The diagram consists of three mathematically similar shapes - OCR - GCSE Maths - Question 17 - 2018 - Paper 5

Given the heights of the shapes in the ratio 1 : 4 : 5, since the areas of similar shapes are proportional to the square of their corresponding heights, we can determine the areas as follows:

Let the heights of the shapes be represented as follows:

• Height 1 (Shape 1): $h_1 = k$ (height ratio 1)
• Height 2 (Shape 2): $h_2 = 4k$ (height ratio 4)
• Height 3 (Shape 3): $h_3 = 5k$ (height ratio 5)

Calculating the areas for each shape gives:

• Area of Shape 1: $A_1 = k^2$
• Area of Shape 2: $A_2 = (4k)^2 = 16k^2$
• Area of Shape 3: $A_3 = (5k)^2 = 25k^2$

Now, summing the areas, we find:

• Total shaded area (Area of Shape 1): $15k^2$
• Total unshaded area (Area of Shape 2 + Area of Shape 3): $16k^2 + 25k^2 = 41k^2$

Thus, the ratio of total shaded area to total unshaded area is:

$\frac{15k^2}{41k^2} = \frac{15}{41}$

In its simplest form, the ratio is 15 : 41.