Problem
Find the whole from a fraction of a fraction
A herdsman drives 70 cattle past. Those 70 are two-thirds of one third of the whole herd. The rest are still out in the field. Find how big the whole herd is.
Reasoning · Grade 6-2 Fraction Calculation in Action
Your answer
Watch out The 70 look like two-thirds, but they are two-thirds of a third — two-ninths.
How to solve
Strategy Draw a Diagram — The hard part of this problem is not the arithmetic, it is the phrase 'two thirds of one third'. A bar drawing settles it instantly: draw the whole herd as one long bar, cut it into three equal thirds, then cut a third into three smaller blocks and shade two of them. Once every third is cut the same way, the bar is nine equal blocks and the mysterious fraction of a fraction turns into the plain statement 'the 70 cattle are 2 of the 9 blocks'. That is a much easier related problem, because now nothing is left but whole-number thinking: 2 blocks hold 70, so 1 block holds 35, so 9 blocks hold 9 times 35. Working backwards from the part to the whole is exactly what a bar model is built for, and dividing by a fraction becomes a check at the end rather than the only route in.
1STEP 1
Draw the whole herd as one bar and cut it into thirds
Draw the herd as a bar cut into thirds.
whole herd → 1/3 ∣ 1/3 ∣ 1/3, one 1/3 → □ ∣ □ ∣ □ (shade 2)
A fraction of a fraction is nothing more mysterious than a cut inside a cut. If you can share a chocolate bar into three equal strips and then share one strip with a friend, you have already done the whole idea; the drawing just keeps track of which cut came first.
5.NF.B.4Draw A Diagram2STEP 2
Cut every third the same way, so the whole bar is 9 equal blocks
Cutting again makes nine blocks.
2/3 × 1/3 = (2 × 1)/(3 × 3) = 2/9
Cutting every third the same way is the whole trick behind multiplying fractions: three cuts one way and three cuts the other way make a 3 × 3 array of nine identical blocks, which is why the denominators multiply. Once the pieces match, comparing them is just counting.
5.NF.B.4Solve An Easier Related ProblemCutting every third the same way turns the whole bar into nine blocks that are all the same size.
Why?
A fraction only means anything when the whole is cut into pieces all the same size, and cutting each third alike delivers exactly that.
🧱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?
Cutting each piece into the same number of smaller pieces multiplies the count of pieces without changing any amount.
🧱Scaling top and bottom keeps a fraction's valueMultiplying or dividing a fraction's top and bottom by the same number only renames it — the pieces change size but the amount you hold does not.
3STEP 3
Two blocks hold 70 cattle, so one block holds 35
The 70 fill two blocks.
70 ÷ 2 = 35 cattle in one block
Asking 'what times 2 gives 70?' is the same as asking '70 divided by 2', which is why division undoes the doubling. A young solver does not need fraction division here at all, only the fact that equal blocks hold equal numbers.
3.OA.B.6Work Backwards4STEP 4
Nine blocks make the whole herd, so multiply by 9
One block is 35 cattle.
9 × 35 = 9 × 30 + 9 × 5 = 270 + 45 = 315
Once the bar is cut into equal blocks, finding the whole is pure repeated addition: nine equal groups of 35. Splitting 35 into 30 and 5 keeps the multiplication inside numbers a fourth grader already multiplies in their head.
4.NBT.B.5Draw A Diagram5STEP 5
Check by undoing the fraction directly
Nine blocks make 315 cattle.
70 ÷ 2/9 = 70 × 9/2 = 630/2 = 315
Multiplying by 9/2 says out loud what the picture did: divide by 2 to get one block, then multiply by 9 to get all nine. Turning the division upside down is a shortcut for those two moves, not a new idea.
6.NS.A.1Work Backwards6STEP 6
Read the answer back into the story
Read back into the story it checks out.
315 ÷ 3 = 105, 105 × 2/3 = 70, 315 - 70 = 245
Putting the answer back into the original sentence is the safest check there is, because it uses the story rather than the algebra. It also shows the numbers were chosen kindly: every third of the herd is a whole number of animals, as it must be for real cattle.
5.NF.B.6Draw A DiagramAnswer
315 cattle
70 × 9 ÷ 2 = 315
The units are right: the answer counts animals, and 315 is a whole number, as a herd must be. The magnitude is right too. The 70 cattle are only 2/9 of the herd, a bit less than a quarter of it, so the total should be roughly four to five times 70; 315 is 4.5 times 70, right in that range, and it sits sensibly between 70 (if he had brought them all) and 9 × 70 = 630 (which would be the answer if the 70 were only one ninth). The internal consistency also holds: 315 divides by 3 to give 105 and by 9 to give 35, both whole numbers, so the thirds and ninths of the herd are all whole animals, and 315 × 2/9 = 70 exactly. A common wrong turn is to multiply instead of divide and answer 70 × 2/9, but that gives a number smaller than 70, which cannot be a herd containing the 70.
Takeaway
A fraction of a fraction is just a cut inside a cut: chop the whole herd into 9 equal blocks and the riddle shrinks to 70 divided by 2.
- Draw the whole herd as one bar and cut it into thirds
- Cut every third the same way, so the whole bar is 9 equal blocks
- Two blocks hold 70 cattle, so one block holds 35
- Nine blocks make the whole herd, so multiply by 9
- Check by undoing the fraction directly
- Read the answer back into the story