Newton's Cows

Newton's Cows

Newton's cow problem

• 3 cows eat in 2 weeks all the grass in 1 acre, together with all the grass that grows there in the 2 weeks.

• 2 cows eat in 4 weeks all the grass in 1 acre, together with all the grass that grows there in the 4 weeks. 

• How many cows, then, will eat in 6 weeks all the grass in 3 acres together with all the grass that grows there in the 6 weeks?

Unknown cow number Z, Two unknown rates and Unknown starting grass amount per acre

• Cow eating rate X (Unit: pound per cow per week)

• Grass growing rate Y per acre (Unit: pound per week)

• Number of cows need to finish 3 acres grass in 6 weeks: Z

• Starting grass amount R per acre (Unit: pound)

Open Ended

Question:

Cow eating rate X (Unit: pound per cow per day)

Grass growing rate Y (Unit: pound per day)

Starting grass per acre R(Unit: pound)

How to use a Equation to describe 3-cow 2-week case

3 cows eat in 2 weeks all the grass in 1 acre, together with all the grass that grows there in the 2 weeks.

Write down your Equation 1 (unscored open poll)

Poll

Answers:

Cow grazing rate X (Unit: pound per cow per day)

Grass growing rate Y (Unit: pound per ace per day)

Starting grass amount R(Unit: pound per ace)

How to use a Equation to describe 3-cow 2-week case

3 cows eat in 2 weeks all the grass, together with all the grass that grows there in the 2 weeks.

Here is my Equation 1 (unscored poll, both are correct. click both)

• 3*2*X=R+2*Y

• i.e. 6X=R+2Y

Open Ended

Question:

Cow eating rate X (Unit: pound per cow per day)

Grass growing rate Y (Unit: pound per day)

Starting grass amount R(Unit: pound)

How to use a Equation to describe 2-cow 4-week case:

2 cows eat in 4 weeks all the grass in 1 acre, together with all the grass that grows there in the 4 weeks.

Write down your Equation 2 (unscored open poll)

Poll

Answer:

Cow eating rate X (Unit: pound per cow per day)

Grass growing rate Y (Unit: pound per day)

Starting grass amount R(Unit: pound)

How to use a Equation to describe 2-cow 4-week case:

2 cows eat in 4 weeks all the grass in acre, together with all the grass that grows there in the 4 weeks.

Here is my equation (unscored poll, both are correct)

• 2*4*X=R+4*Y

• i.e. 8X=R+4Y

Open Ended

Question:

Cow eating rate X (Unit: pound per cow per day)

Grass growing rate Y (Unit: pound per day)

Starting grass amount R(Unit: pound)

How to use a Equation to describe Z-cow 6-week case:

Unknown Z cows eat in 6 weeks all the grass in 3 acre, together with all the grass that grows there in the 6 weeks.

Write down your equation (open poll)

Poll

Answer:

Cow eating rate X (Unit: pound per cow per day)

Grass growing rate Y (Unit: pound per day)

Starting grass amount R(Unit: pound)

How to use a Equation to describe ?-cow 4-week case:

Unknown Z cows eat in 6 weeks all the grass in 3 acre, together with all the grass that grows there in the 6 weeks.

Here is my Question 3

(unscored poll, both are correct)

• Z*6X=3*R+3*6*Y

• Z*6X=3R+18Y

• Let's divide both by 3. 2ZX=R+6Y

Combining Equation 1 and 2

• Equation 1: 6X=R+2Y

• Equation 2: 8X=R+4Y

• Equation 3: 6ZX=3R+3*6Y

• Remember two equations can subtract each other

• Let's try Equation 2-Equation1:

• 2-cow 4-week case: 2*4X=R+4Y

• 3-cow 2-week case: 3*2X=R+2Y

• Z-cow 6-week 3-acre case: 6ZX=3R+3*6Y

• By algebra, we have learn X=Y. We can replace all Y by X.

• We also learn R=4X

• Hence Z-cow 6-week 3-acre Equation becomes

Back to Arithmetic

• Let's compare both 3-cow 2-week and 2-cow 4-week cases.

• 2 cows eat 8 (single-cow) weekly supplies of grass in 4 weeks, which is 2 weekly supplies more than 6 weekly supplies of grass eaten by 3 cows in 2 weeks.

• Therefore, The 2 single-cow weekly supplies of grass equal to grass grown in 1 acre in 2 extra weeks. Cut everything by half. Singe-cow weekly supply equals to grass grown in 1 acre over 1 week.

Now, we are ready to solve the original question

• "How many cows will eat in 6 weeks all the grass in 3 acres together with all the grass that grows there in the 6 weeks?"

• First, let's try to solve the question in 1 acre. Since cows eating in the same time and grass over 3 acres also grows simultaneously

• If we solve the problem in 1 acre, we just need to multiply the answer by 3.

• 
  

• Over 6 weeks, grass grown in 1 acre equals to 6 single-cow weekly supplies.

• Plus starting grass amount 4 weekly supplies

• Together we have 6+4=10 weekly supplies.

• How many cows can finish them in 6 weeks?