Question

The average heights of x number of girls and 15 boys is 123. if the average heights of boys is 125 and that of girls is 120.find number of girls.​

Answer 1

Answer:

$x = 10$. In other words, there number of girls is $10$.

Step-by-step explanation:

The average of a number of measurements is equal to the sum of these measurements over the number of measurements.

$\displaystyle \text{Average} = \frac{\text{Sum of measurements}}{\text{Number of measurements}}$.

Rewrite to obtain:

$\begin{aligned}& \text{Sum of measurements}= (\text{Number of measurements}) \times (\text{Average}) \end{aligned}$.

For this question:

$\begin{aligned}& \text{Sum of heights of boys} \\ &= (\text{Number of boys}) \times (\text{Average height of boys}) \\ &= 15 \times 125 = 1875\end{aligned}$.

$\begin{aligned}& \text{Sum of heights of girls} \\ &= (\text{Number of girls}) \times (\text{Average height of girls}) \\ &= x \times 120 = 120\, x\end{aligned}$.

Therefore:

$\begin{aligned}& \text{Sum of boys and girls} \\ &= \text{Sum of heights of boys} + \text{Sum of heights of girls}\\ &= 1875 + 120\, x\end{aligned}$.

On the other hand, there are $(15 + x)$ boys and girls in total. Using the formula for average:

$\begin{aligned}& \text{Average height of boys and girls} \\ &= \frac{\text{Sum of heights of boys and girls}}{\text{Number of boys and girls}} \\ &= \frac{1875 + 120\, x}{15 + x}\end{aligned}$.

From the question, this average should be equal to $123$. In other words:

$\displaystyle \frac{1875 + 120\, x}{15 + x} = 123$.

Solve this equation for $x$ to obtain:

$1875 + 120\, x= 123\, (15 + x)$.

$(123 - 120)\, x = 1875 - 123 \times 15$.

$x = 10$.

In other words, the number of girls here is $10$.