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Casting out nines

Any number can be reduced to a single digit: add its digits and repeat until one digit is left. 47 gives 4 + 7 = 11 and then 1 + 1 = 2.

The whole point of the check is that the reduction does not break addition or multiplication. If the calculation is right for the numbers, it must also be right for their reduced forms:

47 × 23 = 1081 → 2 × 5 = 10 → 1, and 1081 → 1. It matches.

If those two do not match, the answer definitely has a mistake in it. The check itself is easy, because all of the work is done on single digits.

An honest warning. The check finds mistakes but does not prove correctness. A wrong answer can happen to have the same reduced form, and one case is especially common: transposed digits. Both 1081 and 1801 reduce to one, so this check cannot see that mistake.

That is exactly why it pairs with estimating the answer. The estimate shows whether the answer is the right size, and casting out nines catches many of the smaller errors that leave the size unchanged.

The same digit sum appears in the divisibility rules, where it tests divisibility by nine.

Practise

The exercise gives a completed calculation and asks for four reductions: both factors, the product of the two reduced digits, and the answer. The last two must come out the same, and that is the whole check.

Mental arithmetic techniques

Addition and subtraction

• Number bonds to tenThe foundation of mental arithmetic. Learn the five pairs and practise them right here.
• Bridging through tenWhen a sum goes past ten, always stop at ten on the way. Make the first number up to ten and add the rest.
• Adding from the leftIn your head you add starting from the largest place value, not from the smallest as you do on paper. Start with the tens of the second number.
• Subtracting in partsTake the tens off, then the units. The same move as in addition, only in the other direction.
• Subtracting by counting upInstead of borrowing, count up from the smaller number. The first step always uses a number bond to ten.
• Round and adjustWhen a number you add or the number you take away sits just below a round one, compute with the round one and fix it at the end. Take away when adding, add back when subtracting.
• Constant differenceShift both numbers by the same amount until the number being taken away is round. The difference does not change and no correction is needed.

Multiplication

• Multiplying by eightThree doublings in a row, and you get the twos and the fours for free along the way.
• Squares as anchorsLearn the nine squares by heart and use them to reach the neighbouring products. The hardest corner of the times table sits right next to the squares.
• Multiplying by nineReally just multiplying by ten and subtracting once.
• Multiplying in partsSplit the bigger factor into its tens and its units, multiply each separately, then add.
• Multiplying two-digit numbersSplit one factor and multiply the other factor by each part. Nothing new here, just one more layer.
• Rounding in multiplicationMultiply by the nearby round number and correct afterwards. The correction is the gap times the other factor.
• Halving and doublingMultiplying by five is multiplying by ten and halving. Halving one factor while doubling the other leaves the product unchanged.
• Multiplying by elevenPull the digits of a two-digit number apart and put their sum in the middle. When the sum goes past nine, only its units digit goes in the middle and one is carried to the left-hand digit.
• Squaring any two-digit numberChoose the nearest multiple of ten. Move one factor to it and the other the same distance in the opposite direction, multiply the new factors, and add the square of the shift.
• Difference of two squaresWhen two factors sit the same distance from a round number, square the number halfway between them and subtract the square of the distance.
• Squaring numbers ending in fiveDrop the final five, multiply what is left by the next number up, and put 25 on the end. For example, 35 squared: 3 × 4 = 12 and 25 on the end, giving 1225.

Division

• Dividing in partsDivision is a multiplication question asked backwards. Split a bigger dividend into parts that divide easily.
• Division with a remainderWhen a division does not come out exactly, the answer is a quotient and a remainder. A round multiple of the divisor alone already tells you roughly how big the answer is.
• Halving in divisionDividing by four is two halvings, by eight it is three. Dividing by five is doubling and then dividing by ten.
• Factorising the divisorWhen the divisor breaks down into factors, divide twice in a row, each time by a number smaller than the divisor. Dividing by six is dividing by two and then by three.
• Simplifying divisionDivide both numbers by the same factor. The quotient stays the same and you carry on with smaller numbers.
• Divisibility rulesBefore dividing, you can see almost at a glance whether the division will come out exactly.

Checking

• Estimating the answerRound, get a rough answer, and only then work it out exactly. That way a big mistake does not slip through.
• Casting out ninesReduce every number to a single digit by repeatedly adding its digits. If an addition or multiplication is right, it must also be right for the reduced forms.

Mental arithmetic