Find an expression for the difference between the areas of the two figures below.
Please note that before starting this question, the figures are not to scale. The area of any rectangle is calculated by multiplying the length by the width. In this case, each of the two figures has been split into four rectangles. We have two options here. We could work out the area of each of the four rectangles and then add them together to work out the total area or we could work out the total length and total width and then multiply these values.
The first method we will use is working out the area of each of the small rectangles. In figure one, we can work out the area of the top left rectangle by multiplying 10𝑥 by 𝑥. This is equal to 10𝑥 squared. The top right rectangle can be calculated by multiplying five 𝑥 by 𝑥. This is equal to five 𝑥 squared. The bottom left rectangle is calculated by multiplying 10𝑥 by seven. 10𝑥 multiplied by seven is equal to 70𝑥. Finally, we calculate the area of the bottom right rectangle by multiplying five 𝑥 by seven. This is equal to 35𝑥. We therefore have four rectangles: 10𝑥 squared, five 𝑥 squared, 70𝑥, and 35𝑥. In order to calculate the total area, we need to add these four terms. 10𝑥 squared plus five 𝑥 squared is equal to 15𝑥 squared and 70𝑥 plus 35𝑥 is equal to 105𝑥. Therefore, the area of figure one is 15𝑥 squared plus 105𝑥.
We now need to repeat this process with figure two. The top left rectangle here has an area of 12𝑥 as four multiplied by three 𝑥 equals 12𝑥. The top right rectangle is equal to six 𝑥 squared as two 𝑥 multiplied by three 𝑥 is equal to six 𝑥 squared. We multiply four by five to work out the area of the bottom left rectangle. This is equal to 20. Finally, the area of the bottom right rectangle is equal to 10𝑥 as two 𝑥 multiplied by five equals 10𝑥. Once again, we need to add these four terms to calculate the total area. The only part of this expression that can be simplified is 12𝑥 plus 10𝑥. This is equal to 22𝑥. Therefore, the total area for figure two is six 𝑥 squared plus 22𝑥 plus 20.
The question asked us to find the difference between the areas. This means that we need to subtract the area of figure two from the area of figure one. This can be written as 15𝑥 squared plus 105𝑥 minus six 𝑥 squared plus 22𝑥 plus 20. Subtracting the 𝑥 squared terms gives us nine 𝑥 squared as 15𝑥 squared minus six 𝑥 squared equals nine 𝑥 squared. Subtracting the 𝑥 terms gives us 83𝑥 as 105𝑥 minus 22𝑥 is equal to 83𝑥. There is no number term in the first expression. Therefore, we have zero minus 20, which equals negative 20. This means that the expression for the difference between the areas of figure one and figure two is nine 𝑥 squared plus 83𝑥 minus 20.
We did mention at the start that there was an alternative method to work out the expression for the area of figure one and figure two. Figure one has a total length of 15𝑥 as 10𝑥 plus five 𝑥 equals 15𝑥. It has a total width of 𝑥 plus seven. We could therefore multiply 15𝑥 by 𝑥 plus seven to work out the area of figure one. 15𝑥 multiplied by 𝑥 is equal to 15𝑥 squared, and 15𝑥 multiplied by seven is equal to 105𝑥 as 15 times seven is 105. This gives us the same expression as when we split the rectangle into four smaller rectangles. The length of figure two is two 𝑥 plus four, and the width of it is three 𝑥 plus five. This means that we could multiply two 𝑥 plus four and three 𝑥 plus five to calculate the expression for the area of figure two. We can expand these parenthesis using the FOIL method.
Multiplying the first terms gives us six 𝑥 squared as two 𝑥 multiplied by three 𝑥 is six 𝑥 squared. Multiplying the outside terms gives us 10𝑥. Multiplying the inside terms gives us 12𝑥. And finally, multiplying the last terms gives us 20. We can collect the middle two terms, 10𝑥 and 12𝑥, to give us 22𝑥. Therefore, the area of figure two using this method is six 𝑥 squared plus 22𝑥 plus 20. Whichever method be used to calculate the area of both figures, we will end up with the same answers. The final expression for the difference between the two areas will be nine 𝑥 squared plus 83𝑥 minus 20.