Mathematics · Linear Equations

On a particular day, Danish, Ethan, and Farhan sold three types of pens from their respective shops. Danish and Ethan sold an identical number of pens of Type A while Farhan sold 32 times as many pens of Type A as Danish and Ethan together sold. The ratio of the numbers of pens of Type B sold by Danish, Ethan, and Farhan was 4 : 3 : 1 respectively. Danish and Farhan sold an identical number of pens of Type C each, while Ethan sold 52 times as many pens of Type C as Danish or Farhan sold. The three sellers sold each type of pen at different prices per unit. Assertion (A): It is possible that Danish sold each pen of Type A at a loss of Re 1, each pen of Type B at a profit of Rs 5, and each pen of Type C at a loss of Rs 3 and made an overall profit of Rs 114; Ethan sold each pen of Type A at a profit of Rs 4, each pen of Typ e B at a loss of Rs 3, and each pen of Type C at a profit of Rs 5 and made an overall profit of Rs 267; and Farhan sold each pen of Type A at a profit of Rs 8, each pen of Type B at a profit of Rs 5, and each pen of Type C at a loss of Rs 7 and made an overall profit of Rs60. Reason (R): Framing and solving the three possible linear equations we will find that we get a unique solution.