Algebra word problems

[Bollinger]

I'm having trouble figuring out a couple of word problems, I figure my issue is likely the same with all of them, so I'll post them all I suppose anyone that wants to provide assistance can just pick one.

The room numbers of two adjacent classrooms are two consecutive even numbers. If their sum is 710, find the classroom numbers.

I ended up just brute forcing the answer, but I don't know how to find it the proper way. I would just do something like

n + (n + 1) = 710

But that is going to give me two consecutive numbers, not two consecutive even numbers, right?

The cost of living last year went up 6%. Fortunately, Alice Swanson got a 6% raise in her salary from last year. This year she is earning $26,330. How much did she make last year?

They even give me the formula for figuring this one out

Last year's salary + amount of raise = current salary

Obviously I can't just add .06 to n, so I figured it must be

n + .06n = 26330

So then I multiply both sides by the reciprocal of .06/1, which is 1/.06

2/.06 * n = 26330/1 * 1/.06
33.33n = 438833

Dividing both sides by 33.33 gives me

n = $13,166.30

Which doesn't seem like it's only 6% away from $23,660, according to my calculations it's closer to a 45% raise.

Any help for either of these problems would be greatly appreciated.

 

[Quaid]

The room numbers of two adjacent classrooms are two consecutive even numbers. If their sum is 710, find the classroom numbers.

n + (n + 1) = 710

But that is going to give me two consecutive numbers, not two consecutive even numbers, right?

No, that equation leads to the fraction 709/2, for the first room number.

Consecutive, even numbers always differ by two.

n + (n + 2) = 710

Alice Swanson got a 6% raise in her salary from last year. This year she is earning $26,330. How much did she make last year?

Obviously I can't just add .06 to n

You can add 0.06 to 1. In other words, factor n out of the expression n+0.06n, first.

n + .06n = 26330

So then I multiply both sides by the reciprocal of .06/1, which is 1/.06

2/.06 * n = 26330/1 * 1/.06

That's not correct. Multiplying both sides by 1/0.06 gives

n/0.06 + 1 = 26330/0.06

 

[stapel]

The room numbers of two adjacent classrooms are two consecutive even numbers. If their sum is 710, find the classroom numbers.

I ended up just brute forcing the answer, but I don't know how to find it the proper way. I would just do something like

n + (n + 1) = 710

For what does "n" stand? For what does "n + 1" stand? How far apart are two consecutive EVEN numbers?

The cost of living last year went up 6%. Fortunately, Alice Swanson got a 6% raise in her salary from last year. This year she is earning $26,330. How much did she make last year?

They even give me the formula for figuring this one out

Last year's salary + amount of raise = current salary

Obviously I can't just add .06 to n, so I figured it must be

n + .06n = 26330

So then I multiply both sides by the reciprocal of .06/1, which is 1/.06

What happened to the TWO terms on the left-hand side of the equation? You'd had:

. . . . .n + 0.06n = 26,330

This simplifies as:

. . . . .1n + 0.06n = 26,330

. . . . .1.06n = 26,330

Where did the "2/.06" come from?

 

[Ishuda]

...
n + (n + 1) = 710

But that is going to give me two consecutive numbers, not two consecutive even numbers, right?...

if n is an even number, what is the next even number [That is, if n is even, n+1 is odd].

...
Obviously I can't just add .06 to n, so I figured it must be

n + .06n = 26330

So then I multiply both sides by the reciprocal of .06/1, which is 1/.06 ...

n + .06n = 1.06 n = 26330
so why the reciprocal of just the .06 part?

 

[Bollinger]

Yeah, that first one should have been obvious, I was thinking about that in class and it shouldn't have given me as much trouble as it did.

Turning n + .06n into 1.06n also makes a lot of sense and would have made my life easier. My reasoning for getting 2n/0.6 was that I would have to multiple both "n"s by the reciprocal of .06, so 2n/1 * 1/0.6 = 2n/0.6

Having a hard time understanding where my logic falls apart there, but certainly turning it into 1.06n = 26330 would have solved my problem much more simply.

 

[Ishuda]

...My reasoning for getting 2n/0.6 was that I would have to multiple both "n"s by the reciprocal of .06, so 2n/1 * 1/0.6 = 2n/0.6

Having a hard time understanding where my logic falls apart there, but certainly turning it into 1.06n = 26330 would have solved my problem much more simply.

When you divide an equation (on both sides) by something, that means dividing each term by that something. That was your mistake. Starting with your

n + .06n = 26330

and dividing by .06 gives

n/.06 + (.06 n)/.06 = 26330 / .06
= 16.66667 n + n = 438833.333
or
17.66667 n = 438833.333
or
n=24839.62