Scaling by Fractions | Fifth Grade

Complete Notes & Formulas

1. Scaling Whole Numbers by Fractions: Justify Your Answer

Definition: Understand and explain whether multiplying a whole number by a fraction will make it larger, smaller, or stay the same WITHOUT calculating the actual answer.

🔑 The Golden Rules of Scaling:

Rule 1: Fraction < 1

When you multiply by a fraction LESS THAN 1:

→ The product will be SMALLER than the original number (scaled down)

Rule 2: Fraction = 1

When you multiply by a fraction EQUAL TO 1:

→ The product will be EQUAL to the original number (stays the same)

Rule 3: Fraction > 1

When you multiply by a fraction GREATER THAN 1:

→ The product will be LARGER than the original number (scaled up)

✏️ Example 1: Is 8 × 3/4 greater than, less than, or equal to 8?

Step 1: Compare 3/4 to 1 whole

3/4 < 1 (numerator is less than denominator)

Step 2: Apply the rule

Since 3/4 < 1, the product will be smaller than 8

Answer: 8 × 3/4 < 8 (less than 8)

Justification: Multiplying by 3/4 means taking "three-fourths of 8," which must be less than all of 8.

✏️ Example 2: Is 5 × 7/3 greater than, less than, or equal to 5?

Compare: 7/3 > 1 (numerator is greater than denominator)

Since 7/3 > 1, the product will be larger than 5

Answer: 5 × 7/3 > 5 (greater than 5)

Justification: 7/3 equals 2 1/3, which is more than 1 whole, so it will scale up the value.

2. Scaling Whole Numbers by Fractions

Definition: Calculate the actual product when multiplying a whole number by a fraction to scale it up or down.

📐 Formula:

Whole Number × a/b = (Whole Number × a)/b

Multiply the whole number by the numerator, then divide by the denominator

📝 Steps:

Write the whole number as a fraction (n/1)
Multiply numerators: n × a
Multiply denominators: 1 × b = b
Simplify the resulting fraction

✏️ Example 1: Calculate 12 × 2/3

Method 1: Convert whole to fraction

12/1 × 2/3 = (12 × 2)/(1 × 3) = 24/3 = 8

Method 2: Direct multiplication

(12 × 2)/3 = 24/3 = 8

Answer: 8

Note: 8 < 12 because we multiplied by 2/3, which is less than 1

✏️ Example 2: Calculate 6 × 5/4

6 × 5/4 = (6 × 5)/4 = 30/4 = 15/2 = 7 1/2

Answer: 7 1/2

Note: 7 1/2 > 6 because we multiplied by 5/4, which is greater than 1

3. Scaling Fractions by Fractions

Definition: Multiply one fraction by another fraction to scale it. The same scaling rules apply: compare the scaling fraction to 1.

📐 Formula:

a/b × c/d = (a × c)/(b × d)

🔑 Scaling Rules for Fractions:

If scaling fraction < 1 → product is smaller than original fraction
If scaling fraction = 1 → product equals original fraction
If scaling fraction > 1 → product is larger than original fraction

✏️ Example 1: 3/4 × 2/5 (Scaling Down)

Predict: Since 2/5 < 1, the product will be less than 3/4

Calculate:

(3 × 2)/(4 × 5) = 6/20 = 3/10

Answer: 3/10

Verify: 3/10 (0.3) < 3/4 (0.75) ✓

✏️ Example 2: 2/3 × 5/4 (Scaling Up)

Predict: Since 5/4 > 1, the product will be greater than 2/3

Calculate:

(2 × 5)/(3 × 4) = 10/12 = 5/6

Answer: 5/6

Verify: 5/6 (0.83) > 2/3 (0.67) ✓

4. Scaling Mixed Numbers by Fractions

Definition: Multiply a mixed number by a fraction. Convert the mixed number to an improper fraction first, then apply scaling rules.

