math
posted by gordon on .
The perimeter of a rectangular picture is 240 inches. the length is twice the width. Find the length and the width

4 w + 2 w = 240

Take a look at this. Here is your picture frame:
____
 
 
 
____
OK. It's not PERFECT, but what are you dealing with here? You're dealing with a rectangle. That means that every side is straight. It also means that one side is the same size as the side across from it. Make sure you understand those two points before reading further. Ask if you don't.
Now, if you measure the top, bottom, and two sides, then add them up, you get 240 inches. That is what we call the perimeter. That's why it says the perimeter is 240 inches. Again, ask if you don't understand that part.
The next thing I see from your problem is this:
"The length is twice the width."
So the length is (=) twice (2x) the width.
We can write it like this:
length = 2 x width
Remember, there are 2 sides that make up the length and 2 sides that make up the width. So if we take the:
length + length + width + width, we get the perimeter, or 240 inches.
To make this easier, I'm going to get rid of the "ength" in "length" and get rid of the "idth" in "width." I'm very lazy when I type, so I am just going to say:
l + l + w + w = 240 inches
But wait...the l is 2 times the w. We already saw that. So let's get rid of all those "l"s and just have "w"s.
2w + 2w + w + w = 240 inches.
See what I did? Since:
l = 2w (or length = two times the width), I just replaced every "l" with a "2w." That makes it easier for me since they are all now the same thing.
That means I now have 6 "w"s. 6 "w"s equals 240 inches. So let me rewrite this:
6w = 240 inches.
Well...if 6 "w"s is 240 inches, how long is one w?
If you have 6 friends who have 240 cookies, how do you find out how much each one has? (Assuming they all have the same amount). You divide.
6w divided by 6 is one w.
240 inches divided by 6 is 40.
So each piece that is the width of the frame is 40 inches. So we know what the width of the frame is...it's 40 inches.
What about the length? Let's go back to our equation.
The length is twice the width.
Well...the width is 40 inches. What's twice of 40? (Or 40 times 2?)
We easily see it's 80.
AHA! You say. So the length is 80 inches and the width is 40 inches? Can you prove this works?
Sure.
Take a piece of paper. On the longer side, write that it is 80 inches (remember to do that on both sides). The shorter sides (top and bottom), write that they are both 40 inches.
Now...add up the total around it.
80 + 40 + 80 + 40 = 240....
AHA! It works.
So there, my friend, is an explanation of this. I hope it has helped. 
Note:
For some reason, my rectangle did not turn out too well. Sorry about that. 
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