24. Pete's pies all cost an integer number of pounds. A cherry pie costs the same as two apple pies. A blueberry pie costs the same as two damson pies. A cherry pie and two damson pies cost the same as an apple pie and two blueberry pies. Paul buys one of each type of pie.
Which of the following could be the amount he spends?
A $£ 16$
B $£ 18$
C $£ 20$
D $£ 22$
E $£ 24$
24. B Let the costs, in pounds, of apple, blueberry, cherry and damson pies be $a, b, c, d$ respectively. Then $c=2 a, b=2 d$ and $c+2 d=a+2 b$.
Substituting for $b$ and $c$ in the third equation gives $2 a+2 d=a+4 d$. Therefore $a=2 d$.
So $c=2 \times 2 d=4 d$.
Now the cost, in pounds, of one of each type of pie is $a+b+c+d=2 d+2 d+4 d+d=9 d$. So the amount in pounds which Paul spends is a multiple of 9 .
Of the given options, only $£ 18$ satisfies this requirement.