Video: Multiplying Mixed Numbers

In this video, we will learn how to multiply mixed numbers by both fractions and mixed numbers by either breaking apart the numbers or writing them as improper fractions.

17:37

Video Transcript

In this video, we will learn how to multiply mixed numbers by both fractions and mixed numbers by either breaking apart the numbers or writing them as improper fractions. We will begin by recalling how we convert from a mixed number to an improper fraction and vice versa.

To show how we can convert from a mixed number to an improper fraction, we will use the example two and three-quarters. In our diagram, we have two whole squares shaded. We also have three-quarters of the third square shaded. We can split the two whole squares into four-quarters as shown. Four-quarters is equal to one whole one as four divided by four is equal to one. We can work out the total number of quarters by counting the squares or, alternatively, adding four-quarters, four-quarters, and three-quarters. When adding any fractions with the same denominator, we simply add the numerators. Four plus four plus three is equal to 11. Two and three-quarters written as an improper fraction is eleven-quarters or 11 over four.

This method is useful when we are dealing with small mixed numbers. When the mixed number’s large, however, it would not be practical to draw out the shapes for each whole number. This means that we need a short cut to convert a mixed number into an improper fraction. Our first step is to multiply the whole number by the denominator. In this example, we would multiply two by four, which gives us eight. Our second step is to add the numerator to this answer. The numerator in our example is three, and eight plus three is equal to 11. This answer is the numerator of our improper fraction. And the denominator stays the same. This is a quicker way of working out that two and three-quarters is equal to eleven-quarters.

We will now recall how we can convert an improper fraction back to a mixed number. Let’s consider the same example, the improper fraction eleven-quarters or 11 over four. We know that the line in a fraction means divide. Our first step is to therefore divide the numerator by the denominator, in this case, 11 divided by four. We write down the whole-number answer. There are two fours that go into 11. We then write any remainder as the numerator, and the denominator stays the same. As 11 divided by four is equal to two remainder three, eleven-quarters is the same as two and three-quarters. We will now use these conversion techniques to multiply mixed numbers by fractions and other mixed numbers.

Evaluate one-sixth multiplied by three and three-sevenths.

There are a couple of ways of approaching this problem. One way would be to convert three and three-sevenths into an improper fraction first. To convert a mixed number to an improper fraction, we begin by multiplying the whole number by the denominator. Three multiplied by seven is 21. We do this because there are seven-sevenths in one whole one. This means there are 21 sevenths in three whole ones. We then add the numerator to this answer. 21 plus three is equal to 24. As the denominator stays the same, three and three-sevenths is equal to twenty-four sevenths.

We need to multiply one-sixth by twenty-four sevenths. We recall that to multiply two fractions π‘Ž over 𝑏 and 𝑐 over 𝑑, we simply multiply the numerators and separately multiply the denominators. π‘Ž over 𝑏 multiplied by 𝑐 over 𝑑 is equal to π‘Žπ‘ over 𝑏𝑑. In this question, we could multiply one by 24 and then six by seven. In order to make our life easier, it is always worth checking if we can cross simplify or cross cancel first. As both six and 24 are divisible by six, we can simplify our calculation. We are now left with one multiplied by four on the top and one multiplied by seven on the bottom. One-sixth multiplied by three and three-sevenths is equal to four-sevenths.

An alternative method to solve this question would be to split our mixed number into two parts. We could split it into three and then three-sevenths. We would then multiply these individually by one-sixth. We know that any integer or whole number can be written as this number over one. Therefore, three is the same as three over one. In both of these calculations, we can divide three and six by three. This means that one-sixth multiplied by three is equal to one-half. One-sixth multiplied by three-sevenths is equal to one fourteenth.

We now need to find the sum of these two fractions. In order to add two fractions, we need to ensure we have a common denominator. As two multiplied by seven is equal to 14, we can multiply the numerator and denominator of the first fraction by seven. This gives us seven over 14 plus one over 14. Adding the numerators gives us eight over 14 which, once again, simplifies to four-sevenths. This confirms that one-sixth multiplied by three and three-sevenths is equal to four-sevenths.

In our next question, we will multiply two mixed numbers.

Calculate three and two-thirds multiplied by one and two-thirds.

In order to answer this question, we will begin by converting both of our mixed numbers into improper fractions. We know that one whole one is equal to three-thirds. This means that three whole ones is equal to nine-thirds. From the diagram, we can see that three and two-thirds can be written as the improper fraction eleven-thirds. A quick way of calculating the 11 is to multiply the whole number by the denominator and then adding the numerator. Three multiplied by three is nine, and adding two gives us 11. In the same way, we can see that the mixed number one and two-thirds is equal to the improper fraction five-thirds. We need to multiply eleven-thirds by five-thirds.

In order to multiply two fractions, we simply multiply the numerators and separately the denominators. 11 multiplied by five is 55, and three multiplied by three is nine. Eleven-thirds multiplied by five-thirds is equal to fifty-five ninths or 55 over nine. As the line in a fraction means divide, we can now convert this back to a mixed number by dividing 55 by nine. Nine multiplied by six is equal to 54. Therefore, 55 divided by nine is equal to six remainder one. The improper fraction fifty-five ninths is the same as the mixed number six and one-ninth. Therefore, three and two-thirds multiplied by one and two-thirds is also equal to six and one-ninth.

