How Do You Set Up an Equation for the Ages of John, Peter, and Alice?

[harpazo]

Homework Statement

Set up the correct equation leading to Peter's age now.

Relevant Equations

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

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? Do not solve. Just set up the correct equation.

This one involves more people.
I think a table helps out.

......Age Now...Age In 5 Years

John......2x.....(2x + 5)...

Peter......x......(x + 5)...

Alice......(x - 5).....(x - 5) + 5

Equation:

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

Sorry but it did not display in LaTex form. I tried.

 

[PeroK]

Are you sure about that?

 

[harpazo]

Are you sure about that?

This is why I joined this site. If my work is not right, I need to be corrected. Be back later. Time for dinner.

 

[PeroK]

This is why I joined this site. If my work is not right, I need to be corrected. Be back later. Time for dinner.

I would say that $(x - 5) + 5 = x$, which simplifies your equation.

 

[gmax137]

There are three people
John
Peter
Alice

Pick variable names that help you keep track of what they mean. I would say
J= John's age now
P=Peter's age now
A=Alice's age now.

The first thing we are told, is "John is twice as old as his friend Peter." what does that look like in equation form?
J=2P

Can you write out equations for the other facts we are given?

"Peter is 5 years older than Alice. "
This will be an equation with P and A

In 5 years, John will be three times as old as Alice.
This will be an equation with J and A

You should end up with three equations (one for each fact). This is good, because there are three unknowns.

 

[harpazo]

Are you sure about that?

I checked the answer online. I am right.

 

[Mark44]

Homework Statement:: Set up the correct equation leading to Peter's age now.
Relevant Equations:: 2x + 5 = 3(x - 5) + 5

Equation:
$2x+5=3(x-5)+5$

Sorry but it did not display in LaTex form. I tried.

It displayed just fine. You might have needed to refresh the screen to get the LaTeX to render.

I checked the answer online. I am right.

I don't see how. Your equation is not right. If you managed to get the correct answer starting with an incorrect equation, you probably made another error.

Using your equation (shown above), I get the ages for John, Peter, and Alice as 30, 15, and 10, respectively. In 5 years they will be 35, 20, and 15. Notice that John's age is not 3 times Alice's age.

If you correct Alice's age in 5 years to x, and multiply this by 3, you get the right answer.

 

[harpazo]

It displayed just fine. You might have needed to refresh the screen to get the LaTeX to render.

I don't see how. Your equation is not right. If you managed to get the correct answer starting with an incorrect equation, you probably made another error.

Using your equation (shown above), I get the ages for John, Peter, and Alice as 30, 15, and 10, respectively. In 5 years they will be 35, 20, and 15. Notice that John's age is not 3 times Alice's age.

If you correct Alice's age in 5 years to x, and multiply this by 3, you get the right answer.

I hate to see how the author solves the problem. I like to do the math work myself, right or wrong. BRB.

 

[harpazo]

It displayed just fine. You might have needed to refresh the screen to get the LaTeX to render.

I don't see how. Your equation is not right. If you managed to get the correct answer starting with an incorrect equation, you probably made another error.

Using your equation (shown above), I get the ages for John, Peter, and Alice as 30, 15, and 10, respectively. In 5 years they will be 35, 20, and 15. Notice that John's age is not 3 times Alice's age.

If you correct Alice's age in 5 years to x, and multiply this by 3, you get the right answer.

Here is what the author stated:

In 5 years, John will be three times as old as Alice.

2x + 5 = 3(x – 5 + 5)
2x + 5 = 3x

Isolate variable x
x = 5

Answer: Peter is now 5 years old.

 

[Mark44]

In 5 years, John will be three times as old as Alice.

2x + 5 = 3(x – 5 + 5)

Yes, and your mistake was having 3(x - 5) + 5 on the right side of the equation instead of 3(x -5 + 5). Better yet, as was suggested, it's better to simplify Alice's age in 5 years as just plain x, rather than x-5 + 5, which would make the right side of the equation above 3x.

