Depth · one concept from first principles · step 2 of 4
Match numerators to bound the denominator
Find how many natural numbers n make 12/n lie strictly between 3/7 and 4/3.
FractionsNumber system
Your answer
How to solve
Strategy Organize Information in More Ways — The unknown sits in a denominator, which is awkward to compare. But 3 and 4 both divide 12, so all three fractions can be given the same numerator 12. With equal positive numerators, the fraction with the larger denominator is the smaller one, so the inequality becomes a plain range for n.
1STEP 1
Give every fraction the numerator 12
Multiply top and bottom of 3/7 by 4 and of 4/3 by 3.
3/7 = 12/28, 4/3 = 12/9
Equivalent fractions do not change value, so the inequality is unchanged.
4.NF.A.2Organize Information In More Ways2STEP 2
Compare by denominators
With the same numerator 12, a bigger denominator gives a smaller fraction, so the order flips: n is between 9 and 28.
12/28 ＜ 12/n ＜ 12/9 ⇔ 9 ＜ n ＜ 28
12/n must be smaller than 12/9, so n ＞ 9; and larger than 12/28, so n ＜ 28.
6.NS.C.7Organize Information In More Wayslies between and exactly when lies between and .
Why?
Rewriting as and as multiplies top and bottom by the same number, so the bounds keep their values.
🧱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.
Why?
With the same top , a bigger bottom number cuts the whole into more, smaller pieces, so of them make less; the order of the bottoms is reversed.
🧱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.
3STEP 3
Count the natural numbers
The natural numbers strictly between 9 and 28 are 10 through 27.
n = 10, 11, …, 27: 27 - 10 + 1 = 18
Inclusive count: last minus first plus one.
6.NS.C.7Make A Systematic ListAnswer
18
Test the edges: n = 10 gives 12/10 = 1.2 ＜ 1.33, which works; n = 27 gives 12/27 = 4/9 ≈ 0.444 ＞ 0.429, which works. n = 9 and n = 28 give exactly 4/3 and 3/7, which the strict inequalities exclude.
Takeaway
If the unknown is in the denominator, make the numerators match and compare the bottoms instead.
- Give every fraction the numerator 12
- Compare by denominators
- Count the natural numbers
Where next?
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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 →