How to Check Your Arithmetic Answers
Even experienced mathematicians make arithmetic mistakes. The habit of checking your work is what separates careful, reliable maths from careless errors. Here are four proven methods.
The "casting out nines" technique described below is centuries old — versions of it appear in both medieval Indian and European arithmetic manuscripts, used long before calculators existed as a fast way to catch multiplication and division errors by hand. It works because of a special property of base-10 place value: a number and its digit sum always leave the same remainder when divided by 9.
Method 1 – Use the Inverse Operation
After any calculation, apply the inverse operation to verify the result. If you get back to your starting number, the answer is correct.
| Original calculation | Check using inverse | Correct? |
|---|---|---|
| 48 + 35 = 83 | 83 − 35 = 48 ✓ | Yes |
| 7 × 9 = 62 | 62 ÷ 9 = 6.8... ✗ | No — should be 63 |
| 144 ÷ 12 = 12 | 12 × 12 = 144 ✓ | Yes |
Method 2 – Estimation Check
Before or after calculating, estimate the answer using rounded numbers. If your exact answer is far from the estimate, you probably made an error.
Method 3 – Digit Sum (Casting Out Nines)
Add all digits of each number until you reach a single digit. Apply the same operation to these single digits and check the result matches the digit sum of your answer.
Method 4 – Redo the Calculation a Different Way
Rearrange the calculation or use a different strategy. If both approaches give the same answer, confidence is high.
When to Check
- Always check in examinations where marks are at stake.
- When the answer will be used in further calculations — one error cascades.
- When dealing with money, measurements or important quantities.
Key Takeaways
- The most reliable check is to apply the inverse operation.
- An estimation check quickly reveals wildly incorrect answers.
- Digit sums catch many multiplication errors.
- Build checking into your workflow — do not treat it as optional.
Practice: Check It
A claimed calculation is shown below — it might be right or wrong. Use the inverse operation to work out the correct result. If you get it wrong (or leave it blank), we'll check it together.