Video: Solving Word Problems by Multiplying a Decimal Number by Another

A person accidentally left a tap running for 4.25 hours. Given that 4.64 litres of water were wasted each hour, how much water was wasted in total?

06:10

Video Transcript

A person accidentally left a tap running for 4.25 hours. Given that 4.64 litres of water were wasted each hour, how much water was wasted in total?

In this problem, we’ve got the example of somebody who’s left a tap running for 4.25 hours. And with each hour that goes by, 4.64 litres of water are wasted. So, how can we find the total amount of water that’s been wasted altogether? So, the calculation we need to work out is 4.25 multiplied by 4.64. How do we multiply two decimals together like this? Wouldn’t it be a lot easier if we were multiplying two whole numbers together without any decimal points?

Well, we can change our decimal numbers into whole numbers as long as we convert them back at the end. If we write 4.25 without a decimal point, it’s the same as if we’d multiplied the number by 100. We can see that the digits have shifted two places to the left.

And if we write 4.64 without a decimal point, it’s as if we’ve multiplied that number by 100 too. Again, the digits have shifted two places to the left. So, in total the digits have moved four places. We can now work out the multiplication as long as we shift the digits back four places at the end.

So, we’ll start by using long multiplication to calculate 425 multiplied by 464. First, we’ll multiply 425 by four. Five times four equals 20. Two times four equals eight, plus the two that we’ve exchanged equals 10. And finally, four fours are 16, plus the one that we’ve exchanged equals 17. So, 425 multiplied by four equals 1700.

Now we need to multiply 425 by this six digit, which is worth six 10s, or 60. Remember, multiplying by 60 is the same as multiplying by 10 and then by six. By writing a zero in the placeholder, we shift all the digits one place to the left. So, that’s the same as multiplying by 10. Now all we need to do is to multiply them by six. Five multiplied by six equals 30. Two sixes are 12, plus the three that we’ve exchanged equals 15. And finally, four sixes equal 24, plus the one that we’ve exchanged equals 25. And so, we know 425 multiplied by 60 equal 25500.

Finally, we need to multiply 425 by the four digit, which is worth 400. Multiplying by 400 is the same as multiplying by 100 and then by four. If we write two zeros, one in the ones place and one in the tens place, we shift the digits two places to the left. And that’s the same as multiplying by 100. So, now all we need to do is to multiply by four. But we already know what 425 multiplied by four is. So, we can write this answer in straightaway. 425 multiplied by 400 equals 170000.

Now all we need to do is add our three amounts to find the overall total. We have zero ones, zero 10s. And in the hundreds column, we have seven and five. That gives us 12 hundreds, which is the same as one thousand two hundred. In the thousands column, we have a five and a one, which takes us 6000, plus the one that we’ve exchanged equals 7000. Two lots of 10000 plus seven lots of 10000 equals nine lots of 10000. And in the hundred thousands column, we only have one. And so, the answer to 425 multiplied by 464 equals 197200.

But remember, we didn’t want to find the answer to this calculation. We multiplied both of our decimals by 100 to make the calculation easier for us. So, we need to divide by a 100 and then a 100 again, which is the same as shifting the digits four places to the right, to find the answer to our decimal multiplication.

Let’s look at how our number changes as it shifts four places to the right, one, two, three, four. And we know 19.7200 is the same as 19.72 litres. We found the answer by multiplying both numbers by 100 to start with. This had the effect of shifting the digits two places to the left. And it meant that we didn’t have any decimal points to worry about.

Next, we multiplied the two whole numbers together. And we used long multiplication to do this. But we knew that the answer we got needed to be adjusted. We’d multiplied both numbers by 100, so we needed to divide by 100 and then by 100 again. In other words, we needed to shift the digits four places to the right. So, the amount of water that was wasted altogether was 19.72 litres.

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