MHB What is Peter's current age in this age problem?

[mathdad]

John is twice as old as his friend Peter. Peter is 5 years older than Alice. In 5 years, John will be three times as old as Alice. How old is Peter now?

Set up:

John = 2(x + 5)

Peter = x + 5

Alice = x

2(x + 5 + 5) = 3(x + 5)

Is this the correct equation?

 

[Greg]

To be honest I don't understand how you arrived at that last equation but it doesn't give the correct result for Alice's age.

Try this:

Let John be J years old, let Peter be P years old and let Alice be A years old. Then from the given information

1) J = 2P
2) P = A + 5
3) J + 5 = 3(A + 5)

Use 1) and 2) to get 3) in terms of P:

2P + 5 = 3P $\implies$ P = 5

So Peter is 5 years old, John is 10 and Alice is a newborn!

 

[mathdad]

To be honest I don't understand how you arrived at that last equation but it doesn't give the correct result for Alice's age.

Try this:

Let John be J years old, let Peter be P years old and let Alice be A years old. Then from the given information

1) J = 2P
2) P = A + 5
3) J + 5 = 3(A + 5)

Use 1) and 2) to get 3) in terms of P:

2P + 5 = 3P $\implies$ P = 5

So Peter is 5 years old, John is 10 and Alice is a newborn!

My original equation was 2x + 5 = 3x. For some reason, it did not makes sense. I often rush through applications thinking the problem has been completely understood.

 

[harne90]

My original equation was 2x + 5 = 3x. For some reason, it did not makes sense. I often rush through applications thinking the problem has been completely understood.

Yeah now the equation is correct. But i am still wondering it would give right answer.

 

[HallsofIvy]

"John is twice as old as his friend Peter. Peter is 5 years older than Alice. In 5 years, John will be three times as old as Alice. How old is Peter now?

Set up:

John = 2(x + 5)

Peter = x + 5

Alice = x

2(x + 5 + 5) = 3(x + 5)

Is this the correct equation?"


In the first place, "John", "Peter", and "Alice" are names, not numbers and it makes no sense to set them equal to "2(x+ 5)" "x+ 5", and "x"- especially since you have not said what "x" represents.

If you say "let x, y, and z be John's age, Peter's age, and Alice's age, respectively" then it makes sense to say
1) x= 2y (from "John is twice as old as his friend Peter").

2) y= z+ 5 (from "Peter is 5 years older than Alice").

The third condition is a little more complicated because it talks about "in five years". In five years, John's age will be x+ 5 and Alice's age will be z+ 5. Saying "In 5 years, John will be three times as old as Alice" gives
3) x+ 5= 3(z+ 5).

Since we are specifically asked to find Peter's age, y, I would use (1) x= 2y to write the (3) x+ 5= 3(z+ 5) as 2y+ 5= 3(z+ 5). That is equivalent to 2y+ 5= 3z+ 15 or 2y= 3z+ 10. We also have (2)y= z+ 5, so that z= y- 5 and then 2y= 3(y- 5)+ 10= 3y- 15+ 10= 3y- 5. From 2y= 3y- 5, y= 5. Peter is 5 years old.

Though it is not asked, John is twice as old, 10 years old. Alice's age is apparently 0 meaning, I presume, that she is a new born. These numbers, 10, 5, and 0 fit all of the information:
"John is twice as old as his friend Peter": 10= 2(5).
"Peter is 5 years older than Alice": 5= 0+ 5.
"In five years, John's age will be three times as old as Alice". Since John is 10, in five years, he will be 15. In five years, Alice will be 5 and 15= 3(5)".

 

[HallsofIvy]

IF we were to interpret "In 5 years, John will be three times as old as Alice" to mean that, in 5 years, John will be three times as old as Alice is now" (which I did not before because it seems a strange interpretation) then our three equations would be:
x= 2y
y= z+ 5
x+ 5= 3z.

Now, since x= 2y, x+ 5= 2y+ 5= 3z so 2y= 3z- 5. Since y= z+ 5, z= y- 5 so 2y= 3z- 5= 3(y- 5)- 5= 3y- 20. Then y=20. In this interpretation, Peter is 20 years old. Then "x= 2y" becomes x= 40 so John is 40 years old and y= z+ 5 becomes 20= z+ 5 so Alice is z= 20- 5= 15 years old. Quite a difference!

 

[Wilmer]

Hey Halls, that won't work next year :)