 

[pbuk]

You are making this very difficult. The first step to making it easy is to use sensible letters: you are being asked to find the age of someone called Peter, why on Earth would you call that $x$? The next step is to write down one equation for each statement in the question, directly translating words into symbols:

John is twice as old as his friend Peter.

$j = 2p \quad (1)$ - write this down as the first line of your workings

Peter is 5 years older than Alice.

$p = a + 5 \quad (2)$ - write this down as the second line

In 5 years, John will be three times as old as Alice.

$j + 5 = 3 (a + 5) \quad (3)$ - and this is the third line.

Now we are trying to find $p$. Notice that equation (1) tells us that instead of j we can write 2p, and equation (2) says instead of a + 5 we can write p, so substitute equations (1) and (2) into (3) to give $2p + 5 = 3p$, write this down and now we can easily solve to find $p = 5$.

 

[harpazo]

You are making this very difficult. The first step to making it easy is to use sensible letters: you are being asked to find the age of someone called Peter, why on Earth would you call that $x$? The next step is to write down one equation for each statement in the question, directly translating words into symbols:$j = 2p \quad (1)$ - write this down as the first line of your workings$p = a + 5 \quad (2)$ - write this down as the second line$j + 5 = 3 (a + 5) \quad (3)$ - and this is the third line.

Now we are trying to find $p$. Notice that equation (1) tells us that instead of j we can write 2p, and equation (2) says instead of a + 5 we can write p, so substitute equations (1) and (2) into (3) to give $2p + 5 = 3p$, write this down and now we can easily solve to find $p = 5$.

I used the table method to help me set up the equation. There are several ways to answer math questions. You method may be easy for but perhaps not for others. I do agree that learning how to solve the problem via different methods is not a bad idea.

Question:

Say you came across this problem and you knew nothing about the substitution method for a system of linear equations, can this problem be solved without using algebra?

 

[pbuk]

I used the table method to help me set up the equation. There are several ways to answer math questions. You method may be easy for but perhaps not for others. I do agree that learning how to solve the problem via different methods is not a bad idea.

No, the reason you are taught using questions like this is to learn the basic steps of creating a mathematical model, which is to write down one equation representing each relationship or constraint specified in the problem. Writing some of the facts down in a table and hoping to extract an equation linking them all is not a viable way to solve problems in general.

In an exam there would probably be 2 marks for this question, and if you write down the equations correctly but mess up the algebra and get the answer wrong you would still get a mark. You wouldn't get anything for the table you created.

Say you came across this problem and you knew nothing about the substitution method for a system of linear equations, can this problem be solved without using algebra?

Read what I wrote again:

Notice that equation (1) tells us that instead of j we can write 2p, and equation (2) says instead of a + 5 we can write p.

Does this sound as scary as "the substitution method for a system of linear equations"?

 

[harpazo]

No, the reason you are taught using questions like this is to learn the basic steps of creating a mathematical model, which is to write down one equation representing each relationship or constraint specified in the problem. Writing some of the facts down in a table and hoping to extract an equation linking them all is not a viable way to solve problems in general.

In an exam there would probably be 2 marks for this question, and if you write down the equations correctly but mess up the algebra and get the answer wrong you would still get a mark. You wouldn't get anything for the table you created.Read what I wrote again:

Does this sound as scary as "the substitution method for a system of linear equations"?

I never had a problem using the substitution method for solving linear equations EXCEPT when it is 3 equations in 3 unknowns. In fact, I have not solve a system of linear equations in 3 unknowns in many years.

 

[phinds]

... I have not solve a system of linear equations in 3 unknowns in many years.

Then perhaps you should listen to @pbuk when he tells you that you are doing it the hard way. Do you seriously think he doesn't know what he is talking about? Why come here for help and then not listen to the advice you are given?

 

[Delta2]

Your table in The OP is correct. You just did what it seems to me a minor algebra mistake (as pointed out in post #10 too) you should have stated the equation as $2x+5=3((x-5)+5)$

It is algebra stuff that $$3(x-5)+5\neq 3((x-5)+5)$$.

