Solved Exercises on Size and Location of Fractions in Grade 4

Master size and location of fractions: comparing fractions, ordering fractions, and understanding fraction magnitude for denominators 2, 3, 4, 5, 6, 8, 10, and 12 through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Comparing fractions
Exercise 1
Compare these fractions:
\(\frac{2}{3}\) and \(\frac{3}{4}\). Which is larger? Explain your reasoning.
Definition:

Comparing fractions: Determining which fraction is greater or if they are equal

Equivalent fractions: Fractions that represent the same amount

Fraction comparison methods:
  1. Find common denominators: Convert to same denominator and compare numerators
  2. Find common numerators: Convert to same numerator and compare denominators
  3. Convert to decimals: Divide numerator by denominator
  4. Visual models: Draw pictures to compare sizes
Fractions to compare
\(\frac{2}{3}\) and \(\frac{3}{4}\)
Common denominator
12
Result
\(\frac{8}{12} < \frac{9}{12}\)
Step 1: Find the least common denominator (LCD)

Denominators are 3 and 4

LCM of 3 and 4 is 12

Step 2: Convert both fractions to equivalent forms with LCD

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

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

Step 3: Compare the numerators

Since 8 < 9, we have \(\frac{8}{12} < \frac{9}{12}\)

Step 4: State the final answer

Therefore, \(\frac{2}{3} < \frac{3}{4}\)

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

\(\frac{3}{4}\) is larger than \(\frac{2}{3}\).

When converted to equivalent fractions with denominator 12: \(\frac{2}{3} = \frac{8}{12}\) and \(\frac{3}{4} = \frac{9}{12}\).

Since 9 > 8, \(\frac{3}{4} > \frac{2}{3}\).

Applied rules:

• Common denominator method: Convert fractions to have the same denominator

• Comparison: When denominators are equal, compare numerators

• Equivalence: Multiply both numerator and denominator by the same number

2 Ordering fractions
Exercise 2
Order these fractions from least to greatest:
\(\frac{1}{2}\), \(\frac{2}{3}\), \(\frac{3}{4}\), \(\frac{1}{4}\)
Definition:

Ordering fractions: Arranging fractions in ascending or descending order

Fractions to order
\(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{1}{4}\)
Common denominator
12
Order
\(\frac{1}{4} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4}\)
Step 1: Find the least common denominator

Denominators are 2, 3, 4

LCM of 2, 3, 4 is 12

Step 2: Convert each fraction to equivalent form with LCD

\(\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}\)

\(\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12}\)

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

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

Step 3: Compare numerators and order

3 < 6 < 8 < 9

So \(\frac{3}{12} < \frac{6}{12} < \frac{8}{12} < \frac{9}{12}\)

Step 4: Return to original fractions

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

\(\frac{1}{4}\)
\(\frac{1}{2}\)
\(\frac{2}{3}\)
\(\frac{3}{4}\)
0
1
Order: \(\frac{1}{4} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4}\)
Final answer:

The fractions in order from least to greatest are:

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

Applied rules:

• Common denominator: Convert all fractions to same denominator

• Ordering: Arrange by comparing numerators

• Number line: Fractions can be visualized on a number line

3 Fraction location
Exercise 3
Where would \(\frac{5}{6}\) be located on a number line between 0 and 1? How does it compare to \(\frac{1}{2}\)?
Definition:

Number line location: Positioning fractions between whole numbers

Reference point: Using benchmarks like \(\frac{1}{2}\) to estimate size

Fraction to locate
\(\frac{5}{6}\)
Location
Between \(\frac{4}{6}\) and 1
Comparison to 1/2
\(\frac{5}{6} > \frac{1}{2}\)
Step 1: Understand the fraction

\(\frac{5}{6}\) means 5 parts out of 6 equal parts

Step 2: Locate on number line

Divide the segment from 0 to 1 into 6 equal parts

\(\frac{5}{6}\) is the fifth mark from 0

Step 3: Compare to \(\frac{1}{2}\)

\(\frac{1}{2} = \frac{3}{6}\) (common denominator)

Since 5 > 3, \(\frac{5}{6} > \frac{3}{6}\)

Therefore, \(\frac{5}{6} > \frac{1}{2}\)

Step 4: Describe the location

\(\frac{5}{6}\) is located very close to 1, specifically 5/6 of the way from 0 to 1

It is greater than \(\frac{1}{2}\) by \(\frac{2}{6}\) or \(\frac{1}{3}\)

\(\frac{1}{2}\)
\(\frac{5}{6}\)
0
1
5
5
5
5
5
5
\(\frac{5}{6}\) is located at 5/6 of the distance from 0 to 1
\(\frac{5}{6} > \frac{1}{2}\)
Final answer:

\(\frac{5}{6}\) is located at the fifth mark when the distance from 0 to 1 is divided into 6 equal parts.