📐 Formula:

W a/b × c/d = [(W × b + a)/b] × c/d

📝 Steps:

Convert mixed number to improper fraction
Determine if scaling fraction is <, =, or > 1
Multiply the two fractions
Simplify and convert back to mixed number if needed

✏️ Example 1: 2 1/2 × 3/5

Predict: Since 3/5 < 1, product will be less than 2 1/2

Convert: 2 1/2 = 5/2

Multiply: 5/2 × 3/5 = 15/10 = 3/2 = 1 1/2

Answer: 1 1/2

Verify: 1 1/2 < 2 1/2 ✓

✏️ Example 2: 1 1/3 × 6/5

Predict: Since 6/5 > 1, product will be greater than 1 1/3

Convert: 1 1/3 = 4/3

Multiply: 4/3 × 6/5 = 24/15 = 8/5 = 1 3/5

Answer: 1 3/5

Verify: 1 3/5 > 1 1/3 ✓

5. Scaling by Fractions and Mixed Numbers

Definition: Comprehensive scaling problems that combine all types: whole numbers, fractions, and mixed numbers as both the number being scaled and the scaling factor.

🔑 Key Comparison Rules:

Scaling Factor	Effect on Original	Example
< 1	Makes smaller (scales down)	10 × 1/2 = 5
= 1	Stays the same	10 × 4/4 = 10
> 1	Makes larger (scales up)	10 × 1 1/2 = 15

✏️ Example 1: Mixed × Mixed - 2 1/4 × 1 1/3

Predict: 1 1/3 > 1, so product will be greater than 2 1/4

Convert both: 2 1/4 = 9/4, 1 1/3 = 4/3

Multiply: 9/4 × 4/3 = 36/12 = 3

Answer: 3

Verify: 3 > 2 1/4 ✓

✏️ Example 2: Whole × Mixed - 8 × 2 3/4

Predict: 2 3/4 > 1, so product will be greater than 8

Convert: 2 3/4 = 11/4

Multiply: 8 × 11/4 = 88/4 = 22

Answer: 22

Verify: 22 > 8 ✓

✏️ Example 3: Fraction × Mixed - 3/5 × 1 2/3

Predict: 3/5 < 1, so product will be less than 1 2/3

Convert: 1 2/3 = 5/3

Multiply: 3/5 × 5/3 = 15/15 = 1

Answer: 1

Verify: 1 < 1 2/3 ✓

Quick Reference Chart

Scaling Type	Formula	Example
Whole × Fraction	n × a/b = (n × a)/b	12 × 2/3 = 24/3 = 8
Fraction × Fraction	a/b × c/d = (a×c)/(b×d)	3/4 × 2/5 = 6/20 = 3/10
Mixed × Fraction	Convert to improper, then multiply	2 1/2 × 3/5 = 5/2 × 3/5 = 3/2
Any × Mixed	Convert mixed to improper first	8 × 2 3/4 = 8 × 11/4 = 22

💡 Scaling Decision Chart:

Scaling Factor < 1

Product SMALLER

Scale DOWN ⬇

Scaling Factor = 1

Product EQUAL

No Change ↔

Scaling Factor > 1

Product LARGER

Scale UP ⬆

🔑 Key Tips for Success:

Always compare the scaling fraction to 1 to predict if the product will be larger or smaller
To check if a fraction is greater than 1: numerator > denominator
To check if a fraction equals 1: numerator = denominator
To check if a fraction is less than 1: numerator < denominator
For mixed numbers, they are always greater than 1 (unless the whole part is 0)
When scaling down (multiplying by fraction < 1), you're taking "part of" the original
When scaling up (multiplying by fraction > 1), you're taking "more than all of" the original
Convert all mixed numbers to improper fractions before multiplying
After calculating, verify your answer matches your prediction
Simplify final answers and convert back to mixed numbers when appropriate

📚 Fifth Grade Scaling by Fractions - Complete Study Guide

Master these concepts for math excellence! ✨