 

[Mark44]

No, the reason you are taught using questions like this is to learn the basic steps of creating a mathematical model, which is to write down one equation representing each relationship or constraint specified in the problem. Writing some of the facts down in a table and hoping to extract an equation linking them all is not a viable way to solve problems in general.

The table that @harpazo set up was a reasonable and viable approach, IMO. The only place he fell down was in writing the equation, mistakenly thinking that 3(x - 5) + 5 was the age of Alice in 5 years, instead of 3[(x - 5) + 5], or better yet, 3x.

 

[pbuk]

Your table in The OP is correct.

The entries may be correct but they are not complete. See below.

The table that @harpazo set up was a reasonable and viable approach, IMO.

I disagree, and so do educators and examiners of material at this level.

Here is a question and an extract from the marking scheme for the November 2018 Edexcel GCSE Mathematics Foundation Paper 1 (copyright © 2018 Pearson Education Ltd, fair dealing exception claimed).

Azmol, Ryan and Kim each played a game.

Azmol’s score was four times Ryan’s score.
Kim’s score was half of Azmol’s score.

Write down the ratio of Azmol’s score to Ryan’s score to Kim’s score.

(Total for Question 13 is 2 marks)

[Award 1 mark] for a start to express the statements ... as algebraic expressions, two of 4x, x and 2x eg 4x : x, 1x : 4x, 1x : 2x or 2x : 1x with clear and correct link to Azmol, Ryan, Kim (any letter may be used).

There are other methods that would earn partial credit (at GCSE level, age 16 all abilities, the marking schemes are very generous) but in the OP's question no marks would be awarded for the table because it does not include the relationship "John will be three times as old as Alice".

I'll say it again: answering these questions is easy if you always follow the same, correct method.

1. The unknowns in the question are always given different initial letters so use these as symbols unless told otherwise.
2. For each statement in the question in isolation, write down one expression that does nothing more than translate the words into symbols (don't try to combine information at this stage).
3. Work out a way of combining the expressions in step 2 to form an expression which only has one of the unknowns; it doesn't matter which but there is often one way to do it that works out much easier than the others (formally this is substitution but when introducing this concept I find it simpler to talk about 'replacing' j with 2p, or even 'writing 2p instead of j because we know they are the same').

Using any other method is harder, slower, prone to error and will not get partial credit.

 

[jbriggs444]

One thing that I find important is to document your choice of variable names. I see very few students do so adequately. [And choose good names, but that's another discussion].

When learning computer programming in the 70's, we had this drilled into us. It carries over to mathematics. Before you write down an equation involving a variable x, tell us what x is.

This is crucial when you are trying to communicate with someone else (like the teacher checking your homework). It is also important for your own use. You can get buried so deep in the weeds that you can sometimes forget what you set out to accomplish in the first place.

 

[harpazo]

One thing that I find important is to document your choice of variable names. I see very few students do so adequately. [And choose good names, but that's another discussion].

When learning computer programming in the 70's, we had this drilled into us. It carries over to mathematics. Before you write down an equation involving a variable x, tell us what x is.

This is crucial when you are trying to communicate with someone else (like the teacher checking your homework). It is also important for your own use. You can get buried so deep in the weeds that you can sometimes forget what you set out to accomplish in the first place.

Funny that you should say that but it all depends on the question at hand. Most of the time, I let x, y, and sometimes z be what I'm looking for.

Let x = miles driven

Let y = height of kite

Let z = amount of gas per volume

and many more...

 

[jbriggs444]

Mathematicians and computer programmers operate in slightly different environments. Mathematicians tend to use complex formulas with only a few different variables. They save space by using juxtaposition to indicate multiplication. That forces them to single character variable names (and subscripts, primes and greek letters). So mathematicians get used to re-using the same variable names over and over in each successive problem or theorem.

Computer programmers tend to use simpler formulas with many different variable names. Our work gets parsed by compilers that do not mind seeing explicit multiplication signs (and which can't deal well with subscripts, primes and greek letters) and can deal with large variable names without complaint. We use multi-character variable names and tend to pick new ones every time.

