Question

A pencil costs Rs. 0 25, apen costs Rs. 10 and a book costs Rs. 25. How many each, one should buy so that the total number of articles is 100 and the total purchase is Rs. 1000? 

Answer 1

There are three solutions.

Number of pencils = C      Number of books = B      Number of Pens = P

Total Number of articles = 100 = C + P + B
                                 C = 100 – P - B          -- equation 1

Total Purchase:

              0.25 * C + 10 P + 25 * B = 1,000 ,         multiply by 4
                C + 40 P + 100 B = 4,000   --- equation 2

From the above equations, we can infer that maximum value of P is 100, maximum value of B is 40 and Maximum value of C is 100. Minimum of any of them is 1.  If the values are more than those maximum then equation 1 or 2 will not be satisfied.

As       C + 40 P + 100 B = 4,000 ,      substitute value of C from equation 1
          100 - P - B + 40 P + 100 B = 4,000
           39 P + 99 B = 3900                  divide by 39

            P + (33/13) B = 100           --  equation 3

Here, 100 is an integer.  P is an integer.   So  (33 B/13) is also an integer.
  So   B = 13, 26, or 39       only three possibilities.

1)  B = 13,   P + 33 = 100,      P = 67
         C = 100 - 13 - 67 = 20. 
          Total purchase = 20*0.25 +10*67 + 13*25 = Rs 1,000

2) B = 26,   P + 33 * 26/13 = P + 66 = 100,   P = 34
            C = 100  - 26 - 34 = 40 
                    Total purchase = 40*0.25+10*34+25*26 =  Rs 1,000

3) B = 39 ,   P + 99 = 100,     P = 1
               C = 100 - 39 - 1 = 60
       total purchase = 60*0.25 + 1 * 10 + 39 * 25 = Rs 1,000