Explainer: Order of Operations in Numerical Expressions with Decimals

In this explainer, we will learn how to evaluate expressions involving decimals in different arithmetic operations using the order of operations.

The order of operations is an important convention to ensure that numerical expressions have only one value. The order of operations gives us the defined order in which to solve the operations in an expression. Let us recall the order of operations.

The Order of Operations

We solve the separate operations in a calculation in the following order of priority.

  • Parentheses (or Grouping): We always solve items in parentheses or brackets first, for example, (𝑥+3) or [𝑥+3]. We can also have groupings such as 5+34. In a fractional grouping, we always evaluate all of the numerator and all of the denominator before performing the division.
  • Exponents: For example, 10 and 𝑥.
  • Multiplication and Division: Both of these have equal precedence, so when we have an expression that has more than one multiplication and/or division, we perform the order of operations from left to right.
  • Addition and Subtraction: Both of these have equal precedence, so when we have an expression that has more than one addition and/or subtraction, we perform the order of operations from left to right.

The acronym PEMDAS is often used to refer to this order, with the letters representing parentheses, exponents, multiplication/division, and addition/subtraction.

In the same way that we use the order of operations with integers, we also apply them in exactly the same way with decimal numbers. Let us recall the strategies that we use for arithmetic with decimal numbers.

Arithmetic with Decimal Numbers

Addition and Subtraction

  • We can use the same methods of addition and subtraction of decimals that we use for integers; however, the column method can be the most useful as it will preserve the decimal places clearly. It can be helpful to add zeros as place holders in a calculation if necessary.

Multiplication

  • To multiply decimals, remove the decimal points from the numbers to be multiplied and carry out the multiplication. Put the decimal point into the answer so that it has as many decimal places as the sum of the decimal places in the original numbers.

Division

  • To divide decimals, make the divisor a whole number by multiplying it by a power of 10, such as 10, 100, and 1,000. Multiply the dividend by the same number. We can then divide these numbers, which yields the same answer as if we had divided the decimals.

Let us now look at some examples of how to use the order of operations with decimal numbers.

Example 1: Evaluating a Decimal Expression Using the Order of Operations

Determine the value of 2×1.3+1.5 using the order of operations.

Answer

To begin using the order of operations, we first check if there are any parentheses or groupings. As there are no parentheses in the expression 2×1.3+1.5, we move to the next category in the order of operations which is exponents. As there are none, we can move to the next step, which is multiplication or division. As there is just one multiplication, 2×1.3, we do that first.

Recall that to multiply decimals, 2×1.3, we consider the numbers as whole numbers by removing the decimal point, giving 2×13=26. Then, we replace the decimal point so that the answer has as many decimal places as the sum of the decimal places in the original numbers. Here, this would be one decimal place, so we have 2×1.3=2.6, which we can replace in the original calculation to give 2×1.3+1.5=2.6+1.5.

Next, we add 2.6+1.5, which gives the following: 2.6+1.54.1

And so our answer is 2×1.3+1.5=4.1.

Example 2: Evaluating a Decimal Expression Using the Order of Operations

Calculate 68.79.9÷3.32.5.

Answer

To begin using the order of operations, we first check if there are any parentheses. As there are no parentheses in the expression 68.79.9÷3.32.5, we move to the next category in the order of operations which is exponents. As there are none, we can move to the next step in the order of operations which is multiplication or division. As there is just one division, 9.9÷3, we do that first.

To divide the decimals 9.9÷3.3, we must multiply the dividend, 9.9, and divisor, 3.3, by 10 in order to make the divisor a whole number. So we have 9.9÷3.3=9.93.3=9.9×103.3×10=9933=3.

Then, we get 68.79.9÷3.32.5=68.732.5.

We now have just two subtraction calculations left, so following the order of operations, we perform these from left to right, beginning with 68.73. We can use the column method of subtraction. It can be helpful to add an extra decimal point and zero to our value 3 to assist with our calculation: 68.73.065.7

We can replace this in our calculation to give 68.732.5=65.72.5.

Our final calculation, 65.72.5, can be performed using the column method to give the following: 65.72.563.2

Therefore, the final answer is 68.79.9÷3.32.5=63.2.

Example 3: Evaluating a Decimal Expression Using the Order of Operations

Calculate (0.2)×4(13+7)5.

Answer

In this calculation, we have two sets of parentheses. We can observe that the parentheses around (0.2) are helping maintain the clarity of which part of the expression is the base of the exponent. (0.2) has the same meaning as 0.2. As this is an exponent, we will evaluate this in the next step, after any other parentheses.

Using the order of operations, the first operation to calculate is the addition contained in the parentheses (13+7).

As we have 13+7=20, we can replace this in the calculation to give 0.2×4(13+7)5=0.2×4(20)5.

