Solved Exercises on Arithmetic Reasoning Fractions in Grade 3

Master arithmetic reasoning fractions: addition, subtraction, comparison, equivalence, and word problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Adding fractions with same denominator
Exercise 1
Sarah ate 2/8 of a pizza and her brother ate 3/8 of the same pizza. How much pizza did they eat together?
\( \frac{2}{8} + \frac{3}{8} = ? \)
Definition:

Adding fractions with same denominator: Add numerators and keep the same denominator

Addition method:
  1. Check that denominators are the same
  2. Add the numerators
  3. Keep the same denominator
  4. Simplify if possible
Expression
\( \frac{2}{8} + \frac{3}{8} \)
Addition
\( \frac{5}{8} \)

Visual representation: 2 slices + 3 slices = 5 slices out of 8 total slices

Step 1: Check denominators

Both fractions have the same denominator: 8

Step 2: Add the numerators

\(2 + 3 = 5\)

Step 3: Keep the same denominator

Denominator stays 8

Step 4: Write the result

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

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

Sarah and her brother ate \( \frac{5}{8} \) of the pizza together.

Applied rules:

• Addition rule: When denominators are the same, add numerators only

• Same denominator: Both fractions must refer to the same whole

• Simplification: \( \frac{5}{8} \) cannot be simplified further

2 Subtracting fractions with same denominator
Exercise 2
Tom had 7/10 of a chocolate bar. He gave 3/10 to his friend. How much chocolate does Tom have left?
\( \frac{7}{10} - \frac{3}{10} = ? \)
Definition:

Subtracting fractions with same denominator: Subtract numerators and keep the same denominator

Expression
\( \frac{7}{10} - \frac{3}{10} \)
Subtraction
\( \frac{4}{10} \)
Simplification
\( \frac{2}{5} \)

Visual representation: 7 pieces - 3 pieces = 4 pieces out of 10 total pieces

Step 1: Check denominators

Both fractions have the same denominator: 10

Step 2: Subtract the numerators

\(7 - 3 = 4\)

Step 3: Keep the same denominator

Denominator stays 10

Step 4: Simplify if possible

\( \frac{4}{10} = \frac{2}{5} \) (divide numerator and denominator by 2)

\( \frac{7}{10} - \frac{3}{10} = \frac{4}{10} = \frac{2}{5} \)
Final answer:

Tom has \( \frac{2}{5} \) of the chocolate bar left.

Applied rules:

• Subtraction rule: When denominators are the same, subtract numerators only

• Simplification: Reduce fractions to lowest terms when possible

• Same whole: Both fractions must refer to the same quantity

3 Comparing fractions
Exercise 3
Which is larger: 3/5 or 2/3 of a cake? Compare the fractions.
\( \frac{3}{5} \) vs \( \frac{2}{3} \)
Definition:

Comparing fractions: Find a common denominator or convert to decimals to compare

Fractions
\( \frac{3}{5} \) vs \( \frac{2}{3} \)
Common Denominator
15
Comparison
\( \frac{9}{15} > \frac{10}{15} \)

Visual representation: 3/5 = 9/15 and 2/3 = 10/15, so 2/3 is larger

Step 1: Find the least common denominator (LCD)

The LCD of 5 and 3 is 15

Step 2: Convert fractions to equivalent forms

\( \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \)

\( \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \)

Step 3: Compare numerators

\(9 < 10\), so \( \frac{9}{15} < \frac{10}{15} \)

Step 4: State the final comparison

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

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

\( \frac{2}{3} \) of the cake is larger than \( \frac{3}{5} \) of the cake.

