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Terminating and Repeating Decimals

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Terminating and Repeating Decimals

Study Guide

Key Definition

A rational number can be expressed as either a terminating decimal or a repeating decimal by dividing the numerator by the denominator. If the remainder is 0, it is a terminating decimal. Otherwise, it is a repeating decimal.

Important Notes

A terminating decimal has a finite number of digits.
A repeating decimal has one or more repeating digits endlessly.
The bar over the decimal denotes repeating digits, like $0.58\overline{3}$.
To identify if a decimal is terminating, check if the denominator has only the prime factors 2 and 5.
Repeating decimals occur when the division of the numerator by the denominator never resolves to a remainder of 0.

Mathematical Notation

$\frac{a}{b}$ represents a fraction

$\overline{d}$ indicates a repeating decimal

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

When dividing the numerator by the denominator, the remainder determines if a decimal terminates or repeats. Terminating decimals occur when the remainder becomes zero after finite steps.

Remember

A terminating decimal has finite digits, while a repeating decimal has digits that repeat endlessly.

Quick Reference

Terminating Decimal:A decimal that ends, e.g., $0.625$

Repeating Decimal:A decimal that repeats, e.g., $0.58\overline{3}$

Understanding Terminating and Repeating Decimals

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A terminating decimal is a decimal that ends after a few digits, like $0.625$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What type of decimal is $\frac{5}{8}$ when expressed as a decimal?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You have a pizza cut into 12 slices, and you eat 7 slices. What type of decimal is $\frac{7}{12}$ when expressed as a decimal?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine if the fraction $\frac{25}{40}$ is a terminating or repeating decimal.

Challenge Quiz

Single Choice Quiz

Advanced

Which fraction results in a repeating decimal?

Recap

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Review key concepts and takeaways