Annual linear trend of sales from a trend value and yearly rise, and its monthly form

(i) The trend equation

A linear trend has the form Y = a + bX, where a is the trend value at the origin and b the change in the trend per unit of X; it is the straight line that least squares would fit. Here the trend value of 2023 is ₹14,50,000 and the trend rises by ₹36,000 a year, so with 2023 as the origin:

Y = 14,50,000 + 36,000X, origin 2023 (the middle of the year, 1 July 2023); X unit: one year; Y: annual sales (₹).

For example, the trend value for 2025 (X = 2) is 14,50,000 + 36,000 × 2 = ₹15,22,000.

(ii) The monthly trend

Conversion of an annual linear trend to a monthly basis is part of the syllabus, and the DU guideline keeps it to linear trends. Divide a by 12 and b by 144:

• a / 12 = 14,50,000 / 12 = ₹1,20,833.33 (average sales of a month at the origin);
• b / 144 = 36,000 / 144 = ₹250 (rise in monthly sales from one month to the next).

Monthly trend: Ym = 1,20,833.33 + 250x, origin 1 July 2023, x unit one month, Ym monthly sales (₹).

To centre the origin on the middle of July 2023, add half a month's rise: Ym = 1,20,958.33 + 250x (since 1,20,833.33 + 125 = 1,20,958.33).

Why b is divided by 144. The figure ₹36,000 is the rise in annual sales per year. Dividing by 12 turns annual sales into monthly sales (₹3,000 a year in monthly sales), and dividing by 12 again turns the time unit from a year into a month (₹3,000 / 12 = ₹250 a month). The intercept is a level of sales, not a rate, so it is divided by 12 only.

The other reading. If only the time unit is changed and Y is left as annual sales, the equation becomes Y = 14,50,000 + 3,000x (x in months from 1 July 2023): annual sales on trend grow by ₹3,000 for each month that passes. The full monthly trend, with monthly sales as the variable, is the conversion asked for.