Problem 47 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The product is \(\frac{6}{35}\).

Step 1: Identify the Variables

We start by identifying the two variables we need to multiply. From the problem, we have the values of the variables: \(a = \frac{3}{5}\) and \(b = \frac{2}{7}\).

Step 2: Set Up the Multiplication

Next, we set up the arithmetic operation as a multiplication of two fractions. We need to find the product \(a \times b\), which is \(\frac{3}{5} \times \frac{2}{7}\).

Step 3: Multiply the Fractions

To multiply two fractions, multiply the numerators together and the denominators together. For \(\frac{3}{5} \times \frac{2}{7}\), multiply the numerators: 3 and 2, and the denominators: 5 and 7. This gives us:\[ \frac{3 \times 2}{5 \times 7} = \frac{6}{35} \]

Step 4: Simplify the Fraction

Look for any common factors from the numerator and the denominator to simplify the fraction. In this case, there are no common factors, so \(\frac{6}{35}\) is already in its simplest form.

Key Concepts

Multiplying Fractions

When you multiply fractions, the goal is to combine them to form a new fraction. This process is straightforward:

• Start by multiplying the numerators, which are the top parts of the fractions.
• Then multiply the denominators, which are the bottom parts of the fractions.

The result is a new fraction. For example, if you have fractions \(\frac{3}{5}\) and \(\frac{2}{7}\), you would multiply the numerators (3 and 2) to get 6, and the denominators (5 and 7) to get 35. Thus, the product is \(\frac{6}{35}\).

Remember, multiplication of fractions is commutative, meaning you can multiply them in any order and still get the same result. Becoming comfortable with multiplying fractions is essential for tackling larger math problems that involve fractional values.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1.

To simplify a fraction:

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

For instance, consider the fraction \(\frac{6}{35}\). Both numbers do not share any common factors except 1, which means it is already in its simplest form. Simplifying fractions might not always change the numbers much, but it does make them easier to understand and work with in subsequent math problems.
In many cases, you may not need to look for the GCF explicitly if the numbers are small. However, always keep an eye out to ensure you've reached the simplest form possible.

Fraction Arithmetic

Fraction arithmetic involves operations like addition, subtraction, multiplication, and division with fractions. Each operation has its own rules and methods.

Using multiplication as an example, the main operations often involve:

• Multiplying the numerators for the product's numerator.
• Multiplying the denominators for the product's denominator.
• Then simplifying the resulting fraction if necessary.

This applies specifically to multiplication, but each arithmetic operation has its own method. Always ensure you have a solid understanding of one operation before moving to another. For instance, addition requires a common denominator, which can be quite different from just multiplying straight across.
The key is to understand each process and practice frequently. The more you practice fraction arithmetic, the more automatic these operations will become.