12. A three-piece suit consists of a jacket, a pair of trousers and a waistcoat. Two jackets and three pairs of trousers cost $£ 380$. A pair of trousers costs the same as two waistcoats.
What is the cost of a three-piece suit?
A $£ 150$
B $£ 190$
C $£ 200$
D £228
E more information needed
Solution
B
We let the cost of a jacket, a pair of trousers and a waistcoat be $£ J, £ T$ and $£ W$, respectively. From the information given in the question we can deduce that
$$
2 J+3 T=380
$$
and
$$
T=2 W
$$

Note: In this problem we have three unknowns $J, T$ and $W$, but only two equations. We don't have enough information to enable us to find the values of $J, T$ and $W$. However, we can deduce the value of $J+T+W$ which is what we need to know. We give two methods for doing this.

Method 1
If we subtract equation (2) from equation (1) we obtain
$$
2 J+2 T=380-2 W \text {. }
$$

Hence
$$
2 J+2 T+2 W=380 \text {. }
$$

Hence, dividing both sides of this last equation by 2 , we obtain
$$
J+T+W=190 .
$$

Therefore the cost of a three piece suit is $£ 190$.

Method 2
It follows from equation (2) that $W=\frac{1}{2} T$. Hence,
$$
\begin{aligned}
J+T+W & =J+T+\frac{1}{2} T \\
& =J+\frac{3}{2} T \\
& =\frac{1}{2}(2 J+3 T) \\
& =\frac{1}{2}(380), \quad \text { by equation (1) } \\
& =190 .
\end{aligned}
$$

Therefore the cost of a three-piece suit is $£ 190$.