Not understanding algebra word problem.?

Question

I have a word problem in algebra that I am just not getting. My instructor worked it out with me but I still dont understand....in WORDS....how the numbers were found. Can someone PLEASE help me?

The length of a rectangular field is 2 yards more than its width. Find the width if the area of the field is 120 yd2.
1. Explain why the answer of a width of –12 yards is unreasonable.
2. Give a reasonable answer in its place.
3. Explain why your answer is more reasonable.

L=W+2
L*W=AREA
(W+2)W=120
W^2+2W-120=0
(W+12)(W-10)=0
W+12=0
W=-12 YDS. NOT A VALID ANSWER.
W-10=0
W=10 YDS. THE VALID ANSWER FOR THE WIDTH
PROOF
L=10+2=12 YDS. FOR THE LENGTH.
PROOF
10*12=120
120=120


I have no clue where these numbers are coming from and my
instructor will not explain.

Answer 1

Here's how it goes:

L = W + 2
L * W = Area
L * W = 120
(W + 2) W = 120
W² + 2W - 120 = 0
(W +12) (W - 10 ) = 0

So one possible answer is:
W + 12 = 0
W = -12
but as we know there are no negative widths, lengths etc. (i mean, how do u measure something NEGATIVE in value? unless it's temperature)

So the only thing to consider is:
W - 10 = 0
W = 10 yards

And the Length, L is 10 + 2 = 12 yards

Check:
L*W = 10 * 12 = 120 yards²

Answer 2

Hi there.

Call the length "L" and the width "W".

We're told that the length of the field is two yards more than its width. So, if we add two yards onto the width, we get the length, i.e. L=W+2.

Then we're asked a question involving the area of the field, so let's find an expression for the area. The area of a rectangle is its width multiplied by its length. i.e. AREA=L*W.

But we've already found that L=W+2, so the AREA=L*W becomes AREA=(W+2)W (all we've done here is replace L with W+2). So, we've found an expression that gives us the area in terms of just W, the width of the field.

We're told that the area of the field is 120yd2. Let's write this as an equation:

AREA = 120.

Now let's replace AREA with the expression we've just found:

(W+2)W=120.

Now, let's multiply out the brackets in (W+2)W. This gives W*W + 2*W = W^2 + 2W. So the previous expression becomes:

W^2 + 2W = 120.

Subtract 120 from both sides to get:

W^2 + 2W -120 = 0.

Now we have to factorise this to solve this. To factorise an equation like this that involves a W^2, we want to find two numbers that multiply together to make -120 and add together to make 2. After a bit of thought, you'll hopefully be able to see that -10 and 12 fit the bill. Therefore we can rewrite the previous expression as (W+12)(W-10)=0

So, consider when this expression is equal to zero. We've got two things, (W+12) and (W-10), that are zero when multiplied together make zero. This only happens when one of the brackets is zero. This happens when W+12=0 (so W=-12 here) or W-10=0 (W=10 here). So these are the widths we're looking for.

Can a field have a negative length? Does that make sense to you? Hopefully it doesn't, because it can't! If I took you to a field and asked you to measure the width of it, I'm sure you wouldn't tell me it was minus twenty yards or whatever! So the answer of "W=-12" that we found just doesn't make sense. There's no problem with our other answer of W=10 though. After all, a field can be 10 yards wide, there's no problem there!

The equations you've got at the bottom (from "PROOF" downwards) are just there to demonstrate that your answer actually works. They're the steps that you might go through after solving a problem to check that your answer fits the conditions that the question asked for. So, we've found that W=10 and then going back to the original question, the length was 2 more than the width. So L=10+2 = 12 yards. Therefore the area of the field is width x length = 10 * 12 = 120, which we were told it was! So your answer makes sense! Hurrah!

