6 ÷ 2(1 + 2): is the answer 9 or 1?
It is 9 with the standard left-to-right reading, and 1 if implicit multiplication binds first. This is the small-number version of the 8 ÷ 2(2 + 2) puzzle, and it splits people in exactly the same way.
See it worked out
Left to right
A number touching a bracket counts as ordinary ×.
= 9
Written out: 6 ÷ 2 × (1 + 2)
Implicit multiplication first
A number touching a bracket is glued to it as one factor.
= 1
Written out: 6 ÷ (2 × (1 + 2))
Make the hidden × visible
The fastest way to see what is going on is to rewrite the expression with the missing sign put back, or with brackets that show the grouping. The three versions below are the ones people actually compute.
| Written as | Answer | What it means |
|---|---|---|
| 6 ÷ 2 × (1 + 2) | 9 | The hidden × made visible, then left to right. |
| 6 ÷ (2 × (1 + 2)) | 1 | The pair 2(1 + 2) grouped into one factor. |
| 6 ÷ 2 × 1 + 2 | 5 | Ignoring the brackets. Wrong under either convention. |
A quick test for spotting this trap
Look for a division sign followed by a number that touches a bracket. Write the hidden × in, then add brackets around the touching pair. If the answer changes, the expression is ambiguous as written.
Here, 6 ÷ 2 × (1 + 2) is 9 but 6 ÷ (2 × (1 + 2)) is 1, so the expression fails the test. Compare 2(1 + 2) on its own: it is 6 under both conventions, so nothing to argue about.
Why small numbers do not help
With 6, 2 and 3 everything can be checked by hand, so the disagreement cannot be an arithmetic slip. It comes down to which rule you assumed. To see how large the gap can get, try the same pattern with bigger numbers in 48 ÷ 2(9 + 3).
What to do in an exam
Follow your course's convention. Many school courses teach left to right, which gives 9. If a question is printed this way in a test, show your working with brackets so it is clear which way you read it, and ask your teacher if you can. Do not assume your reading is the only one.
Try your own expression
Type anything, including brackets, fractions and exponents, and get the steps.
Questions
Is 6 ÷ 2(1 + 2) equal to 9 or 1?
It is 9 if you treat 2(1 + 2) as 2 × (1 + 2) and work left to right. It is 1 if you treat 2(1 + 2) as one unit and divide 6 by 6. The expression is ambiguous as written.
What is the fastest way to check an expression like this?
Type it into the solver. When an expression can be read two ways, it shows both readings side by side with each step.
Does writing it as a fraction avoid the problem?
Yes. A stacked fraction shows exactly what sits above the line and what sits below it, so there is only one way to read it.