Problem
Insert parentheses to change the result
The expression 55 − 10 + 3 × 4 is printed three times. Each copy takes one pair of parentheses in a different place. The three copies must give three different values. Find those three values.
Reasoning · Grade 5-1 Mixed Operations with Whole Numbers
Your answer
Watch out Parentheses seem always to change things, but wrapping the product changes nothing.
How to solve
Strategy Make a Systematic List — There are only four numbers, so there are only six blocks of neighbouring numbers a pair of parentheses could enclose. That is a small, completely finite list, so instead of hunting for clever placements I will write down all six, evaluate each one, and then throw away the ones whose value is the same as the plain expression. Each evaluation splits neatly into a subproblem — compute the bracket, then finish the rest — which keeps the arithmetic short and honest.
1STEP 1
Find the value with no parentheses at all
With no parentheses the value is 57.
55-10+3 × 4 = 55-10+12 = 45+12 = 57
Knowing the plain answer first gives a yardstick — any way that also gives 57 is a way that did nothing, and can be dropped immediately.
5.OA.A.1Identify Subproblems2STEP 2
List every place a pair of parentheses could go
List every place a pair could go.
Sorting the placements by how many numbers they capture — two, then three, then four — is a bookkeeping trick that guarantees none is skipped and none is counted twice.
5.OA.A.2Make A Systematic List3STEP 3
Evaluate the three placements that wrap two numbers
Wrapping 10 and 3 gives 3.
55-(10+3) × 4 = 55-13 × 4 = 55-52 = 3
Parentheses only matter when they overrule the natural order; bracketing something that was already going to be done first changes nothing, which is exactly what the first two cases show.
5.OA.A.1Identify Subproblems4STEP 4
Evaluate the two placements that wrap three numbers
Wrapping 10 + 3 × 4 gives 33.
(55-10+3) × 4 = 48 × 4 = 192, 55-(10+3 × 4) = 55-22 = 33
Bracketing three numbers pulls a whole chunk of the work in front of the multiplication, which is where the biggest changes come from — one placement makes the answer far bigger, the other makes it smaller.
5.OA.A.1Identify Subproblems5STEP 5
Eliminate the placements that change nothing
Wrapping the first three gives 192.
Crossing out the do-nothing cases from a finite list is safe here because the list really was complete, so what remains is guaranteed to be every genuine answer.
5.OA.A.1Eliminate PossibilitiesAny pair of parentheses that only re-brackets an addition changes nothing, so those placements are struck out.
Why?
Which two numbers you push together first in an addition makes no difference to the total, so brackets around them are idle.
🧱Associativity of additionAdding three piles, it does not matter which two you push together first.
Why?
Once every idle placement is crossed off, the ones still standing are exactly those that really change the value.
🧱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.
6STEP 6
Write out the three ways
Only these three change the value.
55-(10+3) × 4 = 3, 55-(10+3 × 4) = 33, (55-10+3) × 4 = 192
Recording each finished expression next to its value lets a reader recompute any line and confirm it, which is the point of showing the work.
5.OA.A.2Make A Systematic ListAnswer
3, 33, 192
55 − (10 + 3) × 4 = 3
All three values are whole numbers built from the same four numbers, and they spread sensibly around the plain answer of 57: bracketing (10+3) builds a big 52 to subtract, so 3 is small; bracketing (10+3 × 4) builds a 22 to subtract, so 33 sits just under 57; and bracketing (55-10+3) multiplies 48 by 4, so 192 is much bigger. No placement can push the value above 55 × 4=220, and none does. Recomputing each line from scratch reproduces 3, 33 and 192, and all three differ from each other.
Takeaway
Parentheses only change the answer when they force an addition or subtraction to jump ahead of the multiplication — list every spot they could go and check which ones actually do.
- Find the value with no parentheses at all
- List every place a pair of parentheses could go
- Evaluate the three placements that wrap two numbers
- Evaluate the two placements that wrap three numbers
- Eliminate the placements that change nothing
- Write out the three ways