The straight-line trend of the yearly sales (in lakh Rs.) of a company, fitted by the method of least squares to the data of 2018 to 2024, is \(y = 52 + 4.5X\), where X = year − 2021. If this trend continues, the first year in which the trend value of the sales will exceed 80 lakh Rs. is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 2028
The trend value exceeds 80 when
\(52 + 4.5X \gt 80\)
\(4.5X \gt 28\)
\( \displaystyle X \gt \frac{28}{4.5} = 6.22\ldots \)
X counts whole years from 2021, so the smallest X is 7, which is the year 2021 + 7 = 2028. As a check:
\(2027\ (X = 6): \ y = 52 + 27 = 79\)
\(2028\ (X = 7): \ y = 52 + 31.5 = 83.5\)
Why the other options are incorrect
Option (1) 2026 — Incorrect. For 2026, X = 5 and the trend value is 52 + 22.5 = 74.5, below 80.
Option (2) 2027 — Incorrect. For 2027, X = 6 and the trend value is 79, still below 80.
Option (3) 2029 — Incorrect. The trend value is already 83.5 in 2028, so 2029 is not the first such year.