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Chapter 10: Multiplying and Dividing Fractions

Learn Grade 8 Math from beginner foundations through advanced Grade 8 problem solving with detailed explanations, at least five examples per topic, practice exercises, and review questions.

Grade 8Beginner Friendly5+ Examples Per TopicPractice30 Q&A
Estimated reading time0% read
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What this chapter covers

This chapter contains 13 topics. Technical terms are followed by plain-language meanings where they first appear. Each topic includes at least five worked examples. Coding is included only where it naturally supports the Grade 8 coding expectations.

10.1 Fraction times whole number

Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Fraction times whole number.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: In 483,726, what is the value of the first digit 4?

  1. Count the places from the right.
  2. The digit 4 is in the 100,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 400,000

Worked Example 2

Problem: In 5,904,218, what is the value of the first digit 5?

  1. Count the places from the right.
  2. The digit 5 is in the 1,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 5,000,000

Worked Example 3

Problem: In 72,050,601, what is the value of the first digit 7?

  1. Count the places from the right.
  2. The digit 7 is in the 10,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 70,000,000

Worked Example 4

Problem: In 908,004,315, what is the value of the first digit 9?

  1. Count the places from the right.
  2. The digit 9 is in the 100,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 900,000,000

Worked Example 5

Problem: In 1,250,700,049, what is the value of the first digit 1?

  1. Count the places from the right.
  2. The digit 1 is in the 1,000,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 1,000,000,000

Worked Example 6

Problem: In 64,999, what is the value of the first digit 6?

  1. Count the places from the right.
  2. The digit 6 is in the 10,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 60,000

Worked Example 7

Problem: In 7,305,040, what is the value of the first digit 7?

  1. Count the places from the right.
  2. The digit 7 is in the 1,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 7,000,000

Worked Example 8

Problem: In 800,080, what is the value of the first digit 8?

  1. Count the places from the right.
  2. The digit 8 is in the 100,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 800,000

Worked Example 9

Problem: In 39,640,125, what is the value of the first digit 3?

  1. Count the places from the right.
  2. The digit 3 is in the 10,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 30,000,000

Worked Example 10

Problem: In 999,999,999, what is the value of the first digit 9?

  1. Count the places from the right.
  2. The digit 9 is in the 100,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 900,000,000

Practice exercise

Create one new Grade 8 problem involving Fraction times whole number. Show the important steps, include units when needed, and explain how you checked the answer.

10.2 Fraction times fraction

A fraction (a numerator over a denominator) can represent part of a whole, a quotient, a ratio, or a point on a number line. In this section, the focus is Fraction times fraction.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Write 1/2 as a decimal.

  1. Divide numerator 1 by denominator 2.
  2. 1 ÷ 2 = 0.5000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.5

Worked Example 2

Problem: Write 2/3 as a decimal.

  1. Divide numerator 2 by denominator 3.
  2. 2 ÷ 3 = 0.6667.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.6667

Worked Example 3

Problem: Write 3/5 as a decimal.

  1. Divide numerator 3 by denominator 5.
  2. 3 ÷ 5 = 0.6000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.6

Worked Example 4

Problem: Write 5/8 as a decimal.

  1. Divide numerator 5 by denominator 8.
  2. 5 ÷ 8 = 0.6250.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.625

Worked Example 5

Problem: Write 7/10 as a decimal.

  1. Divide numerator 7 by denominator 10.
  2. 7 ÷ 10 = 0.7000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.7

Worked Example 6

Problem: Write 4/9 as a decimal.

  1. Divide numerator 4 by denominator 9.
  2. 4 ÷ 9 = 0.4444.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.4444

Worked Example 7

Problem: Write 5/6 as a decimal.

  1. Divide numerator 5 by denominator 6.
  2. 5 ÷ 6 = 0.8333.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.8333

Worked Example 8

Problem: Write 3/4 as a decimal.

  1. Divide numerator 3 by denominator 4.
  2. 3 ÷ 4 = 0.7500.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.75

Worked Example 9

Problem: Write 2/5 as a decimal.

  1. Divide numerator 2 by denominator 5.
  2. 2 ÷ 5 = 0.4000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.4

Worked Example 10

Problem: Write 7/12 as a decimal.

