Becky has as many dimes as Ryan and Amy have together. Ryan has two more dimes than Amy, and Amy has one third as many dimes as Becky has. How many dimes does each girl have?

Answer 1

Question: Becky has as many dimes as Ryan and Amy have together. Ryan has two more dimes than Amy, and Amy has one third as many dimes as Becky has. How many dimes does each girl have?

Answer: We're only talking about dimes here, so let's assign some labels to Becky, Ryan, and Amy. Let b = number of Becky's dimes, r = number of Ryan's dimes, and a = number of Amy's dimes.

Let's translate ordinary English into algebra:

"Becky has as many dimes as Ryan and Amy have together." ---> b = r + a

"Ryan has two more dimes than Amy." ---> r = 2 + a

"Amy has one third as many dimes as Becky has." ---> a = (1/3)b

Let's make the problem easier for ourselves. We can solve any two equations for two of the variables, and then plug that information into a remaining equation to get the dirt on everyone here.

My plan is to take the first two equations. I know that b = r + a and that r = 2 + a. I can plug in the value of r into Becky's equation. I'll get this:

b = r + a

b = (2 + a) + a

b = 2 + 2a

See how I don't have the r to deal with now? Now, I'm looking at Amy's equation, a = (1/3)b. I just solved for b, and my answer is in terms of a. Let's plug it in and watch the b disappear!

a = (1/3)b

a = (1/3)(2+2a)

a = (2/3) + (2/3)a

(1/3)a = (2/3)

a = 2 ---> Amy has two dimes.

Now I can start back-substituting. Let's go back to Becky's equation that only has a on the right-hand side:

b = 2 + 2a

b = 2 + 2(2)

b = 2 + 4

b = 6 ---> Becky has six dimes.

And one more time, with Ryan's equation:

r = 2 + a

r = 2 + (2)

r = 4 ---> Ryan has four dimes.

Let's check our work:

"Becky has as many dimes as Ryan and Amy have together. Ryan has two more dimes than Amy, and Amy has one third as many dimes as Becky has."

a=2, b=6, r=4

"Becky has as many dimes as Ryan and Amy have together." ---> 6 = 4 + 2 ---> 6 = 6 is a true statement.

"Ryan has two more dimes than Amy." ---> 4 = 2 + 2 ---> 4 = 4 is a true statement.

"Amy has one third as many dimes as Becky has." ---> 2 = (1/3)(6) ---> 2 = 2 is a true statement.

We're correct!

Amy has two dimes, Becky has six dimes, and Ryan has four dimes.

Answer 2

x dimes - Becky, y dimes - Ryan, z dimes - Amy.

Then: x = y + z; y = z + 2 and z = x/3; so x = 3z.

We substitude in 1st equation: 3z = z + 2 + z; z = 2, x = 6, y = 4.

Becky has 6 dimes; Ryan has 4 dimes, Amy has 2 dimes