Problem
Four fours, with a fraction required
Write one expression for each number from 1 to 10. Each uses the digit 4 exactly four times. Every expression must contain a fraction bar. Build all ten expressions.
Reasoning · Grade 6-2 Fraction Calculation in Action
Your answer
Watch out Hitting the value is not enough: no fraction bar, no answer.
How to solve
Strategy Make a Systematic List — Hunting for ten expressions one at a time, from scratch, means a lot of random trying. It is far quicker to build a stock of parts first — solve the easier problem of what two 4s alone can make, then what three 4s over one 4 can make — and then list the targets against those parts. That way the search is organised by family rather than by target: one family produces 1, 3, 5 and 10 in a single sweep, another produces 4, 6 and 8. Guess and Check is kept for the one or two targets the families miss, and every candidate is checked against both rules (exactly four 4s, at least one fraction) before it goes on the list.
1STEP 1
Read the extra rule carefully: a fraction bar is compulsory
Every expression must carry a fraction bar.
1 = 4/(4+4-4)
Knowing a rule before you start searching saves throwing away good-looking answers later — every expression I write from here on will have a fraction bar in it by design.
5.OA.A.2Make A Systematic List2STEP 2
Solve the easier problem first: what can two 4s make?
Two 4s make 0, 1, 8, 16.
4+4=8, 4-4=0, 4 × 4=16, 4/4=1, 44
Five parts is a list a fourth grader can hold in their head, and every four-4s expression is just two of these parts joined by one operation — so the hard puzzle turns into picking pairs from a very short menu.
5.NF.B.3Solve An Easier Related ProblemListing everything two 4s can make gives a short menu, and every four-4s expression is two of those joined by one sign.
Why?
Two 4s and one operation is a small closed world, so writing it out completely leaves nothing that could turn up later.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
Why?
The left half and the right half can be chosen without either limiting the other, so every pairing from the menu is really available.
🧱Independent choices multiplyIf a first choice can go a ways and each of those leaves b ways for a second, the pair can be made a times b ways.
3STEP 3
Family A: put three 4s over a single 4
Three 4s over one give 1, 3, 5, 10.
3=(4+4+4)/4, 5=(4 × 4+4)/4, 10=(44-4)/4, 1=4/(4+4-4)
A fraction is just a division waiting to happen, so "three 4s on top, one 4 underneath" is a machine that turns 12, 20 and 40 into 3, 5 and 10 — all facts from the four times table.
5.NF.B.3Make A Systematic List4STEP 4
Family B: use 4/4 = 1 to nudge 4 + 4 up or down by one
Nudging 8 by 4 over 4 gives 7 and 9.
9 = 4+4+4/4, 7 = 4+4-4/4
Once you know a pair of 4s can quietly become 1, any number you can already reach becomes three numbers — itself, one more, and one less.
5.OA.A.1Make A Systematic List5STEP 5
Family C: build the small pieces 2 and 0 as fractions, then attach the last 4
Attaching small pieces gives 4, 6, 8.
6 = 4+(4+4)/4, 8 = 4×(4+4)/4, 4 = (4-4)/4+4
Working out a small fraction such as (4 + 4)/4 = 2 first and only then attaching the last 4 keeps each expression to two easy steps, and evaluating inside the brackets before dividing is the ordinary order of operations.
5.OA.A.1Make A Systematic List6STEP 6
The last target: 2, found by matching a numerator to a denominator
The last one, 2, is 16 over 8.
2 = (4 × 4)/(4+4) = 16/8 = 2
With only five things two 4s can make, checking every top-and-bottom pairing is a handful of divisions — a search small enough to finish by hand and be sure nothing was missed.
5.OA.A.1Guess And Check7STEP 7
Check all ten against both rules
All ten obey both rules.
4/(4+4-4)=1, (4×4)/(4+4)=2, (4+4+4)/4=3, (4-4)/4+4=4, (4×4+4)/4=5
Re-reading each expression and counting the 4s out loud is the only way to catch the classic slip of writing a lovely expression that uses three 4s or five.
5.OA.A.1Make A Systematic ListAnswer
all ten are possible
(4 + 4 + 4) ÷ 4 = 3
Each expression was evaluated back to its target using the order of operations — brackets first, then the fraction bar as a division, then the remaining addition or multiplication — and all ten come out exactly right: 4/4 = 1, 16/8 = 2, 12/4 = 3, 0 + 4 = 4, 20/4 = 5, 4 + 2 = 6, 8 - 1 = 7, 4 × 2 = 8, 8 + 1 = 9, 40/4 = 10. The four-4s count was checked digit by digit, remembering that 44 is two 4s, and every expression contains at least one fraction bar, so the extra rule of this version is met throughout. A complete machine search over all ways to combine four 4s (including 44 and 444) with +, -, × and ÷ confirms that each of the ten targets really is reachable with at least one division in the expression, so no target was hit by an illegal shortcut.
Takeaway
Find out what two 4s can make first — then most of the ten answers are just one of those little pieces with the last 4 stuck on.
- Read the extra rule carefully: a fraction bar is compulsory
- Solve the easier problem first: what can two 4s make?
- Family A: put three 4s over a single 4
- Family B: use 4/4 = 1 to nudge 4 + 4 up or down by one
- Family C: build the small pieces 2 and 0 as fractions, then attach the last 4
- The last target: 2, found by matching a numerator to a denominator
- Check all ten against both rules