Applied rules:

• Comparison rule: Convert to common denominators to compare fractions

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

• Ordering: Larger numerator means larger fraction when denominators are equal

Rules and methods, laws,...
\( \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c} \)
Fraction Addition
\( \frac{a}{c} - \frac{b}{c} = \frac{a-b}{c} \)
Fraction Subtraction
Equivalent Fractions
\( \frac{a}{b} = \frac{a \times n}{b \times n} \)
Multiplying numerator and denominator by same number
Comparing Fractions
Convert to common denominators
Compare numerators after conversion
Mixed Numbers
\( a\frac{b}{c} = \frac{a \times c + b}{c} \)
Converting mixed to improper fractions
Key definitions:

Numerator: The top number in a fraction (tells how many parts we have)

Denominator: The bottom number in a fraction (tells how many equal parts the whole is divided into)

Proper fraction: Numerator is smaller than denominator

Improper fraction: Numerator is larger than or equal to denominator

Mixed number: A combination of a whole number and a proper fraction

Fraction arithmetic methodology:
  1. Identify operation: Addition, subtraction, multiplication, or division
  2. Check denominators: Same or different denominators
  3. Apply appropriate method: Follow the specific rules for the operation
  4. Simplify: Reduce to lowest terms when possible
Tip 1: Use visual models like pie charts or bars to understand fractions better.
Tip 2: When comparing fractions with different denominators, find the least common denominator (LCD).
Tip 3: Remember that adding fractions with the same denominator is just adding the numerators.
Tip 4: Always check if your final answer can be simplified by dividing numerator and denominator by their GCD.
Common errors: Adding denominators instead of just numerators, forgetting to simplify results, misidentifying common denominators.
Key memory aids: "Bottom number stays the same when adding/subtracting with like denominators", "Cross multiply when comparing fractions".
Solution: Exercises 4 to 5
4 Word problem with fractions
Exercise 4
A bakery made 4/5 of a batch of cookies in the morning. In the afternoon, they made another 2/5 of a batch. How many batches of cookies did they make in total? Then they sold 3/5 of a batch. How many batches do they have left?
Definition:

Fraction word problems: Real-life scenarios requiring fraction operations to solve

Morning + Afternoon
\( \frac{4}{5} + \frac{2}{5} \)
Total - Sold
\( \frac{6}{5} - \frac{3}{5} \)
Final Answer
\( \frac{3}{5} \)

Visual representation: 4/5 + 2/5 = 6/5, then 6/5 - 3/5 = 3/5

Step 1: Add morning and afternoon production

\( \frac{4}{5} + \frac{2}{5} = \frac{4+2}{5} = \frac{6}{5} \)

Step 2: Subtract the amount sold

\( \frac{6}{5} - \frac{3}{5} = \frac{6-3}{5} = \frac{3}{5} \)

Step 3: Interpret the result

\( \frac{3}{5} \) is an improper fraction converted to mixed number: \( 1\frac{1}{5} \)

Step 4: Final answer

The bakery has \( \frac{3}{5} \) of a batch left.

Total made: \( \frac{6}{5} \) batches
Left after sales: \( \frac{3}{5} \) batches
Final answer:

The bakery made \( \frac{6}{5} \) batches in total and has \( \frac{3}{5} \) of a batch left after selling.

Applied rules:

• Word problem solving: Identify the operations needed based on keywords

• Fraction addition: Same denominators allow direct addition of numerators

• Problem interpretation: Connect mathematical results to real-world context

5 Equivalent fractions and simplification
Exercise 5
Show that 6/8 is equivalent to 3/4 by simplifying. Then find two more fractions equivalent to 3/4.
Definition:

Equivalent fractions: Different fractions that represent the same value

Original
\( \frac{6}{8} \)
Simplified
\( \frac{3}{4} \)
More Equivalents
\( \frac{9}{12}, \frac{12}{16} \)

Visual representation: 6/8 = 3/4 because both represent the same portion of a whole

Step 1: Find the GCD of numerator and denominator

GCD of 6 and 8 is 2

Step 2: Divide both numerator and denominator by GCD

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

Step 3: Find equivalent fractions by multiplying

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

\( \frac{3}{4} = \frac{3 \times 4}{4 \times 4} = \frac{12}{16} \)

Step 4: Verify equivalency

All fractions equal 0.75 when converted to decimal

\( \frac{6}{8} = \frac{3}{4} = \frac{9}{12} = \frac{12}{16} \)
Final answer:

\( \frac{6}{8} \) simplifies to \( \frac{3}{4} \). Two other equivalent fractions are \( \frac{9}{12} \) and \( \frac{12}{16} \).

