Problem 50 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The product \( bd \) is \( \frac{2}{21} \).

Step 1: Define Variables and Formula

In the problem, we are given the variables: \(a=\frac{3}{5}, b=\frac{2}{7}, c=\frac{3}{4},\) and \(d=\frac{1}{3}\). We are asked to find the product \(bd\). The formula for finding the product of two fractions \(x\) and \(y\) is \(xy = \frac{n_1}{d_1} \times \frac{n_2}{d_2} = \frac{n_1 \times n_2}{d_1 \times d_2}\), where \(n_1\) and \(d_1\) are the numerator and denominator of the first fraction, and \(n_2\) and \(d_2\) are the numerator and denominator of the second fraction.

Step 2: Substitute Values

Substitute the values of \(b\) and \(d\) into the product formula: \[ bd = \frac{2}{7} \times \frac{1}{3} \]

Step 3: Multiply Numerators and Denominators

Multiply the numerators together and then multiply the denominators together. The numerators are \(2\) and \(1\), and the denominators are \(7\) and \(3\).\[ bd = \frac{2 \times 1}{7 \times 3} = \frac{2}{21} \]

Step 4: Simplified Result

The resulting fraction \( \frac{2}{21} \) is already in its simplest form since it cannot be reduced any further.

Key Concepts

Fractions

Fractions represent parts of a whole. In a fraction like \( \frac{3}{4} \), the numerator is \(3\) and the denominator is \(4\). This means you have three parts out of a total of four parts.
Understanding fractions is essential because they are used everywhere, including in measurements, finance, and scientific calculations. They allow us to express numbers that are less than one whole unit.
When dealing with fractions, you often add, subtract, multiply, or divide them. Today, we'll focus on multiplication, which helps you find parts of parts. For example, if you know half of a cake is left and you eat half of that remaining portion, you eat \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \) of the entire cake.
When multiplying fractions, the process is straightforward. You multiply the numerators together and the denominators together. Let's break this down further in the next section.

Numerators and Denominators

Numerators and denominators are the two main components of every fraction. The numerator is the top number and tells you how many parts you're working with.
The denominator is the bottom number and shows how many total parts make up a whole.

• The numerator can be any integer, positive or negative.
• The denominator is always a positive integer, never zero, because you cannot divide by zero.

When multiplying fractions, you multiply the numerators to get a new numerator and the denominators to get a new denominator. In our original exercise, we find the product of \( b \) and \( d \) where \( b = \frac{2}{7} \) and \( d = \frac{1}{3} \).
You multiply the numerators (\(2\) and \(1\)) to get \(2\). Then, multiply the denominators (\(7\) and \(3\)) to get \(21\).
Thus, the product of \( b \) and \( d \) is \( \frac{2}{21} \). The next step is to see if you can make this fraction simpler.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means making the numerator and denominator as small as possible while still keeping the value of the fraction the same.
To simplify a fraction, you find the greatest common divisor (GCD) of the numerator and the denominator and divide both by this number. However, fractions like \( \frac{2}{21} \), which result from multiplying other fractions, may already be in their simplest form if there’s no common factor apart from 1 between 2 and 21.

• Check each number's divisors to confirm there’s nothing larger than 1.
• If a fraction is already simplified, it reduces your work!

Being able to quickly determine if a fraction needs simplification can save time and make your math problems easier to solve.
By keeping fractions simple, you can more easily compare, add, subtract, and multiply them in future problems.