Multiply any two-digit number by 11 in two seconds

What is 45 × 11? You could write it out, but there is a shortcut that lets you say the answer almost as soon as you read the question. It works for every two-digit number, and it also gives you a quick test for divisibility by 11.

The trick

Write the two digits of the number with a gap between them. For 45 that is 4 _ 5.
Add the two digits: 4 + 5 = 9.
Put the sum in the gap. The answer is 495.

Check: 45 × 11 = 45 × 10 + 45 = 450 + 45 = 495. Same answer.

When the sum is 10 or more

Take 68 × 11. The digits add up to 6 + 8 = 14, which does not fit in one gap. Write 14 in the middle and carry the 1 to the left digit:

6 _ 8 with 14 in the gap gives 6, 14, 8.
Carry the 1 into the 6, so it becomes 7.
The answer is 748.

Check: 68 × 10 + 68 = 680 + 68 = 748.

Why it works

Write the number as 10a + b, where a is the tens digit and b is the units digit. For 68, a = 6 and b = 8. Then:

(10a + b) × 11 = (10a + b) × 10 + (10a + b)
= 100a + 10b + 10a + b
= 100a + 10(a + b) + b

Read the last line carefully. The digit a sits in the hundreds place, the sum a + b sits in the tens place, and the digit b sits in the units place. That is exactly the picture we drew: a, then a + b in the middle, then b. When a + b is 10 or more, the extra ten simply adds one to the hundreds place, which is the carry.

Three-digit numbers too

The same idea works for longer numbers. To find 253 × 11, keep the first digit, add each pair of neighbouring digits, and keep the last digit:

Start with 2.
2 + 5 = 7 and 5 + 3 = 8.
End with 3.

The answer is 2783. Check: 253 × 10 + 253 = 2530 + 253 = 2783.

A test for divisibility by 11

Look at 2783 again. Add the digits in the odd places and the even places separately, then subtract: (2 + 8) - (7 + 3) = 10 - 10 = 0. When this difference is 0 or a multiple of 11, the number is divisible by 11. Our number came from multiplying by 11, so of course it passes. Try 3719: (3 + 1) - (7 + 9) = -12, which is not a multiple of 11, so 3719 is not divisible by 11.

Try it yourself

Find 45 × 11 mentally, then 87 × 11 and 99 × 11.
Find 123 × 11 using the neighbour-sum method.
Why does the trick need a carry for 68 but not for 45?

Answers: 495, 957, 1089 and 1353. The carry is needed when the two digits add up to 10 or more. This idea is built on the same expansions you use in Class 8 algebra, so have a look at the Class 8 chapters, and when you want practice, build a fresh test with the Practice Generator.