Check yourself, and keep the skill
Three checks that catch a wrong answer in seconds, including casting out nines and what it can't see. Then a five-minute daily plan and a mixed challenge across the whole course.
THE THREE CHECKS
Size, last digit, digit sum
Size: round everything and see whether the answer is even in the right neighbourhood. Last digit: multiply or add only the final digits; the answer must end the same way. Digit sum: add up the digits of each number, do the same operation on those, and the digit sums have to agree. Each one catches a different family of mistakes.
For 47 × 38: size says about 50 × 40 = 2,000. Last digit says 7 × 8 = 56, so it ends in 6. Both fit 1,786.
claim: 47 × 38 = 1,786
size: 50 × 40 = 2,000 plausible ✓
last digit: 7 × 8 = 56, ends in 6 ✓
digit sum: 4+7 = 11 → 2
3+8 = 11 → 2
2 × 2 = 4
1+7+8+6 = 22 → 4 ✓all three agree — the answer standsKeep reducing a digit sum until one digit is left. The three checks take about five seconds together, which is less time than you'd spend doubting the answer.
Check yourself
Which of these can be 63 × 47? Use the checks, don't multiply it out.
- 2,941
- 3,041
- 2,961
- 2,860
Show the answer
2,961
Yes. It ends in 1 because 3 × 7 = 21, it's just under 3,000 because 63 × 50 = 3,150 minus three 63s, and 63 is a multiple of 9 so the answer must be too: 2+9+6+1 = 18.
WHY IT WORKS
Casting out nines
Every power of ten leaves a remainder of 1 when divided by 9: 10, 100 and 1,000 are all one more than a multiple of 9. So a number and the sum of its digits leave the same remainder when divided by 9. Remainders survive addition and multiplication, which is why the check works for both.
1,786 and 1+7+8+6 = 22 both leave remainder 4. That's not a coincidence, it's the whole mechanism.
Check yourself
If the digit-sum check agrees, your answer is definitely correct.
Show the answer
False
False. It only tells you the answer has the right remainder when divided by 9. Two digits swapped, or an error of exactly 90, both slip straight through. Run the size check as well, and you'll catch almost everything.
Five minutes a day, and it stays
- Minute 1 — doubles and halves
Pick any three-digit number and double it four times, then halve it back down. This is the engine under ×4, ×8, ÷8 and doubling-and-halving.
- Minute 2 — complements
Take any two-digit number you can see and say what it needs to reach 100. Then any three-digit number to 1,000. Prices on a shelf work perfectly for this.
- Minute 3 — the multipliers
Take one number and hit it with ×5, ×9, ×11 and ×25 in turn. Four answers from one number, and all four shortcuts stay warm.
- Minutes 4 and 5 — real numbers
Use the numbers in front of you: the total on a receipt, a 15% tip, the unit price of two packs, the split of a bill. Beat the phone to the answer, then check with it.
Check yourself
Which check would catch each mistake?
- The answer came out with one digit too few
- An odd number times an odd number came out even
- A carry was dropped, so the answer is 100 too small
- The decimal point slipped one place
- 7 × 8 was recalled as 54
- Two partial products were added 20 short
Show the answer
Size check: The answer came out with one digit too few, The decimal point slipped one place
Last-digit check: An odd number times an odd number came out even, 7 × 8 was recalled as 54
Digit-sum check: A carry was dropped, so the answer is 100 too small, Two partial products were added 20 short
Check yourself
One from each module. Match them to their answers.
Show the answer
- 1000 − 428 → 572
- 9 × 46 → 414
- 11 × 64 → 704
- 15% of 460 → 69
The whole course in five habits
- Go left to right and say the running total out loud in your head.
- Round to something friendly, then undo the difference. Count up instead of taking away.
- Split a multiplication, or reshape it: halve one side and double the other.
- Build percentages from 10%, 5% and 1%, and think in what you pay, not what you save.
- Estimate before, check after: size, last digit, digit sum.
Check yourself
The final challenge. An order of 2,400,000 gets 15% off, and four people split what's left. What does each pay?
- 600,000
- 510,000
- 540,000
- 480,000
Show the answer
510,000
Exactly. 15% off means paying 85%: 10% is 240,000, so 15% is 360,000, and 2,400,000 − 360,000 = 2,040,000. Halve twice: 1,020,000, then 510,000.
Course recap
- Three checks catch almost every slip: is it the right size, does it end right, do the digit sums agree?
- Casting out nines works because a number and its digit sum leave the same remainder on division by 9.
- It can't see swapped digits or an error that is a multiple of 9, so never use it alone.
- Five minutes a day on doubles, complements, the multipliers and the real numbers around you keeps all of this.