Question

A shepherd has 200 sheep with him. Find the number of sheep with him after 3 years if the increase in the number of sheep is 8% every year.

Answer 1

Hint: There is an increase in the number of sheep by 8% every year, so there is an increase in the number of sheep by 8% of 200. So, the number of sheep that the shepherd will have at the end of the first year is equal to (200 + 8% of 200). Now, calculate the total number of sheep at the end of the first year. The number of sheep at the start of the second year will be the same as the number of sheep at the end of the first year. Since there is an increase in the number of sheep by 8% every year, there is an increase in the number of sheep by 8% (200 + 8% of 200). Now, calculate the total number of sheep at the end of the second year. The number of sheep at the start of the third year will be the same as the number of sheep at the end of the second year. Since there is an increase in the number of sheep by 8% every year, there is an increase in the number of sheep by 8% of 8% of (200 + 8% of 200). Now, calculate the total number of sheep at the end of the third year. Since the number of sheep cannot be in decimal point, we have rounded off the number. Now, solve it further, and round off the number.

Complete step by step answer:
According to the question, it is given that A shepherd has 200 sheep with him and there is an increase in the number of sheep by 8% every year.
The number of sheep that the shepherd have initially = 200 …………………………….(1)
At the start of the first year, the shepherd has 200 sheep.
Since there is an increase in the number of sheep by 8% every year, there is an increase in the number of sheep by 8% of 200.
The number of sheep that the shepherd will have at the end of the first year = 200 + 8% of 200 =
\[200+\dfrac{8}{100}\times 200\] = 200 + 16 sheep = 216 sheep …………………………(2)
The number of sheep at the start of the second year will be the same as the number of sheep at the end of the first year.
From equation (2), we have the number of sheep at the end of the first year.
At the start of the second year, the shepherd has 216 sheep.
Since there is an increase in the number of sheep by 8% every year, there is an increase in the number of sheep by 8% of 216.
The number of sheep that the shepherd will have at the end of the second year = 216 + 8% of 216 =
\[216+\dfrac{8}{100}\times 216\] = 216 + 17.28 sheep = 233.28 sheep ………………………………….(3)
The number of sheep at the start of the third year will be the same as the number of sheep at the end of the second year.
From equation (3), we have the number of sheep at the end of the second year.
At the start of the third year, the shepherd has 233.28 sheep.
Since there is an increase in the number of sheep by 8% every year, there is an increase in the number of sheep by 8% of 233.28.
The number of sheep that the shepherd will have at the end of the third year = 233.28 + 8% of 233.28 = \[233.28+\dfrac{8}{100}\times 233.28\] = 233.28 + 18.66 sheep = 251.94 sheep ………………………………….(3)
Since the number of sheep cannot be in decimal point, we have rounded off the number 251.94.
After rounding off the number 251.94, we get the number 252.
Hence, the number of sheep after three years is 252.

Note: In this question, one might get confused because we got the number of sheep at the end of the second year and third year in the decimal point. So, don’t get confused here. We will be rounding off the decimal point at the last.
We can also solve this question by using the formula,
\[\text{Final number of sheeps=Initial number}{{\left( \text{1+}\dfrac{\text{rate of increase}}{\text{100}} \right)}^{\text{time}}}\] ……………………………..(1)
The initial number of sheep = 200 ……………………..(2)
The rate of increase = 8% ……………………………..(3)
Time = 3 years …………………………………(4)
From equation (1), equation (2), equation (3), and equation (4), we get
\[\begin{align}
  & \Rightarrow \text{Final number of sheeps=200}{{\left( \text{1+}\dfrac{8}{\text{100}} \right)}^{3}} \\
 & \Rightarrow \text{Final number of sheeps}=\text{200}{{\left( \dfrac{108}{\text{100}} \right)}^{3}} \\
 & \Rightarrow \text{Final number of sheeps}=\text{200}{{\left( \dfrac{27}{25} \right)}^{3}} \\
 & \Rightarrow \text{Final number of sheeps}=200{{\left( 1.08 \right)}^{3}} \\
 & \Rightarrow \text{Final number of sheeps}=200\times 1.2597=251.94 \\
\end{align}\]
Since the number of sheep cannot be in decimal point, we have rounded off the number 251.94.
After rounding off the number 251.94, we get the number 252.
Hence, the number of sheep after three years is 252.