Practice Finding Equivalent Fractions - Detailed Exercises for 4th Graders

Master finding equivalent fractions through extensive practice: learn to multiply or divide numerators and denominators by the same number to find equivalent fractions through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding missing numerators
Exercise 1
Find the missing numerator: 3/4 = ?/8
Definition:

Equivalent fractions: Fractions that represent the same amount even though they have different numerators and denominators.

Method for finding equivalent fractions:
  1. Look at the denominators to find the relationship
  2. Determine what number the original denominator was multiplied by
  3. Multiply the numerator by the same number
Original fraction
\( \frac{3}{4} \)
Target denominator
8
Relationship
4×2=8
Missing numerator
3×2=6
Step 1: Identify what we know

We have \( \frac{3}{4} \) and need to find a fraction with denominator 8

Step 2: Find the relationship between denominators

To get from 4 to 8: \( 4 \times 2 = 8 \)

Step 3: Apply the same operation to the numerator

Multiply the numerator by 2: \( 3 \times 2 = 6 \)

Step 4: Write the equivalent fraction

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

Step 5: Verify the answer

Cross multiplication: \( 3 \times 8 = 24 \) and \( 4 \times 6 = 24 \)

Since both products are equal, the fractions are equivalent

3/4
6/8
\( \frac{3}{4} = \frac{6}{8} \)
Final answer:

The missing numerator is 6, so \( \frac{3}{4} = \frac{6}{8} \)

Applied rules:

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

• Relationship identification: Find what number connects the denominators

• Cross multiplication: Verify equivalence by checking if cross products are equal

2 Finding missing denominators
Exercise 2
Find the missing denominator: 2/3 = 8/?
Definition:

Proportional relationship: In equivalent fractions, the relationship between numerators is the same as the relationship between denominators.

Original fraction
\( \frac{2}{3} \)
New numerator
8
Relationship
2×4=8
Missing denominator
3×4=12
Step 1: Identify what we know

We have \( \frac{2}{3} \) and the new numerator is 8

Step 2: Find the relationship between numerators

To get from 2 to 8: \( 2 \times 4 = 8 \)

Step 3: Apply the same operation to the denominator

Multiply the denominator by 4: \( 3 \times 4 = 12 \)

Step 4: Write the equivalent fraction

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

Step 5: Verify the answer

Cross multiplication: \( 2 \times 12 = 24 \) and \( 3 \times 8 = 24 \)

Since both products are equal, the fractions are equivalent

2/3
8/12
\( \frac{2}{3} = \frac{8}{12} \)
Final answer:

The missing denominator is 12, so \( \frac{2}{3} = \frac{8}{12} \)

Applied rules:

• Proportional relationship: The factor that changes the numerator also changes the denominator

• Consistency rule: What you do to the numerator, you do to the denominator

• Verification: Use cross multiplication to confirm equivalence

3 Working backwards with division
Exercise 3
Find the missing numerator: 15/20 = ?/4
Definition:

Division method: When the denominator decreases, divide both numerator and denominator by the same number.

Original fraction
\( \frac{15}{20} \)
Target denominator
4
Relationship
20÷5=4
Missing numerator
15÷5=3
Step 1: Identify what we know

We have \( \frac{15}{20} \) and need to find a fraction with denominator 4

Step 2: Find the relationship between denominators

To get from 20 to 4: \( 20 \div 5 = 4 \)

Step 3: Apply the same operation to the numerator

Divide the numerator by 5: \( 15 \div 5 = 3 \)

Step 4: Write the equivalent fraction

\( \frac{15}{20} = \frac{3}{4} \)

Step 5: Verify the answer

Cross multiplication: \( 15 \times 4 = 60 \) and \( 20 \times 3 = 60 \)

Since both products are equal, the fractions are equivalent

15 ÷ 5 = 3
÷ 5
? = 3
———
———
———
20 ÷ 5 = 4
÷ 5
4
\( \frac{15}{20} = \frac{3}{4} \)
Final answer:

The missing numerator is 3, so \( \frac{15}{20} = \frac{3}{4} \)

Applied rules:

• Division method: Divide both numerator and denominator by the same number

• Consistent operation: What you do to the denominator, you do to the numerator

• Simplification: This process reduces fractions to simpler forms

Finding Equivalent Fractions: Complete Guide
\( \frac{a}{b} = \frac{a \times n}{b \times n} = \frac{a ÷ m}{b ÷ m} \)
Equivalent Fractions Formula
Key definitions:

Equivalent fractions: Fractions that represent the same value despite having different numerators and denominators

