Question

The mean and the median of the following ten numbers in increasing order 10, 22, 26, 29, 34, x, 42, 67, 70, y are 42 and 35 respectively, then \[\dfrac{y}{x}\] is equal to: -
(a) \[\dfrac{7}{3}\]
(b) \[\dfrac{9}{4}\]
(c) \[\dfrac{7}{2}\]
(d) \[\dfrac{8}{3}\]

Answer 1

Hint: Use the formula: - \[\overline{x}=\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}\] to calculate the mean of the given numbers, where ‘\[\overline{x}\]’ is the mean and ‘n’ is the number of observations. Substitute the value of mean given in the question to form a linear equation in x and y. Now, for ‘n’ is equal to even, apply the formula: - Median = \[\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}\] to calculate the median. Substitute the value of median and form another linear equation in x and y. Solve these two equations to get the value of x and y. Finally, take the ratio \[\left( \dfrac{y}{x} \right)\] to get the answer.

Complete step by step answer:
Here, we have been provided with the mean and median of ten numbers given in increasing order 10, 22, 26, 29, 34, x, 42, 67, 70, y. We have to determine the value of \[\left( \dfrac{y}{x} \right)\].
So, let us first calculate the values of x and y.
It is given that the mean of these numbers is 42. Therefore, applying the formula of mean, we get,
\[\Rightarrow \overline{x}=\dfrac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{i}}}\], where ‘\[\overline{x}\]’ is the mean and ‘n’ is the number of observations. Therefore, substituting \[\overline{x}\] = 42 and n = 10, we get,
\[\begin{align}
  & \Rightarrow 42=\dfrac{1}{10}\sum\limits_{i=1}^{10}{{{x}_{i}}} \\
 & \Rightarrow 420={{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}+{{x}_{6}}+{{x}_{7}}+{{x}_{8}}+{{x}_{9}}+{{x}_{10}} \\
 & \Rightarrow 420=10+22+26+29+34+x+42+67+70 \\
 & \Rightarrow 420=x+y+300 \\
\end{align}\]
\[\Rightarrow x+y=120\] - (i)
Now, it is given that the median of the given 10 numbers is 35. Therefore, applying the formula for median of n = 10 numbers, which is even, we get,
Median = \[\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}\]
Substituting the value of median = 35 and n = 10, we get,
\[\begin{align}
  & \Rightarrow 35=\dfrac{{{\left( \dfrac{10}{2} \right)}^{th}}term+{{\left( \dfrac{10}{2}+1 \right)}^{th}}term}{2} \\
 & \Rightarrow 70={{5}^{th}}term+{{6}^{th}}term \\
\end{align}\]
On observing the given numbers we can see that \[{{5}^{th}}\] and \[{{6}^{th}}\] terms are 34 and x respectively. Therefore, we have,
\[\Rightarrow 70=34+x\]
\[\Rightarrow \] x = 36 – (ii)
Now, substituting x = 36 from equation (ii) in equation (i), we get,
\[\begin{align}
  & \Rightarrow 36+y=120 \\
 & \Rightarrow y=120-36 \\
\end{align}\]
\[\Rightarrow \] y = 84 – (iii)
So, dividing equation (iii), by equation (ii), we get,
\[\begin{align}
  & \Rightarrow \dfrac{y}{x}=\dfrac{84}{36} \\
 & \Rightarrow \dfrac{y}{x}=\dfrac{7}{3} \\
\end{align}\]

Hence, option (a) is the correct answer.

Note:
One may note that, we have used the formula of median as: - Median = \[\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}\]. This is because the number of observations are 10, which is even. If number of terms would have been odd, then the median formula would have been given as: - Median = \[{{\left( \dfrac{n}{2}+1 \right)}^{th}}term\]. So, note that before applying the formula of median, we have to check the number of observations.