Balance & Equivalence: Scales, Swaps & Made-up Money

A balance question shows weights on two pans and offers three methods for making them level. Begin by totalling each side and writing down the gap — here twenty-six against nineteen, a gap of seven. Then test each method against the gap, not against the picture. Adding the same amount to both sides changes nothing. Adding to the heavy side makes it worse. A method works only if what it adds to the light side, minus what it adds to the heavy side, is exactly the gap. Sixteen on the right and nine on the left is a difference of seven, so that one works even though it adds weight to both pans.

A swap-rate question gives one fact — three pears weigh the same as five plums — and then describes bags of mixed fruit to compare. Do not compare the bags as they stand; convert each into a single fruit. Choose the fruit whose numbers divide cleanly by the swap: with nine, six and three pears, swap pears in groups of three for five plums each. Bag one becomes twenty plums, bag two twenty plums, bag three seventeen. Now the comparison is plain. The trap is counting pieces of fruit, which the option list quietly rewards; the question asks about weight, and a pear and a plum weigh different amounts.

Made-up money questions give two exchange facts that share one unit and ask you to convert across both. Draw the chain: grozzles to zots to squigs, with zots in the middle because both facts mention them. Then take one link at a time. Twenty grozzles is four groups of five, and each group is worth three zots, so twelve zots. Twelve zots is six groups of two, each worth nine squigs, so fifty-four. At each link you divide by the group size on the side you have and multiply by the number on the side you want. Finish with a sense check: a squig is the smallest unit, so there should be the most of them.

Picture balances show shapes on scales with no numbers except perhaps one weight. Each scale is a fact. One scale usually gives a swap — two triangles balance a square — and the other mixes the shapes with a number. Use the swap to rewrite the second scale in one shape only: a square and a triangle become three triangles, which weigh nine kilograms, so a triangle is three and the square is six. The trap is to halve the nine because two objects sit on the pan; that only works if the objects are the same. Always check your numbers back against every scale in the picture.

Put it together as a routine. Total each side and write the gap with its direction. Test every method by asking whether it closes that exact gap; write a tick or a cross next to each. Only then read the options and choose the one that matches your ticks precisely. The tempting wrong methods are the ones that add more to the light side but not the right amount, and the ones that add the same to both sides. The same discipline — convert first, decide second — runs through fruit bags and money chains, where the temptation is to compare before converting. Quiz and exercises now!