Video Transcript
In this video, we will learn how to
convert between fractions, terminating decimals, and percentages. We will begin by making sure we are
familiar with the vocabulary we will be using.
In mathematics, a fraction
describes a proportion that compares a part 𝑎 to a whole 𝑏. We call 𝑎 the numerator and 𝑏 the
denominator, then write the fraction as the ratio 𝑎 over 𝑏. A fraction can also be understood
as a quotient, 𝑎 divided by 𝑏. As an example, let’s consider the
fraction two over five, or two-fifths. The denominator of five gives us
the number of equal portions in the whole, whereas the numerator of two tells us to
consider the part of the whole made of two portions. This can be visually demonstrated
by two out of five equal portions being shaded as shown.
Let’s now recall the definition of
a terminating decimal. We know that a decimal is another
way to write a fraction, and a terminating decimal is one where the numbers end, or
terminate, at some point. Terminating decimals are equivalent
to fractions with denominators of 10 or 100 or 1000 and so on, where the number of
digits to the right of the decimal place tells us which denominator the equivalent
fraction has. A decimal with only one digit after
the decimal point is the number of portions out of 10. We call this the number of
tenths. For example, the decimal 0.7 is
equal to seven-tenths. Likewise, a decimal with two digits
after the decimal point is the number of portions out of 100. We call this the number of
hundredths. The decimal 0.19 is equal to
nineteen hundredths. This process can be repeated for
thousandths. 0.073 is equal to 73 over 1000, or
seventy-three thousandths.
It is also worth noting that any
digits to the left of the decimal point represent the whole number in a mixed
number. For example, 8.3 is equal to
eighty-three tenths, which can be written as the mixed number eight and
three-tenths, as 83 divided by 10 is equal to eight remainder three.
Let’s now recall our definition of
a percentage. We write a percentage as 𝑝
percent, meaning 𝑝 out of 100. This means that a percentage is a
way to express a given proportion 𝑎 over 𝑏 such that 𝑎 over 𝑏 is equal to 𝑝
over 100. Let’s begin by considering the
fraction 11 over 50, or eleven fiftieths. To convert this into a percentage,
we begin by finding an equivalent fraction where the denominator is equal to
100. 50 multiplied by two is 100. Since we’ve multiplied the
denominator by two, we must do the same to the numerator. 11 multiplied by two is 22. Since eleven fiftieths is
equivalent to twenty-two hundredths, this is equal to 22 percent. In the same way, we see that the
fraction four twenty-fifths is equivalent to sixteen hundredths. We multiply the numerator and
denominator by four. Four twenty-fifths is therefore
equal to 16 percent.
It is also worth noting at this
stage that when our fraction is written with a denominator of 10, 100, 1000, and so
on, we can easily write it as a terminating decimal. In the two examples shown,
twenty-two one hundredths and sixteen one hundredths are equal to 0.22 and 0.16,
respectively. Let’s now consider a practical
example where we need to convert a fraction to a decimal.
A woman bought four twentieths of a
kilogram of apples. She places these apples on a
digital scale that displays their weight, in kilograms, as a decimal. What numbers should appear on the
display of the digital scale?
In this question, we want to
convert the given fraction to a decimal. The easiest way to do this is to
simply divide the numerator four by the denominator 20 on a calculator. This gives us an answer of 0.2. However, it is also important to
have a method we can use to convert from fractions to decimals by hand. We recall that the first three
decimal places to the right of the decimal point are tenths, hundredths, and
thousandths. This means that in order to convert
a fraction to a terminating decimal, we can find an equivalent fraction where the
denominator is equal to 10, 100, or 1000. When finding equivalent fractions,
we must multiply or divide the denominator and numerator by the same amount. Since 20 divided by two is equal to
10, we must also divide the numerator by two. This means that the fraction four
twentieths is equivalent to two-tenths. And as a decimal, we can write this
as 0.2.
An alternative method would be to
multiply the denominator by five to give an equivalent fraction with a denominator
of 100. Since four multiplied by five is
20, four twentieths is equivalent to twenty hundredths. As a decimal, this is written as
0.20. And since twenty hundredths is
equivalent to two-tenths, our final answer is 0.2. Four twentieths of a kilogram is
equivalent to 0.2 kilograms.
