Two farmers, Fred and Violet, were taking their sheep to market. Fred said, “Give me one of your sheep so we will both have the same number.” Violet answered, “No! Instead, you better give me one of yours, so I’ll have twice as many as you have.”

How many sheep did each of them have?

This problem can be solved by children in early grades using blocks or small stones for sheep. Do you see how they might do it? See the arithmetic solution below.

AN ALGEBRAIC SOLUTION

Name the unknown numbers:

Let F be the number of sheep that Fred has.

Let V be the number of sheep that Violet has.

Write 2 equations:

F + 1 = V – 1         Fred says, Give me one of your sheep, so we will have the same number.

V + 1 = 2(F – 1)    Violet says, No, you give me one, so I will have twice as many as you have.

Solve the equations:

F + 2 = V

      V = 2(F – 1) – 1

      V = 2*F-3              (*)

Substituting F + 2 for V in equation (*), we have

F + 2 = 2*F – 3

Combining terms,

2 + 3 = 2*F – F , so

      5 = F                       Fred has 5 sheep.

And

F + 2 = V,

5 + 2 = 7                       Violet has 7 sheep.

Check the solution mentally.

Write the answer:

The first farmer, Fred, had 5 sheep, and the second, Violet, had 7.

AN ARITHMETIC SOLUTION

If giving away one sheep makes their number equal, then their difference is 2. So Violet has 2 more sheep than Fred. If Fred gives one sheep to Violet, the differences increases to 4. But then Violet has twice as many as Fred. So then the numbers are 4 and 8. (See below.)

Thus the original numbers are 5 and 7!

Violet has 2 more sheep than Fred.

So Violet has 7 and Fred has 5!

Young kids can figure this out using blocks or pennies or … for sheep.