Year 7 Mathematics · AC9M7N06

Four Operations with Positive Rational Numbers

We are learning to use the 4 operations with positive rational numbers including fractions, decimals and percentages to solve problems using efficient calculation…

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Learning goalsSay it simply

We are learning to use the 4 operations with positive rational numbers including fractions, decimals and percentages to solve problems using efficient calculation strategies.

Positive rational numbers include positive fractions, decimals and percentages. Choosing an equivalent form can make one of the four operations more efficient without changing the value.

Addition and subtraction of fractions require common-sized parts, while multiplication and division can often be interpreted as scaling or sharing. Percentage problems require a clearly identified base quantity.

Multi-step calculations follow operation order and must be interpreted in context. An estimate, inverse operation or equivalent representation provides an independent reasonableness check.

Success criteria

  • I can choose useful fraction, decimal or percentage forms for a calculation.
  • I can apply the four operations to positive rational numbers accurately.
  • I can follow operation order and verify a contextual result.
Key conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Combine a percentage and fraction

  1. Find 25% of 64 as 1/4×64=16.
  2. Find 3/8 of 64 as 24.
  3. Add the two parts: 16+24=40.
  4. Check that 5/8×64=40.

Final answer: 25% of 64 plus 3/8 of 64 equals 40.

Check: The combined fraction is 1/4+3/8=5/8.

Example 2

Apply operation order

  1. Evaluate 3+2.5×4.
  2. Multiply first: 2.5×4=10.
  3. Add 3 to obtain 13.
  4. Check by repeated addition: 3+2.5+2.5+2.5+2.5=13.

Final answer: 3+2.5×4=13.

Check: Multiplication precedes addition.

Example 3

Application problem 1

Problem: A jacket costs $80. It is reduced by 15%, then a $6 delivery fee is added. Find the final cost and explain why adding 15 to 80 would be incorrect.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: 15% of $80 is $12, so the discounted price is $68; adding $6 gives $74. A percentage is a proportion of the original amount, not a raw dollar amount.
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Final answer: 15% of $80 is $12, so the discounted price is $68; adding $6 gives $74. A percentage is a proportion of the original amount, not a raw dollar amount.

Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Application problem 2

Problem: A 2.5 L drink is shared equally among 8 people after 10% spills. How much each?

  1. Plan: Find 90% of 2.5 L, then divide.
  2. Work: 2.25 L remains; 2.25÷8=0.28125 L, or 281.25 mL each.
  3. Interpret: Apply the percentage loss before division.

Final answer: 2.25 L remains; 2.25÷8=0.28125 L, or 281.25 mL each.

Check: Apply the percentage loss before division.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7N06 - Four Operations with Positive Rational Numbers
Example 1

Example 1 Combine a percentage and fraction Find 25% of 64 as 1/4×64=16. Find 3/8 of 64 as 24. Add the two parts: 16+24=40. Check that 5/8×64=40. Final answer: 25% of 64 plus 3/8 of 64 equals 40. Check: The combined fraction is 1/4+3/8=5/8.

Example 2

Example 2 Apply operation order Evaluate 3+2.5×4. Multiply first: 2.5×4=10. Add 3 to obtain 13. Check by repeated addition: 3+2.5+2.5+2.5+2.5=13. Final answer: 3+2.5×4=13. Check: Multiplication precedes addition.

Example 3

Example 3 Application problem 1 Problem: A jacket costs $80. It is reduced by 15%, then a $6 delivery fee is added. Find the final cost and explain why adding 15 to 80 would be incorrect. Plan: Represent the information first, then calculate, interpret and independently check the result. Work: 15% of $80 is $12, so the discounted price is $68; adding $6 gives $74. A percentage is a proportion of the original amount, not a raw dollar amount. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. Final answer: 15% of $80 is $12, so the discounted price is $68; adding $6 gives $74. A percentage is a proportion of the original amount, not a raw dollar amount. Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Example 4 Application problem 2 Problem: A 2.5 L drink is shared equally among 8 people after 10% spills. How much each? Plan: Find 90% of 2.5 L, then divide. Work: 2.25 L remains; 2.25÷8=0.28125 L, or 281.25 mL each. Interpret: Apply the percentage loss before division. Final answer: 2.25 L remains; 2.25÷8=0.28125 L, or 281.25 mL each. Check: Apply the percentage loss before division.

