Problem 37 - Prealgebra

Question

Write each of the following fractions as an equivalent fraction with denominator 12. $$\frac{56}{84}$$

Step-by-Step Solution

Answer

The fraction \( \frac{56}{84} \) is equivalent to \( \frac{8}{12} \).

Step 1: Simplify the Fraction

First, simplify the given fraction \( \frac{56}{84} \) by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 56 and 84 is 28. Dividing both the numerator and the denominator by 28, we get: \[ \frac{56 \div 28}{84 \div 28} = \frac{2}{3}. \]

Step 2: Find the Equivalent Fraction with Denominator 12

Next, we need to convert \( \frac{2}{3} \) to a fraction with a denominator of 12. Determine what to multiply the denominator 3 by to get 12. Since \( 3 \times 4 = 12 \), multiply both the numerator and the denominator of \( \frac{2}{3} \) by 4: \[ \frac{2 \times 4}{3 \times 4} = \frac{8}{12}. \]

Step 3: Verify the Results

Ensure the operations are correct. Simplify \( \frac{8}{12} \) to check that it equals \( \frac{2}{3} \). The GCD of 8 and 12 is 4: \[ \frac{8 \div 4}{12 \div 4} = \frac{2}{3}. \] This verifies that our equivalent fraction \( \frac{8}{12} \) matches the simplified form.

Key Concepts

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value or proportion. For example, \( \frac{2}{3} \) and \( \frac{8}{12} \) are equivalent because they both describe the same part of a whole. To find equivalent fractions, you can either multiply or divide both the numerator and denominator of a fraction by the same non-zero number.

Finding equivalent fractions involves knowing what you want the new denominator to be. For instance, if you need an equivalent fraction with a specific denominator, like 12, you'll calculate what you must multiply the original denominator by to reach the desired denominator. Once you have that factor, apply it to both the numerator and the denominator.

• This ensures the overall value of the fraction remains constant.
• Equivalent fractions are useful in adding, subtracting, and comparing fractions.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. Understanding the GCD is essential for simplifying fractions. It helps you express a complex fraction in its simplest form.

To discover the GCD of two numbers, you list all their divisors and find the highest number common to both lists. Alternatively, an efficient method involves using the Euclidean algorithm, a systematic approach for calculating the GCD.

• For example, to simplify \( \frac{56}{84} \), the GCD is 28.
• Divide the numerator and the denominator by 28 to simplify the fraction \( \frac{56}{84} \) to \( \frac{2}{3} \).

Understanding and using the GCD ensures you work with the simplest form of a fraction, which can make problem-solving easier and clearer.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the only divisor common to both the numerator and denominator is 1. This process makes fractions easier to understand and work with in computations.

Simplification is achieved using the GCD. First, you identify the GCD of the numerator and denominator. Then, divide both terms by their GCD. The result is the fraction in simplest form, meaning it cannot be reduced further.

• For instance, using \( \frac{56}{84} \), the GCD is 28. Dividing by 28, you get \( \frac{2}{3} \).
• This simplified form is the most concise way of expressing the fraction.

Consistently simplifying fractions aids in accurate mathematical operations such as addition, subtraction, and comparison. It maintains the clarity and elegance of mathematical expressions.