A square tile measures 20 centimeters by 20 centimeters. A rectangular tile is three centimeter longer and two centimeters narrower than the square tile. What is the difference in area between the two tiles in square centimeters?
To find the difference between two areas, we simply need to take one away from the other. But in order for us to get to this calculation, we need to find out what those two areas are. And so there are several steps for us to solve this problem. Let’s start by labelling each tile with the measurements that we know. Firstly, we’re told that the square tile measures 20 centimeters by 20 centimeters. Next, we’re told that the rectangular tile is three centimeters longer and two centimeters narrower than the square tile. 20 plus three equals 23 and so we know the long side of the rectangle is 23 centimeters long. And 20 take away two equals 18. So the shorter side that’s two centimeters narrower than the square tile is 18 centimeters.
Now we can find the areas of both shapes. The area of the rectangle, and this includes squares because squares are a special type of rectangle, is found by multiplying the length by the width. And if the measurements are in centimeters, which they are here, then the area will be found in centimeters squared or square centimeters. Let’s start by calculating the area of the square tile. Its length is 20 centimeters and its width is 20 centimeters. So we need to find the answer to 20 multiplied by 20. We know that 20 multiplied by 10 equals 200. And so 20 multiplied by a number that’s double this equals 400. And so the area of the square tile is 400 centimeters squared.
Now let’s think about the rectangular tile. Again, we need to multiply the length by the width. The length of the tile is 23 centimeters and the width is 18 centimeters. 18 is very close to 20. So we could work out 23 times 20 in the same way that we’ve just worked out multiplying by 20 and then take away two lots of 23. Or, we could also use a written method, long multiplication. Let’s use this method here.
First of all, we’ll multiply 23 by eight. Three eights are 24. Two eights are 16, plus the two that we’ve exchanged equals 18. Now we need to multiply 23 by 10. We don’t need to go through each digit with this one because we know the answer. 23 times 10 equals 230. Now we just need to add each column. Four plus zero equals four. Eight plus three equals 11. And finally, one plus two plus the one that we’ve exchanged equals four. And so the area of the rectangular tile — which we found by multiplying the length by the width, 23 by 18 — is 414 centimeters squared. It’s slightly larger than the square tile.
Now we can find the difference between the two areas. And we can do this mentally just by looking at the numbers. We can see that the difference is going to be 14. 414 centimeters squared take away 400 centimeters squared equals 14 centimeters squared.
We knew that to find the difference in area between the two tiles, we had to calculate the area. And the formula for finding the area of a square or a rectangle is to multiply the length by the width. We knew the length and the width of the square tile. So we used a known fact to help us multiply 20 by 20. But we needed to use a little bit of mental maths before we were able to work out the length and the width of the rectangular tile. But once we’d found them, we knew that we had to multiply 23 by 18 to find out the answer. Then finally, we subtracted one area from the other to find out the difference in area between the two tiles. And the difference in area is 14 centimeters squared.