Curriculum examplesCopied content

The original AC9M7N06 curriculum coverage remains intact.

  • Content description: use the 4 operations with positive rational numbers including fractions, decimals and percentages to solve problems using efficient calculation strategies.
  • E1: solve addition and subtraction problems involving fractions and decimals using arrays, algebra tiles, digital tools or informal jottings.
  • E2: choose an efficient representation such as 12.5%, 1/8, 0.125 or 125/1000.
  • E3: use properties, place value, patterns and facts for multiplication and division with fractions and decimals.
  • E4: solve multiplicative problems using models, calculators or informal jottings.
  • E5: use regrouping, partitioning and properties for additive problems.
  • E6: solve practical problems efficiently, including land use, nutrition and energy-account contexts.
Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. Calculate 2/3 + 1/6.

    Check answer

    Answer: 5/6.

    Hint: Rename 2/3 as sixths.

    Why: Common sixths make the parts comparable.

  2. 2. Calculate 4.8−1.75.

    Check answer

    Answer: 3.05.

    Hint: Line up decimal points.

    Why: Aligned place values support decimal subtraction.

  3. 3. Find 30% of 90.

    Check answer

    Answer: 27.

    Hint: Multiply 90 by 0.3.

    Why: Thirty percent is 0.30 of the quantity.

  4. 4. Evaluate 3+2.5×4.

    Check answer

    Answer: 13.

    Hint: Calculate 2.5×4 first.

    Why: Operation order requires the multiplication to be completed before the addition.

  5. 5. A recipe uses 3/4 cup twice. How much altogether?

    Check answer

    Answer: 3/2 cups, or 1 1/2 cups.

    Hint: Double 3/4.

    Why: Two equal fraction amounts are added.

  6. 6. A $120 jacket is discounted by 15%. Find the sale price.

    Check answer

    Answer: Discount $18; sale price $102.

    Hint: Find 0.15×120 first.

    Why: Subtract the percentage amount from the original price.

  7. 7. Calculate 2.4÷0.6 and justify by multiplication.

    Check answer

    Answer: 4 because 0.6×4=2.4.

    Hint: Ask how many 0.6s fit into 2.4.

    Why: Multiplication checks division.

  8. 8. Mia spends 35% of $80, then 1/4 of the remainder. How much remains?

    Check answer

    Answer: After $28, $52 remains; 1/4 of $52 is $13, so $39 remains.

    Hint: Update the base after the first spend.

    Why: The second fraction uses the remainder, not the original amount.

  9. 9. A 2.5 L drink is shared equally among 8 people after 10% spills. How much each?

    Check answer

    Answer: 2.25 L remains; 2.25÷8=0.28125 L, or 281.25 mL each.

    Hint: Find 90% of 2.5 L, then divide.

    Why: Apply the percentage loss before division.

  10. 10. A jacket costs $80. It is reduced by 15%, then a $6 delivery fee is added. Find the final cost and explain why adding 15 to 80 would be incorrect.

    Check answer

    Answer: 15% of $80 is $12, so the discounted price is $68; adding $6 gives $74. A percentage is a proportion of the original amount, not a raw dollar amount.

    Hint: Represent the information first, then calculate, interpret and independently check the result.

    Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Practice and reviewReady for practice

Common mistake: Fractions added by adding denominators.

Correction: Rename with common-sized parts.

Common mistake: Decimal division leaves divisor unchanged.

Correction: Scale both dividend and divisor equally.

Common mistake: Percentage change confused with percentage points.

Correction: Compare rates carefully.

Curriculum alignmentStart here

We are learning to use the 4 operations with positive rational numbers including fractions, decimals and percentages to solve problems using efficient calculation strategies.

Positive rational numbers include positive fractions, decimals and percentages. Choosing an equivalent form can make one of the four operations more efficient without changing the value.

Addition and subtraction of fractions require common-sized parts, while multiplication and division can often be interpreted as scaling or sharing. Percentage problems require a clearly identified base quantity.

Multi-step calculations follow operation order and must be interpreted in context. An estimate, inverse operation or equivalent representation provides an independent reasonableness check.

Success criteria

  • I can choose useful fraction, decimal or percentage forms for a calculation.
  • I can apply the four operations to positive rational numbers accurately.
  • I can follow operation order and verify a contextual result.
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