  1. Divide numerator 7 by denominator 12.
  2. 7 ÷ 12 = 0.5833.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.5833

Practice exercise

Create one new Grade 8 problem involving Fraction times fraction. Show the important steps, include units when needed, and explain how you checked the answer.

10.3 Simplifying before multiplying

Simplifying before multiplying is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: An item costs $50.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 50.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $6.50

Worked Example 2

Problem: An item costs $100.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 100.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $13.00

Worked Example 3

Problem: An item costs $75.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 75.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $9.75

Worked Example 4

Problem: An item costs $120.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 120.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $15.60

Worked Example 5

Problem: An item costs $200.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 200.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $26.00

Worked Example 6

Problem: An item costs $150.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 150.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $19.50

Worked Example 7

Problem: An item costs $250.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 250.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $32.50

Worked Example 8

Problem: An item costs $80.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 80.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $10.40

Worked Example 9

Problem: An item costs $300.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 300.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $39.00

Worked Example 10

Problem: An item costs $60.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 60.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $7.80

Practice exercise

Create one new Grade 8 problem involving Simplifying before multiplying. Show the important steps, include units when needed, and explain how you checked the answer.

10.4 Cross-cancelling

Cross-cancelling is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate 1/2 × 1/3.

  1. Multiply numerators.
  2. Multiply denominators.
  3. Simplify the result. Cross-cancel first when possible.

Very beginner explanation: Fraction multiplication does not require a common denominator.

Answer: 1/6

Worked Example 2

Problem: Calculate 2/3 × 1/4.

  1. Multiply numerators.
  2. Multiply denominators.
  3. Simplify the result. Cross-cancel first when possible.

Very beginner explanation: Fraction multiplication does not require a common denominator.

Answer: 1/6

Worked Example 3

Problem: Calculate 3/5 × 2/7.

  1. Multiply numerators.
  2. Multiply denominators.
  3. Simplify the result. Cross-cancel first when possible.

Very beginner explanation: Fraction multiplication does not require a common denominator.

Answer: 6/35

Worked Example 4

Problem: Calculate 5/8 × 1/6.

  1. Multiply numerators.
  2. Multiply denominators.
  3. Simplify the result. Cross-cancel first when possible.

Very beginner explanation: Fraction multiplication does not require a common denominator.

Answer: 5/48

Worked Example 5

Problem: Calculate 7/10 × 3/5.

  1. Multiply numerators.
  2. Multiply denominators.
  3. Simplify the result. Cross-cancel first when possible.

Very beginner explanation: Fraction multiplication does not require a common denominator.

Answer: 21/50

Worked Example 6

Problem: Calculate 4/9 × 5/12.

  1. Multiply numerators.
  2. Multiply denominators.
  3. Simplify the result. Cross-cancel first when possible.

Very beginner explanation: Fraction multiplication does not require a common denominator.

Answer: 5/27

Worked Example 7

Problem: Calculate 5/6 × 1/8.

  1. Multiply numerators.
  2. Multiply denominators.
  3. Simplify the result. Cross-cancel first when possible.

Very beginner explanation: Fraction multiplication does not require a common denominator.

Answer: 5/48

Worked Example 8

Problem: Calculate 3/4 × 7/9.

  1. Multiply numerators.
  2. Multiply denominators.
  3. Simplify the result. Cross-cancel first when possible.

Very beginner explanation: Fraction multiplication does not require a common denominator.

Answer: 7/12

Worked Example 9

Problem: Calculate 2/5 × 4/15.

  1. Multiply numerators.
  2. Multiply denominators.
  3. Simplify the result. Cross-cancel first when possible.

Very beginner explanation: Fraction multiplication does not require a common denominator.

Answer: 8/75

Worked Example 10

Problem: Calculate 7/12 × 5/18.

  1. Multiply numerators.
  2. Multiply denominators.
  3. Simplify the result. Cross-cancel first when possible.

Very beginner explanation: Fraction multiplication does not require a common denominator.

Answer: 35/216

Practice exercise

Create one new Grade 8 problem involving Cross-cancelling. Show the important steps, include units when needed, and explain how you checked the answer.

