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C2.2 · Evaluate expressions with whole numbers and decimals

Learn to evaluate expressions with whole numbers and decimals through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Algebra

Use the order of operations to find the value of an expression

An expression is a group of numbers and operation signs. It does not have an equals sign. To evaluate an expression means to find its value. For example, a shopping total can be shown with an expression: add the prices, then multiply by the number of identical items. The order matters. If you calculate in a different order, you may get a different answer. In this lesson, you will use the order of operations with whole numbers and decimals.

What you will learn

  • Explain what it means to evaluate an expression.
  • Use the order of operations to evaluate expressions with whole numbers and decimals.
  • Check whether an answer is reasonable.

Start with a concrete model

Suppose one notebook costs CAD 2.40. You buy 3 notebooks and a pen that costs CAD 1.20. The notebooks cost 3 groups of CAD 2.40. Then you add the pen. The expression is 3×2.40+1.203 \times 2.40 + 1.20.
First find the cost of the notebooks: 3×2.40=7.203 \times 2.40 = 7.20. Then add the pen: 7.20+1.20=8.407.20 + 1.20 = 8.40. The total is CAD 8.40.
The multiplication comes before the addition. If you added 2.40+1.202.40 + 1.20 first, you would be treating the pen as if it were part of the price of each notebook. That does not match the situation.
3×2.40+1.20=8.403 × 2.40 + 1.20 = 8.40
  • Evaluating means finding the value of an expression.
  • Use the order of operations so the expression matches its meaning.

The order of operations

Use these steps when an expression has more than one operation. First, calculate inside brackets. Next, do multiplication and division, working from left to right. Last, do addition and subtraction, also working from left to right.
Brackets are symbols that group part of an expression, such as (4+2)(4 + 2). They tell you to calculate the grouped part first. Multiplication and division have the same priority. Addition and subtraction have the same priority. When operations in one of these pairs appear together, work from left to right.
A decimal is a number with a decimal point, such as 3.63.6. When adding or subtracting decimals, line up the decimal points. When multiplying, multiply as with whole numbers and place the decimal point so the answer has the correct value. You can check by estimating: round numbers to nearby whole numbers and see whether your answer is close.
A simple way to stay organized is to copy the expression and change only the part you calculate. Keep the other numbers and signs in their original order. Do not do every operation from left to right without first checking its priority.
(4+2)×3(4 + 2) × 3
  • Brackets first.
  • Then multiplication and division from left to right.
  • Then addition and subtraction from left to right.

Work carefully with decimals

The order of operations does not change when decimals appear. The same rules apply to whole numbers and decimals. Keep the decimal point in each number easy to see as you calculate.
For example, 8.4−2.4÷38.4 - 2.4 \div 3 includes division and subtraction. Divide first: 2.4÷3=0.82.4 \div 3 = 0.8. Then subtract: 8.4−0.8=7.68.4 - 0.8 = 7.6. Estimating also helps: 2.4÷32.4 \div 3 is less than 1, so subtracting it from 8.4 should give an answer a little below 8.4.
When an expression includes brackets, the brackets can change what is done first. Compare 2×(3+4)2 \times (3 + 4) with 2×3+42 \times 3 + 4. In the first expression, add inside the brackets before multiplying. In the second, multiply before adding. Brackets are part of the expression, so do not remove or ignore them.
8.4−2.4÷3=7.68.4 - 2.4 \div 3 = 7.6
  • Decimals follow the same order of operations as whole numbers.
  • Estimate to check whether a decimal answer makes sense.

A steady method for any expression

Read the whole expression before calculating. Notice its brackets, numbers, and operation signs. Then find the highest-priority operation that is ready to calculate. Replace just that part with its value.
If multiplication and division both appear, move from left to right. Do not assume that multiplication must always happen before division. If addition and subtraction both appear, move from left to right for the same reason.
After each step, check that the expression still means the same calculation. At the end, estimate or use the context to check the answer. A reasonable answer is not a substitute for careful calculation, but it can reveal a misplaced decimal or an operation done in the wrong order.
  • Work one operation at a time.
  • Keep the original order when operations have equal priority.
  • Check the final value.

