Q.The mean weight of a group of students, measured in kilograms, is 52.4 kg. Weight is converted to pounds using the relation $y = 2.2x$ (where $x$ is the weight in kg and $y$ in pounds). Find the mean weight of the group in pounds.

Since $y=2.2x$ is a linear transformation of $x$ with no constant term, the mean transforms the same way: $\bar{y}=2.2\bar{x}$. …

The relation $y = 2.2x$ is a linear transformation $y=ax+b$ with $a=2.2$ and $b=0$. By the linear-transformation rule, $\bar{y} = a\bar{x}+b$.

Step 1 — Identify the known mean. $\bar{x} = 52.4$ kg.

Step 2 — Apply the rule.

Step 3 — Compute. $2.2 \times 52.4 = 2.2\times 52 + 2.2\times 0.4 = 114.4 + 0.88 = 115.28$. …

Trying to convert every individual student's weight to pounds and re-averaging (which would give the identical, correct answer but wastes the whole point of the shortcut) shows a misunderstanding of the rule — the transformation rule is precisely what lets us skip r …