Problem
Common denominators reveal fraction order
I need to find every fraction whose denominator is 72 that sits strictly between 3/8 and 4/9.
Fractions
Your answer
How to solve
Strategy Make a Systematic List — If I rewrite both boundary fractions with the denominator 72, the whole problem turns into a simple question about numerators: which whole numbers fall between the two boundary numerators? Then I just list those numerators in order over 72. Both 8 and 9 divide evenly into 72, which is why 72 is the natural common denominator to use.
1STEP 1
Convert the lower boundary to /72
Since 72 = 8 x 9, I multiply 3/8 by 9/9 to give it a denominator of 72.
3/8 = (3 × 9)/(8 × 9) = 27/72
Multiplying top and bottom by the same number gives an equal fraction, just cut into more pieces.
4.NF.A.1Solve An Easier Related Problem2STEP 2
Convert the upper boundary to /72
I multiply 4/9 by 8/8 to give it a denominator of 72.
4/9 = (4 × 8)/(9 × 8) = 32/72
Same trick: an equivalent fraction lets me compare both boundaries against the same size of piece.
4.NF.A.1Solve An Easier Related Problem3STEP 3
Compare using equal-size pieces
Now every fraction has the same denominator, so I only need the numerator to be bigger than 27 and smaller than 32.
27/72 ＜ n/72 ＜ 32/72
When the pieces are all the same size, more pieces simply means a bigger fraction.
4.NF.A.2Make A Systematic ListOnce every fraction is written over 72, only the numerators need to be compared.
Why?
A denominator of 72 means the whole is cut into 72 equal pieces, so each fraction is just a count of those same-size pieces.
🧱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?
When the pieces are all the same size, having more pieces always means having more of the whole.
Why?
Setting the pieces side by side pairs each piece here with one piece there, so whichever side has pieces left over is the larger amount.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
4STEP 4
List the whole-number numerators between 27 and 32
The whole numbers strictly between 27 and 32 are 28, 29, 30, and 31, so each becomes a fraction over 72.
n = 28, 29, 30, 31
Listing the in-between whole numbers in order guarantees I do not skip any and do not include the endpoints.
4.NF.A.2Make A Systematic ListAnswer
, , ,
Each answer is between 27/72 and 32/72, so each is bigger than 3/8 and smaller than 4/9. The endpoints 27/72 (=3/8) and 32/72 (=4/9) are correctly excluded because the problem says greater than and less than, not equal to.
Takeaway
Give every fraction the same bottom number, and then the top numbers tell you the order at a glance.
- Convert the lower boundary to /72
- Convert the upper boundary to /72
- Compare using equal-size pieces
- List the whole-number numerators between 27 and 32
Where next?
Level upsuggested
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