SOLUTION: The length of a rectangle is 8 cm more than its width. If the length is decreased by 9 and the width is tripled, the area is increased by 50%. What was the area of the original tri

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Question 1048957: The length of a rectangle is 8 cm more than its width. If the length is decreased by 9 and the width is tripled, the area is increased by 50%. What was the area of the original triangle?

Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
What was the area of the original - did you mean to say, RECTANGLE?

L=w%2B8
wL=A, original area

Changed rectangle, %28L-9%29%283w%29=%283%2F2%29A, the 50% area increase.
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3wL-27w=%283%2F2%29A
3A-27w=%283%2F2%29A
3A-%283%2F2%29A=27w
A-%28A%2F2%29=27w
A%2F2=27w
highlight_green%28A=54w%29-------steps might be not the best path, but this equation is still useful.

Re-substitute for A.
wL=54w
%28L%2B8%29L=54%28L-8%29------substituted for w;
L%5E2%2B8L=54L-54%2A8
L%5E2%2B8L-54L%2B54%2A8=0
L%5E2-46L%2B8%2A54=0
-
Maybe the constant term is better factorable.
8%2A54=2%2A2%2A2%2A27%2A2=2%2A2%2A2%2A2%2A3%2A3%2A3-----or maybe this not too useful yet.
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L=%2846%2B-+sqrt%2846%5E2-4%2A8%2A54%29%29%2F2-----general solution of quadratic formula;
L=%2846%2B-+sqrt%28388%29%29%2F2
L=L=46%2B-+sqrt%284%2A97%29%29%2F2
L=%2846%2B-+2%2Asqrt%2897%29%29%2F2
L=23%2B-+sqrt%2897%29---------97 is a prime number.

Approx, L=23%2B-+9.849, and w=23%2B-+9.849-8


If taking the PLUS form of L,
system%28L=32.9849%2Cw=24.849%29
Original area, 820

If taking the MINUS form of L,
system%28L=13.151%2Cw=5.151%29
Original area, 68

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
The length of a rectangle is 8 cm more than its width. If the length is decreased by 9 and the width is tripled,
the area is increased by 50%. What was the area of the original highlight%28cross%28triangle%29%29 rectangle?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

L=w%2B8
wL=A, original area

Changed rectangle, %28L-9%29%283w%29=%283%2F2%29A, the 50% area increase.
-
3wL-27w=%283%2F2%29A
3A-27w=%283%2F2%29A
3A-%283%2F2%29A=27w
%283%2F2%29A=27w
%283%2F2%29LW=27W ---> %283%2F2%29L=27 ---> L = 2%2A27%29%2F3 = 18 cm.

Then W = L - 8 = 18-8 = 10 cm. Hence, the area was 10*18 = 180 cm%5E2.

Answer. The dimensions were 10 cm and 18 cm. The area was 180 cm%5E2.