Charlie and Frankie have sent their clothes, consisting of 30 pieces, to the wash, Frankie calls for the same and explains that as his bundle contains half of the shirts and but one-third of the pants, it shouldn't cost but 27 cents. As 4 shirts cost the same as 5 pants. The laundryman who is a poor mathematician wants to know how much he must charge Charlie for the other package.
I find the problem to be hard to understand (and written in old-fashioned language, as well). I wonder where it comes from?
Are the 30 pieces the total for both people? What does "the same" refer to? Do the fractions refer to the whole, or to Charlie's only? Why say "shouldn't cost but ..." (that is, "should cost only ...")? And why wouldn't the "laundryman" just apply the posted prices?
I just played with numbers to get a feel for what the problem might mean. If we guess (keeping whole numbers and knowing they have to be small) that a shirt costs 5 cents and pants cost 4 cents, then the only way to get 27 cents total would be 3 shirts and 3 pants. If this is 1/2 and 1/3 of the total, respectively, then the total order would be 6 shirts and 9 pants, for a total of only 15 items. So we have to double the number of items and halve the cost per item, so that a shirt costs 2.5 cents and pants cost 2 cents. Considering that this appears to be a very old problem, those prices are probably reasonable.
Now starting fresh with a sense of the problem (number of items is whole but price may not be, and the 30 items are indeed the total for two sets of clothes), we can try again with equations. Given a total of x shirts and y = 30-x pants, and a cost of c cents per shirt (and 4/5 c per pants), we have (x/2)c + (30-x)/3*4/5 c = 27, or xc/2 + (30-x)c*4/15 = 27. Clearing fractions, 15xc + 8c(30-x) = 810. Simplifying, 240c + 7xc = 810, and 240 + 7x =
810/c. Since c need not be an integer, but x must be a non-negative integer less than 30, we can just try x = 1, 2, ... . But also, x is a multiple of 2 and y = 30-x is a multiple of 3; so the only choices for (x, y) are (0, 30), (6, 24), (12, 18), (18, 12), (24, 6), (30, 0). Only three of these yield simple values for c (my answer above, and the two extremes, which can probably be eliminated). But all of them are theoretical possibilities.
(Halls made a small mistake near the end of his work.)
Searching for the original, which I figured must be from around the time of Sam Loyd, I eventually found it (with even more archaic language)
here:
Charlie and Freddie having sent their lingerie, consisting of thirty pieces, to the wash, Freddie calls for the same and explains that as his bundle contains half of the cuffs and but one-third of the collars, it should cost but twenty-seven cents. As four cuffs cost the same as five collars. Hop Lee, who is a poor mathematician, wants to know how much he must charge Charlie for the other package.
And with that, I can search and find many more copies, such as
this.