10.5 Mixed-number multiplication

Mixed-number multiplication is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: An item costs $50.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 50.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $6.50

Worked Example 2

Problem: An item costs $100.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 100.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $13.00

Worked Example 3

Problem: An item costs $75.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 75.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $9.75

Worked Example 4

Problem: An item costs $120.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 120.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $15.60

Worked Example 5

Problem: An item costs $200.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 200.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $26.00

Worked Example 6

Problem: An item costs $150.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 150.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $19.50

Worked Example 7

Problem: An item costs $250.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 250.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $32.50

Worked Example 8

Problem: An item costs $80.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 80.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $10.40

Worked Example 9

Problem: An item costs $300.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 300.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $39.00

Worked Example 10

Problem: An item costs $60.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 60.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $7.80

Practice exercise

Create one new Grade 8 problem involving Mixed-number multiplication. Show the important steps, include units when needed, and explain how you checked the answer.

10.6 Reciprocal

Reciprocal is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate 1/2 ÷ 1/3.

  1. Keep 1/2.
  2. Change division to multiplication.
  3. Use the reciprocal 3/1.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 3/2

Worked Example 2

Problem: Calculate 2/3 ÷ 1/4.

  1. Keep 2/3.
  2. Change division to multiplication.
  3. Use the reciprocal 4/1.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 8/3

Worked Example 3

Problem: Calculate 3/5 ÷ 2/7.

  1. Keep 3/5.
  2. Change division to multiplication.
  3. Use the reciprocal 7/2.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 21/10

Worked Example 4

Problem: Calculate 5/8 ÷ 1/6.

  1. Keep 5/8.
  2. Change division to multiplication.
  3. Use the reciprocal 6/1.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 15/4

Worked Example 5

Problem: Calculate 7/10 ÷ 3/5.

  1. Keep 7/10.
  2. Change division to multiplication.
  3. Use the reciprocal 5/3.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 7/6

Worked Example 6

Problem: Calculate 4/9 ÷ 5/12.

  1. Keep 4/9.
  2. Change division to multiplication.
  3. Use the reciprocal 12/5.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 16/15

Worked Example 7

Problem: Calculate 5/6 ÷ 1/8.

  1. Keep 5/6.
  2. Change division to multiplication.
  3. Use the reciprocal 8/1.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 20/3

Worked Example 8

Problem: Calculate 3/4 ÷ 7/9.

  1. Keep 3/4.
  2. Change division to multiplication.
  3. Use the reciprocal 9/7.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 27/28

Worked Example 9

Problem: Calculate 2/5 ÷ 4/15.

  1. Keep 2/5.
  2. Change division to multiplication.
  3. Use the reciprocal 15/4.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 3/2

Worked Example 10

Problem: Calculate 7/12 ÷ 5/18.

  1. Keep 7/12.
  2. Change division to multiplication.
  3. Use the reciprocal 18/5.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 21/10

Practice exercise

Create one new Grade 8 problem involving Reciprocal. Show the important steps, include units when needed, and explain how you checked the answer.

10.7 Fraction divided by fraction

A fraction (a numerator over a denominator) can represent part of a whole, a quotient, a ratio, or a point on a number line. In this section, the focus is Fraction divided by fraction.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate 1/2 ÷ 1/3.

  1. Keep 1/2.
  2. Change division to multiplication.
  3. Use the reciprocal 3/1.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 3/2

Worked Example 2

Problem: Calculate 2/3 ÷ 1/4.

  1. Keep 2/3.
  2. Change division to multiplication.
  3. Use the reciprocal 4/1.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 8/3

Worked Example 3

Problem: Calculate 3/5 ÷ 2/7.

  1. Keep 3/5.
  2. Change division to multiplication.
  3. Use the reciprocal 7/2.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 21/10

Worked Example 4

Problem: Calculate 5/8 ÷ 1/6.

  1. Keep 5/8.
  2. Change division to multiplication.
  3. Use the reciprocal 6/1.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 15/4

Worked Example 5

Problem: Calculate 7/10 ÷ 3/5.

  1. Keep 7/10.
  2. Change division to multiplication.
  3. Use the reciprocal 5/3.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 7/6

Worked Example 6

Problem: Calculate 4/9 ÷ 5/12.

