9 − 3 ÷ ⅓ + 1: is the answer 1, 9 or 19?
It is 1 when ⅓ is a genuine fraction. The step that trips people up is 3 ÷ ⅓, because dividing by a third gives 9, not 1. Type it on one line as 3 ÷ 1/3 and the slash changes the meaning, so the whole expression comes out as 9.
See it worked out
⅓ as a single fraction
The fraction is one number, so 3 ÷ ⅓ is 9.
= 1
Written out: 9 − 3 ÷ (1 ÷ 3) + 1
Typed on one line as 1/3
The slash is just another ÷, so 3 ÷ 1/3 becomes (3 ÷ 1) ÷ 3.
= 9
Written out: 9 − 3 ÷ 1 ÷ 3 + 1
Dividing by a fraction
Dividing by ⅓ asks how many thirds fit into 3. There are three thirds in every whole, so 3 wholes hold 9 of them. That is why 3 ÷ ⅓ = 9, which is the same as 3 × 3 (multiply by the reciprocal).
Then the rest is straightforward: 9 − 9 + 1. Subtraction and addition share a rank and go left to right, so 9 − 9 = 0 and 0 + 1 = 1.
Where 9 and 19 come from
| Answer | How you get there |
|---|---|
| 1 | Divide first: 3 ÷ ⅓ = 9, then 9 − 9 + 1. |
| 9 | Treating 3 ÷ 1/3 as (3 ÷ 1) ÷ 3 = 1, so 9 − 1 + 1. This is what a one-line 1/3 does. |
| 19 | Subtracting before dividing: (9 − 3) ÷ ⅓ + 1 = 6 × 3 + 1. |
The lesson about slashes
A slash on a keyboard is not a fraction bar. In one-line typing, ÷ and / have the same rank as ×, so they go left to right and 1/3 after a ÷ is read as a separate step. When you type a fraction into a calculator or spreadsheet, put it in brackets: 9 − 3 ÷ (1/3) + 1 gives 1.
Try your own expression
Type anything, including brackets, fractions and exponents, and get the steps.
Questions
Is the answer 1 or 9?
It is 1 when ⅓ is written as a true fraction, or typed in brackets as (1/3). If you type 3 ÷ 1/3 with no brackets, the calculator reads it left to right and the whole expression comes out as 9.
Why does dividing by a fraction make the number bigger?
Because you are asking how many small pieces fit into the number. A third is smaller than one, so more of them fit: 3 ÷ ⅓ = 9.