How Can You Multiply 98 × 97 in Your Head?

The trader’s trick

Here is what the trader did in his head, for 98 × 97:

1. How far is each number below 100? 98 is 2 below and 97 is 3 below.
2. Take one number’s distance from the other number. 98 − 3 = 95. (Or 97 − 2: that is 95 too.) This is the number of hundreds: 9,500.
3. Multiply the two distances. 2 × 3 = 6.
4. Put them together. 9,500 + 6 = 9,506.

Try it on a few more before you trust it. Check each one in the widget.

Numbers	Distances below 100	Hundreds	Distances multiplied	Answer
99 × 98	1 and 2	97	2	9,702
97 × 96	3 and 4	93	12	9,312
99 × 96	1 and 4	95	4	9,504
97 × 97	3 and 3	94	9	9,409
92 × 95	8 and 5	87	40	8,740

A pattern that gives the trick away

Look again at the table. The hundreds are always 100 minus the two distances, and the last two digits are always the distances multiplied. It is as if the answer were 10,000, with 100 taken away for every unit of distance, and then the small product added on. That is just what the widget’s square showed.

The square, cut up

The area of a rectangle 98 across and 97 down is 98 × 97, the number we want. Draw it inside a square of 100 by 100, whose area, 10,000, is easy.

The rectangle is the square with two strips cut off: a strip 2 wide and 100 long down the right side (area 200), and a strip 3 wide and 100 long along the bottom (area 300). So the rectangle is 10,000 − 200 − 300, which is 9,500… nearly.

Look at the bottom right corner. It is a little piece 2 by 3, and it lies in both strips. When you took away both strips, you took that corner away twice, but it is only one piece of the square. So add it back once:

$$98 \times 97 = 10{,}000 - 200 - 300 + 6 = 9{,}506$$

That is the trader’s trick, read off the picture. The 10,000 − 500 is 95 hundreds, “100 minus both distances”, and the corner is the distances multiplied. The 6 that people forget when they guess “500 less” is that corner.

The same thing, in letters

A picture with 98 and 97 proves the trick for 98 and 97. To prove it for every pair at once, give the distances letters: call them a and b. The numbers are 100 − a and 100 − b, and the picture works for any a and b just as it did for 2 and 3: the square, minus two strips of 100a and 100b, plus the corner a × b. Written out,

$$(100 - a)(100 - b) = 10{,}000 - 100a - 100b + ab = 100 \times (100 - a - b) + ab$$

The first step is expanding the brackets: each part of one bracket multiplies each part of the other, four products, just as the picture has four pieces (the square, two strips and the corner). It works because of the distributive law, the rule that 7 × (10 + 3) is 7 × 10 + 7 × 3, used twice. The second step only groups the hundreds together.

A statement like this, true whatever numbers the letters stand for, is called an identity. That is why the trick can be trusted. Trying it on five pairs, as you did, made it believable; topic 16 showed how a pattern can hold many times and still fail. The picture and the letters show it can never fail, for any two numbers whatever.

If you have met $(a - b)^2 = a^2 - 2ab + b^2$ in Grade 8, you have met this already: it is the case where both numbers are the same, with 100 in the place of that rule’s a and the distance in the place of its b. For a distance of 6 it gives 94 × 94 = 10,000 − 1,200 + 36 = 8,836, the answer to the last “Your turn”.

When the corner is too big

Numbers just above 100 work the same way, except the strips are added to the square instead of cut off. In the widget, set 103 and 104: the rectangle sticks out past the square, with a green strip on each side and a corner beyond both. So

$$103 \times 104 = 10{,}000 + 300 + 400 + 12 = 10{,}712$$

and the trick becomes: 103 + 4 = 107 hundreds, and 3 × 4 = 12, which gives 10,712.

What if one number is over 100 and the other under? Set 103 and 97. Now one strip is added and one taken away, and the corner, 3 by 3, is part of the added strip but sticks out beyond the rectangle, so it is taken away:

$$103 \times 97 = 10{,}000 + 300 - 300 - 9 = 9{,}991$$

The two strips cancel completely, and the answer is just 10,000 − 9. This is the case your Grade 8 book writes as $(a + b)(a - b) = a^2 - b^2$: the product of two numbers the same distance either side of a round number is that round number squared minus the distance squared. So 101 × 99 = 9,999, and 47 × 53 = 2,500 − 9 = 2,491.

Any round number will do

Nothing in the picture needs the square to be 100 by 100. Use whatever round number is near your two numbers:

• Near 1,000. 998 × 997: 2 and 3 below a thousand, so 995 thousands and 6, written in three places now: 995,006.
• Near 50. 48 × 47: 2 and 3 below 50, so 45 fifties and 6: 2,250 + 6 = 2,256.

