Algebra: Age word problem/Linear equation

[hackedagainanda]

Homework Statement

When asked her age, Miss Grundy refused to tell. After being begged for a hint she finally admitted that in 12 years she hoped to be three times as old as she was 72 years ago.

Write and solve an equation for this.

How old was she 72 years ago?

How old is she now?

Relevant Equations

None.

I haven't gotten to multiple variables yet in algebra, but I tried to solve it this way. x - 12 = 3y -72 with y = - 12 + x

Then I multiply 3x - 36 -72 = 3x -108

Next step is +108 to both sides to get 3x= 108, and then x = 36

So she was 36 72 years ago and is 96 now.

The math seems to checkout but I'm unsure.

 

[Simon Bridge]

"The math seems to checkout but I'm unsure."

Well you got, "So she was 36 72 years ago and is 96 now." ... so let's check that: $$36 + 72 \neq 96$$

If she is 96 now, how old is she 72 years ago?
Nice try tho ... you got tripped up, I suspect, by trying to say too many things at once and not keeping track of your reasoning.

Let's see if I can demonstrate what is expected:

If she had age x 72 years ago, and her age now is y, then y = x + 72 ... which is eq.(1)
In 12 years [y+12] she will be 3 times "as old as she was 72 years ago" [x] ... so y + 12 = 3x ... which is eq.(2)

See how defining terms as you go makes things clear?
You also put each word sentence translated directly into a maths sentence... at this stage it is just direct translation between languages.

The result is two equation that must both be true at the same time ... use algebra to solve for y.
ie. solve for x in (1) and substitute into (2), then solve for y.

You say you have not done multi-variable, so quicky lesson: after solving for x in (1) you get an equation like
x = my + c ... (you work out what m and c are)

To substitute into (2), every time you see an x in equation (2) you replace it with whatever my + c ... taking care when x is multipled by something to use brackets.

So (2) is y + 12 = 3x
So, after substitution, it becomes y + 12 = 3(my + c)

See?

 

[PeroK]

Homework Statement:: When asked her age, Miss Grundy refused to tell. After being begged for a hint she finally admitted that in 12 years she hoped to be three times as old as she was 72 years ago.

Write and solve an equation for this.

How old was she 72 years ago?

How old is she now?
Relevant Equations:: None.

I haven't gotten to multiple variables yet in algebra, but I tried to solve it this way. x - 12 = 3y -72 with y = - 12 + x

Then I multiply 3x - 36 -72 = 3x -108

Next step is +108 to both sides to get 3x= 108, and then x = 36

So she was 36 72 years ago and is 96 now.

The math seems to checkout but I'm unsure.

Hint: Mrs Grundy is almost the oldest person in the world!

 

[morrobay]

Only one variable: x+12=3(x-72)

 

[Delta2]

This is of the sort of problems where the algebraic equation is sort of simple, but expressing with words what the algebraic equation tell us (or more precisely, trying from the description with words to reach the expression of the algebraic equation), gets things complicated.

The problem tell us that in 12 years from now her age will be three times the age she had 72 years ago. So its natural to assume that $x$ is her age NOW. In 12 years from now her age will be $y=x+12$. The problem also tell us that $y=3(x-72)$ and i believe here is where things might get stuck in a sort of circular reasoning.

The key is to keep in mind that her age is x now, therefor her age in 12 years will be x+12 and her age 72 years ago was x-72.

 

[symbolipoint]

One is supposed to take the description exactly the way it is stated. Essentially the problem is this:

In 12 years she will be three times as old as she was 72 years ago.

Her age now, x

Her age in twelve years, x+12

Her age seventy-two years ago, x-72

Transform the description with the described phrases directly into the symbolic numerical statement, which should be fairly comfortable: x+12=3(x-72)
and from here, solving the equation is very easy.

 

[Mark44]

@hackedagainanda, except for extremely trivial problems, it's always a good idea when you're working with word problems, to define, in words, what your variables mean. What I quoted below is an example of what I'm talking about. A common objection from students is that it takes extra time to do this, but the time you save by not doing this has to be balanced against the time you waste being confused by the problem or getting the wrong answer.

Her age now, x
Her age in twelve years, x+12
Her age seventy-two years ago, x-72

Another point is that you should always do a sanity check on your solution. @Simon Bridge showed in his post why your answer is incorrect.

 

[symbolipoint]

This is a bit of an understatement:

working with word problems, to define, in words, what your variables mean.

A common objection from students is that it takes extra time to do this, but the time you save by not doing this has to be balanced against the time you waste being confused by the problem or getting the wrong answer.

Instead, doing so is an essential, and for academic purposes, often a required part of analyzing the problem.

 

[hackedagainanda]

This is of the sort of problems where the algebraic equation is sort of simple, but expressing with words what the algebraic equation tell us (or more precisely, trying from the description with words to reach the expression of the algebraic equation), gets things complicated.

The problem tell us that in 12 years from now her age will be three times the age she had 72 years ago. So its natural to assume that $x$ is her age NOW. In 12 years from now her age will be $y=x+12$. The problem also tell us that $y=3(x-72)$ and i believe here is where things might get stuck in a sort of circular reasoning.

The key is to keep in mind that her age is x now, therefor her age in 12 years will be x+12 and her age 72 years ago was x-72.

This is where I got tripped up.I get x + 12 = 3(x - 72) So x = 114 and he age 72 years ago was 42 and in 12 years she'll be 126

I didn't know that the 3 had to multiply the -72 also. Thanks for the help!

 

[morrobay]

Isn't basic algebra a prerequisite for this course? And I don't see where you originally got, x + 12 = 3(x-72) ?

 

[hackedagainanda]

This is a problem from an elementary algebra textbook. 114 + 12 = 126, 3(114-72)=3(42)=126 What's wrong?

 

[morrobay]

Only one variable: x+12=3(x-72)

Right which is obtained from the correct set up above. What is missing in your posts is showing the algebra that gives the solution: x +12 = 3(x-72) , x+12 = 3x - 216, -2x = - 228, x=114

 

[hackedagainanda]

Sorry, I omitted some steps that I did mentally.

 

[symbolipoint]

Right which is obtained from the correct set up above. What is missing in your posts is showing the algebra that gives the solution: x +12 = 3(x-72) , x+12 = 3x - 216, -2x = - 228, x=114

This is not tough. Not any bit tough.

Using pure text only,
x+12=3(x-72)
x+12=3x-3*72
12+3*72=3x-x
12+3*72=2x
6+3*36=x
x=114

You can substitute this value into the original equation and find if the solution works.

 

[Simon Bridge]

OK well done.
The takeaway lesson here is to write it out... don't make people guess what you did.

The correct answer is not as important as how you got there ... and even an incorrect answer can be useful if we have that working and reasoning.