Straight-line trends are fitted by the method of least squares to the yearly sales (in lakh Rs.) of two products P and Q for the years 2020 to 2024:
P: \(y = 40 + 6X\), Q: \(y = 64 + 2X\)
where X = year − 2022 in both. If these trends continue, the first year in which the trend value of the sales of P will be more than that of Q is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2029
The trend value of P is more than that of Q when
\(40 + 6X \gt 64 + 2X\)
\(4X \gt 24\)
\(X \gt 6\)
X takes whole-number values, so the smallest is X = 7, that is, the year 2022 + 7 = 2029.
In 2028 (X = 6) both trend values are 76, so P is not yet more than Q; in 2029 (X = 7) P is 82 and Q is 78.
Why the other options are incorrect
Option (1) 2028 — Incorrect. In 2028 the two trend values are equal (76 each), not more.
Option (3) 2030 — Incorrect. P is already more than Q in 2029, so 2030 is not the first such year.
Option (4) 2026 — Incorrect. In 2026 (X = 4) P is 64 and Q is 72, so P is still less.