The smaller rectangle in the figure above represents the original size

Hi EMPOWERgmatRichC,

Thank you for replying to my post.
Yes, I agree my method will take longer as compared to factorizing this method with just intuition. I guess not having a Quant background does bite back.

You said, "1) We know that we're dealing with integers - and the answers are fairly 'nice' numbers to work with, so the solution will probably be a nice, round number".
My question is how can you be so sure?
Integer + Integer = Integer
Integer - Integer = Integer
Integer * Integer = Integer
However, Integer/ Integer = (might / might not) be an Integer.
Is it because we have the product -15,000 and all the answer options are a factor of -15,000?
I would like to know what is your thinking approach?

Can someone please explain for me why we don't subtract the smaller rectangle when calculating the enlarged lot area?

Thank you.

Original area was \(100*150 = 15000\)

Increased Area is \((100 + w)(150 + w) = 30000\)

Check the options \(30000 = 150 * 200\), hence Answer must be (B)

From the given information it follows that
(100 + w)(150 + w) = 2(100)(150), or
(100 + w)(150 + w) = (200)(150). This is a
quadratic equation that can be solved by several
methods. One method is by inspection. The left
side is clearly equal to the right side when w =50.
Another method is by factoring. Expanding the left
side gives (100)(150) + 250w + v} =(200)(150), or
v} + 250w - (100)(150) = 0. Factoring the left
side gives ( w - 50)( w + 300) = 0, which has
w =50 as its only positive solution.
The correct answer is B

2 x 100 x 150 = (100 + w) x (150 + w)
200 x 150 = (100 + w) x (150 + w)

Put w = 50, and voila solved:
200 x 150 = (100 + 50) x (150 + 50)

Bunuel JeffTargetTestPrep I factorized the equation w^2 + 250w - 15000 = 0 into w^2 + 150w + 100w + 15000 = 0 instead of w^2 + 300w - 50w - 15000 = 0
and got w = -150 or 100. Where did I go wrong?

Hello Anubhav21,

I see you are not comfortable with solving quadratic equations. So, let me first explain the general way of splitting the X-coefficient to solve quadratic equations and then we shall solve this question.

“SPLITTING THE X-coefficient METHOD” TO SOLVE QUADRATIC EQUATION

The steps to solve a quadratic equation under this method involve:

• Write the quadratic equation in the standard form, ax\(^2\) + bx + c = 0.
• Identify the values of a, b and c.
  
  1. Example: In 2x\(^2\) + 5x + 2 = 0, a = 2, b = 5 and c = 2.
  2. Example: 2x\(^2\) -5x + 2 = 0. Here, a = 2, b = -5 and c = 2.
• To simplify the quadratic equation, the coefficient of x (b) must be split into two terms b\(_1\) and b\(_2\), such that:
  
  1. b\(_1\) + b\(_2\) = b and
  2. b\(_1\)b\(_2\) = ac.
• Now, the equation will have changed to ax\(^2\) + b\(_1\)x + b\(_2\)x + c = 0. Finally, we take “pair-wise common factors out” to get a completely factored form of the quadratic equation. (Read the explanation below on your equation to best understand this)

APPLYING THIS METHOD ON THE QUESTION IN HAND

In the solution to this question, we arrived at the quadratic equation: w\(^2\) + 250w – 15000 = 0.

Let me first discuss what you did incorrectly and then show the correct simplification.

• You wrote w\(^2\) + 250w – 15000 as w\(^2\) + 150w + 100w – 15000.
  
  • So, you took b\(_1\) = 150 and b\(_2\) = 100.
  • Now, although the sum of 150 and 100 correctly gives us 250 (= b), their product does NOT give -15000 (= ac) as required.
• You most likely missed the sign that the constant term came with.
  
  • In this equation, a = 1 and c = - 15000. Hence, product of a and c (ac) = -15000
• To do this correctly, b (=250) had to be split such that product of b\(_1\) and b\(_2\) would be -15000 and not 15000.
  
  • Another try should give you that b can be split into 300 and -50.

Hence, correct split of equation w\(^2\) + 250w – 15000 = 0 is w\(^2\) + 300w -250w – 15000 = 0.

Now, continue solving the question. Reach back to us if you still have any doubt. 😊


Hope this helps!

Best Regards,
Ashish Arora
Quant Expert, e-GMAT

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