Since the term 4(20) represents 4×20, we can write the calculation as 0.2×4(13+7)5=0.2×4(20)5=0.2×4×205.

The next step in the order of operations is to calculate the exponents. Since 0.2=0.2×0.2, recall that when we multiply decimals, we remove the decimal point and consider the numbers as whole numbers and multiply. Here, we calculate 2×2=4. To find the answer to 0.2×0.2, we put the decimal point into the answer, 4, so that it has the same number of decimal places as the sum of the decimal places in the original numbers. This means that our answer must have two decimal places, so the digit 4 will be in the hundredths column.

Therefore, 0.2=0.2×0.2=0.04.

Replacing this in our calculation, we have 0.2×4×205=0.04×4×205.

We have two more exponents to calculate, 20 and 5. Since we have 20=4005=25,and we have 0.04×4×205=0.04×4×40025.

In the order of operations, after exponents we have multiplication and division. Since we have 0.04×4×400 we calculate this in order from left to right. To evaluate 0.04×4, we remove the decimal point from 0.04 and consider the calculation 4×4=16. Since our answer will have two decimal places, we have 0.04×4=0.16.

We can replace this in our calculation to give 0.04×4×40025=0.16×40025.

To evaluate 0.16×400, again we remove the decimal point to calculate 16×400=6,400 and insert the decimal point so that 6,400 has 2 decimal places, giving 0.16×400=64.00=64.

Replacing this in our calculation gives us 0.16×40025=6425.

So, we can evaluate this last calculation to give 6425=39.

This gives us our final answer: (0.2)×4(13+7)5=39.

In the next example, we will see how a grouping within a calculation has the same priority as parentheses.

Example 4: Evaluating a Decimal Expression Involving a Grouping

Calculate 3.4+9.24.20.3×2.

Answer

As we begin to use the order of operations to calculate this question, it is important to note that the first grouping, 3.4+9.24.2, will fall into the category of parentheses in the order of operations, as this part would be calculated first. The parentheses in this are implied, but since we calculate all of the numerator and denominator first, we could insert parentheses in the following way: 3.4+9.24.2=((3.4+9.2)÷4.2). So, to begin solving 3.4+9.24.20.3×2, we first calculate ((3.4+9.2)÷4.2). When we have two sets of parentheses, we calculate the innermost first; here we must add 3.4 and 9.2 first, which gives the following: 3.4+9.212.6

So we have 3.4+9.24.2=12.64.2.

Recall that, to divide the decimals, 12.6÷4.2, we make the divisor a whole number by multiplying it by a power of 10, and we multiply the dividend by the same number.

This will give us 12.64.2=12.6×104.2×10=12642.

To calculate 12642, we have the following:

Since we have evaluated 3.4+9.24.2=3, we can input this into the original calculation, giving 3.4+9.24.20.3×2=30.3×2.

Since we are left with a multiplication and a subtraction, the order of operations means that we should calculate the multiplication part, 0.3×2, first. We remove the decimal point and consider this as 3×2=6, and our answer will have as many decimal places as the sum of the decimal places in the numbers multiplied, in this case, 1 decimal place.

Therefore, we have 0.3×2=0.6, which we can replace in our calculation as 30.3×2=30.6.

Finally, we can calculate 30.6 using the column method. It can be helpful to add a decimal point and a 0 to our value 3 to make the subtraction clearer: 3.00.62.4

So, our final answer is 3.4+9.24.20.3×2=2.4.

Example 5: Finding the Missing Operators in a Decimal Calculation

Insert the appropriate symbols from +, , ×, and ÷ to make the calculation correct: 80.540=28.

Answer

Often, the best approach to this type of problem is to check the easier options of addition and subtraction first. We can see that there is no combination of two additions or an addition and a subtraction that will result in an answer of 28. Note that two subtractions would give a negative answer, rather than a positive value of 28.

If we check two multiplications, we could consider that the factors of 28 are 4 and 7, 14 and 2, and 1 and 28. We cannot make these factors from the numbers 8, 0.5, and 40.

Next, let us try a single multiplication. To calculate 0.5×40, recall that we remove the decimal point to calculate 5×40=200 and insert the decimal point so that the answer has the same number of decimal places as the sum of the decimal places in the original numbers. Therefore, 0.5×40=20.0=20.

Looking at the original calculation, we can see that adding on the additional value 8 would give the required answer. Therefore, we have 8+0.5×40=28.

Note that since the order of operations means that we would perform the multiplication first, there is no need to add parentheses to any part of the calculation.

Key Points

  • To use the order of operations with decimals, we use the same order of operations that we do with integers. We can use the acronym PEMDAS to remind ourselves that the operations should be calculated in the following order: parentheses, exponents, multiplication/division, and addition/subtraction.
  • 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 for addition and subtraction in the next stage of calculations.

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