Typed-Answer Drills 2: Working Backwards and Two Unknowns

To work backwards, write the chain forwards first, then walk back from the answer doing the opposite of each step in the opposite order. A number divided by four, then with two point six added, giving nine point one: take off two point six to get six point five, then multiply by four to get twenty-six. Check forwards — twenty-six divided by four is six point five, plus two point six is nine point one. The order matters as much as the operations: undo the last thing done first.

A rectangle has two long sides and two short ones, so to find a missing side from the perimeter, take both known sides off and halve what is left. Perimeter thirty-four with a length of twelve: thirty-four take twenty-four is ten for the two short sides, so each is five. Check by adding all four back to thirty-four. There is a neater route as well — half the perimeter, seventeen, is one long plus one short, so seventeen take twelve is five.

On a standard die, opposite faces add to seven: one with six, two with five, three with four. So a die showing two has a five underneath. Three dice showing two, five and three hide five, two and four beneath — eleven altogether. It is also worth knowing that the six faces of one die add to twenty-one, so three dice carry sixty-three in total, which answers the harder version: if the tops show ten, everything else totals fifty-three. Read which faces the question means.

When two unknowns appear, deal with the part you already know. Gold is worth five and silver three; a team with three golds and twenty-four points has fifteen points from gold, leaving nine, which is three silvers. Check the other row as well — four golds and two silvers is twenty plus six, twenty-six — because that confirms the values you were given. If neither value is stated, look for a row with only one kind of medal in it, because that row tells you what one is worth.

All four of these are the same move: take away the part you already know, then undo what is left. Subtract the two long sides and halve; subtract the golds' points and divide; take off the number that was added and multiply back. And every one of them can be checked by running the question forwards from your answer, which takes about five seconds and catches nearly every slip. Quiz and exercises now.