Two observations, picked by activity, give the line

A bill that will not sort itself

Northlake Nordic Centre Inc.'s snowmaking-and-hydro cost is mixed (L9.2): a connection charge that arrives in a month with no snow, plus power that rises with the skiers on the trails. The ledger records only the total. Five months of it:

To predict the bill at 7,000 skier-days the two parts must be separated, and the cost list does not say where the line runs. The high-low method draws it through two of the observations.

Pick by activity, never by cost

The two points are the months with the highest and lowest activity — January (9,000) and March (3,000). Not February: it has the highest cost (a storm month, an ice-damaged pump), but the line is being fitted to activity, and the x-axis decides. Between those two points, every dollar of difference is variable, because the fixed part is the same in both months:

$$\text{Variable cost per unit}=\frac{58,000-34,000}{9,000-3,000}=\frac{24,000}{6,000}=4.00\text{ per skier-day}$$

The fixed part is what is left at either point once the variable part is removed:

$$\text{Fixed}=58,000-4.00\times 9,000=22,000$$

and the check is the other point: 34,000 − 4.00 × 3,000 = 22,000 — the same, as it must be, since the line passes through both. The cost equation is 22,000 + 4.00 × skier-days, and at 7,000 skier-days the bill should be 22,000 + 28,000 = 50,000. In Excel the two-point line is =SLOPE and =INTERCEPT over the two rows.

What the wrong pair does

Take February as the high point and the slope becomes (59,000 − 34,000) ÷ (8,000 − 3,000) = 5.00, and the fixed part 59,000 − 40,000 = 19,000. The storm's extra cost has been read as if it were caused by skiers, the slope is a dollar too steep, and every prediction above 5,000 skier-days is too high. On the paper, a table is always arranged so the highest cost is not the highest activity; the trap is the point of the question.

What the method does not tell you

High-low uses two of the five months and ignores the other three. If either chosen month is odd — a storm, a closure, a price change — the line is wrong and nothing in the arithmetic will say so. A scatter diagram — all five months plotted over skier-days — is the check the method skips: if the points sit roughly on a line, the two chosen ones are representative; if February sits far above it, that month should be excluded before the line is drawn. Regression fits all the points at once; a case at this level asks for high-low, and expects you to say what the scatter diagram would add.

Cedar Ridge Golf Club Ltd.

Clubhouse maintenance by month: April 2,000 rounds / 14,600; May 4,500 / 20,100; June 6,000 / 23,400; July 7,500 / 26,700; August 7,000 / 27,100; September 5,000 / 21,200. Highest activity is July, not August. Slope (26,700 − 14,600) ÷ (7,500 − 2,000) = 2.20 a round; fixed 26,700 − 16,500 = 10,200; checked at April 14,600 − 4,400 = 10,200. At 6,500 rounds: 10,200 + 14,300 = 24,500.