Proportional relationship: The relationship between numerators is the same as between denominators

Multiplication method: Multiply both parts by the same number to create equivalent fractions

Division method: Divide both parts by the same number to simplify or find equivalent forms

Complete methodology for finding equivalent fractions:
  1. Identify known values: Determine what information you have
  2. Find the relationship: Determine how one numerator or denominator relates to another
  3. Apply the operation: Perform the same operation on the unknown part
  4. Verify: Use cross multiplication to confirm equivalence
Tip 1: Look at denominators first to find the relationship.
Tip 2: Whatever you do to the bottom, do to the top (and vice versa).
Tip 3: Use cross multiplication to check your answers.
Tip 4: When denominators increase, you're multiplying; when they decrease, you're dividing.
Tip 5: Always keep the operation consistent between numerator and denominator.
Common errors: Forgetting to apply the same operation to both parts, misidentifying the relationship, not verifying the answer.
Memory aid: "Same operation, same result" - the fraction value stays the same.
Essential rules to remember:

• Multiplication rule: \( \frac{a}{b} = \frac{a \times n}{b \times n} \) where n ≠ 0

• Division rule: \( \frac{a}{b} = \frac{a ÷ n}{b ÷ n} \) where n ≠ 0

• Cross multiplication: \( \frac{a}{b} = \frac{c}{d} \) if \( a \times d = b \times c \)

• Consistency: Operations must be applied equally to numerator and denominator

\( \frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc \)
Cross Product Property
Solution: Exercises 4 to 5
4 Complex missing value
Exercise 4
Find the missing values: 5/? = ?/18, given that the fractions are equivalent to 1/3.
Definition:

Multi-step equivalence: Finding multiple missing values by establishing relationships through a known equivalent fraction.

Known relationship
Both = 1/3
First equation
5/? = 1/3
Second equation
?/18 = 1/3
Step 1: Set up the first relationship

\( \frac{5}{?} = \frac{1}{3} \)

Step 2: Use cross multiplication for first equation

\( 5 \times 3 = ? \times 1 \)

\( 15 = ? \)

So the first missing value is 15: \( \frac{5}{15} = \frac{1}{3} \)

Step 3: Set up the second relationship

\( \frac{?}{18} = \frac{1}{3} \)

Step 4: Use cross multiplication for second equation

\( ? \times 3 = 18 \times 1 \)

\( ? \times 3 = 18 \)

\( ? = 6 \)

So the second missing value is 6: \( \frac{6}{18} = \frac{1}{3} \)

Step 5: Verify the complete relationship

\( \frac{5}{15} = \frac{6}{18} = \frac{1}{3} \)

Cross check: \( 5 \times 18 = 90 \) and \( 15 \times 6 = 90 \) ✓

5
=
6
———
———
———
15
=
18
\( \frac{5}{15} = \frac{6}{18} = \frac{1}{3} \)
Final answer:

The missing values are 15 and 6, so \( \frac{5}{15} = \frac{6}{18} \), both equivalent to \( \frac{1}{3} \).

Applied rules:

• Cross multiplication: Use to solve for unknown values in equivalent fractions

• Transitive property: If A = C and B = C, then A = B

• Step-by-step approach: Solve one unknown at a time

5 Real-world application
Exercise 5
A recipe calls for 2/3 cup of sugar. If you want to make 4 times the recipe, how many cups of sugar do you need? Express as an equivalent fraction.
Definition:

Scaling fractions: Multiplying both numerator and denominator by the same number to scale quantities.

Original amount
\( \frac{2}{3} \)
Scale factor
4
Scaled amount
\( \frac{2×4}{3×4} \)
Result
\( \frac{8}{12} \)
Step 1: Identify the original fraction

Recipe calls for \( \frac{2}{3} \) cup of sugar

Step 2: Determine the scaling factor

You want to make 4 times the recipe, so multiply by 4

Step 3: Apply the scaling to both parts

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

Step 4: Interpret the result

\( \frac{8}{12} \) cup of sugar is equivalent to \( \frac{2}{3} \times 4 = \frac{8}{3} \) cups

Step 5: Verify the equivalence

Cross multiplication: \( 2 \times 12 = 24 \) and \( 3 \times 8 = 24 \) ✓

Step 6: Convert to mixed number (optional)

\( \frac{8}{3} = 2\frac{2}{3} \) cups of sugar needed

Original: 2/3
Scaled: 8/3
\( \frac{2}{3} \times 4 = \frac{8}{3} \)
Final answer:

You need \( \frac{8}{3} \) cups of sugar, which is equivalent to \( 2\frac{2}{3} \) cups.