  1. Keep 4/9.
  2. Change division to multiplication.
  3. Use the reciprocal 12/5.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 16/15

Worked Example 7

Problem: Calculate 5/6 ÷ 1/8.

  1. Keep 5/6.
  2. Change division to multiplication.
  3. Use the reciprocal 8/1.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 20/3

Worked Example 8

Problem: Calculate 3/4 ÷ 7/9.

  1. Keep 3/4.
  2. Change division to multiplication.
  3. Use the reciprocal 9/7.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 27/28

Worked Example 9

Problem: Calculate 2/5 ÷ 4/15.

  1. Keep 2/5.
  2. Change division to multiplication.
  3. Use the reciprocal 15/4.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 3/2

Worked Example 10

Problem: Calculate 7/12 ÷ 5/18.

  1. Keep 7/12.
  2. Change division to multiplication.
  3. Use the reciprocal 18/5.
  4. Multiply and simplify.

Very beginner explanation: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Answer: 21/10

Practice exercise

Create one new Grade 8 problem involving Fraction divided by fraction. Show the important steps, include units when needed, and explain how you checked the answer.

10.8 Whole number divided by fraction

Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Whole number divided by fraction.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: In 483,726, what is the value of the first digit 4?

  1. Count the places from the right.
  2. The digit 4 is in the 100,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 400,000

Worked Example 2

Problem: In 5,904,218, what is the value of the first digit 5?

  1. Count the places from the right.
  2. The digit 5 is in the 1,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 5,000,000

Worked Example 3

Problem: In 72,050,601, what is the value of the first digit 7?

  1. Count the places from the right.
  2. The digit 7 is in the 10,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 70,000,000

Worked Example 4

Problem: In 908,004,315, what is the value of the first digit 9?

  1. Count the places from the right.
  2. The digit 9 is in the 100,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 900,000,000

Worked Example 5

Problem: In 1,250,700,049, what is the value of the first digit 1?

  1. Count the places from the right.
  2. The digit 1 is in the 1,000,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 1,000,000,000

Worked Example 6

Problem: In 64,999, what is the value of the first digit 6?

  1. Count the places from the right.
  2. The digit 6 is in the 10,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 60,000

Worked Example 7

Problem: In 7,305,040, what is the value of the first digit 7?

  1. Count the places from the right.
  2. The digit 7 is in the 1,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 7,000,000

Worked Example 8

Problem: In 800,080, what is the value of the first digit 8?

  1. Count the places from the right.
  2. The digit 8 is in the 100,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 800,000

Worked Example 9

Problem: In 39,640,125, what is the value of the first digit 3?

  1. Count the places from the right.
  2. The digit 3 is in the 10,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 30,000,000

Worked Example 10

Problem: In 999,999,999, what is the value of the first digit 9?

  1. Count the places from the right.
  2. The digit 9 is in the 100,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 900,000,000

Practice exercise

Create one new Grade 8 problem involving Whole number divided by fraction. Show the important steps, include units when needed, and explain how you checked the answer.

10.9 Fraction divided by whole number

Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Fraction divided by whole number.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: In 483,726, what is the value of the first digit 4?

  1. Count the places from the right.
  2. The digit 4 is in the 100,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 400,000

Worked Example 2

Problem: In 5,904,218, what is the value of the first digit 5?

  1. Count the places from the right.
  2. The digit 5 is in the 1,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 5,000,000

Worked Example 3

Problem: In 72,050,601, what is the value of the first digit 7?

  1. Count the places from the right.
  2. The digit 7 is in the 10,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 70,000,000

Worked Example 4

Problem: In 908,004,315, what is the value of the first digit 9?

  1. Count the places from the right.
  2. The digit 9 is in the 100,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 900,000,000

Worked Example 5

Problem: In 1,250,700,049, what is the value of the first digit 1?

  1. Count the places from the right.
  2. The digit 1 is in the 1,000,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 1,000,000,000

Worked Example 6

Problem: In 64,999, what is the value of the first digit 6?

  1. Count the places from the right.
  2. The digit 6 is in the 10,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 60,000

Worked Example 7

Problem: In 7,305,040, what is the value of the first digit 7?