\(\frac{5}{6}\) is greater than \(\frac{1}{2}\) because \(\frac{5}{6} = \frac{5}{6}\) and \(\frac{1}{2} = \frac{3}{6}\), so 5 > 3.

Applied rules:

• Number line: Divide into equal parts based on denominator

• Position: Numerator indicates how many parts from 0

• Comparison: Use common denominators to compare

Fraction Size and Location Fundamentals
\(\frac{a}{b} \text{ where } a \text{ is the numerator and } b \text{ is the denominator}\)
Fraction Structure
Key definitions:

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

Numerator: The top number representing how many parts are taken

Denominator: The bottom number representing total number of equal parts

Proper fraction: Numerator is less than denominator

Improper fraction: Numerator is greater than or equal to denominator

Equivalent fractions: Fractions that represent the same amount

Number line: A visual representation showing fractions between whole numbers

Reference fractions: Benchmark fractions like \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{3}{4}\) for comparison

Fraction comparison strategies:
  1. Common denominators: Convert to same denominator, compare numerators
  2. Common numerators: Convert to same numerator, compare denominators
  3. Decimal conversion: Divide numerator by denominator
  4. Visual models: Draw fraction bars or circles
  5. Reference points: Compare to benchmark fractions like \(\frac{1}{2}\)
Tip 1: When denominators are the same, compare numerators directly.
Tip 2: When numerators are the same, the larger denominator means a smaller fraction.
Tip 3: Use \(\frac{1}{2}\) as a reference point: fractions > \(\frac{1}{2}\) are larger than half.
Tip 4: Draw pictures to visualize fraction comparisons when in doubt.
Common errors: Comparing numerators without considering denominators, confusing size relationships.
Key insight: Larger denominators mean smaller parts, assuming same numerator.
Fraction comparison rules:

• Same denominator: \(\frac{a}{c} > \frac{b}{c}\) if \(a > b\)

• Same numerator: \(\frac{a}{b} > \frac{a}{c}\) if \(b < c\)

• Common denominator: Convert fractions to have same denominator, then compare numerators

• Decimal method: Convert to decimals and compare decimal values

Benchmark Fractions

Reference Points

Key benchmark fractions for comparison:

\(\frac{1}{4} = 0.25\) (one quarter)

\(\frac{1}{2} = 0.5\) (half)

\(\frac{3}{4} = 0.75\) (three quarters)

\(\frac{1}{3} ≈ 0.33\) (one third)

\(\frac{2}{3} ≈ 0.67\) (two thirds)

Number Line Positions

0 ———— \(\frac{1}{4}\) ———— \(\frac{1}{2}\) ———— \(\frac{3}{4}\) ———— 1

Use these positions to estimate fraction locations

Fractions closer to 1 are larger

Fractions closer to 0 are smaller

Solution: Exercises 4 to 5
4 Equivalent fractions
Exercise 4
Find three fractions equivalent to \(\frac{2}{3}\). How do these fractions compare in size?
Definition:

Equivalent fractions: Fractions that represent the same value but have different numerators and denominators

Original fraction
\(\frac{2}{3}\)
Equivalent fractions
\(\frac{4}{6}, \frac{6}{9}, \frac{8}{12}\)
Size comparison
All equal
Step 1: Multiply numerator and denominator by the same number

\(\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}\)

Step 2: Multiply by another number

\(\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}\)

Step 3: Multiply by yet another number

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

Step 4: Verify equivalence

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

\(\frac{6}{9} = \frac{6 ÷ 3}{9 ÷ 3} = \frac{2}{3}\) ✓

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

\(\frac{2}{3}\)
\(\frac{4}{6}\)
\(\frac{6}{9}\)
\(\frac{8}{12}\)
2
2
2
4
4
4
4
Equivalent fractions: \(\frac{4}{6}, \frac{6}{9}, \frac{8}{12}\)
All are equal in size to \(\frac{2}{3}\)
Final answer:

Three fractions equivalent to \(\frac{2}{3}\) are: \(\frac{4}{6}, \frac{6}{9}, \frac{8}{12}\)

All of these fractions represent the same amount and have the same size.

