 Succeed with maths – Part 1

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# 2.1 Multiplying mixed numbers and fractions

Say you want to calculate .

The way to handle this is first by changing the mixed numbers to improper fractions, as you learned in Week 3, then you can once again multiply the numerators together followed by the denominators: , so . , so .

Now you can perform the calculation just as before, looking for ways to cancel first as you did when multiplying fractions in the previous section. Note that you should write the answer as a mixed number, if appropriate. You usually do this if the original numbers were given as mixed numbers.

Now try multiplying with fractions in Activity 4. If you would like to watch somebody working through multiplying mixed numbers, have a look at this video. Note again that the presenter refers to ‘fourths’ instead of quarters, and uses a dot at some points to represent multiplication rather than the more usual cross symbol (×).

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#### Transcript

Narrator
Multiply 1 and 3/4 times 7 and 1/5. Simplify your answer and write it as a mixed fraction. So the first thing we want to do is rewrite each of these mixed numbers as improper fractions. It's very difficult, or at least it's not easy for me, to directly multiply mixed numbers. One can do it, but it's much easier if you just make them improper fractions. So let's convert each of them. So 1 and 3/4 is equal to- it's still going to be over 4, so you're still going to have the same denominator, but your numerator as an improper fraction is going to be 4 times 1 plus 3.
And the reason why this makes sense is 1 is 4/4, or 1 is 4 times 1 fourths, right? 1 is the same thing as 4/4, and then you have three more fourths, so 4/4 plus 3/4 will give you 7/4.
So that's the same thing as 1 and 3/4. Now, let's do 7 and 1/5. Same exact process. We're going to still be talking in terms of fifths. That's going to be the denominator. You take 5 times 7, because think about it.
7 is the same thing as 35/5. So you take 5 times 7 plus this numerator right here. So 7 is 35/5, then you have one more fifth, so you're going to have 35 plus 1, which is equal to 36/5. So this product is the exact same thing as taking the product of 7/4 times 36/5. And we could multiply it out right now. Take the 7 times 36 as our new numerator, 4 times 5 as our new denominator, but that'll give us large numbers. I can't multiply 7 and 36 in my head, or I can't do it too easily. So let's see if we can simplify this first. Both our numerator and our denominator have numbers that are divisible by 4, so let's divide both the numerator and the denominator by 4. So in the numerator, we can divide the 36 by 4 and get 9. If you divide something in the numerator by 4, you need to divide something in the denominator by 4, and the 4 is the obvious guy, so 4 divided by 4 is 1. So now this becomes 7 times 9, and what's the 7 times 9? It's 63, over 1 times 5. So now we have our answer as an improper fraction, but they want it as a mixed number or as a mixed fraction. So what are 63/5? So to figure that out- let me pick a nice colour here—we take 5 into 63. 5 goes into 6 one time. 1 times 5 is 5. You subtract. 6 minus 5 is 1. Bring down the 3. 5 goes into 13 two times. And you could have immediately said 5 goes into 63 twelve times, but this way, at least to me, it's a little bit more obvious. And then 2 times 5 is 12, and then we have sorry! 2 times 5 is 10.
That tells you not to switch gears in the middle of a math problem. 2 times 5 is 10, and then you subtract, and you have a remainder of 3.
So 63/5 is the same thing as 12 wholes and 3 left over, or 3/5 left over. And if you wanted to go back from this to that, just think: 12 is the same thing as 60 fifths, or 60/5. 60/5 plus 3/5 is 63/5, so these two things are the same thing. These two things are equivalent. This is as an improper fraction. This is as a mixed number or a mixed fraction. But this is our answer right there: 12 and 3/5.
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## Activity 4 Multiplying mixed numbers and fractions

Timing: Allow approximately 10 minutes

Remember to cancel before multiplying, and convert mixed numbers to improper fractions if necessary.

• a. ### Comment

You need to change the mixed number into an improper fraction first.

• a. , so  • b. • b. • c. • c. The examples in Activity 4 helped you develop your understanding of multiplying fractions. Now apply these new skills in a more practical situation (Activity 5).

In the previous activity, you might have approached the problem differently. Perhaps you found what one-fifth of the group was first by using division, and then used this portion to find three of those sets. Once you had this value, which indeed is 126 people, you could have then found the number of people with overdraft protection and debit cards as shown above. Both approaches are valid and will give you the correct answers. Choose whichever method is easier for you. Now you’ve dealt with multiplication of fractions, you’ll move onto the last of the four basic operations of maths: division.