In our next question, we will multiply three mixed numbers.

Calculate one and a half multiplied by three and three-quarters multiplied by one and seven-ninths.

We will begin this question by converting each of our mixed numbers into improper or top-heavy fractions. We do this by multiplying the whole number by the denominator and then adding the numerator. One and a half is therefore equal to three-halves or three over two. Three multiplied by four is equal to 12, and adding three to this gives us 15. This means that three and three-quarters is equal to fifteen-quarters or 15 over four. Finally, one and seven-ninths is equal to sixteen-ninths. There are nine-ninths in one whole one, and adding seven to this gives us 16.

We now need to multiply three over two, 15 over four, and 16 over nine. When multiplying fractions, we can just multiply the numerators and then separately the denominators. So we could multiply three by 15 by 16 and then two by four and by nine. These calculations would be quite complicated. Therefore, it is easier to see if we can cross cancel or cross simplify first. Two and 16 are both divisible by two. This gives us a one on the bottom and an eight on the top. We can also divide four and eight by four. This simplifies to one on the bottom and two on the top. We can repeat this process with the three and the nine as there are three threes in nine. Finally, we can divide three and 15 by three.

The calculation is simplified to one over one multiplied by five over one multiplied by two over one. Multiplying the numerators gives us an answer of 10, and multiplying the denominators gives us an answer of one. Any integer divided by one is equal to itself. Therefore, 10 divided by one is equal to 10. One and a half multiplied by three and three-quarters multiplied by one and seven-ninths is equal to 10.

We will now look at some practical real-life problems involving mixed numbers.

A man weighs 97 and four-fifths of a kilogram on Earth, and his weight on the Moon is one-sixth of its value on Earth. Find the man’s weight on the Moon.

We are given the man’s weight on Earth, and we are told that on the Moon it is one-sixth of this. We therefore need to multiply 97 and four-fifths by one-sixth. Our first step is to convert the mixed number 97 and four-fifths into an improper or top-heavy fraction. We do this firstly by multiplying 97 by five, which gives us 485. We then add the numerator to this, giving us 489. 97 and four-fifths is equal to four hundred and eighty-nine fifths.

When multiplying two fractions, we need to multiply the numerators and separately the denominators. We can cross cancel first, though, as six and 489 are both divisible by three. Six divided by three is two, and 489 divided by three is 163 as shown in the bus stop short division method. Multiplying the numerators gives us 163, and multiplying the denominators gives us 10. We can now convert this improper fraction back into a mixed number by dividing the numerator by the denominator. 163 divided by 10 is equal to 16 remainder three. Therefore, 163 over 10 is the same as 16 and three-tenths. The man’s weight on the Moon is 16 and three-tenths of a kilogram. This could also be written in decimal form as 16.3 kilograms.

Our final question is another word problem.

The dimensions of a wall are eight and five-eighths feet and 20 and one-fifth feet. If a gallon of paint can cover about 230 square feet, will one gallon of paint be enough to cover the wall?

We are told that the dimensions of the wall are eight and five-eighths and 20 and one-fifth feet. The area of any rectangle can be calculated by multiplying the length by the width. This means that we need to multiply eight and five-eighths by 20 and one-fifth. We will begin by converting each of these mixed numbers into top-heavy fractions. We do this by multiplying the whole number by the denominator and then adding the numerator. Eight and five-eighths is equal to sixty-nine eighths or 69 over eight. Using the same method, 20 and one-fifth is equal to 101 over five. We need to multiply 69 over eight by 101 over five.

At this stage, we could try and cross cancel. However, in this case, there are no common factors apart from one. Multiplying the denominators gives us 40. And on the top, we need to multiply 69 by 101. 69 multiplied by 100 is 6900. This means that 69 multiplied by 101 is 6969. The area of the wall is therefore 6969 over 40 square feet. We need to convert this back into a mixed number and compare it to 230 square feet. 6960 divided by 40 is equal to 174. As we have a remainder of nine, 6969 over 40 is equal to 174 and nine fortieths. This is less than 230 square feet. We can therefore conclude that, yes, one gallon of paint would be able to cover the wall.

We could have worked out the answer without performing the final calculation by recognizing that 8000 over 40 is equal to 200. As 6969 over 40 is less than this, it must be less than 230 square feet.

We will now summarize the key points from this video. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and then add the numerator. To convert an improper fraction to a mixed number, we divide the numerator by the denominator. This leads us to a general formula that the mixed number π‘Ž and 𝑏 over 𝑐 is equal to π‘Žπ‘ plus 𝑏 over 𝑐. We saw in this video that in order to multiply two mixed numbers, we firstly convert them into improper fractions and then multiply the numerators and denominators separately.

Before multiplying, it is always worth checking whether we can cross cancel or cross simplify first, as this will make the calculations easier. Our final step in most questions is to convert our improper fraction answer back into a mixed number.

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