Applied rules:

• Equivalence: Multiply numerator and denominator by the same number

• Size preservation: Equivalent fractions have the same value

• Verification: Divide numerator and denominator by their GCD to check

5 Real-world application
Exercise 5
Sarah ate \(\frac{3}{4}\) of her sandwich and John ate \(\frac{5}{6}\) of his sandwich. Who ate more? Explain your reasoning using multiple methods.
Definition:

Real-world application: Using fraction comparison in practical situations

Fractions to compare
\(\frac{3}{4}\) and \(\frac{5}{6}\)
Common denominator
12
Result
\(\frac{9}{12} < \frac{10}{12}\)
Method 1: Common denominator approach

LCM of 4 and 6 is 12

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

\(\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}\)

Since 9 < 10, \(\frac{3}{4} < \frac{5}{6}\)

Method 2: Decimal conversion

\(\frac{3}{4} = 3 ÷ 4 = 0.75\)

\(\frac{5}{6} = 5 ÷ 6 ≈ 0.83\)

Since 0.75 < 0.83, \(\frac{3}{4} < \frac{5}{6}\)

Method 3: Comparison to \(\frac{1}{2}\)

\(\frac{3}{4} > \frac{1}{2}\) and \(\frac{5}{6} > \frac{1}{2}\) (both are more than half)

\(\frac{3}{4} = \frac{1}{4}\) more than \(\frac{1}{2}\)

\(\frac{5}{6} = \frac{1}{6}\) more than \(\frac{1}{2}\)

Since \(\frac{1}{4} > \frac{1}{6}\), \(\frac{3}{4}\) is further from \(\frac{1}{2}\) than \(\frac{5}{6}\) is

Wait, let's recalculate: \(\frac{1}{2} = \frac{3}{6}\), \(\frac{5}{6} - \frac{3}{6} = \frac{2}{6} = \frac{1}{3}\)

\(\frac{1}{2} = \frac{2}{4}\), \(\frac{3}{4} - \frac{2}{4} = \frac{1}{4}\)

Since \(\frac{1}{3} > \frac{1}{4}\), \(\frac{5}{6}\) is larger

3
4
3
3
3
3
<
5
6
5
5
5
5
5
5
John ate more: \(\frac{5}{6} > \frac{3}{4}\)
Final answer:

John ate more of his sandwich.

\(\frac{5}{6} > \frac{3}{4}\) as shown by multiple methods:

• Common denominator: \(\frac{9}{12} < \frac{10}{12}\)

• Decimal conversion: 0.75 < 0.83

• Comparison to \(\frac{1}{2}\): \(\frac{5}{6}\) is further from \(\frac{1}{2}\) than \(\frac{3}{4}\)

Applied rules:

• Multiple methods: Use different approaches to verify answers

• Real-world context: Apply fraction comparison to practical situations

• Verification: Check results using alternative methods

Advanced Fraction Concepts
\(\frac{a}{b} = \frac{a \times n}{b \times n} \text{ for any non-zero } n\)
Equivalent Fractions Formula
Key definitions:

Fraction magnitude: The actual size or value of a fraction

Unit fraction: A fraction with numerator 1

Like fractions: Fractions with the same denominator

Unlike fractions: Fractions with different denominators

Simplest form: A fraction reduced to lowest terms

Common denominator: A shared denominator for multiple fractions

Least common denominator: The smallest common denominator

Fraction families: Sets of equivalent fractions

Advanced comparison methods:
  1. Cross multiplication: For \(\frac{a}{b}\) and \(\frac{c}{d}\), compare \(a \times d\) with \(b \times c\)
  2. Percentage conversion: Convert fractions to percentages
  3. Distance from whole: Compare how far each fraction is from 1
  4. Midpoint comparison: Compare to midpoint between two reference points
Tip 1: The larger the denominator, the smaller each part becomes (with same numerator).
Tip 2: Fractions closer to 1 are larger than those farther from 1.
Tip 3: Use cross multiplication to quickly compare two fractions.
Tip 4: Practice with visual models to build intuitive understanding.
Advanced insight: For fractions \(\frac{a}{b}\) and \(\frac{a+k}{b+k}\) where k > 0, the second fraction is larger if a < b.
Mathematical connection: Fraction comparison connects to rational number concepts.
Advanced relationships:

