Multiply Decimals by Whole Numbers

Fifth Grade Math - Complete Guide

🎯 Estimate Products

Why Estimate?

Estimation helps you quickly check if your answer is reasonable before doing the exact calculation. It's a great way to catch mistakes!

✅ Estimation Rule:

Round the decimal to the nearest whole number, then multiply!

📝 Steps to Estimate

Round the decimal to the nearest whole number
Keep the whole number as is
Multiply the rounded numbers
Compare your estimate with the exact answer

💡 Examples

Example 1: Estimate $6.8 \times 4$

Step 1: Round 6.8 → 7 (since 8 ≥ 5, round up)
Step 2: Keep 4 as is
Step 3: $7 \times 4 = 28$
Actual answer: $6.8 \times 4 = 27.2$ ✓ Very close!

Example 2: Estimate $12.3 \times 15$

Step 1: Round 12.3 → 12 (since 3 < 5, round down)
Step 2: Keep 15 as is
Step 3: $12 \times 15 = 180$
Actual answer: $12.3 \times 15 = 184.5$ ✓ Close estimate!

✖️ The Main Rule: Multiplying Decimals by Whole Numbers

The Golden Rule

✅ Multiply as if both were whole numbers

Then count decimal places and place the decimal point in the product

Steps:

Ignore the decimal point and multiply normally
Count the decimal places in the original decimal number
Place the decimal point in the product with the same number of decimal places

💡 Visual Explanation

Example: $2.5 \times 6$

Step 1: Ignore decimal → Multiply 25 × 6 = 150
Step 2: Count decimal places in 2.5 → 1 decimal place
Step 3: Place decimal in 150 with 1 decimal place → 15.0 or 15

✓ Answer: $2.5 \times 6 = 15$

1️⃣ Multiply by One-Digit Whole Numbers

Multiplying Tenths

When multiplying decimals with one decimal place (tenths like 0.3, 2.5, 6.7):

Example: $3.6 \times 4$

Step 1: Multiply without decimal → $36 \times 4 = 144$
Step 2: Count decimal places → 1 place in 3.6
Step 3: Place decimal → 14.4

✓ Answer: $3.6 \times 4 = 14.4$

Multiplying Hundredths

When multiplying decimals with two decimal places (hundredths like 0.25, 1.75, 8.43):

Example: $2.45 \times 3$

Step 1: Multiply without decimal → $245 \times 3 = 735$
Step 2: Count decimal places → 2 places in 2.45
Step 3: Place decimal → 7.35

✓ Answer: $2.45 \times 3 = 7.35$

📐 Using the Distributive Property

What is the Distributive Property?

The distributive property lets you break down the decimal into smaller, easier parts, multiply each part, then add them together.

\[a \times (b + c) = (a \times b) + (a \times c)\]

💡 Strategy:

Break the decimal into whole number + decimal part, then distribute!

💡 Examples

Example 1: $2 \times 7.4$

Step 1: Break down 7.4 → $7 + 0.4$
Step 2: Distribute → $2 \times (7 + 0.4) = (2 \times 7) + (2 \times 0.4)$
Step 3: Multiply each part → $14 + 0.8$
Step 4: Add → $14.8$

✓ Answer: $2 \times 7.4 = 14.8$

Example 2: $5 \times 3.25$

Step 1: Break down 3.25 → $3 + 0.25$
Step 2: Distribute → $5 \times (3 + 0.25) = (5 \times 3) + (5 \times 0.25)$
Step 3: Multiply each part → $15 + 1.25$
Step 4: Add → $16.25$

✓ Answer: $5 \times 3.25 = 16.25$

🔢 Multiply by Multi-Digit Whole Numbers

Standard Algorithm Method

When multiplying by two-digit or larger whole numbers, use the standard multiplication algorithm (just like multiplying whole numbers!).

Steps:

Line up the numbers on the right (ignore decimal for now)
Multiply by each digit of the whole number
Add the partial products
Count decimal places and place the decimal point

💡 Example

Example: $3.6 \times 24$

Step 1: Multiply ignoring decimal → $36 \times 24$
36
× 24
------
144 (36 × 4)
720 (36 × 20)
------
864 Step 2: Count decimal places in 3.6 → 1 place
Step 3: Place decimal in 864 → 86.4

✓ Answer: $3.6 \times 24 = 86.4$

🟦 Using Area Models

What is an Area Model?

