Problem
Match fraction forms to compare sizes
I need to count how many whole numbers can replace the star so that the improper fraction star/8 sits strictly between the mixed numbers 4 and 3/8 and 5 and 1/8.
Fractions
Your answer
Watch out It's easy to also count the endpoints themselves (35 and 41) — but the inequality is strict (no equals sign), so star/8 can never equal 4 3/8 or 5 1/8.
How to solve
Strategy Organize Information in More Ways — To compare fairly, rewrite both mixed-number endpoints as improper fractions over 8 so every quantity has the same denominator. Then the numerators can be compared directly and the valid whole numbers listed.
1STEP 1
Rewrite the left endpoint over 8
Convert 4 3/8 to eighths: 4 wholes is 32 eighths, plus 3 eighths, giving 35/8.
4\tfrac{3}{8} = 4× 8 + 3/8 = 35/8
Matching denominators lets us compare fractions just by their numerators.
3.NF.A.3Organize Information In More WaysThe mixed number 4 3/8 is the same amount as the improper fraction 35/8.
Why?
The 4 whole ones by themselves are exactly 32 eighths.
Why?
One whole is 8 eighths, because cutting a whole into 8 equal parts fills it with no gaps or overlaps.
🧱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?
4 wholes is 4 equal groups of those 8 eighths, and 4 groups of 8 is 32.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
4 3/8 is those 32 eighths joined with the 3 eighths it already names, which is 35 eighths in all.
Why?
A mixed number is its whole part and its fraction part put together with no overlap, so the two parts add back to one amount.
🧱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?
The 32 eighths and the 3 eighths are the very same size of piece, so counting them together just gives 35 eighths.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Rewrite the right endpoint over 8
Convert 5 1/8 to eighths: 5 wholes is 40 eighths, plus 1 eighth, giving 41/8.
5\tfrac{1}{8} = 5× 8 + 1/8 = 41/8
Now all three quantities are eighths, so only the top numbers matter.
3.NF.A.3Organize Information In More Ways3STEP 3
Compare numerators and list
35/8 less than star/8 less than 41/8, so star is a whole number between 35 and 41: 36, 37, 38, 39, 40.
35 less than \star less than 41 → \star \in \{36,37,38,39,40\}
With equal denominators, comparing fractions is the same as comparing whole-number numerators.
3.NF.A.3Make A Systematic List4STEP 4
Count them
There are five whole numbers in the list.
\{36,37,38,39,40\} → 5
Just count the items in the systematic list.
3.OA.A.1Make A Systematic ListAnswer
5
{36, 37, 38, 39, 40} → 5
The endpoints 35 and 41 are 6 apart; excluding both ends leaves the 5 whole numbers in between, which matches the count. Each candidate, like 36/8 = 4.5, indeed lands between 4 3/8 and 5 1/8.
Takeaway
This only needs Grade 3 fraction sense: make the bottoms match, then just compare the top numbers!
- Rewrite the left endpoint over 8
- Rewrite the right endpoint over 8
- Compare numerators and list
- Count them
Where next?
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