Depth · one concept from first principles · step 2 of 4
Smaller denominator means larger unit fraction
Find the greatest natural number that can go in .
Fractions
Your answer
How to solve
Strategy Solve an Easier Related Problem — First multiply each side to turn the scary two-fraction comparison into a comparison of two single unit fractions — an easier related problem. Once both sides are unit fractions, the rule 'smaller denominator means larger fraction' tells us exactly which way the inequality points. Then we guess-and-check the boundary whole number to find the biggest one that still works.
1STEP 1
Multiply the left side
Multiply the two unit fractions on the left by multiplying the denominators: 5 times 12 is 60. So the left side becomes one-sixtieth.
1/5 × 1/12 = 1/60
Multiplying two unit fractions just multiplies the bottom numbers — a fifth of a twelfth is one out of sixty equal pieces, which a 5th grader can picture.
5.NF.B.4Solve An Easier Related Problem2STEP 2
Multiply the right side
The right side becomes one over (8 times the box), another unit fraction.
1/(■) × 1/8 = 1/(8 × ■)
The same denominator-times-denominator rule turns the right side into a single unit fraction, so now both sides have a 1 on top.
5.NF.B.4Solve An Easier Related Problem3STEP 3
Compare two unit fractions
For unit fractions a bigger denominator means a smaller fraction, so 8 times the box must stay under 60.
1/60 ＜ 1/(8 × ■) ⇔ 8 × ■ ＜ 60
Cut a whole into more pieces and each piece is smaller; so the fraction with the bigger denominator is the smaller fraction. That flips the comparison into one about the denominators.
4.NF.A.2Solve An Easier Related ProblemBetween two unit fractions, the one with the smaller denominator is the larger.
Why?
A unit fraction is one piece of a whole cut into as many equal pieces as the denominator says, so cutting the whole less finely leaves each piece bigger.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Comparing one piece against one piece is therefore just comparing how finely each whole was cut.
Why?
The single piece taken from each side pairs straight across, so the side cut into fewer pieces holds the bigger one.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
4STEP 4
Find the biggest whole number
8 x 7 = 56 clears 60 but 8 x 8 = 64 does not, so the largest that works is 7.
8 × 7 = 56 ＜ 60 ✓ 8 × 8 = 64 ＞ 60 ×
Testing the two numbers right at the edge confirms exactly where 'works' turns into 'too big', so we are sure 7 is the largest.
3.NF.A.3Guess And CheckAnswer
7
Plug in 7: right side is one-over (8 times 7) = one-fifty-sixth. Is one-sixtieth less than one-fifty-sixth? Yes, because 60 pieces are smaller than 56 pieces. And 8 would give one-sixty-fourth, which is smaller than one-sixtieth, so the inequality would fail. So 7 is right.
Takeaway
Multiply each side first, then remember: for unit fractions, a smaller bottom number means a bigger fraction.
- Multiply the left side
- Multiply the right side
- Compare two unit fractions
- Find the biggest whole number
Where next?
Level upsuggested
Another one like thissuggested
Same look, different logicsuggested
Step back and solidifysuggested
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The same shape shows up later on the AMC, as Multiplicative Ratio with Integrality Constraints, Small-Domain Constrained Search. Why the shapes repeat →