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 𝑎 over 𝑏 and 𝑐 over 𝑑, we simply multiply the numerators and
separately multiply the denominators. 𝑎 over 𝑏 multiplied by 𝑐 over 𝑑
is equal to 𝑎𝑐 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
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
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
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
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 𝑎 and 𝑏 over 𝑐 is equal to 𝑎𝑐 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.