Solved Exercises on Comparing Each Fraction to the Benchmark in Grade 4

Master comparing fractions to benchmarks: fraction comparison, estimation, and relative position on number line through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Comparing to Benchmark 1/2
Exercise 1
Compare each fraction to the benchmark 1/2:
3/8, 5/8, 1/4
Definition:

Comparing to benchmark 1/2: Determining whether a fraction is less than, greater than, or equal to 1/2

Comparison method:
  1. Convert fractions to equivalent fractions with the same denominator as 1/2
  2. Compare numerators to 1/2's numerator
  3. Or use cross multiplication to compare directly
Fractions
3/8, 5/8, 1/4
Compare to 1/2
<, >, <
Results
3/8 < 1/2, 5/8 > 1/2, 1/4 < 1/2
Step 1: Compare 3/8 to 1/2

Convert 1/2 to eighths: 1/2 = 4/8

Since 3 < 4, then 3/8 < 1/2

Step 2: Compare 5/8 to 1/2

Convert 1/2 to eighths: 1/2 = 4/8

Since 5 > 4, then 5/8 > 1/2

Step 3: Compare 1/4 to 1/2

Convert 1/2 to fourths: 1/2 = 2/4

Since 1 < 2, then 1/4 < 1/2

Step 4: Verify using cross multiplication

For 3/8 vs 1/2: 3×2 = 6 and 1×8 = 8, since 6 < 8, then 3/8 < 1/2 ✓

3/8 < 1/2
5/8 > 1/2
1/4 < 1/2
Final answer:

3/8 is less than 1/2, 5/8 is greater than 1/2, 1/4 is less than 1/2

Applied rules:

• Equivalent fractions: Convert to same denominator to compare

• Cross multiplication: For comparing a/b vs c/d, compare a×d vs b×c

• Benchmark reasoning: Use 1/2 as reference point

2 Estimating with Benchmarks 0, 1/2, 1
Exercise 2
Estimate each fraction using benchmarks 0, 1/2, or 1:
7/12, 1/8, 9/10
Definition:

Estimating with benchmarks: Using 0, 1/2, and 1 as reference points to approximate fraction values

Fractions
7/12, 1/8, 9/10
Benchmarks
≈1/2, ≈0, ≈1
Estimates
1/2, 0, 1
Step 1: Estimate 7/12

Half of 12 is 6, and 7 is close to 6

So 7/12 ≈ 1/2 (slightly more than 1/2)

Step 2: Estimate 1/8

1/8 is much smaller than 4/8 (which is 1/2)

It's closer to 0 than to 1/2

So 1/8 ≈ 0

Step 3: Estimate 9/10

9/10 is very close to 10/10 (which is 1)

It's much larger than 5/10 (which is 1/2)

So 9/10 ≈ 1

Step 4: Verify estimates

7/12 ≈ 0.58 (close to 0.5 = 1/2) ✓

1/8 = 0.125 (close to 0) ✓

9/10 = 0.9 (close to 1) ✓

7/12 ≈ 1/2
1/8 ≈ 0
9/10 ≈ 1
Final answer:

7/12 is approximately 1/2, 1/8 is approximately 0, 9/10 is approximately 1

Applied rules:

• Benchmark estimation: Use 0, 1/2, 1 as reference points

• Relative position: Compare to nearest benchmark

• Decimal conversion: Convert to decimals to verify estimates

3 Ordering with Benchmarks
Exercise 3
Order from least to greatest using benchmarks:
2/3, 1/4, 3/4, 1/10
Definition:

Ordering with benchmarks: Using 0, 1/2, 1 to help order fractions from least to greatest

Fractions
1/10, 1/4, 2/3, 3/4
Benchmarks
≈0, ≈0, ≈1/2, ≈1
Order
Least to Greatest
Step 1: Estimate each fraction

1/10 ≈ 0 (very small)

1/4 ≈ 0 (but larger than 1/10)

2/3 ≈ 1/2 (actually slightly more than 1/2)

3/4 ≈ 1 (but less than 1)

Step 2: Group by benchmarks

Near 0: 1/10, 1/4

Near 1/2: 2/3

Near 1: 3/4

Step 3: Order within each group

Between 1/10 and 1/4: 1/10 < 1/4

Between 2/3 and 3/4: 2/3 < 3/4

Step 4: Combine groups

From least to greatest: 1/10, 1/4, 2/3, 3/4

1/10 < 1/4 < 2/3 < 3/4
Final answer:

The order from least to greatest is: 1/10, 1/4, 2/3, 3/4

Applied rules:

• Benchmark grouping: Group fractions by their benchmark

• Within-group ordering: Order fractions in each group

• Overall ordering: Combine groups in order

Rules and methods, laws,...
\(a/b \text{ compared to } c/d: \text{ if } a \times d > b \times c, \text{ then } a/b > c/d\)
Cross Multiplication Rule
Benchmark 0
Very Small Fractions
1/10, 1/8, 1/5
Benchmark 1/2
Fractions Near Middle
4/8, 3/6, 5/10
Benchmark 1
Fractions Near Whole
9/10, 7/8, 11/12
Key definitions:

Benchmark fractions: Reference fractions (0, 1/2, 1) used to estimate and compare other fractions

Equivalent fractions: Fractions that represent the same value

Comparison: Determining if one fraction is greater than, less than, or equal to another

Complete methodology:
  1. Identify the benchmark: Choose 0, 1/2, or 1 as reference
  2. Compare the fraction: Determine if it's closer to the benchmark
  3. Estimate the value: Use the benchmark to approximate
  4. Verify: Check your estimate with calculations
Tip 1: Remember that 1/2 means the numerator is half the denominator.
Tip 2: Use cross multiplication to compare two fractions quickly.
Tip 3: Group fractions by benchmarks when ordering multiple fractions.
Tip 4: Convert to equivalent fractions with common denominators for precise comparison.

Common errors: Misidentifying benchmarks, incorrect cross multiplication, not converting to same denominator.
Estimation strategies: Use benchmarks for quick comparisons, verify with calculations.
Formulas to know by heart:

• Cross multiplication: For a/b vs c/d, compare a×d vs b×c

• Equivalent fractions: a/b = (a×n)/(b×n) for any non-zero n

• Benchmark identification: Use 0, 1/2, 1 as reference points

• Ordering: Group by benchmarks, then order within groups

Solution: Exercises 4 to 5
4 Real-World Application
Exercise 4
Sarah ate 3/5 of her pizza, John ate 2/3 of his pizza, and Emma ate 1/8 of hers. Who ate the most pizza? Use benchmarks to estimate first.
Definition:

Real-world application: Using benchmark fractions to solve practical problems involving comparisons

Fractions
3/5, 2/3, 1/8
Benchmarks
≈1/2, ≈1/2, ≈0
Order
Emma < Sarah < John
Step 1: Estimate each fraction using benchmarks

3/5: Since 3 is more than half of 5, 3/5 > 1/2

2/3: Since 2 is more than half of 3, 2/3 > 1/2

1/8: Since 1 is much less than half of 8, 1/8 < 1/2

Step 2: Compare 3/5 and 2/3 precisely

Cross multiply: 3×3 = 9 and 2×5 = 10

Since 9 < 10, then 3/5 < 2/3

Step 3: Order all fractions

1/8 < 3/5 < 2/3

Step 4: Answer the question

John ate the most pizza (2/3)

Emma (1/8) < Sarah (3/5) < John (2/3)
Final answer:

John ate the most pizza (2/3), followed by Sarah (3/5), then Emma (1/8)

Applied rules:

• Real-world context: Apply mathematical concepts to practical situations

• Benchmark estimation: Use 0, 1/2, 1 to quickly assess values

• Precise comparison: Use cross multiplication for exact ordering

5 Advanced Comparison
Exercise 5
Which fraction is closest to 1/2: 7/12, 5/12, or 9/16? Show your work using benchmarks and precise calculations.
Definition:

Advanced comparison: Determining which of several fractions is closest to a benchmark using both estimation and precise calculation

Fractions
7/12, 5/12, 9/16
Compare to 1/2
+, -, +
Closest
7/12
Step 1: Estimate each fraction

7/12: 6/12 = 1/2, so 7/12 is slightly more than 1/2

5/12: 6/12 = 1/2, so 5/12 is slightly less than 1/2

9/16: 8/16 = 1/2, so 9/16 is slightly more than 1/2

Step 2: Calculate exact distances from 1/2

7/12 - 1/2 = 7/12 - 6/12 = 1/12

1/2 - 5/12 = 6/12 - 5/12 = 1/12

9/16 - 1/2 = 9/16 - 8/16 = 1/16

Step 3: Compare the distances

1/12 vs 1/12 vs 1/16

Since 1/16 < 1/12, then 9/16 is closest to 1/2

Step 4: Verify the result

9/16 = 0.5625, which is very close to 0.5 (1/2)

7/12 ≈ 0.583, which is 0.083 away from 0.5

5/12 ≈ 0.417, which is 0.083 away from 0.5

9/16 is only 0.0625 away from 0.5

9/16 is closest to 1/2
Final answer:

9/16 is closest to 1/2 (distance of 1/16)

Applied rules:

