Farmer Bob of the Martian Colony counted his sheep and chickens in an unusual way. When he finished, he reported that he had counted 86 feet and 26 heads. Use a guess and check table to find how many sheep and chickens are owned by the famer.

name: monica
grade level: 6th grade
student is asking the question

Hi Monica,

I have two solutions for you.

Penny's solution

I like this problem a lot but I don't like the guess-and-check method for solving it. I want to show you another, more constructive method.

I am going to imagine "constructing" sheep and chickens from marshmallows and toothpicks. A marshmallow with two toothpicks in it is a chicken



and a marshmallow with four toothpicks is a sheep.

You have 26 heads (marshmallows) and 86 feet (toothpicks). Put 2 toothpicks into each marshmallow to make 26 chickens. You have 86 - 2x26 = 34 toothpicks remaining, that is 17 pairs of toothpicks remaining. Put these 17 pairs of toothpicks into 17 of the "chickens" to turn them into "sheep". Now you have 17 sheep and 26 - 17 = 9 chickens. [Check: 17 + 9 = 26 heads and 4x17 + 2x9 = 86 feet.]

Penny

Paul's solution

26 heads means that the number of sheep plus the number of chickens is 26

Lets start with 10 sheep. This means 16 chickens. The number of feet is

10x4 + 16x2 = 72. This is too low so we need to increase the sheep to get more feet.

Lets now try 11 sheep. This means 15 chickens. The number of feet is 11x4 + 15x2 = 74. This is too low so we need to increase the sheep to get more feet.

Lets now try 12 sheep. This means 14 chickens. The number of feet is 12x4 + 14x2 = 76. This is too low so we need to increase the sheep to get more feet.

Have you noticed a pattern? The number of feet is going up by 2 for each sheep we add. To get from 76 to 86 feet, we need to go by 2 five more times. This suggests that we try 12+5=17 sheep. With 17 sheep, there are 9 chickens, and the number of feet is 17x4+9x2=86. Therefore, there are 17 sheep and 9 chickens.

Paul
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