Video Transcript
In this video, we will learn how to
multiply rational numbers including fractions, decimals, and percentages. We will begin by recalling what we
mean by a rational number and how we can convert between decimals, fractions, and
percentages.
A rational number is a number that
can be written as a fraction in the form 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers and
𝑞 is not equal to zero. This means that all whole numbers
or integers are rational numbers. It doesn’t matter if they’re
positive or negative, as nine can be written as nine over one and negative seven as
negative seven over one. It follows that all positive and
negative fractions are rational numbers as they are written in the form 𝑝 over
𝑞.
Any mixed number is also a rational
number. For example, two and a half can be
rewritten as the improper fraction five over two. Decimals of the form 0.2 and 4.31
are also rational, likewise any percentage, for example, 25 percent. 0.2 is equal to two-tenths. This can be simplified to one-fifth
by dividing the numerator and denominator by two. It is also equal to 20 percent. We can convert any decimal into a
percentage by multiplying by 100.
In this video, we will need to
convert between decimals, fractions, and percentages to find the most appropriate
value. We do this in order to make our
calculation as simple as possible. Our first question involves
multiplying two fractions.
Evaluate negative seven-fifths
multiplied by three-quarters.
We recall that when multiplying two
fractions, we simply multiply the numerators and separately multiply the
denominators. Where possible, we can cross cancel
or cross simplify first. We also recall that multiplying a
negative number by a positive number gives us a negative answer. In this question, we’re multiplying
negative seven-fifths by positive three-quarters. This means that our answer must be
negative. Multiplying the numerators gives us
21 as seven multiplied by three is 21. Five multiplied by four is equal to
20, so the denominator equals 20. Negative seven-fifths multiplied by
three-quarters is equal to negative twenty-one twentieths or negative 21 over
20. We could convert this into a mixed
number by dividing 21 by 20. This is equal to one remainder
one. Therefore, negative 21 over 20 is
the same as negative one and one twentieths.
Our next question involves
multiplying a mixed number by a negative fraction.
Calculate two and three-fifths
multiplied by negative two-sevenths. Give your answer as a fraction in
its simplest form.
Our first step in this question is
to convert the mixed number two and three-fifths to a top heavy or improper
fraction. In the bar shown, we have shaded
two and three-fifths. Each complete bar is equal to
five-fifths. This means that altogether, we have
thirteen-fifths shaded. The fraction two and three-fifths
is equal to thirteen-fifths. A quicker way of calculating this
is to multiply the whole number by the denominator and then adding the
numerator. Two multiplied by five is 10, and
adding three gives us 13. This is the numerator of our
improper fraction. We, therefore, need to multiply
thirteen-fifths by negative two-sevenths.
We recall that when multiplying two
fractions 𝑎 over 𝑏 and 𝑐 over 𝑑, we simply multiply the numerators and
separately multiply the denominators. We also need to remember that when
we multiply a positive number by a negative number, we get a negative answer. 13 multiplied by two is 26. Five multiplied by seven is 35. Multiplying positive
thirteen-fifths by negative two-sevenths is equal to negative 26 over 35 or negative
twenty-six thirty-fifths.
Our next question is a worded
problem in context.
Sara works in a supermarket. She earns seven dollars per
hour. How much will she get paid if she
puts in 35 and one-quarter hours per week? Write your answer as a decimal.
Sara’s pay will be equal to her
wage per hour multiplied by the number of hours she works. We are told she earns seven dollars
per hour. We are also told that she works for
35 and a quarter hours per week. We need to multiply this by
seven. There are lots of ways of working
out this calculation. One way would be to multiply seven
by 30, seven by five, and seven by a quarter. As seven multiplied by three is 21,
seven multiplied by 30 is 210. Seven multiplied by five is 35. Seven multiplied by one-quarter is
seven-quarters.
As seven divided by four is equal
to one remainder three, seven-quarters is the same as the mixed number one and
three-quarters. We know that three-quarters is
equal to the decimal 0.75. Therefore, one and three-quarters
is equal to 1.75. We need to add 210, 35, and
1.75. This is equal to 246.75. If Sara earns seven dollars per
hour and works for 35 and a quarter hours, she will earn 246 dollars and 75
cents.
