Problem 61 - Algebra for College Students

Question

0 . \overline{123} \quad \frac{41}{333}

Step-by-Step Solution

Answer

Both values are equivalent; \( 0.\overline{123} = \frac{41}{333} \).

Step 1: Understand the problem statement

The problem presents two values - a repeating decimal \( 0.\overline{123} \) and a fraction \( \frac{41}{333} \). We need to establish if these two values are equivalent.

Step 2: Express the repeating decimal as a fraction

To express \( 0.\overline{123} \) as a fraction, let \( x = 0.\overline{123} \). Multiply both sides by 1000 to shift the decimal point three places: \[1000x = 123.\overline{123}\] Subtract the original equation from this new one:\[1000x - x = 123.\overline{123} - 0.\overline{123}\] This simplifies to:\[999x = 123\] Solving for \( x \) gives:\[x = \frac{123}{999}\]

Step 3: Simplify the fraction

We need to simplify \( \frac{123}{999} \). First, find the greatest common divisor (GCD) between 123 and 999:Factorize both numbers:- 123: 3 and 41 (since 123 = 3 x 41)- 999: 3^3 x 37 (since 999 = 3 x 3 x 3 x 37)The GCD is 3, so divide both the numerator and the denominator by 3:\[\frac{123}{999} = \frac{123 \div 3}{999 \div 3} = \frac{41}{333}\]

Step 4: Compare with the given fraction

Now, compare the simplified fraction from the repeating decimal conversion, \( \frac{41}{333} \), to the original given fraction. Since they are identical, the original repeating decimal and the fraction \( \frac{41}{333} \) represent the same value.

Key Concepts

Fractions

Fractions are numbers that express parts of a whole. They are written with two numbers separated by a line. The number on top is called the numerator, and it represents the number of parts you have. The number below the line is called the denominator, and it tells you the number of parts the whole is divided into. For example, in the fraction \( \frac{41}{333} \), 41 is the numerator and 333 is the denominator.

• The numerator indicates how many parts are being considered.
• The denominator shows the total number of equal parts in one whole.

Fractions can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). This process makes the fraction easier to interpret without changing its value.

Decimal Conversion

Converting between decimals and fractions is a useful skill. To convert a repeating decimal into a fraction, identify what part of the decimal repeats. For \( 0.\overline{123} \), the sequence '123' repeats indefinitely. Let \( x = 0.\overline{123} \). Multiplying by 1000 shifts the decimal point three places to the right to get rid of the decimal part when it is subtracted. This gives:

• \( 1000x = 123.\overline{123} \)
• Subtract \( x = 0.\overline{123} \) from this to isolate the repeating section.
• You end up with \( 999x = 123 \), leading to \( x = \frac{123}{999} \).

The fraction can then be simplified to its lowest terms to perform calculations or comparisons.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that divides two numbers without leaving a remainder. It's essential for simplifying fractions. To find the GCD:

• Factorize both the numerator and the denominator.
• Identify all common factors.
• Choose the highest common factor as the GCD.

For instance, in the exercise, the GCD of 123 and 999 was 3, allowing for the division of both parts of the fraction \( \frac{123}{999} \) by 3. This simplification process led to \( \frac{41}{333} \). Simplifying fractions helps us to ensure they are represented in their most understandable form, equivalent to their original value.