Question 1207310: The length of a rectangle is 3 meters more than 4 times the width. If the area is 115 square meters, find the width and the length.
Found 3 solutions by MathLover1, greenestamps, ikleyn:
Answer by MathLover1(20849)   (Show Source):
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
If a formal algebraic solution is required, then do something like what the other tutor shows in her response.
But note the formal algebraic solution requires you to do some factoring that involves finding two numbers whose product is 115 --which is what the original problem requires. So, except as an exercise in practicing formal algebra, the formal algebra doesn't make finding the answer easier.
Informally, 115 = 23*5; and 23 and 5 satisfy the condition that the larger number is 3 more than 4 times the smaller. So
ANSWER: width 5, length 23
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
The length of a rectangle is 3 meters more than 4 times the width.
If the area is 115 square meters, find the width and the length.
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I will show an elegant way to solve this problem MENTALLY
in your mind by the way, which is 179% unexpected to you.
Let W be the width, in meters; then the length is 4W+3 meters,
and the area equation is W*L = 115 square meters, or
W*(4W+3) = 115. (1)
A standard way is to reduce it to quadratic equation and solve it
using the quadratic formula. This way is boring; but there is
a brilliant trick.
Multiply both sides of equation (1) by 4. You will get
4W*(4W+3) = 460.
So, now you need to find two numbers, 4W and (4W+3) with the product of 460.
These numbers are really close to each other, and they should be close to the
square root of 460, which, in turn, is close to 20.
+-----------------------------------------------------+
| As soon as the number 20 enters into your mind, |
| you MOMENTARILY get the numbers 20 and 23, |
| which are your 4W and 4W+3: 20*23 = 460. |
+-----------------------------------------------------+
So, 4W = 20; hence, W = 20/4 = 5 meters.
At this point, the problem is just solved.
ANSWER. The dimensions of the rectangle are W= 5 meters; L = 4*5+3 = 23 meters.
Solved (mentally).
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