Applied rules:

• Equivalent fractions: Multiply or divide both numerator and denominator by the same non-zero number

• Simplification: Divide by the greatest common divisor to get lowest terms

• Verification: Equivalent fractions have the same decimal value

Fraction Arithmetic Laws, Methods, Rules, and Definitions
\( \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c} \)
Fraction Addition
\( \frac{a}{c} - \frac{b}{c} = \frac{a-b}{c} \)
Fraction Subtraction
Key definitions:

Fraction: A number representing part of a whole, written as \( \frac{a}{b} \) where a is numerator and b is denominator

Unit fraction: A fraction with numerator 1, such as \( \frac{1}{4} \)

Like fractions: Fractions with the same denominator

Unlike fractions: Fractions with different denominators

Reciprocal: For fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \)

Complete fraction methodology:
  1. Identify fraction type: Proper, improper, or mixed number
  2. Check denominators: Are they the same or different?
  3. Select operation: Addition, subtraction, multiplication, or division
  4. Apply algorithm: Follow the specific method for the operation
  5. Simplify result: Reduce to lowest terms if possible
Tip 1: When adding/subtracting fractions with different denominators, find the LCD first.
Tip 2: To compare fractions quickly, cross multiply: \( \frac{a}{b} \) vs \( \frac{c}{d} \) means compare \( a \times d \) vs \( b \times c \).
Tip 3: Use visual models like fraction bars or circles to verify your calculations.
Tip 4: Always check if the final answer makes sense in the context of the problem.
Common errors: Adding denominators instead of numerators, forgetting to find common denominators, not simplifying final answers.
Exam preparation: Practice converting between mixed numbers and improper fractions, memorize multiplication tables, master finding LCD.
Fraction laws and rules:

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

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

• Multiplication: \( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \)

• Division: \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \)

• Equivalent: \( \frac{a}{b} = \frac{an}{bn} \) for any non-zero n

Questions & Answers

Question: I don't understand why we can add numerators when denominators are the same, but we can't add denominators. Can you explain?

Answer: Great question! Think of fractions like slices of pizza:

  • If you have 2 slices of a pizza cut into 8 pieces, that's \( \frac{2}{8} \)
  • If you eat 3 more slices from the same pizza, that's \( \frac{3}{8} \)
  • Total eaten: 2 + 3 = 5 slices out of 8 total slices = \( \frac{5}{8} \)

The denominator (8) represents the total number of equal pieces the whole pizza was cut into. That doesn't change when you add more slices from the same pizza!

The numerator tells you HOW MANY pieces you have. So when adding, you're combining the number of pieces you have, not changing the size of the pieces.

If we added denominators, we'd be saying the pizza got cut into more pieces, which doesn't happen when you eat more slices!

Question: My child is struggling with comparing fractions like 2/3 and 3/4. How can I help them visualize this better?

Answer: Here are some effective visualization strategies:

  • Pie charts: Draw two identical circles. Shade 2/3 of one and 3/4 of the other to compare visually
  • Number line: Mark 0, 1/2, and 1 on a line. Plot 2/3 and 3/4 to see which is closer to 1
  • Bar models: Draw rectangles of equal length, divide one into 3 parts and shade 2, divide another into 4 parts and shade 3
  • Real objects: Use actual food like pizza, chocolate bars, or paper folding

For 2/3 vs 3/4 specifically, you can also convert to equivalent fractions with common denominators: 2/3 = 8/12 and 3/4 = 9/12, so 3/4 is larger.

The key is helping children understand that the size of the whole matters - 2/3 of a small pizza might be less than 3/4 of a large pizza!

Question: What are some common misconceptions students have about fractions that I should watch out for?

Answer: Here are the most common fraction misconceptions:

  • Larger denominators mean larger fractions: Students think 1/8 is bigger than 1/4 because 8 > 4
  • Add denominators too: Thinking 1/4 + 1/4 = 2/8 instead of 2/4
  • Only consider numerators: Believing 2/3 is always greater than 1/5 just because 2 > 1
  • Fraction as two separate numbers: Not understanding the relationship between numerator and denominator
  • Equal parts aren't necessary: Thinking any 2 parts of a shape represent 2/3 regardless of size

To address these, use concrete manipulatives, emphasize the concept of equal parts, and provide plenty of visual representations.

Always connect fractions back to the whole and encourage students to ask "What is the whole?" and "Are the parts equal?"