• Cross multiplication: If \(ad > bc\), then \(\frac{a}{b} > \frac{c}{d}\)

• Reciprocal relationship: If \(\frac{a}{b} > \frac{c}{d}\), then \(\frac{b}{a} < \frac{d}{c}\)

• Transitive property: If \(\frac{a}{b} > \frac{c}{d}\) and \(\frac{c}{d} > \frac{e}{f}\), then \(\frac{a}{b} > \frac{e}{f}\)

• Additive comparison: The difference between fractions can be calculated

Fraction Families

Equivalent Fraction Families

\(\frac{1}{2}\) family: \(\frac{1}{2}, \frac{2}{4}, \frac{3}{6}, \frac{4}{8}, \frac{5}{10}, \frac{6}{12}\)

\(\frac{1}{3}\) family: \(\frac{1}{3}, \frac{2}{6}, \frac{3}{9}, \frac{4}{12}\)

\(\frac{1}{4}\) family: \(\frac{1}{4}, \frac{2}{8}, \frac{3}{12}\)

All fractions in each family represent the same value

Denominator Patterns

Larger denominators create smaller individual parts

\(\frac{1}{12} < \frac{1}{8} < \frac{1}{6} < \frac{1}{4} < \frac{1}{3} < \frac{1}{2}\)

For the same numerator, larger denominators mean smaller fractions

Questions & Answers

Question: I'm confused about why \(\frac{2}{3}\) is larger than \(\frac{1}{2}\). Doesn't 2 being bigger than 1 make \(\frac{2}{3}\) larger?

Answer: Great question! You're thinking about the numerator correctly, but you also need to consider the denominator. Let's think about it:

When you have \(\frac{2}{3}\), you're taking 2 parts out of 3 equal parts.

When you have \(\frac{1}{2}\), you're taking 1 part out of 2 equal parts.

To compare them, let's convert to a common denominator:

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

Now we can see: \(\frac{4}{6} > \frac{3}{6}\), so \(\frac{2}{3} > \frac{1}{2}\)

Another way to think about it: \(\frac{2}{3}\) means you have 2 parts out of 3, which is more than half. \(\frac{1}{2}\) is exactly half. So \(\frac{2}{3}\) is larger!

Question: My child is struggling to understand fraction location on a number line. How can I help them visualize this better?

Answer: Here are effective visualization strategies:

  • Physical models: Use fraction strips, circles, or bars to show parts
  • Number line practice: Start with simple fractions like \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{3}{4}\)
  • Real objects: Use pizza slices, chocolate bars, or paper folding
  • Interactive tools: Online fraction manipulatives and games
  • Reference points: Always compare to 0, \(\frac{1}{2}\), and 1

Start by having your child practice locating \(\frac{1}{2}\) on a number line, then add other fractions. Ask questions like "Is this fraction closer to 0, \(\frac{1}{2}\), or 1?" This builds intuitive understanding.

Practice with denominators 2, 3, 4, 6, and 8 first, as these are easier to visualize.

Question: How does understanding fraction size and location connect to future mathematical concepts?

Answer: Fraction size and location understanding is foundational to many advanced concepts:

  • Decimal conversions: Understanding fraction-to-decimal relationships
  • Percentages: Converting between fractions, decimals, and percents
  • Algebra: Working with rational expressions and equations
  • Geometry: Understanding ratios and proportions
  • Statistics: Working with probabilities and data
  • Calculus: Limits and continuity concepts

The ability to compare and order rational numbers is essential for understanding the real number system. Students who master fraction comparison develop stronger number sense that supports all future mathematical learning.

Additionally, understanding fraction location on number lines prepares students for coordinate geometry and graphing functions.