In our next question, we will
multiply a percentage by a mixed number.
Calculate 50 percent of one and a
half. Give your answer as a fraction.
In order to answer this question,
we firstly need to convert 50 percent into a fraction. As the word “percent” means out of
100, we can convert from a percentage to a decimal by dividing by 100. 50 divided by 100 is 0.5. This is equal to five-tenths. We can then simplify this fraction
by dividing the numerator and denominator by five. 50 percent is equal to
one-half. Next, we need to convert 1.5 into a
top heavy or improper fraction. There are two-halves in one whole
one. Therefore, one and a half is equal
to three-halves or three over two. We know that the word “of” in
mathematics means multiply. We need to multiply one-half by
three-halves.
When multiplying two fractions, we
multiply the two numerators and then the two denominators separately. One multiplied by three is three,
and two multiplied by two is four. We can, therefore, conclude that 50
percent of one and a half is three-quarters. We could also have shown this
pictorially. We began with one and a half and
wanted to calculate 50 percent or a half of this. One-half of a whole one is equal to
one-half, and one-half or 50 percent of a half is a quarter. Adding a half and a quarter once
again gives us a final answer of three-quarters.
The penultimate question in this
video involves multiplying a percentage by a decimal.
Calculate 25 percent multiplied by
0.2. Give your answer as a decimal
number.
As we need to give our answer as a
decimal number, we firstly need to convert 25 percent into either a decimal or a
fraction. The word “percent” means out of
100. Therefore, 25 percent is equal to
25 over or out of 100. As the line in the fraction means
divide, this is equal to 0.25. The fraction can also be simplified
by dividing the numerator and denominator by 25. 25 percent is, therefore, equal to
one-quarter. This means that we have two options
to proceed. We can either multiply one-quarter
by 0.2 or 0.25 by 0.2.
Multiplying by a quarter is the
same as dividing by four. We need to divide 0.2 by four. This is equal to 0.05. When multiplying the two decimals
0.25 and 0.2, we know that 25 multiplied by two is 50. As there are three digits
altogether after the decimal point in the question, there needs to be three digits
after the decimal point in the answer. 0.25 multiplied by 0.2 is equal to
0.050. This is the same as 0.05. 25 percent multiplied by 0.2
written as a decimal is 0.05.
Our final question involves
multiplying an integer, a fraction, and a decimal.
Calculate 25 multiplied by
one-sixth multiplied by 0.08. Give your answer as a fraction in
its simplest form.
As we need to give our answer as a
fraction, our first step is to convert 0.08 into a fraction. As the eight is in the hundredths
column, this is equal to eight hundredths or eight over 100. Both the numerator and denominator
are divisible by four, so this fraction simplifies to two over 25 or two
twenty-fifths. We need to multiply 25, one-sixth,
and two twenty-fifths. Any integer or whole number, in
this case, 25, can be written over one. Before multiplying the fractions,
we can now cross simplify or cross cancel.
There is a 25 on the numerator and
denominator, so these will cancel. The numbers two and six are both
even, so they are divisible by two. On the numerators, we are left with
one multiplied by one multiplied by one. And on the denominator, we are left
with one multiplied by three multiplied by one. 25 multiplied by one-sixth
multiplied by 0.08 is, therefore, equal to one-third.
We will now summarize the key
points from this video. A rational number is any number
that can be written as a fraction where the numerator and denominator are integers
and the denominator cannot be equal to zero. Rational numbers include positive
and negative integers, fractions, and any recurring or terminating decimal. They also include percentages. Where there is a mixture of these,
we need to convert them all into either fractions or decimals. When multiplying two or more
fractions, we multiply the numerators and denominators separately. However, it is useful to cross
cancel or cross simplify first. Finally, we need to remember that
when we multiply two positive or two negative numbers, we get a positive answer,
whereas when we multiply a positive by a negative in either order, we get a negative
answer.
