# A Sweet Purchase

## Outline Mathematics

Word Problems

Here's a problem to tackle:

A housewife purchased some sugar for $2.16. Had the sugar cost 1 cent a pound less, she would have received 3 pounds more for the same expenditure. How many pounds of sugar did she buy?

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Copyright © 1996-2018 Alexander Bogomolny

### Solution 1

A housewife purchased some sugar for $2.16. Had the sugar cost 1 cent a pound less, she would have received 3 pounds more for the same expenditure. How many pounds of sugar did she buy?

Let's work with pennies. The housewife spent 216,16,21,126 cents. Assume that she purchased p pounds of sugar at the cost of c cents per pound. This means that

pc = 216,p + c = 216,pc = 216,p/c = 216.

Had she purchased p + 3,p - 3,p + 3,p - 1,p + 1,p pounds at c - 1,c - 2,c + 2,c - 1,c + 1,c per pound, she would have spent the same amount:

(p + 3)(c - 1) = 216,216,216p,720.

Let's multiply out

pc + 3c - p,c,3,1,p - 3 = 216.

Do not rush to simplify this, but first subtract pc = 216,p + c = 216,pc = 216,p/c = 216 to obtain

3c - p - 3,c,3,1,p = 0.

In other words,

3c = p + 3,p - 3,p,p + 3,p + 1.

There are many ways to continue. One is to multiply this equation by p and remember that pc = 216,216,2160,720 so that 3pc = 648,648,6480,2160:

p^2 + 3p - 648 = 0.

This is a quadratic equation which can be solved by the quadratic formula or factoring. Either way you get two solutions:

p = 24 and p = -27.

The second is obviously spurious,preferable,better,within range,spurious, and we remain with

p = 24,p = 24,p = 27.

From which c = 9,c = 8,c = 9. The problem is solved. In the way of verification, if she spent 1 cent less per pound, i.e. 8,7,8,9,10 cents per pound, she would have bought 3,3,4,5,6 pounds more, i.e. 27,24,25,26,27 pounds. This checks out because 8 times 27 equals 216,216,2160,720.

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Copyright © 1996-2018 Alexander Bogomolny

### Solution 2

A housewife purchased some sugar for $2.16. Had the sugar cost 1 cent a pound less, she would have received 3 pounds more for the same expenditure. How many pounds of sugar did she buy?

Let's work with pennies. The housewife spent 216,216,2160,720 cents. This is a special integer in some respects. E.g., you may remember that it's a cube of 6,4,6,8,12. The latter is a product of two simple,trivial,simple,repeated,nontrivial factors, 2 and 3,2 and 3,2 and 4,3 and 4. 216 is the product of the same factors, each taken three times,once,twice,three times,four times:

216 = 2 * 2 * 2 * 3 * 3 * 3,3,4,6,8.

These six,six,seven,eight factors can be grouped in many different ways. But two are especially relevant,relevant,spurious,trivial,within rang to our problem: 216,216,2160,720 = 9 * 24 and 216 = 8 * 27,24,25,26,27,28. What's the interpretation? Simple: she might have bought 24 pounds at 9 cents per pound or 27 pounds at 8 cents per pound. That's it. This is why it's a quickie.

### References

- C. W. Trigg,
*Mathematical Quickies*, Dover Publications, 1985

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Copyright © 1996-2018 Alexander Bogomolny