Newton's Cow Problem

MediumNumber GamesAlgebra

The classic cow-and-grass problem: grass grows while cows eat. Find the growth rate, how many days 25 cows need, and the largest herd that can graze forever.

The challenge

A pasture grows grass at a steady rate. 10 cows clear it in 20 days, and 15 cows clear it in 10 days. How long do 25 cows take?

More than 300 years ago Newton noticed something odd about a pasture: the cows were eating the grass while the grass was still growing. That is the classic cow-and-grass problem, and it turns up everywhere something is used up while it is being refilled.

Read the problem, work out your own answer, then compare it with the step-by-step solution.

How to play

  1. Read the problem and the two records it gives you. Then work out your own answer and check it.
  2. If you are stuck, ask for a hint, or open the step-by-step solution: total eaten, grass grown per day, grass at the start, answer.
  3. The method is three steps: ① (bigger total − smaller total) ÷ (extra days) = grass grown each day; ② total eaten − grass that grew = grass at the start; ③ starting grass ÷ net eaten per day = days.
  4. One rule to remember: once the herd eats no faster than the grass grows, the pasture never runs out. The Pasture lab lets you try it: change the herd and the growth rate, then run the days forward.