Question

The length of a rectangle is decreased by 20% but its breadth is increased by k%. If the area of the rectangle remains unchanged, find the value of k.

Answer 1

@kryton1212 what is breadth referring to?

Answer 2

width, i think

Answer 3

well start with some basics l = length, b = breadth

Area of the original is A = lb

reduce the length by 20% so the new length is 0.8l
increase the breadth by k% so the new breadth is 1.kb

then the area is A = 0.8l x 1.kb

and you know this is the same as the original area


\[l \times b = (0.8 \times 1.k) l \times b\]

divide both sides of the equation by l x b

so

1 = 0.8 x 1.k

so just solve for k

Answer 4

Ok so let's call the initial with x and initial length y. Then the area of the initial rectangle is then \(\bf xy\). Now the new length is 0.8x and let's call the new width \(\bf W\) and we know that the area is still \(\bf xy\). Hence:\[\bf 0.8x \times W = xy \implies W=\frac{ 5 }{ 4 }y\]Clearly the the new width is (5/4)y and the initial width was y hence the increase is:\[\bf \frac{ 5 }{ 4 }y \div y=\frac{ 5 }{ 4 }\]So k = 125%..

Answer 5

you'll get a decimal when you solve for k. just multiply it by 100 to get the percentage increase.

and just to clear things up... k isn;t 125%

Answer 6

@campbell_st, My calculations are correct which is confirmed here:\[\bf x \times y = xy \ \ and \ \ \frac{4}{5}x \times \frac{5}{4}y=xy\]It's just that I assumed that the question was referring to the number the original width was multiplied by instead of just the increase. Now that I know that it's just the increase then the increase is obviously:\[\bf \frac{ 5 }{ 4 }y-y=\frac{ 1 }{ 4 }y\]So the increase is 25% but the new width is 1.25 (125%) times the original width. The latter was my answer initially because I thought the question was asking about how many times is the new width bigger than the initial width. Now that I realise it's just asking for the 'increase', the answer would just be 25%.

@kryton1212

Answer 7

@genius12

are you sure that k isn't 25%..... if the increase is 125% then the breadth then the length will be 225% of the original...

just a thought

Answer 8

@campbell_st I corrected myself by saying that it is 25% lol. I initally said 125% because I thought the question was referring to the amount by which the original breadth was multiplied by.

Answer 9

oops should read.... k = 25%... so if k = 125% the new breadth would be 225% of original

Answer 10

Did you read the last post I posted?

Answer 11

wow... correction occurred after my observation... so I'm pleased you read it

Answer 12

mhm lol. like i said, misconception of what the question was referring to..

Answer 13

answer should be k= 110

Answer 14

25% is correct