## A framed picture is shown above. The frame, shown shaded, is

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### A framed picture is shown above. The frame, shown shaded, is

by BTGmoderatorLU » Fri Oct 26, 2018 2:26 pm

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Source: Official Guide

A framed picture is shown above. The frame, shown shaded, is 6 inches wide and forms a border of uniform width around the picture. What are the dimensions of the viewable portion of the picture?

1. The area of the shaded region is 24 square inches.
2. The frame is 8 inches tall.

The OA is C.

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by [email protected] » Sat Oct 27, 2018 2:35 am

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## Global Stats

BTGmoderatorLU wrote:Source: Official Guide

A framed picture is shown above. The frame, shown shaded, is 6 inches wide and forms a border of uniform width around the picture. What are the dimensions of the viewable portion of the picture?

1. The area of the shaded region is 24 square inches.
2. The frame is 8 inches tall.

The OA is C.
Say the uniform width of the frame = a inches and the height of the frame = X inches

Dimensions of the picture: Width = 6 - 2a and Height = X - 2a

We have to get the values of (6 - 2a) and (X - 2a).

Question rephrased: What's the value of a and X?

Let's take each statement one by one.

1. The area of the shaded region is 24 square inches.

=> 2[6*a + (X - 2a)*a] = 24
6a + Xa - 2a^2 = 12
Can't get the value of a and X. Insufficient.

2. The frame is 8 inches tall.
=> X = 8.

But, we don't have the value of a. Insufficient.

(1) and (2) together

Plugging-in the value of X = 8 in 6a + Xa - 2a^2 = 12, we get

6a + Xa - 2a^2 = 12 => 6a + 8a - 2a^2 = 12 => 14a - 2a^2 = 12 => a^2 - 7a + 6 = 0 => (a - 6)(a - 1) = 0 => a = 6 or 1.

a cannot be 6 because if it were 6, then there would be no border at all as the width itself = 6 inches.

=> a = 1

So, we have X = 8 and a = 1. Sufficient.

Hope this helps!

-Jay
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by GMATGuruNY » Sat Oct 27, 2018 3:12 am

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## Global Stats

BTGmoderatorLU wrote:Source: Official Guide

A framed picture is shown above. The frame, shown shaded, is 6 inches wide and forms a border of uniform width around the picture. What are the dimensions of the viewable portion of the picture?

1. The area of the shaded region is 24 square inches.
2. The frame is 8 inches tall.
Statement 1:
Test one case that also satisfies Statement 2.

Case 1: The frame is 8 inches tall, with the result that the total area of the frame and picture combined = 6*8 = 48
Area of the picture = (total area) - (shaded region) = 48-24 = 24.
Implication:
The width of the frame is 1 inch, yielding for the picture a base of 4 and a height of 6.

Cases 2: The frame is MORE than 8 inches tall
Here, the area of the picture will be MORE than 24, with the result that the dimensions of the picture will NOT be 4 inches by 6 inches.

Since the dimensions of the picture can be different values. INSUFFICIENT.

Statement 2:
Since the width of the frame is unknown, the area of the picture can be different values.
As a result, the dimensions of the picture can be different values.
INSUFFICENT.

Statements combined:
Only Case 1 satisfies both statements, with the result that the picture has a base of 4 and a height of 6.
SUFFICIENT.

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### Re: A framed picture is shown above. The frame, shown shaded, is

by [email protected] » Fri Jun 11, 2021 3:50 pm

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## Global Stats

BTGmoderatorLU wrote:
Fri Oct 26, 2018 2:26 pm
Source: Official Guide

A framed picture is shown above. The frame, shown shaded, is 6 inches wide and forms a border of uniform width around the picture. What are the dimensions of the viewable portion of the picture?

1. The area of the shaded region is 24 square inches.
2. The frame is 8 inches tall.

The OA is C.
Solution:

We need to determine the dimensions of the viewable portion of the picture. If we let y be the length of the frame and x be the width of the uniform border of the frame and A be the area of the viewable portion of the picture, we can create the equation:

A = (6 - 2x)(y - 2x)

Therefore, to determine the dimensions of the viewable portion of the picture, we either need to know the values of x and y or the value of A and y (so that we can determine the value of x).

Statement One Alone:

This does not allow us to determine the value of A, x, or y. Statement one alone is not sufficient.

Statement Two Alone:

This means y = 8. However, this does not allow us to determine the value of A or x. Statement two alone is not sufficient.

Statements One and Two Together:

Notice that A, the area of the viewable portion of the picture, is the difference between the overall area and the area of the frame (i.e., the shaded region). The overall area is 6 x 8 = 48 and since the area of the frame is 24, A = 48 - 24 = 24. Therefore, we rewrite the equation from our stem analysis:

24 = (6 - 2x)(8 - 2x)

We see that we can determine the value of x. Therefore, both statements together are sufficient.

(Note: since the equation is quadratic, there are two solutions to the equation, i.e., two values of x. However, only one value of x makes sense since it’s really a geometry problem, i.e., only that value of x will make the dimensions positive (the other value of x will make the dimensions negative, which is not possible).)