CAT 2025 Lesson : Averages - GM & HM

The population of Karnavati grew by $20 \%, 25 \%$ and $10 \%$ in $2016, 2017$ and $2018$ respectively. What is the compound annual growth rate during this period?

($1$) $16$%            ($2$) $18$%            ($3$) $20$%           ($4$) $22$%           

Solution

Let the population of Karnavati in $2015$ be $p$.

Population of Karnavati in $2018 = p \times 1.2 \times 1.25 \times 1.1 = 1.65 p$

CAGR is the constant compound rate at which the same growth of $65 \%$ could have been achieved over $3$ years. This is the geometric mean of the three multiplication factors. Let the CAGR be $r$.

$p \times {\left(1 + \dfrac{r}{100} \right)}^{3} = 1.65 p$

⇒ $1 + \dfrac{r}{100} = {(1.65)}^{\frac{1}{3}}$

Going through the options, we note that $(1.18)^{3} = 1.18 \times 1.18 \times 1.18 \approx 1.643$

(Note: In most CAT exams, the onscreen calculators only have the basic arithmetic functions. As the exponential function is not present, we need to work with the options)

∴ $1 + \dfrac{r}{100} = 1.18$

⇒ $r = 18 \%$

Answer: ($2$) $18 \%$

The arithmetic mean of $2$ positive numbers is 6 while the geometric mean is $3 \sqrt{3}$. What is the difference between the two numbers?

Solution

Let the two numbers be $a$ and $b$

$\dfrac{a + b}{2} = 6$ ⇒ $a + b = 12$ ⇒ $b = 12 - a \longrightarrow (1)$

$\sqrt{ab} = 3 \sqrt{3}$ ⇒ $ab = 27 \longrightarrow (2)$

Substituting $(1)$ in $(2)$,
⇒ $a(12 - a) = 27$

⇒ $a^{2} - 12a + 27 = 0$

$a = 9$ or $3$ , so $b = 3$ or $9$

Difference between $a$ and $b = 9 - 3 = 6$

Answer: $6$

A trader earned Rs. $500$ by selling at Rs. $5$ per unit, Rs. $500$ by selling at Rs. $10$ per unit and Rs. $500$ by selling at Rs. $20$ per unit. What was his average price per unit (rounded to $2$ decimals)?

Solution

As the revenues are equal here, the average price per unit is the harmonic mean of individual price per units.

Average Price per unit $= \dfrac{3}{\dfrac{1}{5} + \dfrac{1}{10} + \dfrac{1}{20}} = \dfrac{3}{\dfrac{4 + 2 + 1}{20}} = \dfrac{60}{7} = 8.57$

Note: The revenue of Rs. $500$ does not matter while calculating the average. Only the fact that the revenues earned were equal matters.

Alternatively

When in doubt you can always apply the general average principle.

Average Price per unit $= \dfrac{\text{Total Revenue}}{\text{Total units sold}}$

Number of Units sold $= \dfrac{500}{5} + \dfrac{500}{10} + \dfrac{500}{20} = 100 + 50 + 25 = 175$

Average Price per unit $= \dfrac{500 + 500 + 500}{175} = \dfrac{60}{7} = 8.57$

Answer: $8.57$