# Quantitative Reasoning – Practice – Greatest and Least number of pens!

PSA_PRACTICE_QNR49. A certain store sells two types of pens: one type for \$2 per pen and the other type for \$3 per pen. If a customer can spend up to \$25 to buy pens at the store and there is no sales tax,
a) what is the greatest number of pens the customer can buy?
b) what is the least number of pens the customer can buy?

Ans: Let the number of pens the customer buys for each type be ‘m’ and ‘n’ respectively.

Cost of first type =\$2
Cost of second type =\$3

According to the question,

2m + 3n =25

Rearranging the equation,

2m = 25 – 3n

=> 2m = (24 + 1) – (2n + n)
=> 2m = (24 + 2n) + (1 – n)

LHS, 2m is a multiple of 2.
RHS should also be a multiple 2; (24+ 2n) is already a multiple of 2, thus (1 – n) will be a multiple of 2.

One such value of n will be 1. [ 1- n will become 0 then which is divisible by 2].

If n = 1; m= 11

Once we have got one such value for (m,n) as (16,1), next sets can easily be determined by noting that the values of m will be in arithmetic progression with common difference equal to
the coefficient of ‘n’ and vice versa.

(m,n) =
(11 , 1),
(8 , 3),
(5, 5)
(2, 7)

a) what is the greatest number of pens the customer can buy? —> 12 pens (11 of first type and 1 of the second type)

b) what is the least number of pens the customer can buy? —> 9 pens (2 of first type and 7 of the second type)