Math Competition Problems - Problem of the Week - 16

Updated: Jul 30, 2019

A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inches high and 10 inches wide. What is the area of the border, in square inches? - AMC 8 2012

Let's start by drawing a diagram of the photograph. Visualising a problem always helps us think straight and avoid mistakes.

Now that we have the diagram, we need to find the area of the frame (the outer part without the photograph).


Method 1:

Consider the photo and the frame as 2 rectangles. If we find the area of the bigger rectangle and take out the area of the smaller rectangle, we'll be left with the area of the frame!

Hence, Area of the frame = Area of the bigger rectangle - Area of the smaller rectangle.

How to find the area of the bigger rectangle? Do we knows its dimensions? Can we calculate it?

As you can see the width of the outer rectangle is equal to the width of the photo plus twice the width of the frame. Twice because there is a 2inch frame on the left and right side. Hence the width equals 10 + 2 + 2 = 14in.


Similarly, the height of the outer rectangle is equal to the height of the photo plus twice the width of the frame. Hence the height equals 8+2+2 = 12in.


Now the area of the outer rectangle = width* height = 14*12 = 168sq.in

The area of the inner rectangle (photo) = 10*8 = 80sq.in


Hence, the area of the frame = 168-80 = 88sq.in


A common mistake to watch out for:

A common mistake that students make in this question is how they calculate the width and height of the outer rectangle. They forget to add the width of the frame twice.

Hence they would calculate the width to be 10+2=12in and 8+2=10in and area = 120sq.in, which would be quite wrong. So watch out for it!


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