Rates, Work & Packing: Layers, Limits & Tournaments

A wall is built one layer at a time, six bricks per layer, and you are asked which layer the twenty-sixth brick lands in. Divide: twenty-six by six is four remainder two. The four counts complete layers; the remainder two means two bricks have been laid on a fifth layer, so the last brick is on the fifth. Whenever the remainder is not zero, the answer is the quotient plus one. Only when it divides exactly does the last item finish the layer the quotient names. The same reasoning answers 'which bus', 'which box' and 'which page' questions.

Packing questions have a limit per container and ask for the smallest number of containers. There are two divisions, and they round in opposite directions. First, how many items fit in one container: fifteen hundred grams divided by four hundred is three and three-quarters, and you round down because a fourth box would break the limit. Second, how many containers: twenty boxes at three per bag is six with two left over, and you round up because the leftovers still need a bag. The common wrong route divides the total mass by the bag limit, which pretends boxes can be split.

When two people share one job, the test gives how long each takes alone and asks for the finishing time when they work together. Do not average the times and do not add them. Instead choose a length of time both alone-times divide into, count how many whole jobs each person would do in that time, and add. Ana at thirty minutes and Ben at sixty minutes do two jobs and one job in an hour, three jobs together, so one job takes twenty minutes. If both take forty minutes alone, together they take twenty. Check that the answer is quicker than the faster person alone; if it isn't, something went wrong.

In a tournament where every player meets every other player once, count matches by letting each new player play only the people after them. The first of five players has four matches, the next has three new ones, then two, then one: ten in all. Six players give fifteen, eight give twenty-eight. The tempting error multiplies players by opponents, which counts every match twice; halving fixes it. Watch for 'each pair plays twice', which doubles the answer. Handshakes at a party and lines joining dots are the same count in different clothes.

A savings-jar question gives running totals after certain days, says that certain days had the same deposit, and asks about the odd one out. Running totals are not deposits; subtract neighbouring totals to find what went in between. A running total of twelve when Monday finishes and thirty when Wednesday finishes means Tuesday and Wednesday together added eighteen — nine each, since they were equal. Friday is also nine. Thirty to eighty across Thursday and Friday is fifty, so Thursday was forty-one. Rebuild the whole week to check it reaches eighty. Quiz and exercises now!