Is the Answer Roughly Right?

The last digit

38 pens at Rs 15 each. You don’t need the whole product to know how it ends: only the last digits matter. 8 × 5 = 40, which ends in 0, so 38 × 15 must end in 0 as well. The bill says Rs 507, which ends in 7. It is wrong, whatever the rest of it says. (The right answer is Rs 570. Bikash swapped the 7 and the 0.)

Why do only the last digits matter? Write 38 as 30 + 8 and 15 as 10 + 5. Multiplying out gives 30 × 10, 30 × 5, 8 × 10 and 8 × 5. The first three each have a 10 in them, so each ends in 0. Only 8 × 5 can change the last digit.

The last digit check is very fast, but it only looks at one place. A slip anywhere else gets past it.

The rough size

12 geometry boxes at Rs 185 each. Round each number to its first digit (to one significant figure), as Grade 6 rounds off decimals to a place: 12 is about 10 and 185 is about 200. So the bill should be about 10 × 200 = Rs 2,000. The pad says Rs 22,200, which is about ten times too much. Somewhere a 0 was pressed once too often. (The right answer is Rs 2,220.)

This is called estimating: working out a rough answer from rounded numbers. A rough answer can’t tell you the exact bill, but it tells you its size: how many digits it should have, and roughly what it starts with. That is exactly what you need to catch a slip of a whole place, the most expensive kind.

How rough is rough? You can do better than “about 2,000”. Round both numbers down and you get an answer that is surely too small; round both up and you get one that is surely too big. For 12 × 185, the true answer is somewhere between 10 × 100 = 1,000 and 20 × 200 = 4,000. These two numbers are bounds: the answer can’t be outside them, 1,000 ≤ 12 × 185 ≤ 4,000. The bill’s Rs 22,200 is far outside, so it is wrong for certain, not just “probably”.

Decimals and the point

Estimating is at its best with decimals, because a decimal point in the wrong place is a slip of a whole place. Didi’s cloth is 4.5 meters at Rs 240 a meter. About 5 × 200 = 1,000, so Rs 1,080 is believable. If Bikash had missed the point and typed 45 × 240, the calculator would have said Rs 10,800, and the estimate would have caught it at once.

The same works with fractions and percents. Three quarters of Rs 1,196 is about three quarters of 1,200, which is 900 (exactly Rs 897), and 48% of Rs 205 is about a half of 200, so about Rs 100 (exactly Rs 98.40).

The rough size can’t see small slips. The rulers’ bill, 64 rulers at Rs 29, says Rs 1,846. The estimate, 60 × 30 = 1,800, looks fine. Its last digit, 4 × 9 = 36, ends in 6, and so does 1,846. Both checks pass. And yet didi tapped it. For this one she needed the third check.

Casting out nines

Add up the digits of each number until only one digit is left:

• 64: 6 + 4 = 10, then 1 + 0 = 1.
• 29: 2 + 9 = 11, then 1 + 1 = 2.
• Multiply those: 1 × 2 = 2. So the answer’s digits must also come down to 2.
• The bill, 1,846: 1 + 8 + 4 + 6 = 19, then 1 + 9 = 10, then 1.

The answer should come down to 2, and the bill comes down to 1. It is wrong. (The right answer is 1,856: 1 + 8 + 5 + 6 = 20, which comes down to 2. Bikash typed a 4 for a 5.)

This is casting out nines. To “cast out” is to throw away, and what it throws away is nines: a 9 in a number can simply be crossed out, and so can any digits that add up to 9, because they make no difference to the result. In 1,856, the 1 and the 8 make 9, so cross them out, and 5 + 6 = 11 comes down to 2. When the digits come down to 9, count it as 0.

Here is the casting-out-nines check for anything you like. Type a sum, a difference or a product and the answer you got (or one you made up), and it lays the check out as a cross, the way old arithmetic books did: the two numbers’ nines on the left and right, what their answer’s nines must be at the top, and the answer’s own nines at the bottom.

Topic 4, Why Do the Divisibility Tests Work?, showed one fact: a number and its digit sum leave the same remainder when divided by 9. (Each 10, 100 or 1,000 is one more than a number made of nines, so each digit leaves itself over.) Adding the digits again and again until one digit is left just keeps doing that. So the single digit you end with, counting 9 as 0, is the remainder on dividing by 9: 64 leaves 1 (64 = 9 × 7 + 1), and 29 leaves 2 (29 = 9 × 3 + 2).

So casting out nines is really a check on remainders. It works because remainders behave well under adding and multiplying.

For a sum

Take 1,250 + 875 + 2,460. Each number is some nines plus its remainder: 8, 2 and 3. Adding the numbers adds the nines (still nines, with nothing over) and adds the remainders. So the total must leave the same remainder as 8 + 2 + 3 = 13, which is 4. And the total, 4,585, comes down to 4 + 5 + 8 + 5 = 22, then 4. They agree.

For a product: a picture

A product is an area. Draw 64 × 29 as a rectangle 64 wide and 29 tall, and cut it where the nines end: 64 = 63 + 1, and 29 = 27 + 2, with 63 = 9 × 7 and 27 = 9 × 3.

