Statistics
Median for ungrouped data
Grade 11
Question:
<p>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 <span>\(\frac{y}{x}\)</span> is equal to</p>
<p>(a) \(\frac{7}{3}\)</p>
<p>(b) \(\frac{7}{2}\)</p>
<p>(c) \(\frac{8}{3}\)</p>
<p>(d) \(\frac{9}{4}\)</p>
Step-by-Step Solution
Key Concept: For an even number of observations, the median is the average of the middle two terms. Use the mean condition to form an equation relating x and y, then use the median condition to find their individual values.
<p><strong>Step 1:</strong> Given ten numbers are 10, 22, 26, 29, 34, x, 42, 67, 70, y and their mean = 42</p><p>\[\frac{10 + 22 + 26 + 29 + 34 + x + 42 + 67 + 70 + y}{10} = 42\]</p><p>\[\frac{300 + x + y}{10} = 42\]</p><p>\[x + y = 120 \quad \text{...(i)}\]</p><p><strong>Step 2:</strong> Since n = 10 (even), the median is the average of the 5th and 6th terms when arranged in order.</p><p>The 5th term is 34 and the 6th term is x.</p><p>\[\text{Median} = \frac{34 + x}{2} = 35\]</p><p>\[34 + x = 70\]</p><p>\[x = 36\]</p><p><strong>Step 3:</strong> Substituting x = 36 in equation (i):</p><p>\[36 + y = 120\]</p><p>\[y = 84\]</p><p><strong>Step 4:</strong> Therefore,</p><p>\[\frac{y}{x} = \frac{84}{36} = \frac{7}{3}\]</p><p>∴ Answer is (a) \(\frac{7}{3}\)</p>
Correct Answer: A