Worked example

Notebook and pen total

Evaluate 3×2.40+1.203 \times 2.40 + 1.20.
  1. Multiply first
    Multiplication comes before addition. Find the cost of the three notebooks.
    3×2.40=7.203 × 2.40 = 7.20
  2. Add the remaining cost
    Now add the pen's price to the notebook cost.
    7.20+1.20=8.407.20 + 1.20 = 8.40
Answer: The value is 8.408.40, or CAD 8.40 in the shopping situation.
Check: Three items at about CAD 2 each cost about CAD 6. Adding about CAD 1 gives a total near CAD 7. CAD 8.40 is reasonable because each notebook costs CAD 2.40, not CAD 2.

Worked example

Brackets change the first step

Evaluate (5.6+2.4)×3−4(5.6 + 2.4) \times 3 - 4.
  1. Calculate inside brackets
    The brackets group the two decimals, so add them before multiplying.
    5.6+2.4=85.6 + 2.4 = 8
  2. Multiply
    The multiplication is now the highest-priority operation.
    8×3=248 × 3 = 24
  3. Subtract
    Subtraction is the only operation left.
    24−4=2024 - 4 = 20
Answer: The value is 2020.
Check: The bracketed sum is 8. Three groups of 8 make 24, and subtracting 4 gives a result close to 20.

Worked example

Equal-priority operations

Evaluate 18÷3×1.5+2.518 \div 3 \times 1.5 + 2.5.
  1. Move left to right
    Division and multiplication have equal priority. Do the leftmost one first.
    18÷3=618 \div 3 = 6
  2. Continue with multiplication
    Multiply the result by 1.5 before doing the addition.
    6×1.5=96 × 1.5 = 9
  3. Finish with addition
    Add 2.5 to the result.
    9+2.5=11.59 + 2.5 = 11.5
Answer: The value is 11.511.5.
Check: The division gives 6. Multiplying by 1.5 gives 9, and adding 2.5 gives a value between 11 and 12.

Common mistakes and how to avoid them

Calculating every operation from left to right.
Correction: First follow the priority rules. Move left to right only among operations with equal priority.
Doing multiplication before division every time.
Correction: Multiplication and division have equal priority. Choose the one that appears first from the left.
Forgetting to calculate inside brackets first.
Correction: Treat the bracketed part as the first calculation, then continue with the rest.
Placing a decimal point without checking the value.
Correction: Estimate the answer and ask whether its size makes sense.

Lesson summary

  • To evaluate an expression is to find its value.
  • Calculate inside brackets first.
  • Then calculate multiplication and division from left to right.
  • Finish with addition and subtraction from left to right.
  • Use an estimate to check that the answer is reasonable.

Check your understanding

Question 1

What is the value of 4×2.5+34 \times 2.5 + 3?
  1. 13
  2. 22
  3. 10
  4. 15
Show answer and explanation
13
Multiply first: 4×2.5=104 \times 2.5 = 10. Then add 3 to get 13.

Question 2

What is the value of (6+2)÷4(6 + 2) \div 4?
  1. 8
  2. 2
  3. 3.5
  4. 4
Show answer and explanation
2
Add inside the brackets to get 8. Then divide by 4 to get 2.

Question 3

What is the value of 20÷5×220 \div 5 \times 2?
  1. 2
  2. 8
  3. 10
  4. 20
Show answer and explanation
8
Division and multiplication have equal priority, so work left to right: 20÷5=420 \div 5 = 4, then 4×2=84 \times 2 = 8.

Key terms

Expression
Numbers and operation signs written together, without an equals sign.
Evaluate
Find the value of an expression.
Brackets
Symbols that group part of an expression to be calculated first.
Estimate
A close answer used to check whether a calculation is reasonable.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation C2.2. It is a study resource, not an official curriculum publication.

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