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 1.26.
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 operations.
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 first.
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 28.
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.