CAT 2023 Question Paper | Quant Slot 3

CAT 2023 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 10 : The population of a town in 2020 was 100000 . The population decreased by \(y \%\) from the year 2020 to 2021 , and increased by \(x \%\) from the year 2021 to 2022, where \(x\) and \(y\) are two natural numbers. If population in 2022 was greater than the population in 2020 and the difference between \(x\) and \(y\) is 10 , then the lowest possible population of the town in 2021 was

Video Explanation

Explanatory Answer

\( 100,000 \left( 1 - \frac { 4 } { 100 } \right) \left( 1 + \frac { x } { 100 } \right) > 100,000 \)
\( \left( 1 - \frac { 4 } { 100 } \right) \left( 1 + \frac { x } { 100 } \right) > 1 \)
\( ( 100 - y ) ( 100 + x ) > 10,000 \)
\( x > y \). Why? Think \( ( 1 - k ) ( 1 + k ) = 1 - k ^ { 2 } < 1 \)
\( \therefore x - y = 10 ; \Rightarrow x = y + 10 \)
\( ( 100 - y ) ( 100 + y + 10 ) > 10000 \)
\( - 100 y + 100 y - y ^ { 2 } + 1000 - 10 y > 0 \)
\( y ^ { 2 } + 10 y - 1000 < 0 \)
\( y ^ { 2 } + 2 \cdot 5 \cdot y + 25 - 1025 < 0 \)
\( ( y + 5 ) ^ { 2 } < 1025 \)
\( \sqrt { 1025 } \cong 32.015 \)
\( ( y + 5 ) ^ { 2 } \leq 32 ^ { 2 } \)
To calculate the minimum population in 2021, we need y to be as large as possible.
largest \( y = 32 - 5 = 27 \)
Minimum population in 2021 is \( 100,000 \left( 1 - \frac { 27 } { 100 } \right) = 73,000 \).

The question is " The population of a town in 2020 was 100000 . The population decreased by \(y \%\) from the year 2020 to 2021 , and increased by \(x \%\) from the year 2021 to 2022, where \(x\) and \(y\) are two natural numbers. If population in 2022 was greater than the population in 2020 and the difference between \(x\) and \(y\) is 10 , then the lowest possible population of the town in 2021 was "

The answer is '73000'

Choice D is the correct answer.