Problem
Decompose into sums of unit fractions
I need to break the single fraction 7/15 into a sum of three unit fractions (fractions with 1 on top), and all three must have different denominators.
Fractions
Your answer
How to solve
Strategy Identify Subproblems — Splitting 7/15 into three pieces at once is hard, so I break it into smaller jobs: first peel off one easy unit fraction, then split the small leftover into two more unit fractions. To split the leftover I try a sensible guess and check it with a common denominator.
1STEP 1
Peel off the first unit fraction
5 divides 15, so the chunk 5/15 collapses to the single unit fraction 1/3.
7/15 = 5/15 + 2/15 = 1/3 + 2/15
A fraction is just a count of equal pieces, so 7 fifteenths can be regrouped as 5 fifteenths plus 2 fifteenths, and 5 fifteenths is one third.
4.NF.B.3Identify SubproblemsThe chunk 5/15 can be pulled out of 7/15 as the single unit fraction 1/3.
Why?
Five of the fifteenths is the same amount as one third, because 15 equal pieces fall into three equal groups of 5.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Pulling that chunk out leaves the remaining fifteenths behind, and the two pieces put back together are the original 7/15 again.
🧱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
Split the leftover 2/15 into two unit fractions
Guessing 1/10 for the rest leaves 4/30 - 3/30 = 1/30, itself a unit fraction.
2/15 - 1/10 = 4/30 - 3/30 = 1/30
Trying a friendly denominator like 10 and checking the leftover with a common denominator turns a hard split into a quick subtraction a fifth-grader can do.
5.NF.A.1Guess And Check3STEP 3
Put the three pieces together and check
The three denominators 3, 10 and 30 all differ, and over 30 they add back to 7/15.
1/3 + 1/10 + 1/30 = 10/30 + 3/30 + 1/30 = 14/30 = 7/15
Adding the three pieces back over a common denominator is the natural way to verify a fraction was split correctly.
5.NF.A.1Guess And CheckAnswer
All three denominators (3, 10, 30) are different and each fraction has numerator 1, so they are genuinely three different unit fractions. Adding them over a common denominator of 30 gives 10/30 + 3/30 + 1/30 = 14/30 = 7/15, exactly the target, so the answer is correct.
Takeaway
Break a fraction apart one easy piece at a time: peel off a unit fraction you recognize, then split the small leftover.
- Peel off the first unit fraction
- Split the leftover 2/15 into two unit fractions
- Put the three pieces together and check
Where next?
Another one like thissuggested
Same look, different logicsuggested
Step back and solidifysuggested