Question

The average temperature of the first three days of the week is ${{27}^{\circ }}C$ and the next three days is ${{29}^{\circ }}C$. If the weekly average is ${{28.5}^{\circ }}C$, what is the temperature on the last day?
(A) ${{31.5}^{\circ }}C$
(B) ${{28}^{\circ }}C$
(C) ${{21}^{\circ }}C$
(D) ${{42}^{\circ }}C$

Answer 1

We solve this question by first assuming the temperatures on first three days as ${{x}_{1}},{{x}_{2}},{{x}_{3}}$ and temperatures on next three days as ${{x}_{4}},{{x}_{5}},{{x}_{6}}$ and the temperature on last day as $X$. Then we consider the formula for average, $Average=\dfrac{Sum\ of\,observations}{Total\ number\ of\ observations}$ and find the value of sum of temperatures on the first three days and then sum of temperatures on the next three days. Then we add them to find the value of the sum of temperatures on the first six days. Then we use the same formula to find the sum of temperatures on all seven days. Then we substitute the value of the sum of temperatures on the first six days in the equation obtained and find the value of temperature on the last day.

Complete step by step answer:
We are given that the average temperature of the first three days of the week is ${{27}^{\circ }}C$. Let the temperatures on those days be ${{x}_{1}},{{x}_{2}},{{x}_{3}}$.
We are also given that the average temperature of the next three days of the week is ${{29}^{\circ }}C$. Let the temperatures on those days be ${{x}_{4}},{{x}_{5}},{{x}_{6}}$.
We are also given that the weekly average is ${{28.5}^{\circ }}C$.
We need to find the temperature on the last day. Let that temperature be $X$.
Now let us consider the formula for average.
$Average=\dfrac{Sum\ of\,observations}{Total\ number\ of\ observations}$
As we are given that average temperature of first three days of the week is ${{27}^{\circ }}C$, we can write it as,
$\begin{aligned} & \Rightarrow \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3}=27 \\\ & \Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}=81..........\left( 1 \right) \\\ \end{aligned}$
As we are given that average temperature of next three days of the week is ${{29}^{\circ }}C$, we can write it as,
$\begin{aligned} & \Rightarrow \dfrac{{{x}_{4}}+{{x}_{5}}+{{x}_{6}}}{3}=29 \\\ & \Rightarrow {{x}_{4}}+{{x}_{5}}+{{x}_{6}}=87..........\left( 2 \right) \\\ \end{aligned}$
Now let us add the values from equations (1) and (2). Then we get,
$\begin{aligned} & \Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}+{{x}_{6}}=81+87 \\\ & \Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}+{{x}_{6}}=168.........\left( 3 \right) \\\ \end{aligned}$
As we are also given that the weekly average is ${{28.5}^{\circ }}C$, we can write it as,
$\begin{aligned} & \Rightarrow \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}+{{x}_{6}}+X}{7}=28.5 \\\ & \Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}+{{x}_{6}}+X=199.5 \\\ \end{aligned}$
Now let us substitute the value obtained in equation (3) in the above equation. Then we get,
$\begin{aligned} & \Rightarrow 168+X=199.5 \\\ & \Rightarrow X=199.5-168 \\\ & \Rightarrow X=31.5 \\\ \end{aligned}$
So, we get the temperature of the last day as ${{31.5}^{\circ }}C$.

Hence the answer is Option A.

Note:
While solving this question one might wrongly interpret the given information by taking the given information, the average temperature of first three days is ${{27}^{\circ }}C$ as, the temperature on all three days is ${{27}^{\circ }}C$. It gives the same answer but the interpretation is wrong. Here the temperature on the three days can be different but their average is ${{27}^{\circ }}C$.