 

[Vanadium 50]

The first step to making it easy is to use sensible letters: you are being asked to find the age of someone called Peter, why on Earth would you call that x?

Let a = Betty's age
Let b = Charles' age
Let c = Dorothy's age
Let d = Eugene's age
...

 

[harpazo]

Let a = Betty's age
Let b = Charles' age
Let c = Dorothy's age
Let d = Eugene's age
...

Most textbooks advice to let x be the unknown number. I agree that it is best to let the variable represent what we are searching for.

Samples:

Let a = airplane

Let b = balloon

Let c = car

Let d = distance

and so forth...

I did not commit a crime here. This is math. We are having fun with numbers.

 

[phinds]

We are having fun with numbers.

But it stops being fun if you manage to confuse yourself with sloppy terminology.

 

[harpazo]

But it stops being fun if you manage to confuse yourself with sloppy terminology.

I understand. I joined to enjoy math not to argue. We have a choice to agree or not.

 

[phinds]

I understand. I joined to enjoy math not to argue. We have a choice to agree or not.

That's true. You can choose to confuse yourself if you wish.

 

[harpazo]

That's true. You can choose to confuse yourself if you wish.

Moving on.

 

[Mark44]

Azmol, Ryan and Kim each played a game.

Azmol’s score was four times Ryan’s score.
Kim’s score was half of Azmol’s score.

Write down the ratio of Azmol’s score to Ryan’s score to Kim’s score.

(Total for Question 13 is 2 marks)

[Award 1 mark] for a start to express the statements ... as algebraic expressions, two of 4x, x and 2x eg 4x : x, 1x : 4x, 1x : 2x or 2x : 1x with clear and correct link to Azmol, Ryan, Kim (any letter may be used).

The mark scheme uses a single letter, x, as did the OP's work. Also, the OP gave a clear indication in his table what each of the expressions represented, both now and in five years. Although it's a good idea to use different variable names for the ages of John, Peter, and Alice, the problem can be done using a single variable with a single equation, rather than as a system of equations in three variables. Insisting that this must be done seems very pedantic, IMO.

As I've already said, the OP's two failings are not simplifying x - 5 + 5 (which contributed to writing an incorrect equation), and not checking his solution.

 

[pbuk]

The mark scheme uses a single letter, x, as did the OP's work. Also, the OP gave a clear indication in his table what each of the expressions represented, both now and in five years.

This was the mark scheme for a different question than that in the OP, and mark schemes are not model answers (the model answer for question 13 of that paper used A, R and K).

Also, the OP gave a clear indication in his table what each of the expressions represented, both now and in five years.

But the table in the OP had no representation of the sentence "John will be three times as old as Alice".

Although it's a good idea to use different variable names for the ages of John, Peter, and Alice, the problem can be done using a single variable with a single equation, rather than as a system of equations in three variables. Insisting that this must be done seems very pedantic, IMO.

I'm not insisting on anything, I'm simply pointing out that answering these questions is much easier and less prone to error if you use mnemonic variable names and proceed through the steps methodically rather than trying to jump to a solution.

 

[Mark44]

I'm not insisting on anything, I'm simply pointing out that answering these questions is much easier and less prone to error if you use mnemonic variable names and proceed through the steps methodically rather than trying to jump to a solution.

From earlier in the thread:

The first step to making it easy is to use sensible letters: you are being asked to find the age of someone called Peter, why on Earth would you call that x?

And again, the example you posted used x as the variable. If you wanted an example that emphasized your point, you could have picked a better example.

he next step is to write down one equation for each statement in the question, directly translating words into symbols:

Again the relationships are simple enough that 1) a single variable could be used, and 2) the relationships are combined into a single equation.

But the table in the OP had no representation of the sentence "John will be three times as old as Alice".

And so what? The representation appeared immediately below the table, albeit with an error.

The way you've presented your comments, it comes off as saying there is one and only one way to approach these problems, which is definitely not the case.

It's time to move on. Thread closed.