Two variables y and x are related as y = 5x + 4 . The arithmetic mean of x is 10 . Arithmetic mean of y will be

The correct answer is B, 54. This is a direct application of how a linear transformation affects the mean. If y = a + bx, then the mean of y is a + b times the mean of x. The same rule holds for any constant shift and scaling, because the mean is itself a linear operator on the data.

Here a = 4 and b = 5, with the mean of x equal to 10. So the mean of y = 4 + 5 × 10 = 54.

• Option A (50): This takes only the scaling part, 5 × 10, and drops the constant 4. A constant added to every observation shifts the mean by the same amount.
• Option C (29): This does not follow from the rule. It looks like the mean of x was taken as 5 rather than 10, or the scaling and shift were mixed up.
• Option D (46): This is an arithmetic slip, not a property of means. It does not match a + b times the mean of x.

Core concept: effect of change of origin and scale on the arithmetic mean, which later links to standardisation and the coding method for calculating mean and variance in statistics paper and to data interpretation in Mains.

Trap: Candidates multiply the mean by 5 and forget to add the constant 4, which leads straight to option A.