Word Problem

[Guest]

David spent $1908 on shirts and pants. The number of pairs of pants he bought was [tex]\frac{3}{4}[/tex] the number of shirts he bought.
The cost of a shirt was 15 dollars less than the cost of a pair of pants. He spent 36 dollars more on the pants than on the shirts.
How much did a pair of pants cost?

[Guest]

Pretty basic algebra problem. Let x be the cost of a pair of pants and let y be the cost of a shirt, in dollars. Let p be the number of pairs of pants bought and let s be the number of shirts bought.

Each sentence gives an equation.

"David spent $1908 on shirts and pants."
If a pair of pants cost x then the cost of p pairs of pants is px. If a shirt cost y then the cost of s shirts is sy. Do px+ sy= 1908.

"The number of pairs of pants he bought was 3/4 the number of shirts he bought."
So p= (3/4)s.

"The cost of a shirt was 15 dollars less than the cost of a pair of pants."
So y= x- 15.

"He spent 36 dollars more on the pants than on the shirts."
So px- sy= 36.

We have four equations for the for variables x, y, p, and q.
Adding px+ sy= 1908 and px- sy= 36 we have 2px= 1944, px= 972.
Subracting gives 2sy= 1872 so sy= 936.

"p= (3/4)s" so px= (3/4)sx= 972. sx= (4/3)(972)= 1296.

"y= x- 15" so sy= s(x- 15)= sx- 15s= 1296- 15s= 936. 15s= 1296- 936= 360. s= 360/15= 24. He bought 24 shirts.

"p= (3/4)s" so p= (3/4)(24)= 18. He bought 18 pairs of pants.

px- sy= 18x- 24y= 36. We also have "y= x- 15" so 18x- 24(x- 15)= 18x- 24x+ 360= -6x+ 360= 36. 6x= 360- 36= 324. x= 324/6= 54. Each pair of pants cost $54.

y= x- 15= 54- 15= 39. Each shirt cost $39.

Check:
""David spent $1908 on shirts and pants."
He bought 18 pairs of pants at \$54 each so he paid \$972 for pants. He bought 24 shirts out at $39 each so he paid \$936 for shirts. Yes, he paid 972+ 936= \$1908.

"The number of pairs of pants he bought was 3/4 the number of shirts he bought."
Yes, 18 is 3/4 of 24.

"The cost of a shirt was 15 dollars less than the cost of a pair of pants."
Yes, 54- 39= \$15.

"He spent 36 dollars more on the pants than on the shirts."
He spent, as above, \$972 on pant and \$936 on shirts. Yes, 972- 936= \$36.