Problem
Several ways to build unit-fraction sums
A unit fraction has 1 on top. Grab the biggest unit fraction that still fits underneath, and subtract it. If what is left is not a unit fraction, repeat. Write 4/9 and 23/36 as sums of unit fractions.
Reasoning · Grade 5-1 Egyptian Fractions
Your answer
Watch out Grabbing greedily can pick one bigger than the fraction, and the subtraction fails.
How to solve
Strategy Solve an Easier Related Problem — The method is built to make the problem shrink: after each subtraction I am left with a smaller fraction and face exactly the same question again, so the whole task is one easy step repeated until the leftover is a unit fraction. Inside each step the only real decision is which unit fraction to take, and that is a bounded guess-and-check: try 1/2, then 1/3, then 1/4, … and stop at the first one that is smaller than the fraction in hand. I use the worked 3/7 example as the pattern to copy, and I check both finished answers by adding the unit fractions back up.
1STEP 1
How to find the largest unit fraction that fits
A unit fraction shrinks as its bottom grows.
1/2 ＞ 1/3 ＞ 1/4 ＞ 1/5 ＞ …
Comparing a unit fraction with another fraction is a Grade 4 skill, and here you only have to check a couple of candidates before one fits, because the fractions in the problem are not tiny.
4.NF.A.2Guess And CheckThe largest unit fraction that still fits underneath a fraction is found by seeing how many times the fraction goes into one whole.
Why?
A unit fraction is one of the equal pieces a whole is cut into, so asking which one fits is asking how fine the cut must be.
🧱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.
Why?
Counting how many whole times the fraction fits into one, and taking the next size down, is exactly a division with a leftover.
🧱Division with remainderAny number splits into a whole count of equal groups plus a leftover, and the leftover is always smaller than the group size.
2STEP 2
(1) 4/9: the biggest unit fraction underneath it
The biggest fitting under 4/9 is 1/3.
1/3=3/9 ＜ 4/9 ＜ 9/18=1/2 → use 1/3
Rewriting both fractions with the same bottom number turns the comparison into comparing whole numbers on top, which is the safest way to be sure which one is bigger.
4.NF.A.2Guess And Check3STEP 3
(1) Subtract, and see that the job is already done
Subtracting leaves 1/9, already a unit fraction.
4/9-1/3=4/9-3/9=1/9
Because 3 divides 9, the two fractions share the denominator 9 straight away — no hunting for a common denominator is needed, and the subtraction is a single-digit one.
5.NF.A.1Solve An Easier Related Problem4STEP 4
(1) Write the answer and add it back up
So 4/9 = 1/3 + 1/9.
4/9=1/3+1/9
A fraction with 4 on top is 4 copies of 1/9; splitting those 4 copies as 3+1 is exactly what turns 4/9 into 1/3+1/9.
4.NF.B.3Solve An Easier Related Problem5STEP 5
(2) 23/36: the biggest unit fraction underneath it
The biggest fitting under 23/36 is 1/2.
1/2=18/36 ＜ 23/36 ＜ 36/36=1 → use 1/2
Checking a fraction against 1/2 is the quickest first move: double the top and compare with the bottom — 23+23=46 is more than 36, so the fraction beats a half.
4.NF.A.2Guess And Check6STEP 6
(2) First subtraction
Subtracting leaves 5/36.
23/36-1/2=23/36-18/36=5/36
The leftover is much smaller than the fraction you started with, which is why the method always finishes: each round leaves a fraction with a smaller numerator than the last.
5.NF.A.1Solve An Easier Related Problem7STEP 7
(2) The biggest unit fraction underneath 5/36
The biggest fitting under 5/36 is 1/8.
1/8 ＜ 5/36 ＜ 1/7 → use 1/8
Dividing 36 by 5 gives 7 with 1 left over, which is why the answer sits between 1/7 and 1/8 — the remainder tells you to round the 7 up to 8.
4.NF.A.2Guess And Check8STEP 8
(2) Second subtraction ends the process
Subtracting again leaves 1/72 and stops.
5/36-1/8=10/72-9/72=1/72
Finding a common denominator for 36 and 8 is Grade 5 work: 72 is the first number both divide into, and after that the subtraction is 10-9.
5.NF.A.1Solve An Easier Related Problem9STEP 9
Collect and verify both decompositions
Adding back confirms both answers.
1/2+1/8+1/72=36/72+9/72+1/72=46/72=23/36
Reducing 46/72 back to 23/36 by dividing top and bottom by 2 is the equivalence check that proves nothing was lost along the way.
4.NF.A.1Look For A PatternAnswer
1/3 + 1/9, 1/2 + 1/8 + 1/72
23/36 − 1/2 = 5/36
Both answers add back to exactly the fraction they came from — 3/9+1/9=4/9 and 36/72+9/72+1/72=46/72=23/36 — and every piece really is a unit fraction with all pieces different. The sizes are believable: 4/9 is a bit under a half, so its biggest piece should be 1/3 with a small tail, and it is; 23/36 is a bit under 2/3, so its biggest piece should be 1/2 with roughly 1/7 left over, and 1/8+1/72 is just that. The number of rounds also matches the method's own logic: the numerator falls from 23 to 5 to 1 in part (2) and from 4 to 1 in part (1), and the process must end once it reaches 1. Running the same greedy steps on the book's example reproduces 3/7=1/3+1/11+1/231 exactly, which confirms the procedure was applied the way the book intends.
Takeaway
Keep grabbing the biggest unit fraction that still fits and subtracting it — the leftover shrinks every time, so the process always finishes.
- How to find the largest unit fraction that fits
- (1) 4/9: the biggest unit fraction underneath it
- (1) Subtract, and see that the job is already done
- (1) Write the answer and add it back up
- (2) 23/36: the biggest unit fraction underneath it
- (2) First subtraction
- (2) The biggest unit fraction underneath 5/36
- (2) Second subtraction ends the process
- Collect and verify both decompositions