Problem 7

A man bought a watch, a chain, and a locket for $216. The watch and locket together cost three times as much as the chain, and the chain and locket together cost half as much as the watch. Find the price of each item.

Solution

a) Let $W$ , $C$ , and $L$ denote the costs of the watch, chain, and locket, respectively. We are given:

$W plus C plus L equals 216$ , $W plus L equals 3 C$ , $C plus L equals W over 2$

b) From $W plus L equals 3 C comma$ express $L$ as $L equals 3 C minus W$

c) Substitute $L equals 3 C minus W$ into $W plus C plus L equals 216$ :

$W plus C plus open paren 3 C minus W close paren equals 216$ $right double arrow 4 C equals 216$ $right double arrow C equals 54 period$

d) Use $C plus L equals W over 2$ with $L equals number 3 C minus W$ :

$54 plus open paren 3 times 54 minus W close paren equals W over 2$

Compute:

$54 plus 162 minus W equals 216 minus W equals W over 2$

Multiply by 2:

$432 minus 2 W equals W$ $right double arrow 432 equals 3 W$ $right double arrow$ $W equals 144 period$

e) Find $L$ :

$L equals 3 times 54 minus 144 equals 162 minus 144 equals 18 period$

Answer: Watch $144, Chain $54, Locket $18