Problem 145 - College Algebra

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$ \frac{1}{2}+\frac{2}{3} $$

Step-by-Step Solution

Answer

The simplified form of the fraction is \[ \frac{2}{3} \].

Step 1: Multiply the Numerators and Denominators

First, perform the multiplication between the numerators (the top numbers) and between the denominators (the bottom numbers). It would look like this: \[ \frac{5 \times 8}{4 \times 15} \].

Step 2: Calculate the Product

Now, calculate the product of the numerators and likewise for the denominators. This will result in another fraction. In this case it will be \[ \frac{40}{60} \].

Step 3: Reduce to Lowest Terms

Reduce the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 40 and 60 is 20. So when divided by 20, the fraction now becomes \[ \frac{40 ÷ 20}{60 ÷ 20} \] which equals \[ \frac{2}{3} \].

Key Concepts

Reducing Fractions

Reducing fractions is the process of simplifying a fraction so that the numerator and the denominator are as small as possible while still maintaining the same value of the original fraction. It involves dividing both numbers by their greatest common divisor (GCD).
For example, if you have a fraction like \( \frac{40}{60} \) it means that both 40 and 60 can be divided by some common number. To reduce the fraction, you find the largest number that divides both, which in this case is 20. By dividing the numerator and the denominator by 20, the fraction simplifies to \( \frac{2}{3} \).
Reduction is possible whenever the numerator and the denominator have a common factor other than 1. By reducing fractions to their simplest form, calculations involving fractions become much easier.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest positive integer that divides two or more integers without leaving a remainder. Finding the GCD of the numerator and denominator of a fraction allows you to reduce the fraction to its simplest form.
To find the GCD, list out the factors of each number and then identify the largest factor that they share. There are also more systematic methods like the Euclidean algorithm, which is helpful for larger numbers.
For instance, for the numbers 40 and 60, their factors are:

• Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
• Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

The GCD of 40 and 60 is the highest number found in both lists, which is 20. By dividing both the numerator and the denominator by the GCD, you simplify the fraction effectively.

Numerators and Denominators

Fractions are composed of two parts: the numerator and the denominator. The numerator, located above the fraction bar, indicates how many parts are being considered, while the denominator, below the fraction bar, indicates the total number of equal parts that make up a whole.
For multiplication, each numerator is multiplied by the other, and the same goes for the denominators. In the exercise \( \frac{5}{4} \cdot \frac{8}{15} \) the numerators 5 and 8 are multiplied together, and the denominators 4 and 15 are multiplied together, resulting in \( \frac{5 \times 8}{4 \times 15} \).
Understanding the roles of numerators and denominators in operations is crucial as it helps to comprehend how fractions work and how to perform various arithmetic operations with them properly.