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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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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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 © 19962017 Alexander Bogomolny
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