  1. Count the places from the right.
  2. The digit 7 is in the 1,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 7,000,000

Worked Example 8

Problem: In 800,080, what is the value of the first digit 8?

  1. Count the places from the right.
  2. The digit 8 is in the 100,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 800,000

Worked Example 9

Problem: In 39,640,125, what is the value of the first digit 3?

  1. Count the places from the right.
  2. The digit 3 is in the 10,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 30,000,000

Worked Example 10

Problem: In 999,999,999, what is the value of the first digit 9?

  1. Count the places from the right.
  2. The digit 9 is in the 100,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 900,000,000

Practice exercise

Create one new Grade 8 problem involving Fraction divided by whole number. Show the important steps, include units when needed, and explain how you checked the answer.

10.10 Mixed-number division

Mixed-number division is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Mixed-number division: Explain Mixed-number division in one simple sentence.

  1. Look at the words in “Mixed-number division”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Mixed-number division is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Mixed-number division: A student says, “I can use Mixed-number division without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Mixed-number division: What is the first step when solving a problem about Mixed-number division?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Mixed-number division: After solving a Mixed-number division problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Mixed-number division: Give one way Mixed-number division could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Mixed-number division can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Mixed-number division: Which representation could help explain Mixed-number division: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Mixed-number division: A student gets an answer for Mixed-number division but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Mixed-number division: Why can estimation help before a detailed Mixed-number division calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Mixed-number division: How can you test whether your rule for Mixed-number division works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Mixed-number division: How would you teach Mixed-number division to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving Mixed-number division. Show the important steps, include units when needed, and explain how you checked the answer.

10.11 Negative fraction operations

A fraction (a numerator over a denominator) can represent part of a whole, a quotient, a ratio, or a point on a number line. In this section, the focus is Negative fraction operations.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Write 1/2 as a decimal.

  1. Divide numerator 1 by denominator 2.
  2. 1 ÷ 2 = 0.5000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.5

Worked Example 2

Problem: Write 2/3 as a decimal.

  1. Divide numerator 2 by denominator 3.
  2. 2 ÷ 3 = 0.6667.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.6667

Worked Example 3

Problem: Write 3/5 as a decimal.

  1. Divide numerator 3 by denominator 5.
  2. 3 ÷ 5 = 0.6000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.6

Worked Example 4

Problem: Write 5/8 as a decimal.

  1. Divide numerator 5 by denominator 8.
  2. 5 ÷ 8 = 0.6250.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.625

Worked Example 5

Problem: Write 7/10 as a decimal.

  1. Divide numerator 7 by denominator 10.
  2. 7 ÷ 10 = 0.7000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.7

Worked Example 6

Problem: Write 4/9 as a decimal.

  1. Divide numerator 4 by denominator 9.
  2. 4 ÷ 9 = 0.4444.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.4444

Worked Example 7

Problem: Write 5/6 as a decimal.

  1. Divide numerator 5 by denominator 6.
  2. 5 ÷ 6 = 0.8333.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.8333

Worked Example 8

Problem: Write 3/4 as a decimal.

  1. Divide numerator 3 by denominator 4.
  2. 3 ÷ 4 = 0.7500.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.75

Worked Example 9

Problem: Write 2/5 as a decimal.

  1. Divide numerator 2 by denominator 5.
  2. 2 ÷ 5 = 0.4000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.4

Worked Example 10

Problem: Write 7/12 as a decimal.

  1. Divide numerator 7 by denominator 12.
  2. 7 ÷ 12 = 0.5833.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.5833

Practice exercise

Create one new Grade 8 problem involving Negative fraction operations. Show the important steps, include units when needed, and explain how you checked the answer.

10.12 Measurement applications

Measurement applications is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Which is greater: 3.6 or 0.4?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 3.6

Worked Example 2

Problem: Which is greater: 8.25 or 1.5?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 8.25

Worked Example 3

Problem: Which is greater: 12.75 or 2.4?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 12.75

Worked Example 4

Problem: Which is greater: 0.96 or 0.3?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 0.96

Worked Example 5

Problem: Which is greater: 5.04 or 1.2?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 5.04

Worked Example 6

Problem: Which is greater: 14.8 or 2?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 14.8

Worked Example 7

Problem: Which is greater: 7.125 or 0.5?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 7.125

Worked Example 8

Problem: Which is greater: 20.05 or 3.75?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 20.05

Worked Example 9

Problem: Which is greater: 2.4 or 0.06?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 2.4

Worked Example 10

Problem: Which is greater: 100.5 or 4.02?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 100.5

Practice exercise

Create one new Grade 8 problem involving Measurement applications. Show the important steps, include units when needed, and explain how you checked the answer.

