Depth · one concept from first principles · step 2 of 4
Rewrite both ends over one denominator
Count the non-integer fractions k/14 that lie strictly between -9/7 and 1/3.
Number system
Your answer
How to solve
Strategy Organize Information in More Ways — Every candidate is a whole number of fourteenths, so rewrite both ends in fourteenths and count the possible numerators k. Then take away the numerators that make an integer.
1STEP 1
Write the left end in fourteenths
Multiply top and bottom of 9/7 by 2.
-9/7 = -18/14
The left end is exactly -18 fourteenths, and it is not included.
6.NS.C.6Organize Information In More Ways2STEP 2
Trap the right end between fourteenths
1/3 is not a whole number of fourteenths, so find the two fourteenths around it by comparing in forty-seconds.
4/14 = 12/42 ＜ 14/42 = 1/3 ＜ 15/42 = 5/14
1/3 sits between 4 and 5 fourteenths, so the largest allowed numerator is 4.
6.NS.C.7Organize Information In More Wayslies between and .
Why?
Renaming , and as , and multiplies each top and bottom by the same number, so no value changes.
🧱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?
Now all three are counts of the same-sized forty-second pieces, so they compare just by counting: .
🧱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 numerators
The numerator k must be an integer with -18 ＜ k and k/14 ＜ 1/3, so k runs from -17 up to 4.
-18 ＜ k ＜ 14/3, k = -17, -16, …, 4; 4 - (-17) + 1 = 22
Counting from one integer to another, inclusive, is last minus first plus one.
6.NS.C.7Make A Systematic List4STEP 4
Remove the integers
k/14 is an integer only when k is a multiple of 14. In the range, those are k = -14 (giving -1) and k = 0 (giving 0).
22 - 2 = 20
Two of the 22 fractions are really integers, so they drop out.
6.NS.C.6Change Focus Count The ComplementAnswer
20
The interval runs from about -1.29 to about 0.33, a length of about 1.62, and fourteenths are spaced about 0.07 apart, and 1.62 × 14 ≈ 22.7, so about 22 fourteenths fit inside. Removing -1 and 0 leaves 20.
Takeaway
When every candidate shares one denominator, turn the question into counting numerators.
- Write the left end in fourteenths
- Trap the right end between fourteenths
- Count the numerators
- Remove the integers
Where next?
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A parent dashboard for the family lives at sensimlab.com.
The same shape shows up later on the AMC, as Multiplicative Ratio with Integrality Constraints, Small-Domain Constrained Search. Why the shapes repeat →