The big piece, 63 × 27, is a multiple of 9. So is the strip 63 × 2 (63 is), and so is the strip 1 × 27 (27 is). Only the corner, 1 × 2 = 2, is made of the two remainders. So the whole product leaves the same remainder as 1 × 2, which is 2. That is why the rule is “multiply the remainders”: the rest of the rectangle is all nines. It works for any two numbers, because any number can be cut into nines and a remainder.

With letters: if one number is 9a + r and the other is 9b + s, the four pieces are 81ab, 9as, 9br and rs. The first three have a 9 in them. So the product leaves the same remainder as rs.

Subtracting and dividing

Subtraction works the same way, with one care: if the second remainder is bigger, add 9 first. For 5,000 − 1,875 = 3,125: 5,000 comes down to 5 and 1,875 to 21, then 3; 5 − 3 = 2, and 3,125 comes down to 11, then 2.

For division, check it backwards: dividend = divisor × quotient + remainder. Is 7,345 ÷ 23 = 319, remainder 8, right? 23 comes down to 5, and 319 to 13, then 4; 5 × 4 = 20, which comes down to 2; add the remainder 8 to get 10, which comes down to 1. And 7,345 comes down to 19, then 10, then 1. They agree.

The check looks only at the digit sum, so any slip that keeps the digit sum the same gets past.

• Two digits swapped. 1,856 and 1,586 have the same digits, so the same digit sum. A swap is invisible to the nines, wherever the two digits are. Swapping is one of the commonest typing slips.
• A 0 and a 9 mixed up. 9 is counted as 0, so 1,056 and 1,956 look the same to the nines.
• A 0 too many or too few. 22,200 and 2,220 have the same digit sum. Didi’s geometry-box bill passes the nines check.
• A decimal point in the wrong place. Moving the point doesn’t change the digits at all, so 1,080 and 10,800 and 108.0 all look the same.

And a random wrong answer, such as a whole number misread, still leaves one of the nine possible remainders. About 1 time in 9 it is the right one by chance, so about 1 in 9 random mistakes, or 11%, get through.

Estimating can’t see a slip in the last digits, and the last digit check sees only the last place. So the three checks cover for one another:

Slip	Last digit	Rough size	Nines
One digit wrong, in the last place	catches	misses	catches (unless it is 0 for 9)
One digit wrong, in the middle	misses	misses	catches (unless it is 0 for 9)
One digit wrong, the first digit	misses	often catches	catches (unless it is 0 for 9)
Two digits swapped	only if the last digit moves	misses, unless the first digit moves	misses
A 0 or a decimal point in the wrong place	sometimes	catches	misses

Estimating catches the costly slips of a whole place; the nines catch a wrong digit in the middle. The slip that gets past all three is the quiet swap in the middle: Rs 1,586 for 1,856, or Rs 750 for 570.

A swap leaves its fingerprint

The nines can’t catch a swap, but they leave a clue. Swapping two digits never changes the remainder, so the difference between the right number and the swapped one is always a multiple of 9: 570 − 507 = 63 = 9 × 7, and 1,856 − 1,586 = 270 = 9 × 30. If a shop’s cash and its book disagree by an amount that divides by 9, a swap is a good first guess. (Swapping the ones and tens digits changes the number by 9 times the gap between the two digits: 9, 18, … up to 81. Swapping the tens and hundreds changes it by 90 times the gap: 90, 180, … up to 810.)

Nine is not the only number you can cast out: any number works, if you can find its remainders quickly. The best partner for nines is eleven.

Topic 4 showed the test for 11: take the digits from the right, and add, subtract, add, subtract. For 1,856: 6 − 5 + 8 − 1 = 8. So 1,856 leaves 8 when divided by 11. (If you get a negative number, add 11; if you get 11 or more, take 11 away.) It works because 10 is one less than 11, 100 is one more than 99, 1,000 is one less than 1,001, and so on: each place counts as +1 or −1.

The check is the same as with nines. 64 leaves 4 − 6 = −2, which is 9; 29 leaves 9 − 2 = 7; 9 × 7 = 63 leaves 3 − 6 = −3, which is 8. And 1,856 leaves 8. They agree.

Now the swap. 1,586 gives 6 − 8 + 5 − 1 = 2, not 8. The elevens catch it. Swapping two neighboring digits moves one onto a plus place and the other onto a minus place, so the alternating sum changes, by twice the difference between the two digits. That is never a multiple of 11 unless the two digits are the same, and then swapping them changes nothing. So casting out elevens catches every swap of two neighbors and every single wrong digit, 0 for 9 included.

Elevens have their own blind spot: swapping two digits two places apart, such as the 1 and the 5 in 1,856, keeps them both on plus places or both on minus places. And no check that works on remainders can catch everything. Try to fool all four checks below.

• How Can You Multiply 98 × 97 in Your Head? Tricks for multiplying near 100, and the algebra that makes them work.
• How Does an Abacus Think? Beads for place value, and adding in your head by picturing them.
• Why Do the Divisibility Tests Work? Why the digit sum knows the remainder by 9, and tests for every number from 2 to 13.
• How Much Rice Does Nepal Eat in a Day? Estimating a number nobody has counted, and how close a good estimate gets.
• How Does a Barcode Catch a Typing Mistake? (Computing, Going Further 6–8) Check digits built to catch the swaps that nines miss, and a program that tries every mistake.