In this video, we will learn how to
evaluate expressions involving decimals in different arithmetic operations using the
order of operations. Let’s begin by recalling what we
mean by the order of operations.
The order of operations is an
important convention to ensure that numerical expressions have only one value. For example, let’s consider the
expression four plus two multiplied by three. If we add the four and two first,
we have six multiplied by three, which is equal to 18, whereas if we calculate two
multiplied by three first, we have four plus six, which is equal to 10. To determine which of these values
is correct, we use the order of operations. The acronym PEMDAS is often used to
refer to this order of operations, with the letters representing parentheses,
exponents, multiplication, division, addition, and subtraction.
In some locations, this is referred
to as BIDMAS. Instead of the word parentheses, we
use brackets. We use indices instead of
exponents. And the division and multiplication
letters are reversed. Either of these acronyms is a good
way of remembering the order. We always evaluate or solve items
in parentheses or brackets first. In fractional groupings, it is also
important to evaluate all of the numerator and all of the denominator before
performing the division.
Next we calculate any exponents,
for example, 10 squared or four to the fifth power. Multiplication and division have
equal precedence. This means that we perform these
from left to right. If a calculation has more than one
multiplication or division sign, we perform the one to the left first. Addition and subtraction also have
equal precedence, which means we also perform these from left to right. We can use this order of operations
to solve problems involving integers. And we can also apply them in the
same way with decimal numbers and fractions.
Before looking at some examples,
let’s recall the strategies we use for arithmetic with decimal numbers. We begin by recalling our methods
for addition and subtraction of decimal numbers. We can use the same methods of
addition and subtraction of decimals that we use for integers. However, the column method is
probably the most useful as it preserves the decimal places clearly. For example, 4.7 plus 3.2 can be
set out as shown. The decimal point remains in the
same place. We then add the columns working
from right to left, giving us an answer of 7.9. It can also be helpful to add zeros
as placeholders where necessary.
Multiplication of two decimals is
slightly more complicated. One method is to remove the decimal
points from the numbers, carry out the multiplication, and then put the decimal
point back in the answer. The answer must have the same
number of decimal places as the sum of the decimal places in the original
members. For example, let’s imagine we wish
to multiply 4.2 and 0.3.
Removing the decimal points, we
have 42 multiplied by three. This is equal to 126. As there were two numbers after
decimal points in the question, there need to be two numbers after the decimal point
in the answer. We have multiplied 4.2 by 10 and
0.3 by 10. This is the same as multiplying by
100. We need to divide 126 by 100 to get
the answer to 4.2 multiplied by 0.3. 4.2 multiplied by 0.3 is equal to
Finally, we have division. To divide decimals, we must make
the divisor a whole number by multiplying it by a power of 10. We must then multiply the dividend
by the same number. We can then divide these numbers,
which gives the same answer as if we had divided the decimals.
As an example, let’s consider 4.8
divided by 0.3. This could be written as 4.8 over
0.3. We then multiply the numerator and
denominator by 10 to ensure the divisor or bottom number is an integer. 4.8 multiplied by 10 is 48 and 0.3
multiplied by 10 is three. As 48 divided by three is equal to
16, then 4.8 divided by 0.3 is also 16. If required, we could use the bus
stop or alternative method at this point. We will now look at some examples
using the order of operations to solve problems involving decimal numbers.
Determine the value of two
multiplied by 1.3 plus 1.5 using the order of operations.
One way of remembering the order of
operations is using the acronym PEMDAS. The letters stand for parentheses,
exponents, multiplication, division, addition, and subtraction. Whilst we perform the operations in
order from top to bottom, it is important to remember that multiplication and
division, along with addition and subtraction, can be done in either order. If we have more than one of these
pairs of signs, we work from left to right.
In this question, we have no
parentheses or brackets and no exponents or indices. This means that our first step is
multiplication. We must multiply two by 1.3. Multiplying by two is the same as
doubling a number. So two multiplied by 1.3 is
2.6. For a more complicated calculation,
we could’ve removed the decimal point and then put it back in afterwards. There are no more multiplication
signs and there is no division sign, so our next step is to add 2.6 and 1.5. Whilst it doesn’t matter with
addition which order we write the numbers, as a general rule, it is important to
keep the numbers in the same order. 2.6 plus 1.5 is equal to 4.1. We could work this out using the
column method if required. The value of two multiplied by 1.3
plus 1.5 is 4.1.
Our next question will involve more
Calculate 68.7 minus 9.9 divided by
3.3 minus 2.5.
In order to solve any problem of
this type, we recall our order of operations, otherwise known as PEMDAS. The letters in the acronym stand
for parentheses, exponents, multiplication, division, addition, and subtraction. Multiplication and division, along
with addition and subtraction, have equal precedence. If we have more than one of these
signs, we work from left to right. Apart from that, we work our way
down the acronym.