• Distance calculation: Find exact difference from benchmark

• Comparison of differences: Smallest difference means closest

• Verification: Convert to decimals to confirm result

Key Concepts: Benchmark Fractions 0, 1/2, 1
\(a \div b = c \text{ means } b \times c = a\)
Division Relationship
Key definitions:

Division: The operation of splitting a number into equal parts

Quotient: The result of a division problem

Dividend: The number being divided

Divisor: The number that divides the dividend

Fact family: A set of related multiplication and division facts

Complete methodology:
  1. Identify the dividend and divisor
  2. Think of the related multiplication fact
  3. Use the known multiple to find the quotient
  4. Verify by multiplying the quotient by the divisor
Tip 1: Always start by thinking of the related multiplication fact.
Tip 2: Use fact families to connect multiplication and division.
Tip 3: Practice your multiplication tables to improve division speed.
Tip 4: Always verify your division by multiplying back.

Common errors: Confusing dividend and divisor, forgetting to verify answers, not knowing multiplication facts.
Mental math strategies: Use known facts, break apart numbers, recognize patterns.
Formulas to know by heart:

• Basic division facts: Know all facts up to 12 ÷ 12

• Fact families: If a × b = c, then c ÷ a = b and c ÷ b = a

• Verification: Quotient × Divisor = Dividend

• Division and multiplication are inverse operations

Questions & Answers

Question: I don't understand how using multiplication helps me divide faster. Can you explain?

Answer: Using multiplication makes division faster because division and multiplication are inverse operations! Here's how:

  • Instead of dividing 28 ÷ 4 directly, think: "What number times 4 equals 28?"
  • You know 7 × 4 = 28, so 28 ÷ 4 = 7
  • This uses your knowledge of multiplication facts to solve division problems

The more multiplication facts you know, the faster you can solve division!

Question: What is a fact family and how does it help with division?

Answer: A fact family is a group of related multiplication and division facts using the same numbers!

For example, with 4, 7, and 28:

  • 4 × 7 = 28
  • 7 × 4 = 28
  • 28 ÷ 4 = 7
  • 28 ÷ 7 = 4

If you know the multiplication facts, you automatically know the division facts!

Question: What if I forget my multiplication facts? How can I still use multiplication to divide?

Answer: If you forget multiplication facts, you can still use multiplication to divide by counting up!

For example, for 42 ÷ 6:

  • Count by 6s: 6, 12, 18, 24, 30, 36, 42
  • How many 6s did you count? 7
  • So 42 ÷ 6 = 7

This takes longer but still uses the concept of multiplication!

Complete Summary: Using Multiplication to Divide
Key Definitions:

Division: The operation of splitting a number into equal parts. It's the opposite of multiplication.

Quotient: The result you get when you divide one number by another.

Dividend: The number you're dividing up (the big number).

Divisor: The number you're dividing by (the number outside).

Fact Family: A group of related multiplication and division facts using the same numbers.

Core Rules and Principles:

• Division and multiplication are inverse operations: If a × b = c, then c ÷ a = b

• Fact families connect multiplication and division: Same numbers create related facts.

• Use known multiples: Think of multiplication facts to solve division problems.

• Verification: Always check your division by multiplying the quotient by the divisor.

Step-by-Step Method:
  1. Identify the dividend and divisor: Find the number being divided and the number dividing it.
  2. Think of the related multiplication fact: What number times the divisor equals the dividend?
  3. Use the known multiple: Apply the multiplication fact to find the quotient.
  4. Verify: Multiply the quotient by the divisor to check your answer.
Simple Examples

Using multiples: 24 ÷ 3 = 8 because 3 × 8 = 24

Fact family: 5 × 6 = 30, so 30 ÷ 5 = 6 and 30 ÷ 6 = 5

Mental math: 35 ÷ 7 = 5 because you know 7 × 5 = 35

Advanced Examples

Multiples of 10: 80 ÷ 8 = 10 (because 8 × 10 = 80)

Pattern recognition: 48 ÷ 6 = 8, 56 ÷ 7 = 8, 64 ÷ 8 = 8

Multiple steps: 72 ÷ 6 = (60 ÷ 6) + (12 ÷ 6) = 10 + 2 = 12

Tips: Always start by thinking of the related multiplication fact. Practice your times tables regularly!
Common Pitfalls: Don't confuse the dividend and divisor. Always verify your answer by multiplying back.
Memorization Tips: Practice fact families together. If you know 8 × 9 = 72, you also know 72 ÷ 8 = 9.
Key Takeaways:

• Using multiplication makes division faster and more accurate.

• Division and multiplication are inverse operations.

• Fact families connect multiplication and division facts.

• Always verify your division by multiplying back.

• Practice multiplication tables to improve division speed.