BTT Mathematics / Primary Mathematics Learning Hub / Unit-Fraction Decomposition
A unit fraction has numerator one. Examples are 1/2, 1/5 and 1/12. A proper fraction can sometimes be represented as a sum of distinct unit fractions. In this guide, unless a question says otherwise, an “Egyptian-fraction decomposition” means a sum of distinct positive unit fractions equal to the original proper fraction.
For example, 2/3 = 1/2 + 1/6 because 1/2=3/6 and 1/6 remains 1/6, giving 4/6=2/3. The equality is what matters. A decomposition is not accepted because the denominators look plausible; it must reconstruct the same quantity.
This guide is Primary Mathematics enrichment. It deepens Fractions of a Quantity, Equivalence and Operations, Fractions, Decimals and the Same Whole and Systematic Listing. The named historical style is optional enrichment; the core mathematics is fraction equivalence and decomposition.
Use the MOE Primary curriculum page and the learner’s school programme for required content.
Unit fractions · Verify decompositions · Build decompositions · Split one unit fraction · Non-uniqueness · 24 questions · Worked answers · Teaching and transfer
1. Unit fractions name one equal part
The denominator tells how many equal parts make one whole; the numerator one says we have one of those parts. As the positive denominator increases, the unit fraction becomes smaller: 1/3>1/4>1/5.
Worked example A: Identify unit fractions
Among 1/7, 2/7, 1/12 and 5/8, the unit fractions are 1/7 and 1/12.
Worked example B: Same whole
1/2 + 1/6 refers to one-half and one-sixth of the same whole. Convert to sixths: 3/6+1/6=4/6=2/3.
Worked example C: Compare candidates
For decomposing 3/5, 1/2 is a sensible first unit fraction because it is below 3/5, while 1/1 is too large. The remainder after taking 1/2 is 3/5−1/2=6/10−5/10=1/10. Hence 3/5=1/2+1/10.
Worked example D: A false decomposition
2/3 is not equal to 1/2+1/5. The sum is 5/10+2/10=7/10. A decomposition must be verified by common denominators or another exact argument.
2. Verification is ordinary fraction addition
Every proposed decomposition can be checked by adding the unit fractions and simplifying.
Worked example E: Two terms
5/6 = 1/2+1/3 because 3/6+2/6=5/6.
Worked example F: Three terms
7/8 = 1/2+1/4+1/8 because 4/8+2/8+1/8=7/8.
Worked example G: Mixed denominators
4/5 = 1/2+1/4+1/20. In twentieths, 10/20+5/20+1/20=16/20=4/5.
Worked example H: Detect an over-count
3/4 is not 1/2+1/3 because 3/6+2/6=5/6, which is larger than 3/4. An estimate can reject the claim before exact common-denominator work.
3. Build a decomposition by taking a unit fraction and decomposing the remainder
One practical strategy is: choose a unit fraction not exceeding the target; subtract it exactly; then represent the positive remainder. The strategy needs checking because different choices can lead to different valid decompositions.
Worked example I: 5/8
Take 1/2=4/8. Remainder=1/8. Therefore 5/8=1/2+1/8.
Worked example J: 7/10
Take 1/2=5/10. Remainder=2/10=1/5. Therefore 7/10=1/2+1/5.
Worked example K: 5/12
Take 1/3=4/12. Remainder=1/12. Therefore 5/12=1/3+1/12.
Worked example L: 7/12
Take 1/2=6/12. Remainder=1/12. Therefore 7/12=1/2+1/12.
Worked example M: 11/12
Take 1/2, leaving 5/12. Then use 1/3=4/12, leaving 1/12. Thus 11/12=1/2+1/3+1/12.
Worked example N: 11/15
Take 1/2. In thirtieths, 11/15=22/30 and 1/2=15/30, leaving 7/30. Take 1/5=6/30, leaving 1/30. So 11/15=1/2+1/5+1/30.
A decomposition is complete when the remainder is zero. If the same denominator repeats and the local convention requires distinct unit fractions, the representation needs another step.
4. A unit fraction can be split into smaller distinct unit fractions
A useful identity is:
1/n = 1/(n+1) + 1/[n(n+1)].
The right side has common denominator n(n+1): n/[n(n+1)] + 1/[n(n+1)] = (n+1)/[n(n+1)] = 1/n.
Worked example O: Split 1/2
1/2 = 1/3+1/6.
Worked example P: Split 1/3
1/3 = 1/4+1/12.
Worked example Q: Split 1/4
1/4 = 1/5+1/20.
Worked example R: Produce another decomposition of 3/4
Start with 3/4=1/2+1/4. Split 1/4 into 1/5+1/20. Then 3/4=1/2+1/5+1/20.
Worked example S: Produce another decomposition of 2/3
Start with 2/3=1/2+1/6. Split 1/6 using the identity: 1/6=1/7+1/42. Hence 2/3=1/2+1/7+1/42.
5. One fraction can have several valid decompositions
Fraction decomposition is generally not unique. For 3/4, both 1/2+1/4 and 1/2+1/5+1/20 are valid. The existence of one correct representation does not make alternatives incorrect.
Worked example T: Compare two forms of 5/6
Form 1: 1/2+1/3.
Form 2: split 1/3 into 1/4+1/12, giving 1/2+1/4+1/12.
Both add to 5/6. The second uses more terms but smaller denominators after the first term.
