Question:
<p>If the mean of the numbers \(27 + x,\ 31 + x,\ 89 + x,\ 107 + x,\ 156 + x\) is 82, then the mean of \(130 + x,\ 126 + x,\ 68 + x,\ 50 + x,\ 1 + x\) is</p>
<p>75</p>
<p>157</p>
<p>82</p>
<p>80</p>
Step-by-Step Solution
Key Concept: Adding a constant to each term increases the mean by that same constant. Therefore, find x from the first set, then recognize the second set's mean follows the same principle based on its constituent values.
<p><strong>Step 1:</strong> Find x using the first set of numbers.</p><p>Mean of (27+x), (31+x), (89+x), (107+x), (156+x) = 82</p><p>$$\frac{(27+x)+(31+x)+(89+x)+(107+x)+(156+x)}{5} = 82$$</p><p>$$\frac{410+5x}{5} = 82$$</p><p>$$410 + 5x = 410$$</p><p>$$x = 0$$</p><p><strong>Step 2:</strong> Find the mean of the second set (130+x), (126+x), (68+x), (50+x), (1+x).</p><p>$$\text{Mean} = \frac{(130+x)+(126+x)+(68+x)+(50+x)+(1+x)}{5}$$</p><p>$$= \frac{375+5x}{5} = \frac{375+0}{5} = 75$$</p><p><strong>Alternative insight:</strong> Since x=0, the second set becomes 130, 126, 68, 50, 1 with mean = (130+126+68+50+1)/5 = 375/5 = 75</p><p>∴ Answer: <strong>75</strong></p>
Correct Answer: A