Helen's age is 1/2 of Ray's age. Ray's age is 4/7 of Danny's age. Given that Danny is 15 years older than Helen, how old is Ray?

Question

 What is the answer to this word problem? I cannot figure it out. 

Answer 1

Let's define Helen's age as H, Ray's age as R, and Danny's age as D. Doing this allows us to set up a system of equations to solve this problem.

 

Since Helen is 1/2 of Ray's age, we can say that:

 

H=1/2*R We can multiply both sides of the equation to make it

 

1) R=2H

 

Doing the same for Ray, we can say:

 

R=4/7*D Multiplying both sides by 7 gives us

 

2) 7R=4D

 

We can't leave Danny out, either!

 

3) D=H+15

 

Since we have three equations and three unknowns, we can solve this system of equations. The easiest method, in this case, would be substitution.

 

We can start by substituting the R in equation 2 with equation 1. That gives us

 

7(2H)=4D

 

Dividing both sides by 2 gives us

 

4) 7H=2D

 

Plugging 3 into 4 gives us

 

7H=2(H+15).

 

Simplifying

 

7H=2H+30

 

5H=30

 

5) H=6:

 

So Helen is 6 years old. Since we know (from equation 1) that Ray is twice as old as Helen, we can plug 5 into 1 to get Ray's age:

 

R=2(6)

 

6) R=12

 

That gives us our answer. Ray is 12 years old.

 

Now, plugging 6 into 2 will give us Danny's age as 21, which also satisfies equation 3. Each person's age is as follows:

 

Danny = 21 years old

Helen = 6 years old

Ray = 12 years old