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

Options

A. `(9)/(2)`
B. `(5)/(3)`
C. `(7)/(3)`
D. `(7)/(2)`

Answer

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

### Step 1: Understand the Median
The median of a dataset is the middle value when the numbers are arranged in increasing order. Since there are 10 numbers in this case, the median will be the average of the 5th and 6th numbers.

Given that the median is \(35\), we can write:
\[ \frac{34 + x}{2} = 35 \]

### Step 2: Solve for \(x\)
To find \(x\), we multiply both sides of the equation by \(2\):
\[ 34 + x = 70 \]
Now, subtract \(34\) from both sides:
\[ x = 70 - 34 = 36 \]

### Step 3: List the Numbers
Now we can list the numbers in increasing order:
\[ 10, 22, 26, 29, 34, 36, 42, 67, 70, y \]

### Step 4: Understand the Mean
The mean of a dataset is the sum of all the numbers divided by the count of numbers. We know the mean is \(42\), so we can set up the equation:
\[ \frac{10 + 22 + 26 + 29 + 34 + 36 + 42 + 67 + 70 + y}{10} = 42 \]

### Step 5: Solve for \(y\)
First, multiply both sides by \(10\):
\[ 10 + 22 + 26 + 29 + 34 + 36 + 42 + 67 + 70 + y = 420 \]
Now, calculate the sum of the known numbers:
\[ 10 + 22 + 26 + 29 + 34 + 36 + 42 + 67 + 70 = 336 \]
So we have:
\[ 336 + y = 420 \]
Subtract \(336\) from both sides:
\[ y = 420 - 336 = 84 \]

### Step 6: Find \(\frac{y}{x}\)
Now we have \(x = 36\) and \(y = 84\). We can find \(\frac{y}{x}\):
\[ \frac{y}{x} = \frac{84}{36} \]
To simplify, divide both the numerator and denominator by \(12\):
\[ \frac{y}{x} = \frac{7}{3} \]

### Final Answer
Thus, the value of \(\frac{y}{x}\) is:
\[ \frac{y}{x} = \frac{7}{3} \]