Video: Calculating the Area of a Rectangle given Its Side Lengths in Decimal Form Then Rounding

Calculate the area of a rectangle whose side lengths are 42.89 cm and 9.3 cm. Give your answer correct to the nearest square centimeter.

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Video Transcript

Calculate the area of a rectangle whose side lengths are 42.89 centimeters and 9.3 centimeters. Give your answer correct to the nearest square centimeter.

This problem starts off by giving us the dimensions of a rectangle. We’re told its side lengths are 42.89 centimeters and 9.3 centimeters. Because it’s clearly the longest side, we could think of 42.89 centimeters as being the length. And we could describe 9.3 centimeters as our rectangle’s width. The problem asks us to calculate the area of this rectangle. We know that the area of a 2D shape is the space inside that shape. And to calculate the area of a rectangle, we need to multiply its length by its width. We’ve already been given both of these measurements. So we know that to calculate the area of the rectangle in our question, we need to multiply 42.89 by 9.3.

A multiplication like this involving two decimals looks like it might be tricky. It would be much easier to work out if we were dealing with whole numbers. What if we multiplied 9.3 by 10? We know that when we multiply any number by 10, the digits shift one place to the left. And so 9.3 becomes 93, which is a whole number. We can also do the same sort of thing with 42.89. But if we look carefully at this decimal, we can see that one shift of the digits to the left is not gonna be enough. If we multiply by 10, the digit shift once to the left and the number becomes 428.9, which is still a decimal. We need to multiply by 10 again. This is the same as multiplying by 100. But if they shift this second time, we get a whole number, 4289.

So we now have a multiplication we can work with involving two whole numbers, 4289 multiplied by 93. But we must remember we’ve altered the numbers in this multiplication. The first number is 100 times greater. And the second number is 10 times greater. So our answer is going to be 1000 times greater than it needs to be. We need to keep this in mind. And when we get to the end, we need to remember it. But for now, let’s calculate the answer, 4289 multiplied by 93.

Let’s begin by multiplying all the digits in the top number by three. 10 threes are 30. So we know nine threes are 27, which is the same as two tens and seven ones. Eight tens multiplied by three are 24 tens. We’ve got two tens that we’ve exchanged. So that’s 26 tens, which is the same as two hundred and six tens. Two hundreds multiplied by three are six hundreds plus the two hundreds we exchanged, eight hundreds. And finally, four thousands multiplied by three are 12 thousands. 4289 multiplied by three is 12,867.

Now, we need to multiply our top number by the nine digit in 93. And of course, the nine digit doesn’t have a value of nine. It’s worth 90. Now, we know 90 is the same as 10 times nine. And to multiply a number by 10, we simply shift the digits one place to the left. Well, we can do this in our answer by writing a zero as a placeholder in the ones place. Now, all the digits are going to be shifted one place to the left. So now, we can just multiply by nine because we’ve already multiplied by 10. Nine nines are 81. Eight nines are 72. We’ve exchanged eight. So that takes us up to 80. Two nines are 18. Again, we’ve exchanged eight. So this is going to take us to 26. And finally, four nines are 36 plus the two we’ve exchanged, 38.

To find the total of this multiplication then, we just need to add our two parts together. Seven plus zero equals seven. Six tens plus one ten equals seven tens. Eight hundreds plus zero hundreds equals eight hundreds. Two thousands plus six thousands equals eight thousands. One lot of ten thousand plus eight lots of ten thousand equals nine lots of ten thousand. And we only have a three in the hundred thousands’ place. So the answer to the multiplication is 398,877.

Now, is this the area of the rectangle should we just write centimeter squared on the end? Of course not. Remember, we altered our numbers to make them into whole numbers. We made the first number 100 times larger and the second number 10 times larger. So our answer must be 1000 times larger than it needs to be. So to adjust it, we need to divide this number by 1000. This is the same as dividing by 10 three times. And so the digits need to shift three times to the right this time. Let’s count three shifts to the right and see how our number changes. One, that’s the same as dividing by 10. Two, that’s the same as dividing by 100. Three, this is the same as dividing by 1000. We’ve adjusted our answer to what it should be. 42.89 multiplied by 9.3 is 398.877.

So now, does this mean we’ve found the area of our rectangle? Can we just write centimeter squared on the end of it? Well, yes and no. Yes, we found the area of the rectangle. And yes, this is the correct area of the rectangle. But no, we can’t answer the question by just writing centimeter squared on the end of it. If we look carefully at the last sentence in the question, we’re told to give our answer correct to the nearest square centimeter. We need to round this number before we finish. And it makes sense because it’s quite a long decimal.

Should we round 398.877 down to 398 or up to 399? If we look at the tenths digit in our number, we can see that it’s eight. Eight tenths is past halfway. And so it’s nearer to 399 centimeters squared. So we need to round our answer up. If the side lengths of a rectangle are 42.89 centimeters and 9.3 centimeters, we can find the area of the rectangle by multiplying these numbers together. The area of the rectangle is 398.877 centimeters squared. But if we give our answer correct to the nearest square centimeter, the answer is 399 centimeters squared.

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