Question Video: Comparing Expressions Involving Multiplication of Decimals without Evaluation Mathematics

Complete 0.011 × 0.15 _ 0.011 × 0.015 using <, = or >.

07:42

Video Transcript

Complete 0.011 multiplied by 0.15 blank 0.011 multiplied by 0.15, using is less than, is equal to, or is greater than.

In this problem, we’re given two multiplications. And both of them involve multiplying two decimals together. We have 0.011 multiplied by 0.15. And we have 0.011 multiplied by 0.015. And in the middle of the two calculations, we have a blank space. This is the part that we need to complete. And we’re told to fill in the gap using one of three symbols ⁠— is less than, we’d use this symbol if the first statement is less than the second statement; is equal to, we’d use the equal sign if both statements are worth the same; and is greater than, and we’d use this symbol if the first statement is greater than the value of the second statement.

And we might look at this question and think to ourselves that we need to work out both multiplications to be able to compare them and then work out which symbol to put in the middle of them. And in a way, that is what the question’s asking us to do. But the numbers that are involved in this particular question are such that we can answer it really quickly without even doing any maths. Let’s go through the really quick way to answer the question. And then, after we’ve done that, we’ll calculate both amounts and answer it the way it’s supposed to be answered.

Let’s start by looking very closely at both multiplications. What do we notice about them? They both begin with the same number, 0.011. Because both multiplication involve multiplying 0.011 by another number, we just need to look at the other number and decide which one’s larger and which one’s smaller. Whichever of the two second numbers is larger will give a larger answer. In the first statement, 0.011 is being multiplied by 0.15. That’s zero ones, one tenth, and five hundredths. In the second calculation, 0.011 is being multiplied by 0.015. Again, that zero ones. This time we have zero tenths, one hundredth, and five thousandths.

So let’s compare the two numbers that we’re multiplying 0.011 by. Both numbers have no ones. So we need now to compare the tenths. The first number contains one tenth, but the second number contains zero tenths. The first number then is larger than the second number. Because both numbers are being multiplied by the same number, 0.011, this means that the greater number will give the greater answer. We don’t even have to work it out. And so the correct symbol to use is is greater than.

Let’s check our answer now, in perhaps the way that the question was intended to be worked out. When we multiply two decimals together, it’s often useful to ignore the decimal point and to think of the number without a small point in it. In other words, that calculation becomes 11 multiplied by 15. This isn’t just a trick. We’ve actually done something to the numbers to turn them into 11 and 15. 0.11 has been multiplied by 1000; the digits have moved three places to the left. That gets rid of the decimal point. And 0.15 has been multiplied by 100. The digits have moved two places to the left. so In total, the digits have moved five places to the left. They moved three places in the first number and two places in the second number.

We’ll make a note of that because once we’ve worked out the answer, we need to move the digits five places back again, this time five places to the right. So first of all, let’s work out. 11 multiplied by 15. We know 10 lots of 15 equals 150. So 11 lots of 15 is one more 15 on top of that, which is 165. But remember, we altered our numbers by moving them in total five places to the left. So we now need to look at the number 165 and move the digits five places back again to the right. There’s 165 and we can shift them once, twice, three times, four times, five times. The answer to our first multiplication is 0.00165. Now, let’s work out the answer to the second multiplication.

First, we can write down both numbers without a decimal point in them. Remember again, this isn’t a trick that we’ve used. What we’ve done is we’ve multiplied both numbers. We’ve multiplied the first number by 1000. We know this because it’s the same numbers we used in the last one. The digits have moved three places to the left and to get from 0.015 to 15 we’ve also multiplied by 1000. So we’ve moved the digits in that number. Another three places to the left. The digits have moved six places to the left in total.

So again, once we’ve finished working out the multiplication, we need to move them back again six places to the right. The multiplication is exactly the same, 11 multiplied by 15 equals 165. But this time, instead of moving their digits five places to the right, we need to move them one more place to the right. So we can say that the answer is definitely going to be 0.000165. And this is smaller than the first number. The first number is greater than the second number. And that’s how to prove that we need to use the is greater than symbol.

We found the answer in two ways. The second method we used was to calculate both statements separately and to compare them to see which was the larger. But the first method we used was a lot quicker. We noticed that both statements involved the same number to start with. So we just looked at the second numbers, compared them and decided which was the larger. We knew that the statement with the largest second number would have a larger total and a larger value. 0.011 multiplied by 0.15 is greater than 0.011 multiplied by 0.015.

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