In this calculation, there are no
parentheses or brackets or exponents. There is also no
multiplication. So our first calculation is the
division 9.9 divided by 3.3. We might recognize straight away
that this is equal to three as three multiplied by 3.3 is 9.9. If we didn’t recognize this, we
could begin by writing our calculation as a fraction. We could then multiply the
numerator and denominator of the fraction by 10. This would give us the integer
values 99 and 33. Both of these numbers are divisible
by 11 as they are in the 11 times table. Therefore, the fraction simplifies
to nine over three. This is equal to three, confirming
that 9.9 divided by 3.3 is three.
Going back to our calculation, we
drop the 68.7 and 2.5 into the next line, along with the two subtraction signs. There are no addition signs, but we
have two subtraction signs. We calculate these working from
left to right. 68.7 minus three is equal to 65.7,
so we’re left with this minus 2.5. This is equal to 63.2. We could’ve worked out either of
these calculations using column subtraction. 68.7 minus 9.9 divided by 3.3 minus
2.5 is equal to 63.2.
Our next question involves
parentheses and exponents.
Calculate 0.2 squared multiplied by
four multiplied by 13 plus seven squared minus five squared.
In order to answer this question,
we recall our order of operations acronym, known as PEMDAS. P stands for parentheses, E for
exponents, M for multiplication, D for division, A for addition, and S for
subtraction. We carry out the operations working
from top to bottom. It is important to remember however
that multiplication and division, along with addition and subtraction, have equal
precedence. And if we have two of these signs,
we can calculate them from left to right.
We begin by carrying out any
calculation inside parentheses or brackets. In this case, we have 13 plus
seven. Whilst the 0.2 is inside
parentheses, this decimal number is being raised to a power or exponent. 13 plus seven is equal to 20, so
we’re left with 0.2 squared multiplied by four multiplied by 20 squared minus five
squared. Three of our terms have
exponents. We have 0.2 squared, 20 squared,
and five squared. Squaring a number involves
multiplying it by itself, so 0.2 squared is 0.2 multiplied by 0.2. As two multiplied by two is equal
to four, 0.2 multiplied by 0.2 is equal to 0.04.
There are two digits after decimal
points in the question, which means there must be two digits after the decimal point
in the answer. 20 squared is equal to 400, so the
middle term becomes four multiplied by 400. Finally, five squared is equal to
25. Our next step is to multiply four
by 400. This leaves us with 0.04 multiplied
by 1600 minus 25. We’re left with a multiplication
sign and a subtraction sign. We must perform the multiplication
There are lots of ways of
calculating 0.04 multiplied by 1600. One way would be to remove the
decimal point from 0.04. Alternatively, we could split 1600
into 100 multiplied by 16. Multiplying by 100 moves all our
digits two places to the left. This means that 0.04 multiplied by
100 is four, and we’re left with four multiplied by 16. This is equal to 64. So our calculation becomes 64 minus
25. As this is equal to 39, we can say
that 0.2 squared multiplied by four multiplied by 13 plus seven squared minus five
squared is 39.
Our final question involves
inserting the correct symbols to make the calculation correct.
Insert the appropriate symbols from
addition, subtraction, multiplication, and division to make the calculation
correct. Eight blank 0.5 blank 40 equals
We could answer this question by
trial and error by substituting each combination of the four symbols in turn. An alternative method would be to
initially consider the order of operations and acronym PEMDAS. Parentheses and exponents will not
be relevant for this question, as we’re only allowed to use addition, subtraction,
multiplication, and division. It’s important to recall that
multiplication and division, along with addition and subtraction, have equal
precedence. However, any multiplication or
division calculation must be done prior to any addition or subtraction.
We might initially notice in our
calculation that the right-hand side is equal to 28, and our first number is
eight. We know that eight plus 20 equals
28. This suggests that the first
missing symbol could be an addition one. 0.5 is the same as one-half, and we
know that one-half of 40 is 20. This means that 0.5 multiplied by
40 is also equal to 20. This indicates that the second
missing sign is a multiplication one.
We can now check to see whether
eight plus 0.5 multiplied by 20 does indeed equal 28. Our first step would be to carry
out the multiplication, 0.5 multiplied by 40. We know this is equal to 20. So we’re left with eight plus
20. Eight plus 20 is indeed equal to
28. The missing symbols in the
calculation are addition and multiplication such that eight plus 0.5 multiplied by
40 is 28.
We will now summarize the key
points from this video. To use the order of operations with
decimals, we use the same order that we do with integers. We can use the acronym PEMDAS to
remind ourselves of the order. The letters stand for parentheses,
exponents, multiplication, division, addition, and subtraction. This is sometimes referred to as
BIDMAS, where the B stands for brackets and the I indices. It is important to remember that
multiplication and division have the same level of priority. So when we have two or more
instances of either of these, we calculate them in order from left to right. The same is true of addition and
subtraction in the next stage of calculations.