An area model is a visual way to multiply by breaking numbers into parts and creating a rectangle divided into sections.

💡 Strategy:

1. Break both numbers into place values
2. Create a rectangle grid
3. Multiply each section
4. Add all sections together

💡 Example

Example: $2.3 \times 12$ using area model

Step 1: Break down → 2.3 = (2 + 0.3) and 12 = (10 + 2)

Step 2: Create grid and multiply sections:
  • $2 \times 10 = 20$
  • $2 \times 2 = 4$
  • $0.3 \times 10 = 3$
  • $0.3 \times 2 = 0.6$

Step 3: Add all sections → $20 + 4 + 3 + 0.6 = 27.6$

✓ Answer: $2.3 \times 12 = 27.6$

📖 Word Problems

🎯 Steps to Solve

READ the problem carefully
IDENTIFY the decimal and whole number
DETERMINE if you need to multiply
SOLVE using the multiplication rules
CHECK if your answer makes sense

Problem 1

A pencil costs $0.75. How much do 8 pencils cost?

Step 1: Identify → $0.75 per pencil, 8 pencils
Step 2: Operation → Multiply $0.75 \times 8$
Step 3: Solve → $75 \times 8 = 600$, 2 decimal places → $6.00$

✓ Answer: 8 pencils cost $6.00

Problem 2

Sarah runs 3.5 kilometers every day. How many kilometers does she run in 5 days?

Step 1: Identify → 3.5 km per day, 5 days
Step 2: Operation → Multiply $3.5 \times 5$
Step 3: Solve → $35 \times 5 = 175$, 1 decimal place → $17.5$

✓ Answer: Sarah runs 17.5 kilometers in 5 days

🔢 Multiply Three or More Numbers

Strategy

When multiplying three or more numbers where one is a decimal, multiply two at a time from left to right.

✅ Important Rule:

Count ALL decimal places from ALL decimal numbers in the problem!

💡 Examples

Example 1: $2 \times 3.5 \times 4$

Method 1: Left to right
  • First: $2 \times 3.5 = 7.0$
  • Then: $7.0 \times 4 = 28.0 = 28$

Method 2: Multiply whole numbers first
  • First: $2 \times 4 = 8$
  • Then: $8 \times 3.5 = 28.0 = 28$

✓ Answer: $2 \times 3.5 \times 4 = 28$

Example 2: $5 \times 2.4 \times 3$

Step 1: Multiply first two → $5 \times 2.4 = 12.0$
Step 2: Multiply result by third → $12.0 \times 3 = 36.0 = 36$

Alternative: Multiply whole numbers first
• $5 \times 3 = 15$
• $15 \times 2.4 = 36$

✓ Answer: $5 \times 2.4 \times 3 = 36$

✏️ Practice Problems

Set 1: One-Digit Multiplication

1. $4.2 \times 5 = ?$
2. $0.68 \times 7 = ?$
3. $9.3 \times 6 = ?$

Answers: 1) 21.0 or 21 | 2) 4.76 | 3) 55.8

Set 2: Multi-Digit Multiplication

1. $1.5 \times 12 = ?$
2. $3.25 \times 20 = ?$
3. $0.8 \times 35 = ?$

Answers: 1) 18.0 or 18 | 2) 65.0 or 65 | 3) 28.0 or 28

Set 3: Three Numbers

1. $2 \times 1.5 \times 4 = ?$
2. $3 \times 2.2 \times 5 = ?$
3. $6 \times 0.5 \times 8 = ?$

Answers: 1) 12 | 2) 33 | 3) 24

📋 Quick Reference Guide

Method	When to Use	Key Steps
Standard	Any problem	Ignore decimal → Multiply → Count places → Place decimal
Distributive Property	Mental math	Break decimal → Multiply parts → Add results
Area Model	Visual learners	Break both numbers → Grid → Multiply sections → Add all
Estimation	Quick check	Round decimal → Multiply → Compare