On which day is the grass exactly eaten up?

Question

A farm in the New Territories has a meadow where the grass grows at a steady rate and every goat eats the same each day.

The grass would feed 8 goats for exactly 15 days, or 6 for exactly 25 days.

Now 20 goats graze on day 1, and one is led away every evening — 19 on day 2, 18 on day 3, and so on.

Accepted Answer

Answer: 5 days. Let one goat's day be 1 part. 8 for 15 days: 120 parts; 6 for 25 days: 150. The extra 150 − 120 = 30 parts grew in 25 − 15 = 10 days, so 3 parts grow a day; at the start there were 120 − 3 × 15 = 75 parts. Let 3 goats eat each day's growth, so the grass never gains; the rest eat the 75 old parts. On day 1 that is 20 − 3 = 17, then one fewer each day. We need 17 + 16 + 15 + … to make exactly 75. n days in a row add to (17 + the last) × n ÷ 2. Try n = 5: the last day has 17 − 4 = 13, and (17 + 13) × 5 ÷ 2 = 75 ✓. (A day fewer gives only 62: not yet.) So the grass is eaten up exactly on day 5. Check: in 5 days the goats eat (20 + 16) × 5 ÷ 2 = 90 parts; the grass is 75 + 3 × 5 = 90 ✓ Common mistake: treating the herd as 20 every day, or forgetting the grass keeps growing. Split the new-grass eaters from the old-grass eaters, and the old grass becomes a sum of consecutive whole numbers.

Work problems: olympiad question 9

The question

Worked solution