1. Key Concepts and Formulas

To solve this problem, we need a clear understanding of the definitions and formulas for the mean and median of ungrouped data.

• Mean (Arithmetic Average): The mean, denoted by $\bar{x}$, is calculated by summing all observations in a dataset and dividing by the total number of observations. It represents the dataset's central tendency. For $n$ observations $x_1, x_2, \ldots, x_n$: $\text{Mean} (\bar{x}) = \frac{\sum_{i=1}^{n} x_i}{n}$

• Median: The median is the middle value of a dataset when the observations are arranged in ascending or descending order. It provides a measure of central tendency that is less sensitive to extreme values than the mean.

  • If the total number of observations ($n$) is odd, the median is the value of the $\left(\frac{n+1}{2}\right)^{\text{th}}$ observation.
  • If the total number of observations ($n$) is even, the median is the average of the two middle observations. Specifically, it is the average of the $\left(\frac{n}{2}\right)^{\text{th}}$ and $\left(\frac{n}{2} + 1\right)^{\text{th}}$ observations. $\text{Median} = \frac{\left(\frac{n}{2}\right)^{\text{th}} \text{ observation} + \left(\frac{n}{2} + 1\right)^{\text{th}} \text{ observation}}{2}$

2. Step-by-Step Solution

Step 1: Understand the Given Data and Identify Key Information

We are provided with ten numbers arranged in increasing order: $10, 22, 26, 29, 34, x, 42, 67, 70, y$.

From this, we extract the following critical information:

• Total number of observations, $n = 10$.
• The dataset is already sorted in increasing order, which is crucial for calculating the median.
• The mean of these numbers is given as 42.
• The median of these numbers is given as 35.
• Our objective is to determine the value of the ratio $\frac{y}{x}$.

Step 2: Determine the value of 'x' using the Median Information

• Why this step? The median calculation for an even number of observations directly involves the two middle terms. In our given dataset, $x$ is one of these middle terms. By utilizing the median information first, we can directly solve for $x$, simplifying the problem by reducing the number of unknowns before proceeding to the mean calculation.

• Applying the Median Formula: Since $n=10$ (an even number), the median is the average of the $\left(\frac{n}{2}\right)^{\text{th}}$ and $\left(\frac{n}{2} + 1\right)^{\text{th}}$ observations. The $\left(\frac{10}{2}\right)^{\text{th}}$ observation is the $5^{\text{th}}$ observation. The $\left(\frac{10}{2} + 1\right)^{\text{th}}$ observation is the $6^{\text{th}}$ observation.

• Identifying the Middle Observations: From the given ordered list: The $5^{\text{th}}$ observation is $34$. The $6^{\text{th}}$ observation is $x$.

• Setting up and Solving the Equation for Median: We are given that the median is 35. Substituting the values into the median formula: $\text{Median} = \frac{5^{\text{th}} \text{ observation} + 6^{\text{th}} \text{ observation}}{2}$ $35 = \frac{34 + x}{2}$ Multiply both sides by 2: $35 \times 2 = 34 + x$ $70 = 34 + x$ Subtract 34 from both sides to solve for $x$: $x = 70 - 34$ $\mathbf{x = 36}$ Self-check: Since the numbers are in increasing order, $34 < x < 42$. Our calculated value $x=36$ fits this condition ($34 < 36 < 42$), confirming its validity.

Step 3: Determine the value of 'y' using the Mean Information

• Why this step? Now that we have successfully found the value of $x$, $y$ is the only remaining unknown. The mean formula involves the sum of all observations. By substituting the known value of $x$ and the given mean, we can establish an equation to solve for $y$.

• Listing the Observations with 'x' Substituted: The ten observations are now: $10, 22, 26, 29, 34, 36, 42, 67, 70, y$.

• Calculating the Sum of Observations ($\sum x_i$): $\sum x_i = 10 + 22 + 26 + 29 + 34 + 36 + 42 + 67 + 70 + y$ Summing the known numerical values: $10+22+26+29+34+36+42+67+70 = 336$ So, the sum of all observations is: $\sum x_i = 336 + y$

• Setting up and Solving the Equation for Mean: We are given that the mean is 42 and $n=10$. Using the mean formula: $\text{Mean} = \frac{\sum x_i}{n}$ $42 = \frac{336 + y}{10}$ Multiply both sides by 10: $42 \times 10 = 336 + y$ $420 = 336 + y$ Subtract 336 from both sides to solve for $y$: $y = 420 - 336$ $\mathbf{y = 84}$

Step 4: Calculate the Required Ratio $\frac{y}{x}$

• Why this step? The question explicitly asks for the value of the ratio $\frac{y}{x}$. Having determined both $x$ and $y$, this is the final step to provide the answer.

• Substitute and Simplify: We found $x = 36$ and $y = 84$. Substitute these values into the ratio: $\frac{y}{x} = \frac{84}{36}$ To simplify the fraction, we find the greatest common divisor (GCD) of 84 and 36, which is 12. Divide both the numerator and the denominator by 12: $\frac{84 \div 12}{36 \div 12} = \frac{7}{3}$ Thus, the required ratio is $\mathbf{\frac{7}{3}}$.

3. Common Mistakes & Tips

• Median for Even 'n': A frequent error is to incorrectly identify the median for an even number of observations. Remember, it's always the average of the two middle terms, not just one of them.
• Order of Data: While this problem states the data is already in increasing order, always verify this condition before calculating the median. If not, the first step must be to sort the data.
• Arithmetic Accuracy: Be meticulous with all calculations (addition, subtraction, multiplication). A small error in summing the numbers or solving the equations can lead to an incorrect final answer.
• Consistency Check: After finding $x$, quickly check if it maintains the increasing order of the given sequence. This provides a useful self-check for your calculation.

4. Summary

This problem effectively tests the fundamental definitions of mean and median. We first utilized the median formula for an even number of observations to find the value of $x$, as it was one of the two middle terms. Once $x$ was determined, we used the mean formula, which involves the sum of all observations, to solve for $y$. Finally, we calculated the ratio $\frac{y}{x}$ and simplified the fraction to its lowest terms. This systematic approach ensures accuracy and efficiency in solving such statistical problems.

The final answer is $\boxed{\frac{7}{3}}$, which corresponds to option (A).