 24 students, so one strip holds 6. Then I work backwards along the same relation the ordinary direction uses — instead of multiplying the whole by a rate to get a part, I divide a part by its rate to get the whole. Finally I guess-and-check the candidate against both sentences of the problem, because the classic error here is to divide 24 by the rate that belongs to the other group.
1STEP 1
Find the percent that belongs to the 24 students
The 24 are 40 % of the whole.
100 % - 60 % = 40 %
Percents of a whole behave like the parts of one full bar: if you know how much of the bar one part covers, the rest of the bar is whatever is left over from 100 %.
6.RP.A.3Work Backwards2STEP 2
Draw the sixth grade as a bar in ten strips
Draw the whole as a bar of ten strips.
4 strips = 40 % = 24 students
Cutting the whole into ten equal tenths turns a percent into something you can count, and 60 % and 40 % both land on strip boundaries, so no strip has to be split.
6.RP.A.3Draw A Diagram3STEP 3
Count one strip, then all ten
Four strips are 24, so one strip is 6.
24 ÷ 4 = 6, 6 × 10 = 60
Finding the value of one equal part and then rebuilding the whole from it is the same unit-rate move used for prices and recipes, only here the unit is 10 % of a grade.
6.RP.A.3Draw A DiagramCounting one strip and then all ten rebuilds the whole grade from the part that is known.
Why?
The bar is cut into ten equal strips, so one strip is one of ten equal shares of the whole.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
Why?
Multiplying one share back by ten undoes the sharing into ten, so the whole is recovered exactly.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
4STEP 4
Do the same step as a single division
Ten strips make 60.
(sixth graders) × 0.4 = 24 → (sixth graders) = 24 ÷ 0.4 = 60
Dividing by 0.4 asks how many groups of four tenths fit into 24, and the answer, 60, is bigger than 24 — which is right, because the whole grade must be bigger than the part still waiting.
6.NS.B.3Work Backwards5STEP 5
Check 60 against both sentences
Putting 60 back in checks out.
60 × 0.6 = 36, 60 - 36 = 24
Pairing each number with the group it describes is the whole safeguard here — a percent means nothing until you say what it is a percent of.
6.RP.A.3Guess And CheckAnswer
60 students
24 ÷ 0.4 = 60
The answer counts students, so it has to be a whole number, and 60 is. It also has to be larger than 24, since the 24 waiting students are only part of the grade, and 60 is comfortably larger. The split it predicts, 36 with notebooks and 24 without, adds back to 60, and 36 out of 60 really is 60 %, since 36/60 = 6/10. A sixth grade of 60 is also a believable size for a school, roughly two or three classes.
Takeaway
When a count is given for the group with no percent attached, name its percent first — then the whole is just that count divided by its own rate.
- Find the percent that belongs to the 24 students
- Draw the sixth grade as a bar in ten strips
- Count one strip, then all ten
- Do the same step as a single division
- Check 60 against both sentences