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Division with a remainder

Many real-world divisions do not come out exactly. If 130 people need buses with 45 seats each, 130 ÷ 45 does not mean 2.89 buses. It means two full buses with 40 people left over, so a third bus is needed. The answer is a quotient and a remainder, and here it is the remainder that decides the matter.

The move is the same as in dividing in parts: take a round multiple of the divisor that fits inside the dividend and see what is left over.

349 ÷ 8 → 8 × 40 = 320, 29 left over → 8 × 3 = 24, 5 left over

The quotient is 40 + 3, which is 43, and the remainder is 5. Written out, 349 = 8 × 43 + 5.

The rough answer arrives with the very first step. As soon as you know that 8 × 40 fits and 8 × 50 no longer does, you also know the answer is forty-something. If no more precision is needed, no more work is needed either. That is why this is also the way to go about estimating the answer in division, just as rounding is in addition and multiplication.

The remainder is always smaller than the divisor, which also makes it a quick check. If what is left over is at least as large as the divisor, the divisor fits in once more and the quotient was too small.

Practise

The exercise gives a multiplication of the divisor by a round number, such as 8 × 40. You work out its result, then the part of the dividend that is left over, and finally the remainder. The quotient is the sum of the two partial quotients, so 40 + 3 = 43.

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