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| Math Central | Quandaries & Queries |
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Question from Kenneth: Hello: Do you know why some "as...as" phrases indicate multiplication? For example, ? is 60% as much as $\$30.$ This is equivalent to 60% X $\$30$ = $\$18.$ How does this make sense? I saw this example in an old textbook on business mathematics, but the author did not explain why it is equivalent to multiplication. I thank you for your reply. |
Kenneth,
This is a good question - this sort of thing shouldn't be taken for granted!
I don't consider "60% as much as" to be very elegant phrasing, but:
"Sixty per cent" is short for "sixty per centum", or "sixty per hundred." This indicates a whole-number ratio of 60:100 or 3:5, which is represented by the lowest-terms fraction 3/5 and the exact decimal 0.6.
"As much as" means that quantities are being compared - "much" is an adjective referring to quantity. So "60% as much as" means "for every hundred units of quantity in $\$30,$ the answer has sixty such units." So we could solve this as
"$\$30$ is thirty times a hundred cents, so the answer is thirty times sixty cents"
or as
"$\$30$ has a hundred thirty-cent chunks, so the answer has sixty thirty-cent chunks"
or in many other ways, all giving the answer $\$18.$
We are really dividing and multiplying here: dividing is the same as multiplying by a unit-numerator fraction of the form 1/n, and multiplying by a fraction m/n (or the equivalent decimal) carries out the multiplication and the division in one go.
Good Hunting!
RD
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