The simplest way to check a solution is to test the answer against the original question using a different route: substitute it back, estimate it, or reverse the calculation. If both routes agree, you can be far more confident.
This habit helps most where you understand the method but lose marks through slips. The original mixed-practice builder gives you varied questions to practise it on.
Which second methods work?
| Type of question | A second method |
|---|---|
| Equations | Substitute the answer into the original equation |
| Percentage change | Reverse the change and see if you return to the start |
| Speed, distance, time | Use a different form of the formula |
| Quadratic roots | Substitute each root back |
| Any numerical answer | Round the numbers and estimate roughly |
Worked example
Question: A jacket costs RM340 after a 15% discount. What was the original price?
First method: After a 15% discount the price is 85% of the original. So original = 340 ÷ 0.85 = RM400.
Second method, reverse it: 15% of 400 = 60, and 400 − 60 = 340. This matches the question, so RM400 is correct.
Estimation as a quick third check: 85% is a little under 90%, and 90% of 400 is 360, so 85% of 400 is a little under 360. The exact value, 340, fits that picture.
The mistake to watch for
A common slip is to think “15% off, so add 15% back” and calculate 340 × 1.15 = 391.
Test it with the reverse method. 15% of 391 = 58.65, and 391 − 58.65 = 332.35. This is not 340, so 391 cannot be the original price.
The discount applies to the original price, which is why dividing by 0.85 is correct.
The check does not only tell you that you were wrong. It shows you why.
How do I build this into my routine?
- After each question, ask “what is one quick way to test this?”
- Write the check on the same page, in one or two lines.
- If the check fails, find the step where the two routes split.
- Keep a short list of the checks you used for each topic.
Self-check
- A student says the roots of x² − 5x + 6 = 0 are 2 and 3. Check them.
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x = 2: 4 − 10 + 6 = 0. x = 3: 9 − 15 + 6 = 0. Both work, so the roots are correct.- A car travels 150 km in 2 hours 30 minutes. A student says the average speed is 60 km/h. Check it.
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Reverse it: 60 × 2.5 = 150 km. This matches, so 60 km/h is correct.- A price increases by 20% to RM90. A student says the original was RM70. Check it.
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Increase 70 by 20%: 70 × 1.2 = 84, not 90. So RM70 is wrong. The correct original is 90 ÷ 1.2 = RM75, and 75 × 1.2 = 90.When might one-to-one tuition be worth considering?
Self-study may be enough when checking catches most of your slips. Tuition may be worth considering when you do check but cannot find the error, or when answers differ and you do not know which method to trust.
A teacher can watch you check a real question and show a faster route. See building an independent route, deciding whether targeted tuition adds value, and the Mathematics tuition page.