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A rectangle with perimeter 28 millimeters is shown in the diagram.
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Find the proportion of the shaded area inside the rectangle.
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The word “proportion” is telling us to find the shaded area as a fraction of the whole shape.
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To do this then, we’ll need to calculate the area of the whole shape and the area of the shaded region.
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To find the area of the shaded region, we’ll first find the area of the rectangle and then we’ll subtract the area of the two triangles.
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Notice how we’re told that the perimeter of the shape is 28 millimeters.
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We can use this then to help us find the width of the rectangle.
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We know that in a rectangle, opposite sides have the same length.
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So we can start by subtracting two lots of six from the perimeter of 28 millimeters.
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That gives us 16 millimeters.
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16 millimeters represents two lots of the width of the rectangle.
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We can halve that then to get that the width of the rectangle is eight millimeters.
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Once we have all these measurements, we can work out the missing dimensions of the two triangles.
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The missing height of this triangle at the top of our diagram is calculated by subtracting four from six.
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Six minus four is two.
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So it has a height of two millimeters.
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Similarly, the missing width of this triangle to the left of our diagram is calculated by subtracting six from eight.
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It’s also two millimeters.
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The formula for the area of a rectangle is its width multiplied by its height.
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We calculated earlier that the width of this rectangle is eight millimeters and it has a height of six millimeters.
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Eight multiplied by six means it has an area of 48 millimeters squared.
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Next, we need to find the area of the two triangles.
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The formula for area of a triangle is a half multiplied by its width and multiplied by its height.
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In the case of this triangle on the left then, that’s a half multiplied by two multiplied by six.
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And a half of two is one.
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So this becomes one multiplied by six, which is six.
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This triangle has an area of six millimeters squared.
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For the triangle at the top of the diagram, its area is a half multiplied by eight multiplied by two.
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A half of eight is four and four multiplied by two is eight.
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So this area is eight millimeters squared.
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Remember we said that to find the shaded area, we’d need to subtract the area of the two triangles from the area of the rectangle.
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48 minus six plus eight is equal to 34 millimeters squared.
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Finally, to write this as a proportion, we have to write as a fraction of the whole shape.
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We said that the whole shape, which is a rectangle, has an area of 48 millimeters squared.
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So the proportion is given by 34 over 48.
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We can simplify this fraction by dividing both the numerator and the denominator by two.
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The proportion of the shaded area inside the rectangle is 17 out of 24.