let x = Susan’s age THEN

let 2x = John’s age NOW

and

let y = John’s age THEN

You set the variables up this way because John is twice as old now (2x) as Susan was then (x) …. see that? …. and you want John’s age then (was) to be represent by the same variable (I’m using “y”) that also represents Susan’s age now…. see that?

Okay…. now deciphering the riddle…. “The sum of John’s and Susan’s ages is 28…” {Note: they mean the sum of their ages “now” equals 28}

That means that (John’s age now) + (Susan’s age now) = 28

OR

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# 2x + y = 28 # (EQUATION 1) *

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You ALSO know that the difference in their ages now should be the same as the difference in their ages then…. (like if you are 4 years older than your sister right now…. then sometime in the past…. you are “still” going to be older than her by 4 years… you know what I mean?)

Okay… so if the difference in ages now is the same as the difference in their “then” ages …. that means…

(John’s Age Now) – (Susan’s Age Now) = (John’s Age Then) – (Susan’s Age Now)

which when translated into an algebraic equation… you have…

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# 2x – y = y – x # (EQUATION 2) **

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And that’s how you get the 2nd equation…

Now… with Equation 2…. move all the x’s to the left side of the “=”s sign and move all the y’s to the right side…

When you do that, you should have…

3x = 2y

OR…. solving for “y”….

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# y = (3/2)x # (EQUATION 3) ***

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2x + y = 28 (EQUATION 1) * (Remember this?)

Well…. rearranging Equation 1 you have…

2x = 28 – y

now solving for “x”, you get…

x = (28 – y) / 2

Now substitute “x” into Equation 3 and you get…

y = (3/2)x = (3/2)(28 – y) / 2 = (3/4)(28 – y) = 21 – (3/4)y

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# y = 21 – (3/4)y # (Equation 4)

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multiply the whole Equation 4 by “4” so that you aren’t dealing with fractions…. so now you should have….

4y = 84 – 3y

add “3y” to both sides… and you get…

7y = 84

divide both sides by 7 and you have…

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# y = 12 #

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Now you know “y”…. y = 12

You also know from Equation 1 that ” 2x + y = 28 ”

so substituting y = 12 into that equation… you get…

2x + 12 = 28

2x = 16

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# x = 8 #

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Now…. remember at the very start of solving this problem we said….

let y = Susan’s age now

let x = Susan’s age then

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# let 2x = John’s age now #

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and

let y = John’s age then

So…. Now that we know that x = 8…. we know that John’s age now which we represent by “2x” means that he is 2 * 8 = 16 years old now…

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# John is 16 years old now #

#######################

Check: we know that y = 12 and x = 8, right?

Earlier assumptions:

let y = Susan’s age NOW …. so we now know she is 12 now

let x = Susan’s age THEN ….. so we now know she was 8 then

let 2x = John’s age NOW ….. so we now know that he is 16 now

and

let y = John’s age THEN …. so we now know that he was 12 then…

So NOW try reading the first part of the stupid math riddle again… and see if it now makes sense…

John is TWICE as old as Susan WAS when John WAS as old as Susan is NOW.

Is it true with the now and then ages that we found?

John is Twice as old (16) as Susan was (8) when John was (12) as old as Susan is now (12)

yup… as you can see 16 = 2 x 8…. and 12 = 12…. AND… the difference between their ages then (12 – 8 = 4) is EQUAL TO the difference in their ages now (16 – 12 = 4)

Let’s check the 2nd sentence of the riddle….

The SUM of John’s and Susan’s ages NOW is 28.

Remember this is what we found out in the end?

let y = Susan’s age NOW …. so we now know she is 12 now

let 2x = John’s age NOW ….. so we now know that he is 16 now

So, yes, the 2nd sentence of the riddle is TRUE…

because 12 + 16 = 28

haha nvm, someone beat me… you CAN guess and check, but that can be really inefficient… simple algebra/variables would work better. one of the equations would be J+S = 28, but i’m still trying to work out what the other would be.

oh got it! okay so we’ll say that J is john’s age right now, and S is susan’s. so you know one equation is J+S = 28, like i said before. the other equation would be:

J = 2(S-(J-S)), because J is john’s present age, and we know that it’s twice something, so you have the 2 multiplied by some quantity. that quantity has to be susan’s age at the time that john was as old as susan is now (you can call this “time x”). this means that you have to figure out the difference between susan’s present age and her age at time “x,” and subtract that difference from her present age to get her actual age at time “x.” if we say that time “x” was a certain number of years ago (call it “z” years ago, for example), we know that the z is the difference between john’s present age and susan’s present age, because john was susan’s age at time “x.” so, “z” years ago can also be known as (J-S) years ago. this means that two figure out susan’s age at time “x,” we subtract (J-S) from susan’s present age, which is S, and we get “susan’s age when john was as old as susan is now,” which is part of what the question asks. and because it says that john is twice as old as this, we multiply that age by two, and get the equation

J = 2(S-(J-S)) along with the first equation of J+S = 28, and then you can use substitution and solve for J and S.

sorry… that came out sounding really complicated… :\ it’s not really that complex, just some logic. lol it just sounds long-winded when you explain every part.

j = 2 *(2s – j)

j + s = 28

s = 28 -j

j = 2 *(2(28-j) – j)

j = 4*28 – 4j – 2j

j = 112 -6j

7j = 112

j = 16 (john’s age)

Susan is 9.333 yrs old now, half of John’s age.