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Chapter 7: Problem 51

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

Short Answer

Expert verified
2.5

Step by step solution

01

Write the Division as Multiplication

To divide fractions, multiply by the reciprocal of the second fraction. Thus, \[\frac{3 b}{8 d} \div \frac{3 b}{20 d} = \frac{3 b}{8 d} \times \frac{20 d}{3 b}\]
02

Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together separately: \[\frac{3 b \times 20 d}{8 d \times 3 b}\]
03

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':\[\frac{20}{8} = 2.5\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

For exercises \(41-44\), the formula \(R=\frac{V C}{T}\) describes the flow rate of fluid \(R\) through an intravenous drip. Is the relationship of the given variables a direct variation or an inverse variation? $$ V \text { and } C \text { are constant; the relationship of } R \text { and } T \text {. } $$

The rulebook of the U.S. Lawn Mower Racing Association describes how to award points. 100 points for registration 100 points for starting a race 100 points for finishing a race 300 points for first place 250 points for second place 200 points for third place 150 points for fourth place 100 points for fifth place Source: www.letsmow.com A lawn mower racer registered for a day of racing. She started and completed three races. She placed fourth in the first race, third in the second race, and first in the final race. Find the total number of points she earned.

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=4, y=5\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=10\).

The relationship of \(x\) and \(y\) is a direct variation. When \(x=2, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this direct variation. c. Find \(y\) when \(x=4\). d. Use slope-intercept graphing to graph this equation. e. Use the graph to find \(y\) when \(x=5\).

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=10\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=5\).
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