24. Four positive numbers $p, q, r$ and $s$ are in increasing order of size. One of the numbers is to be increased by 1 so that the product of the four new numbers is now as small as possible.
Which number should be increased?
A $p$
B $q$
C $r$
D $s$
E either $q$ or $r$
SolUTION
D

The product of the four numbers is pqrs. If each number is increased by 1 in turn, the new products are $(p+1) q r s=p q r s+q r s, p(q+1) r s=p q r s+p r s, p q(r+1) s=p q r s+p q s$ and $p q r(s+1)=p q r s+p q r$. Hence, to minimise the new product, we need to include the minimum of $q r s, p r s, p q s$ and $p q r$. Since $p, q, r$ and $s$ are in increasing order, the minimum value of these is $p q r$. Therefore the number which should be increased by 1 is $s$.