Distinctness is a rule, not an arithmetic necessity
2/3 can also be written 1/3+1/3, but under this guide’s Egyptian-fraction convention that is not accepted because the unit fraction repeats. It is still an arithmetically correct fraction sum.
Shortest is a separate optimisation question
A valid decomposition may use two, three or more terms. If a question asks for the fewest terms, that adds a new objective. This guide usually asks only for a valid distinct-unit representation unless “minimum terms” is stated.
Bounds help choose a first term
For a target below one-half, 1/2 is too large. For 5/12, we can start with 1/3 because 1/3≤5/12 while 1/2>5/12. Comparing sizes prevents a negative remainder.
6. Practice: 24 original questions
Unless stated otherwise, use distinct positive unit fractions.
Questions 1–8: Identify and verify
1. Which of 1/5, 2/5, 1/9 and 3/7 are unit fractions?
2. Verify 2/3=1/2+1/6.
3. Verify 5/6=1/2+1/3.
4. Verify 7/8=1/2+1/4+1/8.
5. Is 2/3=1/2+1/5 correct?
6. Is 3/4=1/2+1/3 correct?
7. Verify 4/5=1/2+1/4+1/20.
8. Verify 11/12=1/2+1/3+1/12.
Questions 9–16: Construct decompositions
9. Decompose 3/5 into two distinct unit fractions.
10. Decompose 5/8 into two distinct unit fractions.
11. Decompose 7/10 into two distinct unit fractions.
12. Decompose 5/12 into two distinct unit fractions.
13. Decompose 7/12 into two distinct unit fractions.
14. Decompose 11/15 using 1/2 as the first term.
15. Decompose 4/5 using 1/2 as the first term.
16. Decompose 11/12 using 1/2 as the first term.
Questions 17–24: Splitting and non-uniqueness
17. Split 1/2 using 1/n=1/(n+1)+1/[n(n+1)].
18. Split 1/3 using the same identity.
19. Split 1/4 using the same identity.
20. Starting from 3/4=1/2+1/4, replace 1/4 by two smaller unit fractions.
21. Starting from 2/3=1/2+1/6, replace 1/6 by two smaller unit fractions.
22. Give two different valid decompositions of 5/6.
23. Is 2/3=1/3+1/3 arithmetically correct? Does it satisfy this guide’s distinct-unit convention?
24. Explain why a proposed first term 1/2 cannot be used by simple subtraction to begin a positive-unit decomposition of 5/12.
7. Worked answers
Answers 1–8
1. 1/5 and 1/9. A unit fraction has numerator one.
2. Correct. 1/2+1/6=3/6+1/6=4/6=2/3.
3. Correct. 3/6+2/6=5/6.
4. Correct. 4/8+2/8+1/8=7/8.
5. No. 1/2+1/5=7/10, not 2/3.
6. No. 1/2+1/3=5/6, not 3/4.
7. Correct. 10/20+5/20+1/20=16/20=4/5.
8. Correct. 6/12+4/12+1/12=11/12.
Answers 9–16
9. 3/5=1/2+1/10.
10. 5/8=1/2+1/8.
11. 7/10=1/2+1/5.
12. 5/12=1/3+1/12.
13. 7/12=1/2+1/12.
14. 11/15=1/2+1/5+1/30. In thirtieths: 15+6+1=22, which is 11/15.
15. 4/5=1/2+1/4+1/20.
16. 11/12=1/2+1/3+1/12.
Answers 17–24
17. 1/2=1/3+1/6.
18. 1/3=1/4+1/12.
19. 1/4=1/5+1/20.
20. 3/4=1/2+1/5+1/20.
21. 2/3=1/2+1/7+1/42.
22. For example, 5/6=1/2+1/3 and 5/6=1/2+1/4+1/12.
23. Yes arithmetically; no under the distinct-unit convention. The unit fraction 1/3 repeats.
24. Because 1/2>5/12. Subtracting it would leave a negative remainder, so it cannot begin a decomposition into positive unit fractions by this method.
8. Teaching and transfer
If a learner treats denominators as pieces to manipulate independently, draw the same whole partitioned into common-sized parts. The decomposition must preserve one total quantity.
When a proposed decomposition is accepted by appearance
Require an exact recombination. Convert the unit fractions to a common denominator and check that the numerator total matches the target fraction.
When a first unit fraction is too large
Compare it with the target before subtracting. A positive-unit decomposition cannot start by removing more than the entire target amount.
When one valid answer is assumed unique
Use 3/4. First write 1/2+1/4, then split 1/4 into 1/5+1/20. Both representations are exact. This distinguishes equivalence from uniqueness.
When distinctness is confused with arithmetic
Separate the mathematical equality from the puzzle rule. 1/3+1/3=2/3 is true, but it does not satisfy a distinct-unit requirement. Constraints sit on top of arithmetic validity.
Connect to optimisation only when asked
Finding any valid decomposition and finding one with the fewest terms are different tasks. This mirrors the distinction between feasibility and optimality in Optimisation, Minimum Moves and Efficient Constructions.
Continue through this enrichment collection
For guaranteed distributions, use Pigeonhole Principle, Guaranteed Repetition and Distribution. For combination counts, use Pascal’s Triangle, Combinations and Path Counting. For whole-preserving geometric rearrangements, use Geometric Dissections, Rearrangement and Area Proofs.
Return to the BTT Primary Mathematics Learning Hub.
Original enrichment guide with 24 original practice questions and separate worked answers. “Egyptian fraction” is used here as a local mathematical convention for distinct positive unit fractions; the focus is equivalence and decomposition.