Solved Exercises on Reasoning with Fractions and Whole Numbers in Grade 3

Master reasoning with fractions and whole numbers: comparing, ordering, adding, subtracting, and problem solving through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Comparing fractions
Exercise 1
Compare the fractions:
Which is larger,
\(\frac{3}{4}\) or \(\frac{2}{3}\)?
Definition:

Fraction: A number representing part of a whole. It has a numerator (top number) and denominator (bottom number).

Numerator: The number of parts taken

Denominator: The total number of equal parts

Comparison method:
  1. Find a common denominator
  2. Convert fractions to equivalent forms
  3. Compare numerators
  4. The fraction with the larger numerator is greater
Original fractions
\(\frac{3}{4}, \frac{2}{3}\)
Common denominator
12
Result
\(\frac{9}{12} > \frac{8}{12}\)
Step 1: Find common denominator

Find LCM of denominators 4 and 3

4: 4, 8, 12, 16...

3: 3, 6, 9, 12...

LCM = 12

Step 2: Convert to equivalent fractions

\(\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)

\(\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)

Step 3: Compare numerators

\(9 > 8\), so \(\frac{9}{12} > \frac{8}{12}\)

Step 4: Write final answer

\(\frac{3}{4} > \frac{2}{3}\)

3
3
3
3
vs
2
2
2
\(\frac{3}{4} > \frac{2}{3}\)
Final answer:

\(\frac{3}{4} > \frac{2}{3}\)

Applied rules:

• Common denominator: To compare fractions, find a common denominator

• Equivalent fractions: Multiply both numerator and denominator by the same number

• Comparison: With same denominator, larger numerator means larger fraction

2 Adding fractions
Exercise 2
Add the fractions:
\(\frac{2}{5} + \frac{1}{5}\)
Definition:

Like fractions: Fractions with the same denominator

Unlike fractions: Fractions with different denominators

Expression
\(\frac{2}{5} + \frac{1}{5}\)
Add numerators
\(\frac{2+1}{5}\)
Result
\(\frac{3}{5}\)
Step 1: Check denominators

Both fractions have the same denominator (5)

Step 2: Add numerators

\(2 + 1 = 3\)

Step 3: Keep denominator

The denominator stays 5

Step 4: Write result

\(\frac{2}{5} + \frac{1}{5} = \frac{3}{5}\)

2
2
2
2
2
+
1
1
1
1
1
=
3
3
3
3
3
\(\frac{2}{5} + \frac{1}{5} = \frac{3}{5}\)
Final answer:

\(\frac{2}{5} + \frac{1}{5} = \frac{3}{5}\)

Applied rules:

• Addition of like fractions: Add numerators, keep denominator

• Denominator stays the same: Only numerators change

• Simplification: Reduce fraction if possible

3 Subtracting fractions
Exercise 3
Subtract:
\(\frac{7}{8} - \frac{3}{8}\)
Definition:

Subtraction of fractions: Finding the difference between two fractional amounts

Expression
\(\frac{7}{8} - \frac{3}{8}\)
Subtract numerators
\(\frac{7-3}{8}\)
Result
\(\frac{4}{8} = \frac{1}{2}\)
Step 1: Check denominators

Both fractions have the same denominator (8)

Step 2: Subtract numerators

\(7 - 3 = 4\)

Step 3: Keep denominator

The denominator stays 8

Step 4: Simplify if possible

\(\frac{4}{8} = \frac{1}{2}\) (divide by GCD 4)

7
7
7
7
7
7
7
7
-
3
3
3
3
3
3
3
3
=
4
4
4
4
4
4
4
4
\(\frac{7}{8} - \frac{3}{8} = \frac{1}{2}\)
Final answer:

\(\frac{7}{8} - \frac{3}{8} = \frac{1}{2}\)

Applied rules:

• Subtraction of like fractions: Subtract numerators, keep denominator

• Keep denominator: Only numerators change in subtraction

• Simplification: Always reduce to lowest terms

Rules and methods, laws,...
\(\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}\)
Adding Like Fractions
Rule 1
\(\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}\)
Adding fractions with same denominator
Rule 2
\(\frac{a}{b} - \frac{c}{b} = \frac{a-c}{b}\)
Subtracting fractions with same denominator
Rule 3
\(\frac{a}{b} = \frac{a \times c}{b \times c}\)
Creating equivalent fractions
Key definitions:

Fraction: A number representing part of a whole (\(\frac{\text{numerator}}{\text{denominator}}\))

Whole number: Counting numbers including zero (0, 1, 2, 3, ...)