Does that help? Hope so, if not, ask about any specific bits that don't make sense :-)

Answer 3

Let

w = width

w + 2 = length

120 = Area

Area formula for a rectangle is

A = L x W

120 = (w + 2)w

120 = w² + 2w

Transpose 120

120 - 120 = w² + 2w - 120

Collect like terms

0 = w² + 2w - 120

The middle term is + 2w

The first term w² = 1

Find the sum of the middle term

Multiply the first term 1 times the last term 120 equals 120 and factor

Factors of 120

1 x 120
2 x 60
3 x 40
4 x 30
5 x 24
6 x 20
8 x 15
10 x 12. . .←. .use these factors

- 10 and + 12 satisfy the sum of the middle term

insert - 10w and + 12w into the equation

0 = w² + 2w - 120

0 = w² - 10w + 12w - 120

Group factor

0 = (w² - 10w) + (12w - 120)

0 = w(w - 10) + 12(w - 10)

0 = (w + 12)(w - 10)

- - - - - - - -

Roots

0 = w + 12

Transpose 12

0 - 12 = w + 12 - 12

Collect like terms

- 12 = w. . .←. .not used
- - - - - - -
Roots

0 = w - 10

Transpose 10

0 + 10 = w - 10 + 10

Collect like terms

10 = w. . ,←. .The width

- - - - - - - - -

The width is 10 yards

- - - - - - - -

The length is w + 2

10 + 2 = 12

The length is 12 yards

- - - - - - - - -s-

Answer 4

Seems like a problem for calculating territory (or area). You have L for length and W for width so:
Length equals Width + 2
Length times Width equals the Area
(Width+2) -which by the way is the length- times Witgh equals 120
That means that Length times Width equals 120
That means that the Area is 120
Not sure what the little ^ sign means so can't help you with that part...
Width = -12 is not a valid answer because width cannot be a negative number
Width equals 10 yards - that is the valid answer for width (that I suspect you did not get when solving the problem)
if Width is 10 yards, then Length = Width + 2 = 12 yards
In order to prove that Width = 10 and Lenght - 12 you need to multiply them and see if they equal 120 (the Area that is given above)
10*12=120 - that is the proof that length and width were calculated correctly.
Hope I was of help.

Answer 5

It is not as much difficult:

L = LENGTH OF THE FIELD
W = WIDTH OF THE FIELD
L IS W + 2 (WIDTH + 2 YARDS)

NOW , the Total Area i.e LENGTH MULTIPLIED BY WIDTH
i.e
L*W = 120 Square Yards.

So it forms into a Quadratic Equation and
gives 2 Answers .
One gives WIDTH AS = -12 (Minus 12) which is negative and not
"practically" correct.
So the correct answer is Width = 10 Yards
and Length as 12 Yards and
The Total Areas as 120 Square Yards.

Answer 6

Answer is w=10
After you factor down to (w+12)(w-10)=0, you have 2 possible answers
(w+12)=0
w= -12 which is impossible

(w-10)=0
w=10

There you go

Answer 7

Well, the key to this problem is realizing that the distance up the mountain = distance down the mountain. We also see some sort of relationship between the times, so we should choose a variable for this and set up an equation. In hours, let's call the time down = t, and then time up = t + 1 We know that distance = speed * time, so we can set speed up * time up = speed down* time down 1.5(t) = 4(t + 1) Solve for t. Then answer the final question, how long was the entire hike. This would be time up + time down = (t + 1) + t = 2t + 1 Hope that helps :)

Answer 8

Mathematicians are very lazy.
They try to put real life problems into mathematical terms then mess abou with the mathematics to find easier solutions.

In this case the problem is about putting the area of the rectangular problem into a mathematical term. And for this problem they have used something called factorisation.

Answer 9

try these sites they might help you

www.hawaii.edu/suremath/intro_algebra.html
math.about.com/library/weekly/aa071002a.htm

www.jamesbrennan.org/algebra
www.helpalgebra.com/onlinebook/generalwordproblems.htm

Answer 10

-_-