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B2.1 · Use properties and order of operations in multi-step number and percent problems

Learn to use properties and order of operations in multi-step number and percent problems through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Number

Ontario Grade 7 Mathematics — B2.1

A multi-step problem needs more than one calculation. The order matters because doing steps in a different order can change the answer. Imagine three items that each cost CAD 4, with a CAD 6 fee added after buying the items. The total is three groups of CAD 4, then CAD 6 more. That is not the same as charging the fee three times. In this lesson, you will learn to read the situation, show its groups clearly, calculate in the agreed order, and check what your answer means.

What you will learn

  • Use the order of operations to evaluate multi-step number expressions.
  • Use addition and multiplication properties to make calculations easier.
  • Find a percent of a number and use it in a multi-step problem.
  • Check that a result answers the question and fits the situation.

1. Follow the order of operations

An operation is an action such as adding, subtracting, multiplying, or dividing. An expression is a number statement without an equals sign. For example, 8+3×48+3\times4 is an expression.
The order of operations is the shared rule for deciding what to calculate first. Calculate inside brackets first. Next, do multiplication and division in the order they appear, from left to right. Last, do addition and subtraction in the order they appear, from left to right.
For example, in 8+3×48+3\times4, multiply first. This gives 8+128+12, which equals 2020. Adding first would give a different result. Following the shared order makes the expression clear to everyone.
Brackets show which calculation belongs together. Three items costing CAD 4 each, with a CAD 6 fee added to each item, can be shown as 3×(4+6)3\times(4+6). If the fee is added only once after buying the three items, the expression is 3×4+63\times4+6.
The two expressions describe different situations. In the first, each of three groups includes both the item cost and fee. In the second, the fee is outside the three groups. Read the story carefully before choosing brackets. The order is: brackets first; multiplication and division from left to right; then addition and subtraction from left to right.
  • Calculate inside brackets first.
  • Do multiplication and division next, from left to right.
  • Do addition and subtraction last, from left to right.

2. Use properties to make calculations clear

A property is a rule about how numbers behave. For addition, changing the order of the numbers does not change the sum. For example, 7+5=5+77+5=5+7. A number being added is called an addend.
For multiplication, changing the order of the numbers does not change the product. A product is the answer to a multiplication. For example, 4×6=6×44\times6=6\times4. These order changes work for addition and multiplication, but not for subtraction or division. For instance, 9−29-2 is not the same as 2−92-9.
The distributive property can make some multiplications easier. It says that a number multiplied by a sum gives the same result as multiplying that number by each part of the sum and then adding. For example, 6×(10+2)6\times(10+2) can be calculated as 6×10+6×26\times10+6\times2.
This is useful when a number can be split into parts that are easy to multiply. Six groups of 12 can be thought of as six groups of 10 and six groups of 2. Add those two amounts to get the total.
A property helps organize a calculation, but it does not change the meaning of the problem. Keep track of which numbers are being added, multiplied, or grouped.
a×(b+c)=a×b+a×ca\times(b+c)=a\times b+a\times c
  • Changing the order of numbers does not change a sum or product.
  • Do not change the order of subtraction or division.
  • The distributive property can split a multiplication across a sum.

3. Find a percent of a number

Percent means “out of 100.” For example, 25% means 25 out of 100. You can write it as a fraction with 100 as the bottom number, or as a decimal: 25%=25100=0.2525\%=\frac{25}{100}=0.25.
To find a percent of a number, multiply the number by the percent written as a decimal. For example, to find 25% of 80, calculate 0.25×800.25\times80. The answer is 20.
In a multi-step percent problem, identify the amount the percent applies to. It may be the starting amount, or it may be a new amount found in an earlier step. Brackets can show the amount being grouped.
An estimate can help you check. Since 25% is one quarter, 25% of 80 should be 20. If the question asks how much remains after a part is used, subtract the used amount from the starting amount.
Read the last question again before writing your answer. The amount used, the amount saved, and the amount remaining are different quantities. p% of n=p100\frac{p}{100}× n
  • Percent means out of 100.
  • Write a percent as a decimal by dividing it by 100.
  • Multiply by the correct amount, then complete any remaining steps.

4. Plan, calculate, and check

First, decide what the question asks you to find. Notice how the quantities are connected. Words such as “each” may point to equal groups, while “in total” may point to addition. Use the whole situation, not just one word.
Next, write an expression. Use brackets when the story groups amounts together. Then follow the order of operations. Keep enough working to show what each result represents.
Finally, check the calculation and the meaning. Does the result fit the situation? Is it the right size? Did you report the amount requested and include the unit when one is needed? A reasonable estimate can help you spot an error.
  • Translate the story into an expression before calculating.
  • Use brackets to show groups that belong together.
  • Check both the arithmetic and what the answer represents.