Before looking at some further
examples, let’s briefly recall how we can convert quickly between fractions,
decimals, and percentages. To express a fraction 𝑎 over 𝑏 as
a percentage 𝑝 percent, we simply multiply 𝑎 over 𝑏 by 100. We can express a percentage 𝑝
percent as the fraction 𝑝 over 100. We can then simplify this fraction
if possible. To convert between decimals and
percentages, we simply multiply or divide by 100. If we have a terminating decimal 𝑑
and a percentage 𝑝 percent, then 𝑝 is equal to 100 multiplied by 𝑑 and 𝑑 is
equal to 𝑝 divided by 100.
Let’s now consider some examples
where we need to use these rules.
Write 5.07 as a fraction in its
simplest form. Then, write 5.07 in percentage
form.
We begin by considering the decimal
5.07. Using a decimal place value table,
we see that 5.07 terminates in the hundredths place. We can therefore write this decimal
as 507 out of 100, or five hundred and seven hundredths. The numerator and denominator of
our fraction have no common factor other than one. However, we can write this fraction
as a mixed number. We begin by writing 507 over 100 as
500 over 100 plus seven over 100. Since 500 divided by 100 is five,
the whole number part is five. This means that the decimal 5.07
written as a fraction in its simplest form is five and seven hundredths.
The second part of this question
asks us to express the decimal 5.07 in percentage form. We recall that we can convert a
decimal 𝑑 to a percentage 𝑝 percent by multiplying 𝑑 by 100. In this question, we have to
multiply 5.07 by 100, and this is equal to 507. We can therefore conclude that the
decimal 5.07 is equivalent to 507 percent.
In our next question, we’re asked
to write a percentage as a fraction in its simplest form.
Write 120 percent as a fraction in
its simplest form.
We begin by recalling that any
percentage 𝑝 percent can be expressed as the fraction 𝑝 over 100. This means that 120 percent can be
written as 120 over 100. In order to simplify this fraction,
we need to find common factors of the numerator and denominator. And whilst we can do this in
several steps, we can do it in one step by finding the highest common factor of 120
and 100. This is equal to 20, so we can
divide both the numerator and denominator by 20. 120 divided by 20 is six, and 100
divided by 20 is five. So the fraction 120 over 100
simplifies to six over five, or six-fifths.
We can actually go one stage
further here as we can convert six-fifths to a mixed number. Since six-fifths is equal to
five-fifths plus one-fifth, it can be written as the mixed number one and
one-fifth. And we can therefore conclude that
120 percent written as a fraction in its simplest form is one and one-fifth.
We will now consider one final
question in this video.
If 0.05 is equal to 𝑎 percent,
which is equal to 𝑏 over 20, find 𝑎 and 𝑏.
In the first part of this question,
we need to express the decimal 0.05 as a percentage. We recall that any decimal 𝑑 can
be written as a percentage 𝑝 percent by multiplying 𝑑 by 100. This means that in this question,
𝑎 is equal to 0.05 multiplied by 100. As multiplying by 100 moves all our
digits two places to the left, 0.05 multiplied by 100 is five. 0.05 is equal to five percent. Therefore, 𝑎 is equal to five. We also need to write this decimal
as a fraction. As 0.05 is equivalent to five
hundredths, we have the fraction five over 100.
We’re asked to write this in the
form 𝑏 over 20. Since 20 multiplied by five is 100,
we’ve divided the denominator of our fraction by five. We must do the same to the
numerator. And as five divided by five is
equal to one, the fraction five hundredths is equivalent to one twentieth. And we can therefore conclude that
𝑏 is equal to one. The decimal 0.05 is equivalent to
five percent and one twentieth as a fraction in its simplest form.
We will now summarize the key
points from this video. We saw in this video that a decimal
place value table is helpful when converting between a decimal and a fraction or
percentage. To convert from a fraction to a
terminating decimal, we find an equivalent fraction over 10, 100, 1000, and so
on. To convert a terminating decimal to
a fraction, we use the digits of the decimal as the numerator of the fraction, with
a denominator of 10 if the decimal terminates in the tenths place, a denominator of
100 if the decimal terminates in the hundredths place, and so on.
To express a fraction as a
percentage, we multiply the fraction by 100. A percentage 𝑝 percent can be
expressed as the fraction 𝑝 over 100, and we can then write the fraction in its
simplest form if necessary. To express a decimal as a
percentage, we multiply the decimal by 100. This involves moving the digits two
places to the left. And finally, to express a
percentage as a decimal, we divide the percentage by 100. This involves moving the digits two
places to the right.
