Math · Stage 6

Newton's grazing (cows eating growing grass)

The grass keeps growing while the cows eat — so you cannot just divide. Two grazing scenarios let you split the daily growth from the grass that was already there, and then you can answer any question about the field.

✦ Math📘 Stage 6 · S6⏱ ~15 min🧒 Ages 10–12★ Free to start

I can solve Newton's-grazing (growing-grass) problems.

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Kavi and the Two Herds 🐄

Kavi 🦉 watched two herds share one meadow 🌿. “10 cows nibbled it bare in 20 days — but 15 cows cleared the very same meadow in just 10 days,” Kavi hooted. “How odd! The secret is that the grass keeps growing while they eat 🌱. Two herds and two times are exactly enough to split the new grass from the grass that was already there. Sir Isaac Newton loved this puzzle!”

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What your child will learn

1Grass grows even while cows eatThis is not a fixed pile of hay. Every single day the field does two things: new grass grows, and the cows eat. Tap round the loop →

2The slow herd eats MORE grass in totalEach herd eats cows × days units of grass before the field is bare. Look how much each herd ate on the same meadow:

3Don't treat it like a fixed pileTap each card. One way forgets the growing grass; the other uses the two scenarios to pull the growth out.

4Grass eaten = grass at start + grass that grewEvery herd on the field obeys this one sentence. Tap a row to hear it. Once you know the start and the growth, you can answer a brand-new question.

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A peek inside

On one meadow, 10 cows eat it bare in 20 days; 15 cows clear the very same meadow in 10 days. How much grass grows each day? (each cow eats 1 unit a day)

grass eaten: 10 × 20 = 200 and 15 × 10 = 150; the slow herd's extra 10 days grew 200 − 150 = 50; so growth = 50 ÷ 10

Try this at home: Growth: 8 cows take 10 days, 12 cows take 6 days on one field → how much grass grows each day?

Ready to play and learn?

Start the free interactive lesson, or print the handout for screen-free play.

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Part of Math · Stage 6 · S6 on KAVI Academy — a secular, whole-child learning platform for ages 5–12.