What these methods are
Long multiplication asks you to hold a column of partial products. Each method below replaces it with two or three steps you can keep in your head. Each uses a fact about the numbers in front of you, so each fits some pairs and not others.
What they do, and what they do not
They make one class of question fast. They measure nothing about you, and they do not replace knowing your tables.
They also rest on recall. Every method ends in a table fact or a small square. If 7 × 8 is not instant, no shortcut helps.
Split one factor
Break one number into tens and units, multiply each part, then add. For 23 × 7: 20 × 7 is 140, 3 × 7 is 21, and 140 + 21 is 161.
This always applies. Use it when nothing else fits. It gets slow when both numbers have two digits, because each part is itself a two digit product, so you end up holding four partial products instead of two.
Multiply by 11
Take ten times the number, then add the number again. For 34 × 11: 340 + 34 is 374.
It applies only when one factor is 11. On a three digit number the addition carries more often, so it stops feeling instant.
Multiply by 5, 25 and 50
Each is a fraction of a round number. Five is ten halved: 86 × 5 is 860 halved, 430. Fifty is a hundred halved: 46 × 50 is 4600 halved, 2300. Twenty five is a hundred quartered: 36 × 25 is 3600 quartered, 900.
The halving is the whole method, so it is easiest when the other number is even.
Squares ending in 5
Take the number without its final 5, multiply it by the next whole number up, then write 25 on the end. For 65 squared: 6 × 7 is 42, then 25 gives 4225. For 115 squared: 11 × 12 is 132, so 13225.
It applies only to a number ending in 5, multiplied by itself. It does not extend to 65 × 75.
Near 100, through the deficits
When both numbers sit just below 100, work with how far below. For 97 × 94: 97 is 3 short, 94 is 6 short. Take one deficit off the other number: 97 − 6 is 91, the hundreds. Multiply the deficits: 3 × 6 is 18, the rest. The answer is 9118.
It applies when both numbers are within about ten of 100. Further out, the deficits multiply past 100 and must be carried, and it stops being quick.
The difference of squares
When two numbers sit the same distance either side of an easy middle, square the middle and take off the square of that distance. For 48 × 52: both sit 2 either side of 50, so 2500 − 4 is 2496.
It applies when the two numbers add to an even total, so the middle is a whole number whose square you know. It fails for 48 × 53, because the middle is 50.5.
Where the numbers come from
The worked examples need no source. Each is an identity you can check with long multiplication.
These methods lean on the tables to 12. In England the national curriculum expects year 4 pupils to recall multiplication and division facts for multiplication tables up to 12 × 12 (Department for Education).
The national check that follows gives pupils 25 questions, 6 seconds to answer each, and a 3 second pause between them (Standards and Testing Agency). The check gives no extra time because it tests fluent recall. If a table fact takes you longer than that, you are probably working it out rather than remembering it.
How to practice
Take one method at a time. Do ten questions where it applies, then mix them. The slow part is deciding which method fits, not the arithmetic.
Say the running total out loud, because losing a partial result is a common slip.
Practice them here
- Fast multiplication sets a level per method family and shows the method under each answer. Lesson: Multiplication shortcuts.
- Times tables drills the facts every method ends in. Lesson: Learn the tables.
- Squares and roots covers the squares the last two methods need. Lesson: Squares near round numbers.
- The same arithmetic under an interview clock: the quant interview math test guide.
Questions people ask
What is the fastest way to multiply in your head?
It depends on the numbers. Splitting one factor into tens and units always works: for 23 × 7, 20 × 7 is 140, 3 × 7 is 21, and together they make 161. Faster methods fit special cases: multiplying by 11, by 5, 25 or 50, squaring a number ending in 5, working near 100, and the difference of squares.
How do you multiply by 11 in your head?
Take ten times the number, then add the number again. For 34 × 11, ten times 34 is 340, and 340 plus 34 is 374. The method applies only when one factor is 11. On a three digit number the addition carries more often, so it stops feeling instant.
How do you square a number ending in 5?
Take the number without its final 5, multiply it by the next whole number up, then write 25 on the end. For 65 squared, 6 × 7 is 42, and writing 25 after it gives 4225. It works only for a number ending in 5 multiplied by itself. It does not extend to a product such as 65 × 75.
How do you multiply two numbers near 100?
Work with how far each one is below 100. For 97 × 94, 97 is 3 short and 94 is 6 short. Take one shortfall from the other number: 97 minus 6 is 91, the hundreds. Multiply the shortfalls: 3 × 6 is 18. The answer is 9118. It stays quick when both numbers are within about ten of 100.
Do you need your times tables to use these methods?
Yes. Every method ends in a table fact or a small square, so if 7 × 8 is not instant, no shortcut helps. In England the national curriculum expects year 4 pupils to recall the tables up to 12 × 12. The national check then gives 6 seconds for each question because it tests fluent recall, so a slower fact is probably being worked out, not remembered.