SOLUTION: Two rectangles have the same area of 24cm^2. The second rectangle is 4cm shorter and 1cm wider than the first. What is the length and the breadth of the first rectangle?

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Question 1166715: Two rectangles have the same area of 24cm^2. The second rectangle is 4cm shorter and 1cm wider than the first. What is the length and the breadth of the first rectangle?
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
let L = length and W = width.

your formula for the area of a rectangle is:
A = L * W

when A = 24, this formula becomes:
24 = L * W

when L is 4 cm less and W is 1 cm more, the formula becomes:
24 = (L - 4) * (W + 1)

you need to solve these two formulas simultaneously.

from the first equation, solve for L to get:
L = 24 / W

in the second equation, replace L with W / 24 to get:
(24 / W - 4) * (W + 1) = 24
simplify this formula to get:
(24 / W * W) + (24 / W * 1) - (4 * W) - (4 * 1) = 24
simplify to get:
24 + 24 / W - 4 * W - 4 = 24
combine like terms to get:
24 / W - 4 * W + 20 = 24
subtract 20 from both sides of this equation to get:
24 / W - 4 * W = 4
multiply both sides of this equation by W to get:
24 - 4 * W ^ 2 = 4 * W
subtract the left side of this equation from both sides of this equation and simplify to get:
0 = 4 * W - 24 + 4 * W ^ 2
switch sides in this equation and order the terms in descending order of degree to get:
4 * W^2 + 4 * W - 24 = 0
divide both sides of this equation by 4 to get:
W^2 + W - 6 = 0
factor this equation to get:
(W - 2) * (W + 3) = 0
solve for W to get:
W = 2 or W = -3
W can't be negative, so:
W = 2

in the first equation of L * W = 24, replace W with 2 to get:
L * 2 = 24
solve for L to get:
L = 12

you have:
L = 12 and W = 2

in your first original equation, L * W = 24 becomes 12 * 2 = 24.
since this is true, it confirms the values for L and W are good in the first equation.

in your second original equation, (L - 4) * (W + 1) = 24 becomes (12 - 4) * (2 + 1) = 24 which becomes 8 * 3 = 24.
since this is true, it confirms the values for L and W are good in the second equation as well.

your solution is that the length and width of the first rectangle is 12 cm for the length and 4 cm for the width.

breadth and width mean the same thing in this problem.





Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.

Let x and y be the dimensions of the first rectangle.


We have  xy = 24 for its area.


The dimensions of the second rectangle are (x-4) and (y+1)  with the equation for the area

    (x-4)*(y+1) = 24,  or

    xy + x - 4y - 4 = 24.


Replacing here xy by 24, based on the very first equation, we get

    24 + x - 4y - 4 = 24, 

or, after collecting/canceling common terms

    x - 4y = 4.    (*)


So, we have now two equations

    xy = 24      (1)

    x = 4 + 4y.  (2)


By substituting (2) to (1), you get a quadratic equation

    (4+4y)*y = 24

    (1+y)*y = 6


At this point, you can solve it as a quadratic equation

or GUESS the solution mentally  y = 2.


ANSWER.  The dimensions of the first rectangle are  2 cm (the width)  and  24/2 = 12 cm (the length).

Solved.



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