How to Check Your Own Maths Without an Answer Key

Why checking is a separate skill

Most maths teaching is about getting an answer. Rather less of it is about knowing whether the answer is right, which is the skill that actually matters when there is no answer key — which is to say, in an exam, and afterwards for the rest of your life.

The three habits below take very little time and catch most errors. They are worth building deliberately rather than hoping accuracy arrives on its own.

Check 1 — substitute back

If you solved an equation, put your answer into the original equation and confirm both sides match.

For x² − 2x − 15 = 0 with x = 5: 25 − 10 − 15 = 0. Correct.

The word "original" is doing real work there. Substituting into a rearranged version only proves your rearrangement was self-consistent, not that it was right. If you made an error in step one, checking against step three will happily confirm it.

This catches nearly every arithmetic slip and every sign error.

Check 2 — estimate before you start

Before calculating anything, decide roughly what the answer should be. Then a wrong answer looks wrong immediately.

"1/2 + 1/3 should be a bit less than 1." So an answer of 2/5 is visibly wrong before you have checked any arithmetic.

"Six people need more flour than four people." So an answer smaller than what you started with is wrong.

"This triangle is roughly 10 by 6, so the area is around 30." An answer of 300 means a factor-of-ten error somewhere.

This is the fastest check there is, and it is the one that catches the biggest errors — the misplaced decimal points and the inverted operations.

Check 3 — use a different method

Two methods agreeing is much stronger evidence than one method feeling right, because they fail differently.

Solve a quadratic by factoring, then check with the formula. Find a determinant by Sarrus, then by cofactor expansion. Convert a number to binary by division, then check with place values.

Repeating the same method is much weaker. People tend to make the same mistake twice, particularly when they half-remember what they did the first time.

Topic-specific quick checks

• Fractions — convert to decimals and check the sum roughly matches.
• Quadratics — the roots should add to −b/a and multiply to c/a.
• Determinants — if two rows are the same or one is a multiple of another, it must be zero.
• Binary — a number ending in 0 is even, ending in 1 is odd.
• Area — check the units. Multiplying two lengths gives square units.
• Proportions — check the direction. Did the answer move the way it should have?

What to do when the check fails

Resist the urge to start again from scratch. You will usually repeat the same mistake and lose the time twice.

Instead, work forward through what you wrote and find the first line that is wrong. Everything before it is fine and does not need redoing.

This is where a calculator that shows its steps earns its place. Comparing your working against a correct working line by line finds the divergence point in seconds, which is far more useful than being told the final answer is different.