The Cross Multiplication Technique

What is 6,942 × 3,141? What about 314,159,265 × 123,456,789? After this short lesson, you will be able to perform cross multiplication and be able to compute these results mentally.

Motivation

Let’s start small, with 12 × 24. Here’s the old school method:

What actually goes under the hood, though, is something else.

What actually happens is the distributive property of multiplication:

But we can go the extra mile:

This is more commonly taught as the cross multiplication technique:

The first digit comes from multiplying the first digits, the third digit comes from multiplying the last digits, and crucially the numbers are multiplied in a cross-like fashion for the second digit.

A more complex example

Let us take 89 × 79.

The old-school method becomes much more complex: 89 × 9, for instance, does not come naturally. Let us use cross multiplication. We omit the image in hope that you can try it yourself.

You will receive three numbers as the result: 56, 135, and 81. You process the carries exactly how you think you do: carry the 8 and get 143, carry the 14 and get 70, then carry the 7. That is how you get 7,031. Since we carry, we usually start from the right and proceed towards the left.

A simplified example

Take 37 × 27. 7 × 7 is 49, so the final digit of the answer is 9. Carry the 4.

Then you have 3 × 7 = 21, then 21 + 4 = 25. Repeat with 7 × 2 = 14, then 25 + 14 = 39. The second to last digit of the answer is hence 9. Carry the 3.

Finally, you have 3 × 2 = 6, and 6 × 3 = 9, which completes our answer: 999.

More digits

Take 23 × 123.

The final digit is 3 × 3 = 9.
The second to last digit is 2 × 3 = 6 plus 3 × 2 = 6 which is 12. Carry the 1.
The third to last digit is 1 plus 2 × 2 = 4 plus 3 × 1 = 3 which is 8.
The fourth to last (i.e. first) digit is 2 × 1 = 2.

Hence, the final answer is 2,829.

Are you ready for 6,942 × 3,141? This time we’ll do without the image.
(We’ll also say first, second, etc. instead of last, etc. to save space.)

• The first digit is simple: 2 × 1 = 2.
• The second digit is 4 × 1 + 2 × 4 = 12. Carry the 1.
• The third digit is 1 + 9 × 1 + 4 × 4 + 2× 1 = 28. Carry the 2.
• The fourth digit is 2 + 6×1 + 9×4 + 4×1 + 2×3 = 54. Carry the 5.
• The fifth digit is 5 + 6 × 4 + 9 × 1 + 4 × 3 = 50. Carry the 5.
• The sixth digit is 5 + 6 × 1 + 9 × 3 = 38. Carry the 3.
• The seventh digit is 3 + 6 × 3 = 21. Carry the 2.
• The eighth digit is 2.

Combining all of that, the result is 2–1–8–0–4–8–2–2: 21,804,822.

The Boss Battle

314,159,265 × 123,456,789. Are you ready?

• First digit is 5 carry 4.
• Second digit is 8 carry 9.
• Third digit is 0 carry 11.
• Fourth digit is 0 carry 18.
• Fifth digit is 0 carry 21.
• Sixth digit is 5 carry 19.
• Seventh digit is 1 carry 20.
• Eighth digit is 9 carry 17.
• Ninth digit is 0 carry 17.
• Tenth digit is 4 carry 13.
• Eleventh digit is 9 carry 9.
• Twelfth digit is 0 carry 7.
• Thirteenth digit is 5 carry 4.
• Fourteenth digit is 8 carry 2.
• Fifteenth digit is 7 carry 1.
• Sixteenth digit is 8. (No carry.)
• Eighteenth digit is 3. (No carry.)

Combining all of that, we get 38,785,094,091,500,085, the answer.