Learn on PengiIllustrative Mathematics, Grade 5Chapter 5: Place Value Patterns and Decimal Operations

Lesson 13: Divide by Decimals

In this Grade 5 Illustrative Mathematics lesson from Chapter 5: Place Value Patterns and Decimal Operations, students learn how to divide by decimals by applying place value understanding and patterns. Students explore how dividing by a decimal such as 0.1 or 0.01 relates to multiplying by 10 or 100, building fluency with decimal division. This lesson strengthens students' number sense and prepares them for more complex decimal operations.

Section 1

Dividing a Whole Number by a Decimal

Property

To divide a whole number by a decimal, convert the divisor into a whole number by multiplying both the divisor and the dividend by the same power of 10. This is equivalent to moving the decimal point in both numbers to the right by the same number of places. Then, perform the division as you would with whole numbers.

a÷b=(a×10n)÷(b×10n)a \div b = (a \times 10^n) \div (b \times 10^n)

Examples

  • To solve 42÷0.742 \div 0.7, multiply both numbers by 10:
42÷0.7→(42×10)÷(0.7×10)→420÷7=6042 \div 0.7 \rightarrow (42 \times 10) \div (0.7 \times 10) \rightarrow 420 \div 7 = 60
  • To solve 15÷0.0315 \div 0.03, multiply both numbers by 100:
15÷0.03→(15×100)÷(0.03×100)→1500÷3=50015 \div 0.03 \rightarrow (15 \times 100) \div (0.03 \times 100) \rightarrow 1500 \div 3 = 500

Explanation

When dividing a whole number by a decimal, the key is to transform the problem into one you already know how to solve: division by a whole number. You can do this by multiplying both the dividend (the number being divided) and the divisor (the number you are dividing by) by the same power of 10 (like 10, 100, or 1000). This process effectively moves the decimal point in both numbers to the right, creating an equivalent division problem with a whole number divisor. After setting up the equivalent problem, you can use long division to find the final answer.

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Dividing a Whole Number by a Decimal

Property

To divide a whole number by a decimal, convert the divisor into a whole number by multiplying both the divisor and the dividend by the same power of 10. This is equivalent to moving the decimal point in both numbers to the right by the same number of places. Then, perform the division as you would with whole numbers.

a÷b=(a×10n)÷(b×10n)a \div b = (a \times 10^n) \div (b \times 10^n)

Examples

  • To solve 42÷0.742 \div 0.7, multiply both numbers by 10:
42÷0.7→(42×10)÷(0.7×10)→420÷7=6042 \div 0.7 \rightarrow (42 \times 10) \div (0.7 \times 10) \rightarrow 420 \div 7 = 60
  • To solve 15÷0.0315 \div 0.03, multiply both numbers by 100:
15÷0.03→(15×100)÷(0.03×100)→1500÷3=50015 \div 0.03 \rightarrow (15 \times 100) \div (0.03 \times 100) \rightarrow 1500 \div 3 = 500

Explanation

When dividing a whole number by a decimal, the key is to transform the problem into one you already know how to solve: division by a whole number. You can do this by multiplying both the dividend (the number being divided) and the divisor (the number you are dividing by) by the same power of 10 (like 10, 100, or 1000). This process effectively moves the decimal point in both numbers to the right, creating an equivalent division problem with a whole number divisor. After setting up the equivalent problem, you can use long division to find the final answer.