10.13 Word problems

Word problems is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Word problems: Explain Word problems in one simple sentence.

  1. Look at the words in “Word problems”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Word problems is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Word problems: A student says, “I can use Word problems without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Word problems: What is the first step when solving a problem about Word problems?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Word problems: After solving a Word problems problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Word problems: Give one way Word problems could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Word problems can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Word problems: Which representation could help explain Word problems: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Word problems: A student gets an answer for Word problems but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Word problems: Why can estimation help before a detailed Word problems calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Word problems: How can you test whether your rule for Word problems works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Word problems: How would you teach Word problems to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving Word problems. Show the important steps, include units when needed, and explain how you checked the answer.

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Chapter 10 Review Questions and Answers

Q1. What is important to remember about Fraction times whole number?

Answer: Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Fraction times whole number.

Q2. What is important to remember about Fraction times fraction?

Answer: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, a ratio, or a point on a number line. In this section, the focus is Fraction times fraction.

Q3. What is important to remember about Simplifying before multiplying?

Answer: Simplifying before multiplying is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q4. What is important to remember about Cross-cancelling?

Answer: Cross-cancelling is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q5. What is important to remember about Mixed-number multiplication?

Answer: Mixed-number multiplication is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q6. What is important to remember about Reciprocal?

Answer: Reciprocal is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q7. What is important to remember about Fraction divided by fraction?

Answer: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, a ratio, or a point on a number line. In this section, the focus is Fraction divided by fraction.

Q8. What is important to remember about Whole number divided by fraction?

Answer: Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Whole number divided by fraction.

Q9. What is important to remember about Fraction divided by whole number?

Answer: Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Fraction divided by whole number.

Q10. What is important to remember about Mixed-number division?

Answer: Mixed-number division is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q11. What is important to remember about Negative fraction operations?

Answer: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, a ratio, or a point on a number line. In this section, the focus is Negative fraction operations.

Q12. What is important to remember about Measurement applications?

Answer: Measurement applications is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q13. What is important to remember about Word problems?

Answer: Word problems is an important Grade 8 concept in Multiplying and Dividing Fractions. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q14. Why should you show your steps?

Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.

Q15. Why is estimation useful?

Answer: Estimation helps you judge whether a calculated answer is reasonable.

Q16. Why do units matter?

Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.

Q17. When should you round?

Answer: Usually round near the end unless the question specifically asks for earlier rounding.

Q18. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.

Q19. What should you identify first in a word problem?

Answer: Identify what is known, what is unknown, and the mathematical relationship connecting them.

Q20. What makes a final answer complete?

Answer: Give the value, correct units when needed, and a brief interpretation in context.

Q21. What should you do if an answer seems unreasonable?

Answer: Re-read the question and check copied values, signs, units, formulas, and calculations.

Q22. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships among lengths, angles, areas, and coordinates.

Q23. Why are tables useful?

Answer: Tables organize values and help reveal patterns, rates, relationships, and missing information.

Q24. Why should you explain your reasoning?

Answer: Reasoning shows why a method works, not just what answer was obtained.

Q25. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent repetition.

Q26. When is a calculator useful?

Answer: A calculator helps with lengthy arithmetic after the mathematical setup is understood and estimated.

Q27. Why compare more than one strategy?

Answer: Different strategies may make a problem easier and provide a way to verify the result.

Q28. What is mathematical communication?

Answer: It is presenting ideas clearly with words, symbols, diagrams, tables, graphs, and justified steps.

Q29. What is mathematical modelling?

Answer: It is representing a real situation with mathematics, testing the model, and revising it if needed.

Q30. Why should assumptions be stated?

Answer: Assumptions show the conditions the model depends on and help readers judge whether it is reasonable.