Quantitative Reasoning – Square or Rectangular Tiles!

PSA 2014_XI_Q21. A small area can be covered by 20 identical square tiles or 9 identical rectangular tiles. The length of the side of each square tile is a whole number, and this is 2 cm shorter than the longer side of each rectangular tile. What is the length of the shorter side of the rectangular tile ?
(1) 5 cm

(2) 1 cm

(3) 3 cm

(4) 4 cm

Ans: Let ‘l’ be the length of the square and ‘b’ be the shorter side of the rectangular tile. Length of the longer side of the rectangular time= l+2.

Area of 20 identical square tiles= Area of 9 identical rectangular tiles

=> 20*l²= 9*b(l+2)

=>20*l² – 9*b*l -18b=0

=> l= +9b ±√(81b²+4*20*18b)/2*20

=> l= +9b ±3√(9b²+4*20*2b)/40

=> l= (1/40)*[9b ±3√k]     where k=9b²+4*20*2b= 9b²+160b

Since, l is a whole number, k needs to be a perfect square.

The options have the values of ‘b’. Let us check which ‘b’ fits in starting with the lowest value.

If b= 1; k= 169, but l= 48/40. Rejecting this.

If b= 3; k= 81+480= 561, k is not perfect square. Rejecting this.

If b= 4; k= 144+640= 784, but l= (36+84)/40= 3. Length of the square is 3 and sides of the rectangular tile are 4,5. Option 4.

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