Problem
Surplus and shortage when sharing equally
A teacher shares one pile of marbles equally among her class. Handing out 2 each leaves 9 over. Handing out 3 each uses the pile up exactly. Find how many marbles there are.
Reasoning · Grade 4-1 Division
Your answer
Watch out The 9 looks like just spare marbles, but one extra each means 9 is the class size.
How to solve
Strategy Draw a Diagram — Two unknowns with no numbers attached to either is confusing in words but obvious in a picture, so I draw the students in a row with their marbles and put the 9 leftovers in a separate pile. The picture then makes one thing plain: moving from 2 each to 3 each costs exactly one extra marble per student, and those extra marbles can only come from the leftover pile. That single comparison gives the number of students, after which the total is one multiplication. I finish by checking the answer against both statements and by listing nearby class sizes to show no other number fits.
1STEP 1
Draw the first hand-out
Draw the 9 leftovers to one side.
Drawing the students as identical circles lets you reason about the class even though nobody has told you how big it is.
4.OA.A.3Draw A Diagram2STEP 2
See what changes in the second hand-out
With 3 each every student gets one more.
3 - 2 = 1
Comparing the two pictures instead of computing each one separately turns two unknown situations into one simple statement about the difference between them.
4.OA.A.3Draw A DiagramComparing the two hand-outs turns two unknown situations into one plain statement about the difference between them.
Why?
The class size is the same in both hand-outs, so it cancels the moment one is subtracted from the other.
🧱Shift both by the same amount and the difference holdsSlide two points the same distance along the line and the gap between them never changes.
Why?
Each hand-out is the sweets given out plus the sweets left over, so the two descriptions of the same pile can be set against each other.
🧱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
Turn the leftovers into the number of students
Those extras use up the 9, so there are 9 students.
9 ÷ 1 = 9
Asking how many groups of 1 fit into 9 is exactly the unknown-factor question that division answers, and here the groups are students.
3.OA.B.6Draw A Diagram4STEP 4
Find the total from the exact hand-out
3 each for 9 students makes 27 marbles.
3 × 9 = 27
When nothing is left over, the total is just equal groups multiplied together — the same as counting 9 groups of 3.
3.OA.A.3Draw A Diagram5STEP 5
Check both statements against 27
2 each uses 18 and leaves 9 — it checks.
2 × 9 = 18, 27 - 18 = 9
A guess only becomes an answer once it has been tested against every sentence in the problem, not just the one used to find it.
4.OA.A.3Guess And Check6STEP 6
Show no other class size works
No other class size can work.
2s + 9 = 3s → s = 9
A short ordered list shows the two totals crossing exactly once, which is why no second answer can be hiding further down the list.
4.OA.A.3Make A Systematic ListAnswer
27 marbles
3 × 9 = 27
The answer is a count of marbles, so it must be a whole number, and 27 is. It also has to be a multiple of 3, since 3 marbles each finishes the pile exactly, and 27 = 3 x 9. Sizes make sense too: 27 marbles for 9 students is small enough for a real classroom, and the 9 left over from the first hand-out is exactly one per student as the reasoning requires.
Takeaway
When the same pile is shared two different ways, compare the two hand-outs instead of solving each one — the extra marble each student gains tells you how many students there are.
- Draw the first hand-out
- See what changes in the second hand-out
- Turn the leftovers into the number of students
- Find the total from the exact hand-out
- Check both statements against 27
- Show no other class size works