Problem 49 - Algebra 2

Question

Find each product if \(a=\frac{3}{5}, b=\frac{2}{7}, c=\frac{3}{4},\) and \(d=\frac{1}{3}\). \(c d\)

Step-by-Step Solution

Answer

The product of \( c \) and \( d \) is \( \frac{1}{4} \).

Step 1: Identify the Values

First, we need to identify the values given for each variable in the problem. Here, we only need the values of \( c \) and \( d \) for this specific product. \( c = \frac{3}{4} \) and \( d = \frac{1}{3} \).

Step 2: Write the Expression for the Product

To find the product of \( c \) and \( d \), write it as a mathematical expression: \( c \cdot d = \frac{3}{4} \times \frac{1}{3} \).

Step 3: Multiply the Numerical Values

To multiply fractions, multiply the numerators together and the denominators together. Compute as follows: \( \frac{3 \times 1}{4 \times 3} = \frac{3}{12} \).

Step 4: Simplify the Fraction

The resulting fraction from the multiplication is \( \frac{3}{12} \). Simplify this fraction by finding the greatest common divisor of 3 and 12, which is 3, and then divide both the numerator and the denominator by this number. \( \frac{3 \div 3}{12 \div 3} = \frac{1}{4} \).

Key Concepts

Simplifying Fractions

Simplifying a fraction means reducing it to its smallest form, where the numerator and the denominator have no common factors other than 1. This makes the fraction easier to comprehend and use in calculations.

To simplify a fraction, such as \( \frac{3}{12} \), you follow these steps:

• Identify the greatest common factor (GCF) of both the numerator and the denominator.
• Divide both the numerator and the denominator by this GCF.

For example, with \( \frac{3}{12} \), the GCF is 3. Dividing both 3 and 12 by 3, you get \( \frac{1}{4} \), which is the simplified form. Simplifying fractions is a crucial skill, especially when dealing with real-life problems where exact numbers are needed.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. It's used in simplifying fractions, as it helps reduce them to their simplest form.

To find the GCD, you can use different methods:

• List out the factors of each number and find the largest number that appears in both lists.
• Use the Euclidean algorithm, which involves repeated division, to efficiently find the GCD.

Finding the GCD is particularly useful when working with fractions like \( \frac{3}{12} \). In this case, listing out the factors:

• Factors of 3: 1, 3
• Factors of 12: 1, 2, 3, 4, 6, 12

The largest common factor is 3, making it the GCD. Dividing both numbers by the GCD simplifies the fraction.

Numerator and Denominator

Every fraction consists of two main parts: the numerator and the denominator. Understanding these terms is essential when performing operations with fractions like addition, subtraction, multiplication, and division.

In a fraction, such as \( \frac{3}{12} \):

• The numerator (3) is the number above the fraction line. It represents how many parts of a whole we have.
• The denominator (12) is the number below. It shows the total number of equal parts the whole is divided into.

Knowing the roles of the numerator and the denominator is crucial when you're simplifying fractions or finding the product of fractions. When multiplying fractions, you multiply the numerators together to find the new numerator and the denominators together for the new denominator, as with \( \frac{3}{4} \) and \( \frac{1}{3} \), resulting in \( \frac{3}{12} \).