Like fractions: Fractions with the same denominator

Unlike fractions: Fractions with different denominators

Proper fraction: Numerator smaller than denominator

Improper fraction: Numerator greater than or equal to denominator

Mixed number: Combination of whole number and fraction

Complete methodology:
  1. Identify fraction types: Like or unlike fractions
  2. For like fractions: Add/subtract numerators, keep denominator
  3. For unlike fractions: Find common denominator first
  4. Compare fractions: Find common denominator and compare numerators
  5. Simplify: Reduce to lowest terms
Tip 1: Always check if denominators are the same before adding or subtracting.
Tip 2: To find a common denominator, multiply the denominators together.
Tip 3: A fraction is in simplest form when numerator and denominator have no common factors except 1.
Tip 4: Draw pictures to visualize fractions when comparing.
Common errors: Adding denominators, forgetting to find common denominators for unlike fractions, not simplifying results.
Visual models: Use circles, rectangles, or number lines to represent fractions.
Formulas to know by heart:

• Addition of like fractions: \(\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}\)

• Subtraction of like fractions: \(\frac{a}{b} - \frac{c}{b} = \frac{a-c}{b}\)

• Equivalent fractions: \(\frac{a}{b} = \frac{a \times c}{b \times c}\)

• Comparing fractions: Find common denominator then compare numerators

• Simplifying fractions: Divide numerator and denominator by their GCD

Solution: Exercises 4 to 5
4 Word problem with fractions
Exercise 4
Sarah ate \(\frac{3}{8}\) of a pizza and Tom ate \(\frac{1}{4}\) of the same pizza. How much pizza did they eat altogether?
Definition:

Word problems: Real-world situations requiring mathematical operations to solve

Expression
\(\frac{3}{8} + \frac{1}{4}\)
Common denominator
8
Result
\(\frac{5}{8}\)
Step 1: Identify fractions

Sarah: \(\frac{3}{8}\), Tom: \(\frac{1}{4}\)

Step 2: Check denominators

Denominators are different: 8 and 4

Step 3: Find common denominator

LCM of 8 and 4 is 8

Step 4: Convert to like fractions

\(\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}\)

Step 5: Add like fractions

\(\frac{3}{8} + \frac{2}{8} = \frac{5}{8}\)

Step 6: Interpret result

They ate \(\frac{5}{8}\) of the pizza altogether

Sarah
Sarah
Sarah
Sarah
Sarah
Sarah
Sarah
Sarah
+
Tom
Tom
Tom
Tom
Tom
Tom
Tom
Tom
=
Total
Total
Total
Total
Total
Total
Total
Total
\(\frac{3}{8} + \frac{1}{4} = \frac{5}{8}\)
Final answer:

Together they ate \(\frac{5}{8}\) of the pizza.

Applied rules:

• Problem solving: Identify key information and operation needed

• Common denominator: Required for adding unlike fractions

• Real-world context: Answer makes sense in the situation

5 Ordering fractions
Exercise 5
Order these fractions from least to greatest:
\(\frac{1}{3}, \frac{2}{5}, \frac{1}{2}\)
Definition:

Ordering fractions: Arranging fractions in ascending or descending order

Fractions
\(\frac{1}{3}, \frac{2}{5}, \frac{1}{2}\)
Common denominator
30
Ordered
\(\frac{1}{3} < \frac{2}{5} < \frac{1}{2}\)
Step 1: List fractions

\(\frac{1}{3}, \frac{2}{5}, \frac{1}{2}\)

Step 2: Find common denominator

LCM of 3, 5, and 2 is 30

Step 3: Convert to equivalent fractions

\(\frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30}\)

\(\frac{2}{5} = \frac{2 \times 6}{5 \times 6} = \frac{12}{30}\)

\(\frac{1}{2} = \frac{1 \times 15}{2 \times 15} = \frac{15}{30}\)

Step 4: Compare numerators

\(10 < 12 < 15\), so \(\frac{10}{30} < \frac{12}{30} < \frac{15}{30}\)

Step 5: Write in original form

\(\frac{1}{3} < \frac{2}{5} < \frac{1}{2}\)

1/3
1/3
1/3
vs
2/5
2/5
2/5
2/5
2/5
vs
1/2
1/2
\(\frac{1}{3} < \frac{2}{5} < \frac{1}{2}\)
Final answer:

The order from least to greatest is: \(\frac{1}{3}, \frac{2}{5}, \frac{1}{2}\)

Applied rules:

• Common denominator: Needed to compare multiple fractions

• LCM calculation: Find smallest common multiple of denominators

• Ordering principle: With same denominator, order by numerator size

Key Concepts: Fraction Reasoning and Whole Number Operations
\(\frac{a}{b} = \frac{a \times c}{b \times c}\)
Equivalent Fractions
Key definitions:

