Video: Solving Word Problems by Adding and Multiplying Decimal Numbers

Michael earns $7.90 per hour working part-time at the public library. Given that he worked for 5.5 hours on Monday, 2.5 hours on Tuesday, and 9.5 hours on Wednesday, determine the total amount of money he earned.

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Video Transcript

Michael earns seven dollars 90 per hour working part-time at the public library. Given that he worked for 5.5 hours on Monday, 2.5 hours on Tuesday, and 9.5 hours on Wednesday, determine the total amount of money he earned.

The first important piece of information that we’re given in this problem is how much Michael earns working part-time at a library. He gets paid seven dollars 90 every hour. We’re then given three lengths of time that Michael works. We’re told that he works 5.5 hours on Monday, 2.5 hours on Tuesday, and also 9.5 hours on Wednesday. And we need to apply all of this information to find the total amount of money that Michael earned. As we’ve been going through a question, we’ve been sketching out a diagram here. It’s an area model of multiplication, and it gives us an idea of how to work out the answer.

To find the amount of money that Michael earns every day, we need to multiply his wage by the number of hours he works. So, for example, on Monday, we’d multiply seven dollars 90 by 5.5. It’s the same as multiplying the length by the width of the rectangle. On Tuesday, we’d need to multiply seven dollars 90 by 2.5. And on Wednesday, we’d need to multiply the same wage again but, this time, by 9.5. And to find the total amount of money he earned, we’d have to add these three results together. So, that’s four different calculations we’d need to do, three multiplications and then an addition.

But there’s actually a more efficient, quicker way to find the answer, and it only involves doing two calculations. Can you see what it is? Well, what if we add together the three lengths of time that Michael works? Well, then, in the second step of the problem, all we’d have to do would be to multiply seven dollars 90 by the total number of hours that he worked. And because multiplying by decimals can be tricky, this looks like a really efficient way to find the answer, much quicker than doing three multiplications involving decimals. So, let’s use this method to find the answer.

In step one, we need to calculate the total number of hours that Michael worked on Monday, Tuesday, and Wednesday altogether. So, that’s 5.5 hours plus 2.5 hours plus 9.5 hours. We can make sure that there’s a decimal point in our answer. And in the tenths place, we have three fives to add. Five plus five plus five equals fifteen-tenths. That’s the same as five-tenths and one whole. We can add the ones digits quickly by combining them into pairs. Nine plus the one we exchanged makes 10, five and two makes seven, so altogether are 17. So, we can say that the total number of hours that Michael worked is 17.5. And we know now what we need to do to find the answer. We need to multiply seven dollars 90 by 17.5.

And we often find it easier to multiply whole numbers than decimals, so we can change our decimals into whole numbers as long as we remember what we’ve done and adjust our answer at the end. The first thing we can do is to multiply 7.90 by 10. When we multiply any number by 10, the digits shift one place to the left. So, 7.90 becomes 79.0. We don’t need to worry too much about the point zero on the end. This is a whole number, and we’ve made it a whole number just by multiplying by 10. We’ve got to remember, though, our answer now is going to be 10 times larger than we want it to be.

To turn 17.5 into a whole number, again we simply need to multiply it by 10. And those digits are going to shift one place to the left again. 17.5 is going to become 175. But by multiplying this number by 10, our answer is going to become 10 times greater again. So, in all, it’s going to be 100 times greater than it needs to be. We need to remember this and adjust it at the end. But now that we’ve converted our decimals into whole numbers, we can work out the answer much more quickly. 175 multiplied by 79. We’ve just switched those two numbers around, so we start with a three-digit number. First, we’ll multiply each of the digits in 175 by nine. Five nines are 45. Seven tens multiplied by nine are 63 tens plus the four tens underneath takes us to 67 tens. And finally, 100 multiplied by nine is 900 plus the 600 we’ve exchanged, 1515 hundreds.

Next, we need to multiply everything by the seven digit in 79. Remember that this seven digit isn’t worth seven, it’s worth seven tens or 70. If we write a zero in the ones place, then our answer is going to shift one place to the left. This makes everything 10 times greater. So, now we can multiply by seven as if it was just seven by shifting all to the left. It’s the same as multiplied by 70. Five lots of seven are 35. Seven sevens are 49 plus the three that we’ve exchanged, 52. And one lot of seven is seven plus the five we’ve exchanged equals 12. To find the overall answer for this whole number multiplication, we need to add each column.

Five plus zero ones equals five. Seven tens plus five tens equals 12 tens. Five hundreds plus two hundreds equals seven hundreds plus the one we’ve exchanged equals eight hundreds. 1000 plus 2000 equals 3000, and we’ve got one lot of 10000 in the ten thousands place. And so, 175 multiplied by 79 equals 13825. But remember, we never did want to multiply 79 by 175. We needed to multiply two decimals. The only reason we turn them into whole numbers was to make the whole thing easier. And the way that we turned the decimals into whole numbers was by multiplying by 10 and then 10 again. Our answer, remember, is 100 times greater than it needs to be.

So, to find the correct answer, we need to make this number 100 times less. We need to divide it by 100. And we’re going to do this by dividing by 10 and then 10 again. This is going to make the digits shift twice to the right. Watch what happens. First, we divide by 10, 1382.5. And then, we divide by 10 again. This is the same as divided by 100 altogether, 138.25. So, if Michael works for 5.5 hours on Monday, 2.5 hours on Tuesday, and 9.5 hours on Wednesday, we know that he works 17.5 hours altogether. And if he earns seven dollars 90 every hour, all we have to do is to multiply 7.9 by 17.5 to find the answer. The total amount of money that Michael earned is 138 dollars 25 cents or 138.25 dollars.

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