Problem
Unit fractions from the denominator's divisors
List all the divisors of the denominator. Pick some that add up to the numerator. Split the fraction along that sum and simplify. Write 3/8 and 8/12 as sums of unit fractions.
Reasoning · Grade 5-1 Egyptian Fractions
Your answer
Watch out One answer seems enough, but several divisor choices can work.
How to solve
Strategy Make a Systematic List — The recipe hands me the structure; the only real work is step 2, choosing divisors that add to the numerator. A denominator like 8 or 12 has only a handful of divisors, so I can list them all and then walk through their combinations in an organised way — smallest first — until I find every group with the right sum. The reason the recipe works at all is a change of viewpoint: splitting the numerator instead of the fraction, since a divisor over the denominator always simplifies to a unit fraction. Each candidate answer is then checked by adding the unit fractions back up.
1STEP 1
See why a divisor on top always makes a unit fraction
A divisor on top always gives a unit fraction.
d/n = (d ÷ d)/(n ÷ d) = 1/(n ÷ d)
Simplifying a fraction by dividing top and bottom by the same number is the one grade-4 skill this rests on, and it explains why the recipe insists on divisors rather than just any numbers.
4.NF.A.1Organize Information In More WaysA fraction whose top is a divisor of its bottom always simplifies to a unit fraction.
Why?
If the top divides the bottom exactly, the same number can be taken out of both, and each divisor brings its partner with it.
🧱Divisors come in pairsEach divisor hands over its partner for free, so divisors pair off two by two unless one is its own partner.
Why?
After the cancelling, one piece is being taken out of however many pieces remain, which is what a unit fraction names.
🧱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.
2STEP 2
Fraction (1): list the divisors of 8
8 has divisors 1, 2, 4, 8.
8 = 1 × 8 = 2 × 4 → divisors 1, 2, 4, 8
Hunting factor pairs from the outside in — 1 × 8, then 2 × 4 — finds every divisor without missing any and without repeating one.
4.OA.B.4Make A Systematic List3STEP 3
Fraction (1): find divisors adding to the numerator 3
Adding to 3 gives 1 and 2.
1+2 = 3
Ruling out every divisor larger than the numerator shrinks the search to two numbers, small enough to settle by inspection.
4.OA.B.4Guess And Check4STEP 4
Fraction (1): split the numerator and simplify
Splitting gives 3/8 = 1/4 + 1/8.
3/8 = (1+2)/8 = 1/8+2/8 = 1/8+1/4 = 1/4+1/8
Splitting a fraction along a sum in its numerator is just counting eighths in two groups instead of one, which is exactly what a fraction with numerator bigger than 1 already means.
4.NF.B.3Organize Information In More Ways5STEP 5
Fraction (2): list the divisors of 12
12 has divisors 1, 2, 3, 4, 6, 12.
12 = 1 × 12 = 2 × 6 = 3 × 4 → divisors 1, 2, 3, 4, 6, 12
Six divisors is still a short list, so the search in the next step stays completely manageable.
4.OA.B.4Make A Systematic List6STEP 6
Fraction (2): find divisors adding to the numerator 8
Adding to 8 gives 2 and 6, or 1, 3 and 4.
2+6 = 8, 1+3+4 = 8
Sweeping the combinations from the largest divisor downwards makes it clear when every possibility has been tried, so the claim that there are exactly two groups is safe.
4.OA.B.4Make A Systematic List7STEP 7
Fraction (2): build both answers
So 8/12 = 1/2 + 1/6.
8/12 = 1/2+1/6 or 8/12 = 1/3+1/4+1/12
Each grouping of the numerator is a different way of sharing the same eight twelfths, so different groupings must give different-looking but equally valid answers.
4.NF.B.3Organize Information In More Ways8STEP 8
Add every answer back to check it
Adding back confirms every answer.
1/4+1/8 = 3/8, 1/2+1/6 = 8/12, 1/3+1/4+1/12 = 8/12
Adding fractions with unlike denominators by rewriting them over a common denominator is a direct check that does not reuse the recipe, so it would catch a mistake in the divisor step.
5.NF.A.1Guess And CheckAnswer
1/4 + 1/8, 1/2 + 1/6
1 + 2 = 3, 2 + 6 = 8
Each answer was added back up and returned the original fraction exactly, so nothing was lost or gained. The sizes are sensible too: 3/8 is a bit under a half and 1/4+1/8 is 3/8, clearly under a half; 8/12 is 2/3, comfortably over a half, and both decompositions start with a piece — 1/2 or 1/3 — big enough to get most of the way there. Every piece has a 1 on top and, within each answer, no two pieces are the same, which is what the problem demanded.
Takeaway
Split the numerator into divisors of the denominator — each piece then simplifies to a unit fraction all by itself.
- See why a divisor on top always makes a unit fraction
- Fraction (1): list the divisors of 8
- Fraction (1): find divisors adding to the numerator 3
- Fraction (1): split the numerator and simplify
- Fraction (2): list the divisors of 12
- Fraction (2): find divisors adding to the numerator 8
- Fraction (2): build both answers
- Add every answer back to check it