Fraction: Represents a part of a whole. Written as \(\frac{\text{numerator}}{\text{denominator}}\)

Numerator: Top number showing how many parts are taken

Denominator: Bottom number showing total number of equal parts

Unit fraction: A fraction with numerator 1 (e.g., \(\frac{1}{4}\))

Equivalent fractions: Different fractions that represent the same amount

Mixed number: A whole number combined with a fraction

Improper fraction: A fraction where numerator ≥ denominator

Complete methodology:
  1. Identify fraction type: Proper, improper, or mixed number
  2. For comparison: Find common denominators or convert to decimals
  3. For addition/subtraction: Make denominators alike first
  4. For multiplication: Multiply numerators and denominators separately
  5. For division: Multiply by reciprocal
  6. Simplify: Reduce to lowest terms
Tip 1: Use visual models like fraction bars or circles to understand relationships.
Tip 2: Remember that larger denominators mean smaller pieces of the whole.
Tip 3: When comparing, think about how close each fraction is to one whole.
Tip 4: Always check your work by estimating whether the answer makes sense.
Common errors: Adding denominators, forgetting to find common denominators, not simplifying answers.
Key relationships: \(\frac{1}{2} = \frac{2}{4} = \frac{3}{6}\), \(\frac{1}{4} < \frac{1}{3} < \frac{1}{2}\).
Formulas to know by heart:

• Addition of like fractions: \(\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}\)

• Subtraction of like fractions: \(\frac{a}{b} - \frac{c}{b} = \frac{a-c}{b}\)

• Equivalent fractions: \(\frac{a}{b} = \frac{a \times c}{b \times c}\)

• Comparing fractions: Find common denominator then compare numerators

• Simplifying fractions: Divide numerator and denominator by their GCD

Fraction Relationships

Equivalent Fractions

\(\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8}\)

1
1
=
2
2
2
2
=
3
3
3
3
3
3
Fraction Size Order

\(\frac{1}{4} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4}\)

1/4
1/4
1/4
1/4
<
1/3
1/3
1/3

Questions & Answers

Question: I don't understand why \(\frac{1}{4}\) is smaller than \(\frac{1}{3}\). Doesn't 4 being bigger than 3 make \(\frac{1}{4}\) bigger?

Answer: That's a great question! The denominator tells us how many equal pieces the whole is cut into. Think of it like this:

  • If you cut a pizza into 4 pieces, each piece is smaller than if you cut it into 3 pieces
  • \(\frac{1}{4}\) means 1 out of 4 pieces
  • \(\frac{1}{3}\) means 1 out of 3 pieces
  • Since 4 pieces are smaller than 3 pieces, \(\frac{1}{4}\) is smaller than \(\frac{1}{3}\)

Think of it as sharing: If you share 1 cookie among 4 friends, each person gets less than if you shared 1 cookie among 3 friends.

The rule is: The larger the denominator (with the same numerator), the smaller the fraction!

Question: When I add \(\frac{2}{5} + \frac{1}{5}\), why don't I add the denominators too? Why does the 5 stay the same?

Answer: Great observation! When adding fractions with the same denominator, think about what the fraction represents:

  • \(\frac{2}{5}\) means 2 pieces out of 5 equal pieces
  • \(\frac{1}{5}\) means 1 piece out of 5 equal pieces
  • When you add them, you're still dealing with the same size pieces (the same whole divided into 5 parts)
  • You're just combining the number of pieces: 2 pieces + 1 piece = 3 pieces

So you get 3 pieces out of 5 total pieces, which is \(\frac{3}{5}\).

The denominator stays the same because the size of the pieces doesn't change - you're just counting more of the same-sized pieces!

Question: My child is struggling with comparing fractions like \(\frac{2}{3}\) and \(\frac{3}{4}\). What's the best way to teach this concept?

Answer: Here are effective strategies for comparing fractions:

  • Visual models: Draw circles or rectangles divided into equal parts. Shade \(\frac{2}{3}\) of one and \(\frac{3}{4}\) of another.
  • Common denominators: Convert both fractions to have the same bottom number: \(\frac{2}{3} = \frac{8}{12}\) and \(\frac{3}{4} = \frac{9}{12}\), so \(\frac{3}{4} > \frac{2}{3}\).
  • Distance from whole: See how far each fraction is from 1. \(\frac{2}{3}\) is \(\frac{1}{3}\) away from 1, while \(\frac{3}{4}\) is \(\frac{1}{4}\) away from 1. Since \(\frac{1}{4} < \frac{1}{3}\), \(\frac{3}{4}\) is closer to 1, so it's larger.

Start with visual representations to build conceptual understanding before moving to abstract methods!