Fractions and recurring decimals

Understanding Fractions

A fraction represents a part of a whole and consists of a numerator and a denominator.
The numerator is the number above the line and indicates the number of considered parts.
The denominator is the number below the line and shows the total number of equal parts.
For example, in the fraction 3/4, 3 is the numerator which signifies that we’re looking at 3 parts, and 4 is the denominator meaning a whole is divided into 4 equal parts.

A simplified fraction is when the numerator and the denominator have no common factors apart from 1. For example, the fraction 2/4 is not simplified as 2 is a common factor, after simplifying we get 1/2.
Remember, the fraction value stays the same when you simplify, you’re just expressing it in the most reduced terms.

Recurring decimals are decimals that have digits or groupings of digits that repeat endlessly.
Recurring decimals can be written using a dot above the repeating digit (or digits), for example, 0.666… can be written as 0.6̅.
Some fractions, when converted into decimal form, give recurring decimals. For example, 1/3 = 0.333…

To convert a fraction to a decimal, divide the numerator by the denominator. If the division ends with a remainder, the decimal is finite. If the division leads to an endlessly repeating pattern, the decimal is recurring.
Converting a recurring decimal to a fraction requires a more complex process. For a simple recurring decimal like 0.3̅, you would equate it to a variable, multiply it to eliminate the recurrence, and solve for the variable.

Problem: Simplify the fraction 72/90.
- Solution: Both numbers are divisible by 18, so after dividing both the numerator and denominator by 18, we get 4/5.
Problem: Convert the fraction 2/5 to a decimal.
- Solution: When we divide 2 by 5, we get 0.4.
Problem: Convert the recurring decimal 0.777… to a fraction.
- Solution: Let the variable x equal 0.777… Now, multiply x by 10 to get 10x = 7.777…, then subtract x from 10x to get 9x = 7. Solving for x gives x = 7/9.

Double-check your calculations when working with fractions and recurring decimals.
Converting between fractions and recurring decimals or simplifying fractions is easier when you have a strong understanding of division and multiplication.
Remember to be aware of repeating patterns and how they affect your outcome when working with recurring decimals.