Foundations path · F02

Before this lesson: No earlier lesson is required. Start here and take your time.

Your goal: Calculate accurately and retain exact values until approximation is requested.

Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.

Fractions, Ratios and Exact Arithmetic: the key idea

Exact arithmetic is part of reasoning. To add fractions, express them using a common denominator; to multiply, multiply numerators and denominators and cancel common factors. Cancellation removes factors, not terms of a sum. Multiplying an inequality by a negative number reverses its direction. A ratio compares quantities measured consistently. Keep a fraction exact unless the question asks for an approximation.

A worked example

Evaluate 3/4 − 2/3 without decimals.

The least common denominator is 12. Thus 3/4 = 9/12 and 2/3 = 8/12, so the difference is 1/12. The positive sign is consistent with 3/4 being larger.

Your turn: change one thing

Is (6 + 9)/3 equal to 6 + 3? Explain.

Try this on paper before opening the explanation.

Compare your reasoning

No. Divide the entire numerator: (6 + 9)/3 = 6/3 + 9/3 = 2 + 3 = 5. Cancelling only one term incorrectly gives 9.

Pause and check

A trap to avoid: State the domain, keep exact values and explain why each step is allowed. A correct answer still needs a reason.

Practise and adjust the level

Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.

Interactive practice loads here. You can also use the complete question set below.

Write a complete argument

Prove that if a/b = c/d, with b and d nonzero, then ad = bc, and prove the converse.

Planning hint

List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.

Read the full solution after your attempt

Multiplying a/b = c/d by the nonzero product bd gives ad = bc. Conversely, dividing ad = bc by bd recovers a/b = c/d. The nonzero assumptions make both operations legal.

My proof notebook

Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.

All nine practice questions

Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.

1. What is 1/2 + 1/3?

Foundation

  1. 2/6
  2. 5/6
  3. 2/5
  4. 1/5
Hint

Use denominator 6.

Answer and reasoning

5/6. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

2. Evaluate −3 × (−4).

Foundation

  1. 12
  2. 11
  3. 13
  4. 14
Hint

A product of two negative numbers is positive.

Answer and reasoning

12. The magnitudes multiply to 12 and the two negative signs give a positive product.

3. Simplify (2x + 6)/2 for real x.

Core

  1. x + 2
  2. x + 3
  3. x + 6
  4. 2x + 3
Hint

Divide both terms by 2.

Answer and reasoning

x + 3. (2x + 6)/2 = 2x/2 + 6/2 = x + 3.

4. If −2x < 8, which condition is equivalent?

Core

  1. x > 4
  2. x < 4
  3. x > −4
  4. x < −4
Hint

Division by a negative reverses order.

Answer and reasoning

x > −4. Dividing by −2 changes < to > and gives x > −4.

5. Two positive numbers are in ratio 3:5 and sum to 40. Find the smaller.

Stretch

  1. 20
  2. 25
  3. 15
  4. 8
Hint

The sum contains eight equal parts.

Answer and reasoning

15. Each part is 40/8 = 5. The smaller number is 3×5 = 15.

6. Which cancellation is valid when x ≠ 0?

Stretch

  1. (x + 3)/x = 3
  2. (x + 2)/(x + 3) = 2/3
  3. (x² + 1)/x = x + 1
  4. (3x)/(2x) = 3/2
Hint

Cancel a common factor of the whole numerator and denominator.

Answer and reasoning

(3x)/(2x) = 3/2. In 3x/(2x), x is a nonzero factor of both. The other proposed cancellations remove only terms.

7. What is 3/4+1/4?

Foundation

  1. 1
  2. 1/2
  3. 4/8
  4. 3/8
Hint

The denominators already match.

Answer and reasoning

1. Add the numerators to get 4/4=1.

8. What is (2/3)÷(4/5)?

Core

  1. 5/6
  2. 8/15
  3. 6/5
  4. 2/5
Hint

Multiply by the reciprocal of 4/5.

Answer and reasoning

5/6. (2/3)(5/4)=10/12=5/6.

9. Why can x not be cancelled from (x+2)/x to leave 2?

Stretch

  1. Because 2 is prime
  2. Cancellation applies to common factors, not terms in a sum
  3. Because x must be negative
  4. Because fractions cannot simplify
Hint

Factorisation is required for cancellation.

Answer and reasoning

Cancellation applies to common factors, not terms in a sum. For x≠0 the expression is 1+2/x, not 2.

Choose your next step

Continue to Powers, radicals and logarithms. If this felt difficult, return to a prerequisite above. Every lesson stays open.

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