# If P, Q, R are physical quantities, having different dimensions, which of the following combinations can never be a meaningful quantity ?

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If P, Q, R are physical quantities, having different dimensions, which of the following combinations can never be a meaningful quantity ?
(a) (P-Q)/R
(b) PQ-R
(c) PQ/R
(d) (PR-Q2)/R
(e)(R + Q)/P

(a, e)
Key concept: Principle of Homogeneity of dimensions: It states that in a correct equation, the dimensions of each term added or subtracted must be same. Every correct equation must have same dimensions on both sides of the equation.
According to the problem P, Q and R are having different dimensions, since, sum and difference of physical dimensions, are meaningless, i.e., (P – Q) and (R + Q) are not meaningful.
So in option (b) and (c), PQ may have the same dimensions as those of R and in option (d) PR and Q2 may have same dimensions as those of R.
Hence, they cannot be added or subtracted, so we can say that (a) and (e) are not meaningful.