How to Multiply Two-Digit Numbers in Your Head

Multiplying 47 by 36 in your head sounds like a party trick, but it is just a handful of small, easy steps done in a sensible order. Once you know the times tables up to 10×10, you already have everything you need. What remains is learning how to break a large problem into pieces you can hold in memory.

Math Racer Learn · About 8 minutes · Multiplication

The idea behind every method: split and add

Multiplication distributes over addition. That one property, a × (b + c) = a × b + a × c, is behind almost every mental multiplication trick. To multiply by a hard number, split it into easy parts, multiply each part, and add the results.

Method 1: Split one number into tens and ones

Pick the number that is easier to split (usually the one with the smaller ones digit) and break it apart. Multiply the tens part first, because it gives the biggest piece of the answer, then add the ones part.

Working left to right like this is the opposite of the paper method, and it matters: the first partial result is already close to the final answer, so even if you run out of time you have a good estimate.

Method 2: Split both numbers (the four-part method)

For a two-digit by two-digit product, split both numbers. That gives four small multiplications, which you add in order from largest to smallest.

Keep a running total instead of memorizing four separate results. After each partial product, say the new total and forget the old one. That is what keeps the method within the reach of ordinary working memory.

Method 3: Round to a friendly number and adjust

When one factor is just below or above a round number, multiply by the round number and correct the difference.

Method 4: Halve and double

If one factor is even, halve it and double the other. The product does not change, and one of the numbers often becomes much friendlier.

Multiplying by 5 is a special case: multiply by 10 and halve, so 48 × 5 = 480 ÷ 2 = 240. Multiplying by 25 works the same way with 100 and quartering: 36 × 25 = 3,600 ÷ 4 = 900.

Special shortcuts worth memorizing

• Squares ending in 5. For n5 × n5, multiply n by (n + 1) and write 25 after it. So 35 × 35: 3 × 4 = 12, giving 1,225. And 85 × 85: 8 × 9 = 72, giving 7,225.
• Multiplying by 11. For a two-digit number, add its digits and place the sum in the middle: 63 × 11 → 6, (6 + 3), 3 → 693. If the digit sum is 10 or more, carry the 1: 78 × 11 → 7, 15, 8 → 858.
• Difference of squares. When two numbers sit evenly around a round midpoint, use (a − b)(a + b) = a² − b². For 47 × 53, the midpoint is 50 and the offset is 3, so the product is 50² − 3² = 2,500 − 9 = 2,491.

Pick the method before you start

The fastest calculators are not faster at arithmetic; they are faster at choosing. A two-second scan usually tells you which method fits: a factor near a round number (adjust), an even factor with a 5 or 25 around (halve and double), the same digit pattern as a shortcut, or nothing special (split one number). The more of these patterns you have seen, the shorter that scan becomes.

Always check the size of the answer

Before you commit to an answer, round both factors and multiply the rounded numbers. For 47 × 36 that is about 50 × 36 = 1,800, or more precisely 47 × 36 is a little less than that. The exact answer, 1,692, is in the right range, while a result like 1,892 or 1,492 would be flagged immediately. The rough check will not catch a small slip, though, so add a last-digit check: 7 × 6 = 42 ends in 2, and the answer 1,692 ends in 2. Together the two checks catch most mistakes in a few seconds.