In this diagram, the shaded rectangles are all of equal width, 𝑤. Calculate the width, 𝑤, of one shaded rectangle in centimeters.
In this problem, we’re given a diagram which is made up of one large rectangle. And it split into several smaller rectangles. In a way, we can look at it as two separate rows. The top row is made up of a large rectangle and then a smaller one that’s been shaded. The long rectangle has a width of 34 centimeters. And this is labelled on the diagram. We don’t know the width of the shaded rectangle. But it’s given a letter to represent the measurement, 𝑤. There are two more rectangles with a width of 𝑤 in the bottom row. And we’re told in the first sentence that all three of these rectangles are of equal width. We need to calculate the width of one of these shaded rectangles. But how can we find out the answer?
We might think that we could measure to find the answer. But we’re told that the diagram is not to scale. And we’re told to calculate the width anyway not measure it. If the diagram showed the width of the whole rectangle, then we could simply subtract 34 to find 𝑤. But we’re not given this measurement. So how can we calculate the answer? Although we need to do some calculation to find the answer, this question is mostly about visualizing what we can do to the shapes to make them easier to understand. We can’t cut out the rectangles and move them around. But we can imagine or visualize doing this.
Firstly, let’s think of the diagram as being made up, as we said already, of two rows. What if we swapped around the rectangle that’s 18 centimeters wide and the shaded rectangle next door to it? The top row would still look the same. But the bottom row would now look like this. In a way, we could ignore these two shaded rectangles on the left-hand side of the diagram because they cancel each other out. And so we can now think of the diagram as being like a bar model. 18 plus nine plus 𝑤 is the same as 34. And now we can solve the bar model to find out the width, 𝑤.
First of all, we can add 18 and nine to find the total of the rectangles we know. 18 plus nine equals 27. So to find the value of 𝑤, we just need to subtract 27 from 34, which is seven. So the width of one shaded rectangle equals seven centimeters. Because the diagram was not to scale and also because we were asked to calculate the width, we knew that a ruler would not be helpful with this question. Also, because we were not given the total width of the whole shape, it wasn’t going to be a straightforward question. In the end, the maths involved was quite easy. We just had to add nine to 18 and then subtract 27 from 34.
The tricky part of the question was being able to visualize what the shape might look like if we moved some of the rectangles around. By swapping these two rectangles, we made a shape where there were now two shaded rectangles on the left-hand side. In a way, they cancelled each other out and we could forget about them. And we can think of the rest of the diagram as being like a bar model. 18 plus nine plus 𝑤 is the same as 34 centimeters. We added the two rectangles we knew; 18 plus nine equals 27. And then we subtracted them from 34 to find the one rectangle, the shaded rectangle that we didn’t know. All of the shaded rectangles have the same width. And we’ve calculated that that width, 𝑤, equals seven centimeters.