Problem 51 - Elementary Algebra

Question

For exercises 39-82, simplify. $$ \frac{3 b}{8 d} \div \frac{3 b}{20 d} $$

Step-by-Step Solution

Answer

2.5

Step 1: Write the Division as Multiplication

To divide fractions, multiply by the reciprocal of the second fraction. Thus, \[\frac{3 b}{8 d} \div \frac{3 b}{20 d} = \frac{3 b}{8 d} \times \frac{20 d}{3 b}\]

Step 2: Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together separately: \[\frac{3 b \times 20 d}{8 d \times 3 b}\]

Step 3: Simplify the Expression

Cancel out the common terms in the numerator and the denominator:\[\frac{(3 b) \times (20 d)}{(3 b) \times (8 d)} = \frac{20 d}{8 d}\]Next, simplify by cancelling out common factor 'd':\[\frac{20}{8} = 2.5\]

Key Concepts

fraction division

To divide fractions, you need to convert it into a multiplication problem by using the reciprocal of the second fraction. The reciprocal of a fraction is created by swapping its numerator and denominator. In this instance, we need to divide \(\frac{3b}{8d}\) by \(\frac{3b}{20d}\).
Therefore, the division becomes multiplication: \(\frac{3b}{8d} \times \frac{20d}{3b}\).
This makes the process easier because multiplication is generally more straightforward than division.

reciprocal

The reciprocal of a fraction is when you flip the fraction. Specifically, if you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). The reciprocal plays a vital role in fraction division. Instead of directly dividing the fractions, you multiply by the reciprocal of the divisor.
For example, in the problem \(\frac{3b}{8d} \div \frac{3b}{20d}\), the reciprocal of \(\frac{3b}{20d}\) is \(\frac{20d}{3b}\). Thus, the division becomes:
\(\frac{3b}{8d} \times \frac{20d}{3b}\).

simplifying expressions

Simplifying an expression involves combining like terms and reducing fractions to their simplest form. During fraction multiplication or division, simplification usually happens after multiplying the numerators and the denominators.
For our example:
\(\frac{3b \times 20d}{8d \times 3b}\),
Notice that \((3b \times 20d)\) and \((8d \times 3b)\) share common terms (3b and d), which can be canceled out to simplify the fraction.
After canceling out these common terms, we are left with \(\frac{20}{8}\), which simplifies further to 2.5.

numerator and denominator

A fraction consists of two main parts: the numerator and the denominator.
The **numerator** is the top number and represents the number of parts you have.
The **denominator** is the bottom number and signifies the total number of equal parts.
In \(\frac{3b}{8d}\), 3b is the numerator, and 8d is the denominator. Similarly in \(\frac{3b}{20d}\), 3b is the numerator, and 20d the denominator.
When performing operations with fractions, both the numerator and the denominator are essential for accurate calculations.