Where the trick stops helping

Try 63 × 58 with the 100 trick: the distances are 37 and 42, so 21 hundreds and a corner of 37 × 42 = 1,554, and 2,100 + 1,554 = 3,654. The answer is right, because the identity is true for any numbers. But working out 37 × 42 in your head is harder than 63 × 58 was! The trick is always true, but it is only useful when the corner is small, that is, when both numbers are close to the round number.

So choose a better square. Both numbers are close to 60: 63 is 3 over and 58 is 2 under. Then 63 − 2 = 61 sixties, and the corner is taken away, 3 × 2 = 6: 61 × 60 − 6 = 3,660 − 6 = 3,654. Choosing the round number is your part of the trick.

Squaring a number, multiplying it by itself, has a trick of its own, with a different cut. Take a square 35 by 35. Cut a strip 5 tall off the bottom, leave its 5 by 5 corner where it is, and stand the rest of the strip on end at the right. The top part, which was 35 across and 30 down, is now 40 across: a rectangle 30 by 40, plus the small 5 by 5 square left over.

So 35 × 35 = 30 × 40 + 25 = 1,200 + 25 = 1,225. Nothing was lost or added; the pieces only moved. For any number n and any strip d,

$$n^2 = (n - d)(n + d) + d^2$$

Choose the strip so that n − d or n + d is round, and the rectangle is an easy product.

Numbers ending in 5

For a number ending in 5, take d = 5: the rectangle runs from the ten below to the ten above. 65² = 60 × 70 + 25 = 4,225; 85² = 80 × 90 + 25 = 7,225. That gives the famous shortcut: multiply the tens digit by one more than itself, then write 25. For 65: 6 × 7 = 42, then 25: 4,225. It works for bigger numbers too: 105² is 10 × 11 = 110, then 25: 11,025.

Numbers near a round number

The strip needn’t be 5. 48² = 46 × 50 + 4 = 2,300 + 4 = 2,304, with a strip of 2 to reach 50. And here is Bhaskara’s own example, from the Lilavati of about 1150: to square “three hundred less three”, he took the strip 3, so that 297² = 294 × 300 + 9 = 88,200 + 9 = 88,209.

Times 11

To multiply a two-digit number by 11, write its first digit, then the two digits added, then its last digit. 52 × 11: 5, then 5 + 2 = 7, then 2, so 572.

Why? 11 is 10 + 1, so 52 × 11 = 520 + 52. Write them one under the other and add: the 2 of 52 stands alone in the ones, the 5 of 52 and the 2 of 520 land together in the tens, and the 5 of 520 is alone in the hundreds. That middle place is where the digits are added.

If the two digits add up to 10 or more, carry: 87 × 11 is 8, then 8 + 7 = 15, so write 5 and carry 1 to the 8: 957. Check: 870 + 87 = 957.

Times 25

25 is a quarter of 100. So to multiply by 25, multiply by 100 and divide by 4, or divide by 4 first if that is easier. 36 × 25 = 3,600 ÷ 4 = 900. 84 × 25: 84 ÷ 4 = 21, so 2,100. Paying 48 people Rs. 25 each costs a quarter of 4,800, which is Rs. 1,200.

The same thinking gives × 5 (multiply by 10 and halve: 68 × 5 = 680 ÷ 2 = 340) and × 99 (multiply by 100 and take one lot away: 47 × 99 = 4,700 − 47 = 4,653). That last one is exactly Bhaskara’s rule in the story below.

Now practice until the tricks are faster than a phone. Pick a kind, or Mixed. The steps of the trick show after each answer, so a mistake teaches you something.

The other use is the one in every school curriculum that asks for mental arithmetic: checking. The US Common Core asks students to “assess the reasonableness of answers using mental computation and estimation strategies”, and England’s national curriculum says that calculators “should not be used as a substitute for good written and mental arithmetic”. If a calculator says 98 × 97 is 9,056 or 95,060, you will know at once that a key was pressed wrongly, because you know it is 9,506. Checking a long answer quickly is the next topic, “Is the Answer Roughly Right?”.

• Is the Answer Roughly Right? Estimating, and checking an answer in seconds by casting out nines.
• How Does an Abacus Think? Beads as place value, and adding in your head by picturing them.
• How Did People Multiply Before Calculators? Doubling and halving, the lattice, Napier’s bones, and Karatsuba’s trick of three products instead of four.
• Can You Solve the Puzzles of the Lilavati? More of Bhaskara’s problems, in verse.
• How Many Ways Can You Prove Pythagoras’ Theorem? More proofs by cutting squares up and moving the pieces.
• Why Do the ‘Vedic Mathematics’ Tricks Work? (Math, Going Further 9–12) Where the famous book of tricks came from, and the algebra behind each one.