Worked example

Evaluate an expression with brackets

Evaluate 18+4×(12−7)18+4\times(12-7). Explain why the operations happen in this order.
  1. Calculate inside the brackets
    The brackets group the subtraction, so calculate it first. The result is the number that will be multiplied by 4.
    12−7=512-7=5
  2. Multiply
    Multiplication comes before addition. Replace the bracketed calculation with 5, then multiply.
    4×5=204\times5=20
  3. Add
    Only addition remains. Add 18 to the product.
    18+20=3818+20=38
Answer: The value of the expression is 38.
Check: The bracketed difference is 5. Four groups of 5 make 20, and adding 18 gives 38.

Worked example

Use a property to multiply

A class sorts 6 packs of cards. Each pack has 12 cards. Find the total number of cards using a useful calculation.
  1. Represent the equal groups
    There are 6 equal packs, with 12 cards in each pack. Multiplication represents equal groups.
    6×126\times12
  2. Split 12 into useful parts
    The number 12 is the sum of 10 and 2. The distributive property lets us multiply each part by 6, then add the products.
    6×(10+2)=6×10+6×26\times(10+2)=6\times10+6\times2
  3. Calculate and combine
    Find both products, then add them. The result counts all the cards in the six packs.
    60+12=7260+12=72
Answer: There are 72 cards.
Check: Six groups of 10 make 60, and six groups of 2 make 12. Together they make 72.

Worked example

Find a percent and the amount left

A school club has a supply budget of CAD 120. It uses 25% of the budget for art supplies. How much money remains?
Hypothetical club budget
Hypothetical club budgetThe stated budget is divided into CAD 30 used and CAD 90 remaining. These values come from the problem.30 CADUsed90 CADRemaining

The stated budget is divided into CAD 30 used and CAD 90 remaining. These values come from the problem.

  1. Find the amount used
    The percent applies to the full budget of CAD 120. Write 25% as 0.25, then multiply by the budget.
    0.25×120=300.25\times120=30
  2. Subtract the amount used
    The question asks for the amount remaining, not the amount spent. Subtract CAD 30 from the original CAD 120.
    120−30=90120-30=90
  3. Check the total
    Twenty-five percent is one quarter. One quarter of CAD 120 is CAD 30. The used amount and the remaining amount should add to the original budget.
    30+90=12030+90=120
Answer: CAD 90 remains.
Check: The amount used and the amount remaining add to the starting budget: CAD 30 plus CAD 90 is CAD 120.

Common mistakes and how to avoid them

Adding before multiplying in an expression such as 8+3×48+3\times4.
Correction: Multiply first, then add. The expression equals 20.
Doing all multiplication before division, even when division appears first.
Correction: Multiplication and division have equal priority. Work through them from left to right.
Changing the order of subtraction because addition can be reordered.
Correction: Changing order works for addition and multiplication, not subtraction. Keep subtraction in its stated order.
Finding a percent of the amount left when the problem says the percent is of the original amount.
Correction: Identify the amount the percent applies to before calculating. In the budget example, 25% applies to CAD 120.
Reporting the amount used when asked for the amount remaining.
Correction: Read the final question carefully. Find the amount used, then subtract it from the starting amount.

Lesson summary

  • Follow the order: brackets, multiplication and division from left to right, then addition and subtraction from left to right.
  • Changing the order of numbers does not change a sum or product; the distributive property can split multiplication across a sum.
  • To find a percent of a number, write the percent as a fraction out of 100 or as a decimal, then multiply.
  • Check that your answer fits the question and the situation.

Check your understanding

Question 1

What is the value of 7+2×67+2\times6?
  1. 54
  2. 19
  3. 42
  4. 27
Show answer and explanation
19
Multiply first: 2×6=122\times6=12. Then add 7, giving 19.

Question 2

What is 15% of 200?
  1. 15
  2. 30
  3. 185
  4. 300
Show answer and explanation
30
Write 15% as 0.15. Then 0.15×200=300.15\times200=30.

Question 3

A team has 40 bottles of water. It gives away 25% of them. How many bottles are left?
  1. 10
  2. 25
  3. 30
  4. 35
Show answer and explanation
30
Twenty-five percent of 40 is 10. Subtract the 10 given away: 40−10=3040-10=30 bottles remain.

Key terms

Expression
A number statement made with numbers and operation signs, without an equals sign.
Order of operations
The agreed rules for deciding which operation to calculate first.
Property
A rule that describes how numbers behave in an operation.
Percent
A way to describe an amount out of 100.
Distributive property
A rule that lets you multiply each part of a sum by the same number, then add the products.

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