48 ÷ 2(9 + 3): is the answer 288 or 2?
It is 288 if you work left to right, and 2 if implicit multiplication comes first. It is the same trap as 8 ÷ 2(2 + 2), but here the two answers are 144 times apart.
See it worked out
Left to right
A number touching a bracket counts as ordinary ×.
= 288
Written out: 48 ÷ 2 × (9 + 3)
Implicit multiplication first
A number touching a bracket is glued to it as one factor.
= 2
Written out: 48 ÷ (2 × (9 + 3))
Why the gap is so big
The two readings of this pattern always differ by the square of the bracket's value. Here the bracket is worth 9 + 3 = 12, and 12² = 144, so 288 ÷ 2 = 144. In the 8 ÷ 2(2 + 2) version the bracket is worth 4, so the readings differ by only 16.
The reason is simple. Reading left to right you get (48 ÷ 2) × 12, so the 12 multiplies. Reading implicit multiplication first you get 48 ÷ (2 × 12), so the 12 divides. Moving a factor from the top to the bottom changes the answer by that factor twice over.
Answers people give
| Answer | How you get there |
|---|---|
| 288 | Left to right: 48 ÷ 2 × 12 = 24 × 12. |
| 2 | Implicit multiplication first: 48 ÷ (2 × 12) = 48 ÷ 24. |
| 219 | Ignoring the brackets: 48 ÷ 2 × 9 + 3. Wrong under either convention. |
A rule of thumb
If you see a ÷ b(c) written on one line, work out both a ÷ b × c and a ÷ (b × c). If they match, there is nothing to argue about. If they differ, the writer needed brackets. As always, the bracket itself goes first: 9 + 3 = 12.
Try your own expression
Type anything, including brackets, fractions and exponents, and get the steps.
Questions
Why do calculators disagree on 48 ÷ 2(9 + 3)?
Some read 2(9 + 3) as a single unit and return 2. Others read it as 2 × (9 + 3) and return 288. Neither is a bug; they use different conventions for multiplication with no sign.
Which answer should I give?
Give the one that matches your course or textbook, and show the bracket you assumed. Most school PEMDAS courses give 288.