Converting Repeating Decimals to Fractions in Simplest Form
In this article, we will explore how to convert repeating decimals to fractions in simplest form using the example of 4.express 2.bar(36) + 0.bar(23).
Understanding Repeating Decimals
A repeating decimal is a decimal that has a sequence of digits that repeats indefinitely. For example, 0.3636... is a repeating decimal where the sequence 36 repeats indefinitely.
Converting Repeating Decimals to Fractions
To convert a repeating decimal to a fraction, we can use the following steps:
- Let the repeating decimal be
x. - Multiply both sides of the equation by
10^n, wherenis the number of digits in the repeating sequence. - Subtract the original equation from the new equation.
- Simplify the resulting equation to get the fraction.
Example: 4.express 2.bar(36) + 0.bar(23)
Let's convert the given repeating decimal to a fraction in simplest form.
Step 1: Let the repeating decimal be x
Let x = 4.3636... + 0.2323...
Step 2: Multiply both sides by 10^n
Since the repeating sequence has 2 digits, we multiply both sides by 10^2 = 100.
100x = 436.3636... + 23.2323...
Step 3: Subtract the original equation from the new equation
Subtract x from both sides of the equation:
99x = 432 + 23
Step 4: Simplify the resulting equation
Simplify the equation by combining like terms:
99x = 455
x = 455/99
Simplifying the Fraction
To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD).
The GCD of 455 and 99 is 1, so the fraction is already in simplest form:
x = 455/99
Therefore, the given repeating decimal 4.express 2.bar(36) + 0.bar(23) is equal to the fraction 455/99 in simplest form.
