Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 6: Decimals

Lesson 1: Arithmetic with Decimals

In this Grade 4 lesson from The Art of Problem Solving: Prealgebra, students learn how the base 10 place value system extends to decimals, covering decimal notation using negative powers of 10 (tenths, hundredths, thousandths) and the rules for adding, subtracting, multiplying, and dividing decimals. Key skills include aligning decimal places when adding and subtracting, accounting for hidden zeros, and correctly shifting the decimal point when multiplying or dividing by powers of 10. This lesson builds the arithmetic foundation needed for AMC 8 problem-solving with decimal quantities.

Section 1

Decimal and Fraction Equivalence

Property

Decimals are another way of writing fractions whose denominators are powers of 10.

\begin{array}{ll} 0.1 = \frac{1}{10} & \text{(one tenth)} \\ 0.01 = \frac{1}{100} & \text{(one hundredth)} \\ 0.001 = \frac{1}{1000} & \text{(one thousandth)} \\ 0.0001 = \frac{1}{10,000} & \text{(one ten-thousandth)} \end{array}

Any decimal can be written as a fraction with a denominator that is a power of 10.

Examples

Section 2

Add and subtract decimals

Property

How to add or subtract decimals:
Step 1. Write the numbers vertically so the decimal points line up.
Step 2. Use zeros as place holders, as needed.
Step 3. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.

Examples

To add $18.5 + 3.45$ , we write the numbers vertically, aligning the decimal points and adding a placeholder zero: $18.50 + 3.45 = 21.95$ .

To subtract $50 - 22.18$ , we treat $50$ as $50.00$ to align the decimal points and use placeholders: $50.00 - 22.18 = 27.82$ .

Section 3

Multiply and Divide by Powers of 10

Property

To multiply a number by a power of ten:

Move the decimal point to the right the same number of places as the exponent on ten.
Write in zeros to fill any empty places at the end of the new number.

To divide a number by a power of ten:

Move the decimal point to the left the same number of places as the exponent on ten.
Write in zeros to fill any empty places at the beginning of the new number.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Decimal and Fraction Equivalence

Property

Decimals are another way of writing fractions whose denominators are powers of 10.

\begin{array}{ll} 0.1 = \frac{1}{10} & \text{(one tenth)} \\ 0.01 = \frac{1}{100} & \text{(one hundredth)} \\ 0.001 = \frac{1}{1000} & \text{(one thousandth)} \\ 0.0001 = \frac{1}{10,000} & \text{(one ten-thousandth)} \end{array}

Any decimal can be written as a fraction with a denominator that is a power of 10.

Examples

Section 2

Add and subtract decimals

Property

Examples

To add $18.5 + 3.45$ , we write the numbers vertically, aligning the decimal points and adding a placeholder zero: $18.50 + 3.45 = 21.95$ .

To subtract $50 - 22.18$ , we treat $50$ as $50.00$ to align the decimal points and use placeholders: $50.00 - 22.18 = 27.82$ .

Section 3

Multiply and Divide by Powers of 10

Property

To multiply a number by a power of ten:

Move the decimal point to the right the same number of places as the exponent on ten.
Write in zeros to fill any empty places at the end of the new number.

To divide a number by a power of ten:

Move the decimal point to the left the same number of places as the exponent on ten.
Write in zeros to fill any empty places at the beginning of the new number.

Examples

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Decimal and Fraction Equivalence

Property

Decimals are another way of writing fractions whose denominators are powers of 10.

\begin{array}{ll} 0.1 = \frac{1}{10} & \text{(one tenth)} \\ 0.01 = \frac{1}{100} & \text{(one hundredth)} \\ 0.001 = \frac{1}{1000} & \text{(one thousandth)} \\ 0.0001 = \frac{1}{10,000} & \text{(one ten-thousandth)} \end{array}

Any decimal can be written as a fraction with a denominator that is a power of 10.

Examples

Section 2

Add and subtract decimals

Property

Examples

To add $18.5 + 3.45$ , we write the numbers vertically, aligning the decimal points and adding a placeholder zero: $18.50 + 3.45 = 21.95$ .

To subtract $50 - 22.18$ , we treat $50$ as $50.00$ to align the decimal points and use placeholders: $50.00 - 22.18 = 27.82$ .

Section 3

Multiply and Divide by Powers of 10

Property

To multiply a number by a power of ten:

Move the decimal point to the right the same number of places as the exponent on ten.
Write in zeros to fill any empty places at the end of the new number.

To divide a number by a power of ten:

Move the decimal point to the left the same number of places as the exponent on ten.
Write in zeros to fill any empty places at the beginning of the new number.