Best Buys and Successive Discounts

A best buy question asks which option gives you more for your money. Compare unit prices: the cost of the same amount of each. A five hundred gram pack costs four dollars, and a seven hundred and fifty gram pack costs five dollars seventy. Five hundred grams is five lots of one hundred grams, so the small pack is eighty cents per hundred grams. Seven hundred and fifty grams is seven and a half lots, and five seventy divided by seven and a half is seventy-six cents per hundred grams. The bigger pack is the better buy here, but you have to check: bigger is not always cheaper per gram.

In Australia, the goods and services tax, GST, is ten percent of the price before GST. To add GST, multiply by one point one: eighty dollars before GST becomes eighty-eight dollars. Going backwards, if a price of sixty-six dollars already includes GST, divide by one point one to get the price before GST, sixty dollars. The GST was six dollars. A quick way: the GST is one eleventh of a GST-inclusive price, because the price is eleven parts and the GST is one of them. Ten percent of sixty-six is six dollars sixty, and that is wrong, because the ten percent was of sixty, not sixty-six.

When one discount follows another, the second discount is taken off the already-reduced price. Take a hundred and fifty dollar item with twenty percent off, then a further ten percent off. Twenty percent off is times nought point eight, giving one hundred and twenty dollars. Ten percent off that is times nought point nine, giving one hundred and eight dollars. Combined, the multiplier is nought point eight times nought point nine, which is nought point seven two. You pay seventy-two percent, so the deal is twenty-eight percent off, not thirty. The second ten percent is smaller because it is ten percent of a smaller price.

A rise of ten percent followed by a fall of ten percent does not bring you back to where you started. Five hundred dollars rises to five hundred and fifty, and then ten percent of five hundred and fifty, fifty-five dollars, comes off, leaving four hundred and ninety-five. The multiplier is one point one times nought point nine, which is nought point nine nine: a one percent decrease overall. The fall is bigger than the rise because it is taken from a bigger amount. Multiplying in the other order gives the same answer.

To compare deals, turn each one into a single multiplier. Deal A is thirty-five percent off, so you pay nought point six five of the price. Deal B is twenty percent off then fifteen percent off: nought point eight times nought point eight five is nought point six eight, so you pay sixty-eight percent, which is thirty-two percent off. Adding twenty and fifteen makes the deals look the same, but Deal A is better, because its multiplier is smaller. Quiz and exercises now.