Problem 38 - Prealgebra

Question

Write each of the following fractions as an equivalent fraction with denominator 12. $$\frac{143}{156}$$

Step-by-Step Solution

Answer

The equivalent fraction with denominator 12 is \(\frac{11}{12}\).

Step 1: Simplify the Fraction First

Before finding a fraction with a certain denominator, it's often useful to simplify the fraction as much as possible. Simplify \(\frac{143}{156}\) by finding the greatest common divisor (GCD) of 143 and 156, which is 13. Thus, \(\frac{143}{156} = \frac{143 \div 13}{156 \div 13} = \frac{11}{12}\).

Step 2: Analyze if Further Adjustment is Necessary

Since we are asked to change the denominator to 12, check if it is already 12 after simplification. Indeed, the denominator we achieved after simplification step is 12. Therefore, \(\frac{11}{12}\) is already in the desired form.

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial skill in math that involves reducing a fraction to its most basic form. This means finding an equivalent fraction where the numerator and the denominator have no common factors except for 1. It makes calculations easier and helps in comparing fractions.

To simplify a fraction, you'll want to find the greatest number that divides both the numerator and the denominator evenly. This is known as the greatest common divisor (GCD). Once you find the GCD, you divide the numerator and the denominator by it.

For example, consider the fraction \(\frac{143}{156}\). To simplify, determine the GCD of 143 and 156, which is 13. Then divide both numbers by 13:

• 143 divided by 13 equals 11.
• 156 divided by 13 equals 12.

This gives you the simplified fraction \(\frac{11}{12}\). Making fractions simpler doesn't change their value, but makes them easier to understand and use.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. Finding the GCD is an essential step when simplifying fractions. It ensures we reduce the fraction completely and accurately.

To find the GCD of two numbers, you can use several methods:

• Prime Factorization: Break down each number into its prime factors and multiply the common factors.
• Euclidean Algorithm: A more efficient method, especially for larger numbers. It involves a series of division steps.

In our example, the GCD of 143 and 156 is 13. By dividing both the numerator and denominator by 13, we simplify the fraction effectively. Utilizing the GCD keeps the process of simplifying fractions straightforward and verifiable.

Denominators

The denominator of a fraction is the lower part that indicates how many equal parts the whole is divided into. Understanding denominators is vital when working with fractions, as they determine the size of each piece represented by the fraction.

When converting a fraction to have a specific denominator, like 12 in our exercise, we ensure the fraction's ratio remains the same. After simplifying \(\frac{143}{156}\) to \(\frac{11}{12}\), the desired denominator was already 12. This illustrates how simplifying can often directly achieve the goal.

Working with denominators allows us to perform operations such as addition, subtraction, and comparison of fractions. Matching them helps in finding common ground when fractions aren't readily compatible, paving the way for further arithmetic operations.