Division with Remainders, Rounding Up or Down

Thirty-four divided by five is six remainder four. Six full groups of five use thirty, and four are left that cannot make another group. Written as a number sentence, thirty-four equals five times six plus four — and that sentence is the check: multiply back and add the remainder, and you must land on the number you started with. The remainder is always smaller than the divisor; a remainder of five or more means another group was possible and you stopped too early.

Lin packs thirty-four cakes into boxes that hold five each. Thirty-four divided by five is six remainder four. Six boxes are full — and four cakes are sitting on the bench. They need a box too, so Lin needs seven boxes. Whenever every item has to be held, carried or covered — boxes, buses, tables, tins of paint — a remainder means one more container. One hundred and fifty-five children in buses of forty: three buses carry one hundred and twenty, and the other thirty-five need a fourth bus. Quotient plus one, whenever there is a remainder.

Now the same thirty-four children make teams of five. Six full teams, and four children left over — but four children are not a team, so the answer is six. Whenever the question asks how many full, complete or whole things can be made, the remainder is dropped. A hundred-metre rope cut into twelve-metre pieces gives eight full pieces and four metres of rope that is not a piece. Notice that it is exactly the same division as the cake boxes, and the opposite decision. Nothing in the arithmetic tells you which way to go; the words do.

Sometimes the remainder is not a nuisance to round away but the thing the question wants. One hundred and seventeen chocolates shared among fourteen people: fourteen goes into one hundred and seventeen eight times, with five left over. Each person gets eight; five are left over; and as a number sentence, one hundred and seventeen equals fourteen times eight plus five. A year of three hundred and sixty-five days is fifty-two weeks and one day, and that one day is the remainder of three hundred and sixty-five divided by seven. One division answers three different questions — so read which one is being asked.

Four hundred students, buses that hold forty-five. Four hundred divided by forty-five is eight remainder forty. How many buses are needed? Nine — the forty left over need a bus. How many buses will be full? Eight. How many students are on the last bus? Forty, the remainder. One division, three answers, and the words choose between them. The same division also runs backwards: if a number divided by nine is twenty-three remainder four, the number is nine times twenty-three plus four, two hundred and eleven. Quiz and exercises now.