The mean and median of `10,22,26,29,34,x,42,67,70,y` (in increasing order) are `42` and `35` respectively then the value of `(y)/(x)` is (A)9/4 (B)7/2 (C)8/3 (D)7/3

Options

A. `9//4`
B. `7//2`
C. `8//3`
D. `7//3`

Answer

To solve the problem, we need to find the values of \(x\) and \(y\) based on the given mean and median of the dataset \(10, 22, 26, 29, 34, x, 42, 67, 70, y\).

### Step 1: Find \(x\) using the median
The median of a dataset is the average of the two middle numbers when the dataset is arranged in increasing order. Since there are 10 numbers, the median will be the average of the 5th and 6th numbers.

In the ordered list, the 5th number is \(34\) and the 6th number is \(x\). The median is given as \(35\).

\[ \text{Median} = \frac{34 + x}{2} = 35 \]

### Step 2: Solve for \(x\)
To find \(x\), we can set up the equation:

\[ 34 + x = 70 \]

Subtract \(34\) from both sides:

\[ x = 70 - 34 = 36 \]

### Step 3: Find \(y\) using the mean
The mean of the dataset is given as \(42\). The mean is calculated as the sum of all the numbers divided by the total number of observations.

The sum of the numbers is:

\[ 10 + 22 + 26 + 29 + 34 + x + 42 + 67 + 70 + y \]

Substituting \(x = 36\):

\[ 10 + 22 + 26 + 29 + 34 + 36 + 42 + 67 + 70 + y \]

Calculating the sum of the known values:

\[ 10 + 22 + 26 + 29 + 34 + 36 + 42 + 67 + 70 = 336 \]

So, the equation for the mean becomes:

\[ \frac{336 + y}{10} = 42 \]

### Step 4: Solve for \(y\)
Multiplying both sides by \(10\):

\[ 336 + y = 420 \]

Subtracting \(336\) from both sides gives:

\[ y = 420 - 336 = 84 \]

### Step 5: Find the ratio \(\frac{y}{x}\)
Now that we have both \(x\) and \(y\):

\[ x = 36, \quad y = 84 \]

The ratio \(\frac{y}{x}\) is:

\[ \frac{y}{x} = \frac{84}{36} \]

### Step 6: Simplify the ratio
To simplify \(\frac{84}{36}\):

\[ \frac{84 \div 12}{36 \div 12} = \frac{7}{3} \]

### Final Answer
Thus, the value of \(\frac{y}{x}\) is \(\frac{7}{3}\).