Problem
Build fractions from number cards
There are four cards: 1, 3, 4 and 5. Use each once to build two proper fractions. Add the two fractions. Find the largest and the smallest possible sum.
Reasoning · Grade 5-1 Fraction Calculations
Your answer
Watch out The biggest card wants to be on top, but 5 cannot make a proper fraction.
How to solve
Strategy Make a Systematic List — There are only four cards, so the whole set of legal arrangements is tiny — small enough to write out completely and be certain of the biggest and smallest. First I cut the search down with one hard fact: the card 5 is the largest, so it can never be a numerator of a proper fraction. That pins 5 to a denominator and leaves just three cases, decided by which card sits above it. Then I add each pair and compare.
1STEP 1
Pin down where the 5 has to go
Proper fractions force the 5 to be a denominator.
5/1, 5/3, 5/4 are all improper → 5 is a denominator
One quick look at "is the top smaller than the bottom?" throws away most of the arrangements before any adding starts — the fastest kind of progress there is.
4.NF.A.2Eliminate PossibilitiesThe card 5 has only one place it can sit, which cuts the whole search down to three cases.
Why?
A denominator names how many equal pieces the whole is cut into, so it controls how much each remaining card is worth.
🧱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?
Every other position for the 5 is ruled out at once, so the search starts from three cases rather than every arrangement.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
2STEP 2
Split into just three cases
That leaves only three cases.
1/5 + 3/4, 3/5 + 1/4, 4/5 + 1/3
Organising by "what is on top of the 5" guarantees the three cases do not overlap and that nothing is left out — much safer than picking arrangements at random.
4.NF.A.2Make A Systematic List3STEP 3
Add case 1
1/5 + 3/4 = 19/20.
1/5 + 3/4 = 4/20 + 15/20 = 19/20
Rewriting with a shared denominator is the standard Grade 5 move: once the pieces are the same size you can just count them.
5.NF.A.1Make A Systematic List4STEP 4
Add case 2
3/5 + 1/4 = 17/20.
3/5 + 1/4 = 12/20 + 5/20 = 17/20
Same denominators as case 1, so the two results can be compared instantly — 17 twentieths against 19 twentieths.
5.NF.A.1Make A Systematic List5STEP 5
Add case 3
4/5 + 1/3 = 1 and 2/15.
4/5 + 1/3 = 12/15 + 5/15 = 17/15 = 1 2/15
Two proper fractions can still add to more than 1 — each is less than 1, so the total is less than 2, but nothing stops it climbing past 1.
5.NF.A.1Make A Systematic List6STEP 6
Compare the three sums
Largest is 1 and 2/15, smallest 17/20.
17/20 ＜ 19/20 ＜ 1 ＜ 1 2/15
Comparing against the benchmark 1 settles case 3 without any common denominator at all, and the other two are already in twentieths.
4.NF.A.2Guess And Check7STEP 7
Sanity-check why the winner wins
As decimals the order matches.
4/5 = 0.8, 1/3 ≈ 0.33, 3/5 = 0.6, 1/4 = 0.25
A fraction gets big when its numerator and denominator are close together and small when the numerator is far below the denominator — the same feel a student gets from pizza slices.
4.NF.A.2Guess And CheckAnswer
1 2/15, 17/20
4/5 + 1/3 = 17/15, 3/5 + 1/4 = 17/20
Both fractions are proper, so each is less than 1 and the sum must lie strictly between 0 and 2 — and 17/20 = 0.85 and 1 2/15 ≈ 1.13 both do. The three cases are genuinely all of them: 5 must be a denominator, its numerator can only be 1, 3 or 4, and the remaining pair has just one proper arrangement, so 3 cases is the complete count and the largest and smallest really are extremes. Decimal estimates back the ordering up: 0.2 + 0.75 = 0.95, 0.6 + 0.25 = 0.85, 0.8 + 0.333… ≈ 1.13.
Takeaway
The biggest card has to hide in a denominator, and that one fact shrinks the whole puzzle to three sums you can just add and compare.
- Pin down where the 5 has to go
- Split into just three cases
- Add case 1
- Add case 2
- Add case 3
- Compare the three sums
- Sanity-check why the winner wins