Grass in a lawn grows equally thick and at a uniform rate. It takes 24 days for 70 cows and 60 days for 30 cows to eat the whole of the grass. How many cows are needed to eat the grass in 96 days?

### Concept & Formula Apply Newton's Grazing Problem formula where initial grass plus growth over time balances the total food consumed by the cows. $$\\text{Initial Grass} + (\\text{Daily Growth} \\times \\text{Days}) = \\text{Cows} \\times \\text{Days}$$ ### Step-by-Step Solution * Let the initial grass be $g$ units, and the daily growth rate of the grass be $r$ units. * Case 1: 70 cows for 24 days. $$g + 24r = 70 \\times 24 = 1680 \\quad \\text{--- (Equation 1)}$$ * Case 2: 30 cows for 60 days. $$g + 60r = 30 \\times 60 = 1800 \\quad \\text{--- (Equation 2)}$$ * Subtract Equation 1 from Equation 2 to eliminate $g$: $$36r = 120 \\implies r = \\frac{120}{36} = \\frac{10}{3}\\text{ units/day}$$ * Substitute $r = \\frac{10}{3}$ back into Equation 2 to find $g$: $$g + 60\\left(\\frac{10}{3}\\right) = 1800 \\implies g + 200 = 1800 \\implies g = 1600\\text{ units}$$ * Case 3: Let $c$ cows be required to finish the grass in 96 days. $$g + 96r = c \\times 96$$ $$1600 + 96\\left(\\frac{10}{3}\\right) = 96c$$ $$1600 + 320 = 96c \\implies 1920 = 96c$$ $$c = \\frac{1920}{96} = 20\\text{ cows}$$ ### Exam Strategy & Shortcut Find the constant consumption rate relative to growth. The difference in total grass consumed between the two scenarios is $1800 - 1680 = 120$ units, which grows over $60 - 24 = 36$ days. This gives the growth rate $r = \\frac{120}{36} = \\frac{10}{3}$. Extending this linear growth safely to 96 days leads straight to $c = 20$. ### Common Pitfall Forgetting that grass continues to grow throughout the 96 days. Ignoring the growth factor entirely leads to completely incorrect inverse proportion estimates. ### Final Answer Therefore, the correct answer is 20.