Applied rules:

• Scaling principle: Multiply both numerator and denominator by the same number

• Real-world application: Equivalent fractions help with proportional adjustments

• Conversion: Improper fractions can be converted to mixed numbers for practical use

Complete Summary: Practice Finding Equivalent Fractions
\( \frac{a}{b} = \frac{a \times n}{b \times n} = \frac{a ÷ m}{b ÷ m} \)
Equivalent Fractions Formula
Key definitions:

Equivalent fractions: Fractions that represent the same value or portion of a whole, even though they have different numerators and denominators.

Proportional relationship: The relationship between numerators is the same as the relationship between denominators in equivalent fractions.

Multiplication method: Multiply both numerator and denominator by the same non-zero number to create equivalent fractions.

Division method: Divide both numerator and denominator by the same non-zero number to create equivalent fractions or simplify.

Cross multiplication: A method to verify that two fractions are equivalent by multiplying diagonally.

Step-by-step methods for finding equivalent fractions:
  1. Identify known values: Determine which parts of the equivalent fractions you know
  2. Find the relationship: Determine how the known parts relate to each other
  3. Apply the operation: Perform the same operation on the unknown part
  4. Calculate the result: Compute the missing value
  5. Verify: Use cross multiplication to confirm the fractions are equivalent
Pattern recognition: Look for multiplication or division patterns between denominators first.
Consistency rule: Whatever operation you perform on the denominator, perform the same on the numerator.
Verification: Always cross multiply to check your answer.
Direction matters: If denominators increase, you're multiplying; if they decrease, you're dividing.
Multiple approaches: You can solve using multiplication, division, or cross multiplication.
Common errors: Forgetting to apply the same operation to both parts, misidentifying the relationship, not checking the answer.
Key insight: The value of the fraction never changes, only its representation.
Practical tip: Use equivalent fractions for scaling recipes, measurements, and proportional problems.
Core rules and principles:

• Conservation principle: The value of the fraction remains unchanged during equivalent fraction creation

• Proportionality rule: The relationship between numerators equals the relationship between denominators

• Equality test: Two fractions are equivalent if their cross products are equal

• Operation consistency: Apply the same operation to both numerator and denominator

• Verification method: Cross multiplication confirms equivalence

\( \frac{a}{b} = \frac{c}{d} \iff ad = bc \)
Cross Product Property
Advanced problem-solving strategies:
  1. For missing numerators: Find the denominator relationship first, then apply to numerator
  2. For missing denominators: Find the numerator relationship first, then apply to denominator
  3. For complex problems: Break into smaller steps and solve one unknown at a time
  4. For verification: Always use cross multiplication to confirm your answer

Questions & Answers

Question: How do I know whether to multiply or divide when finding equivalent fractions?

Answer: Great question! Look at the denominators to decide:

  • Multiply when denominators get bigger: If you go from 4 to 8, you multiply by 2
  • Divide when denominators get smaller: If you go from 12 to 3, you divide by 4
  • Always do the same to the numerator: What you do to the bottom, do to the top!

Example: 3/4 = ?/12

Going from 4 to 12: 4 × 3 = 12, so multiply by 3

Then: 3 × 3 = 9, so 3/4 = 9/12

The key is to figure out the relationship between the denominators first!

Question: My child sometimes gets confused about which number to use when finding the relationship. Any tips?

Answer: Here are some helpful strategies:

  1. Start with the known values: Focus on the denominator you know both values for
  2. Think of multiplication tables: "What times 4 equals 12?" (4 × 3 = 12)
  3. Use division if needed: "What divided by 4 equals 3?" (12 ÷ 4 = 3)
  4. Check your work: Multiply back to see if you get the target number

For 2/5 = ?/15:

Think: "5 times what equals 15?" Answer: 5 × 3 = 15, so multiply by 3

Then: 2 × 3 = 6, so 2/5 = 6/15

Practice with smaller numbers first, then work up to larger ones!

Question: Can I use cross multiplication to find missing values in equivalent fractions?

Answer: Absolutely! Cross multiplication is a very effective method:

For \( \frac{3}{4} = \frac{x}{8} \):

Cross multiply: 3 × 8 = 4 × x

24 = 4x

24 ÷ 4 = x

x = 6

So \( \frac{3}{4} = \frac{6}{8} \)

Cross multiplication works because for equivalent fractions \( \frac{a}{b} = \frac{c}{d} \), we know that \( a \times d = b \times c \).

This method is especially helpful when the relationships aren't immediately obvious!