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UPSC Prelims 2025 CSAT, question 10

Topic: Arithmetic

Q10

The price (p)(p) of a commodity is first increased by k%k\%; then decreased by k%k\%; again increased by k%k\%; and again decreased by k%k\%. If the new price is qq, then what is the relation between pp and qq ?

  1. (a) p(104−k2)2=q×108p(10^4-k^2)^2=q \times 10^8Correct answer
  2. (b) p(104−k2)2=q×104p(10^4-k^2)^2=q \times 10^4
  3. (c) p(104−k2)=q×104p(10^4-k^2)=q \times 10^4
  4. (d) p(104−k2)=q×108p(10^4-k^2)=q \times 10^8

Solution.

Answer: (a) p(104−k2)2=q×108p(10^4-k^2)^2=q \times 10^8

  1. An increase of k%k\% followed by a decrease of k%k\% multiplies the price by (1+k100)(1−k100)=1−k2104=104−k2104(1 + \frac{k}{100})(1 - \frac{k}{100}) = 1 - \frac{k^2}{10^4} = \frac{10^4 - k^2}{10^4}.
  2. Repeating this increase and decrease a second time gives q=p(104−k2104)2=p(104−k2)2108q = p \left(\frac{10^4 - k^2}{10^4}\right)^2 = p \frac{(10^4 - k^2)^2}{10^8}.
  3. Cross-multiplying yields p(104−k2)2=q×108p(10^4 - k^2)^2 = q \times 10^8.