Problem 124 - Beginning and Intermediate Algebra

Question

Write a numerical expression for each phrase, and simplify the expression. The product of \(-\frac{2}{3}\) and \(-\frac{1}{5},\) divided by \(\frac{1}{7}\)

Step-by-Step Solution

Answer

\[ \frac{14}{15} \]

Step 1: Identify the mathematical operations

First, determine the operations required. The phrase 'The product of \(-\frac{2}{3}\) and \(-\frac{1}{5},\) divided by \frac{1}{7}\', includes multiplication and division.

Step 2: Write the expression

Translate the phrase into a numerical expression. This results in: \[ \( -\frac{2}{3} \times -\frac{1}{5}\) \div \frac{1}{7} \]

Step 3: Simplify the multiplication

First, simplify the multiplication part of the expression: \[ \-\frac{2}{3} \times -\frac{1}{5} = \frac{2 \times 1}{3 \times 5} = \frac{2}{15} \]

Step 4: Rewrite the division as multiplication

To divide by \frac{1}{7}, multiply by its reciprocal: \[ \frac{2}{15} \times 7 = \frac{2 \times 7}{15} \]

Step 5: Simplify the final expression

Complete the multiplication: \[ \frac{2 \times 7}{15} = \frac{14}{15} \]

Key Concepts

Multiplication of Fractions

When multiplying fractions, you simply multiply the numerators (top numbers) together and the denominators (bottom numbers) together.
For example: \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \).
As seen in the given problem, to multiply \( -\frac{2}{3} \) and \( -\frac{1}{5} \), follow this rule: \( -\frac{2}{3} \times -\frac{1}{5} = \frac{-2 \times -1}{3 \times 5} = \frac{2}{15} \).
The negative signs cancel each other out, resulting in a positive fraction.
This simplified result, \( \frac{2}{15} \), becomes the output of our multiplication process.

Division of Fractions

Dividing fractions might seem tricky, but it's quite simple when you understand the basic rule: To divide by a fraction, multiply by its reciprocal.
A reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \).
So, when we see a division sign like \( \div \frac{1}{7} \), we know we need to multiply by \( \frac{7}{1} \).
Applying this to our previous result: \( \frac{2}{15} \div \frac{1}{7} \), we instead do \( \frac{2}{15} \times \frac{7}{1} = \frac{2 \times 7}{15} = \frac{14}{15} \).
And thus, our final simplified expression is \( \frac{14}{15} \).

Reciprocal in Mathematics

A reciprocal is essentially flipping a fraction. If you have \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
This comes in handy, particularly with division of fractions.
For example, dividing by \( \frac{1}{7} \) is the same as multiplying by \( 7 \).
Think of it like this: Instead of splitting a whole into even more parts, you see how many times a slightly larger part (like 7, in this instance) can fit.
In our given problem, this step is crucial to turning the division into a multiplication and simplifying our work.
It's these little rules that make working with fractions much simpler and more consistent.