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Chapter 7: Problem 94

Simplify. $$ \frac{\frac{3}{5}}{6} $$

Short Answer

Expert verified
\( \frac{1}{10} \)

Step by step solution

01

Rewrite the Fraction

The given expression is \(\frac{ \frac{3}{5} }{6}\). Start by rewriting the complex fraction. Divide by 6 by multiplying \(\frac{3}{5}\) by the reciprocal of 6, which is \(\frac{1}{6}\).
02

Multiply the Fractions

Next, multiply the fractions: \(\frac{3}{5} \times \frac{1}{6}\). Multiply the numerators and denominators: \(\frac{3 \times 1}{5 \times 6}\).
03

Simplify the Product

Simplify the result: \(\frac{3 \times 1}{5 \times 6} = \frac{3}{30}\).
04

Reduce the Fraction

Reduce \(\frac{3}{30}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3: \(\frac{3 \div 3}{30 \div 3} = \frac{1}{10}\).

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Key Concepts

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

Multiplying Fractions
Simplifying complex fractions involves multiplying fractions. When multiplying fractions, you simply multiply the numerators (top numbers) together and the denominators (bottom numbers) together.
For example, let's say you need to multiply \(\frac{3}{5} \) by \(\frac{1}{6}\):
  • Multiply the numerators: 3 times 1 equals 3
  • Multiply the denominators: 5 times 6 equals 30
The result is \(\frac{3}{30}\).
Multiplying fractions is straightforward once you remember to multiply the tops together and the bottoms together.
Greatest Common Divisor
The greatest common divisor (GCD) is the largest number that can divide both the numerator and the denominator without leaving a remainder.
To reduce fractions, finding the GCD is crucial because it helps simplify the fractions to their lowest terms.
In the example \(\frac{3}{30}\), both 3 and 30 can be divided by their GCD, which is 3.

Steps to find the GCD:
  • List the factors of both the numerator and the denominator.
  • Identify the largest factor that appears in both lists.
For 3 and 30:
  • Factors of 3: 1, 3
  • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The greatest common factor is 3.
Reciprocal
The reciprocal of a number is what you multiply that number by to get 1.
For any nonzero number \(a\), the reciprocal is \(\frac{1}{a}\).
When working with fractions, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
In the exercise, we need to divide \(\frac{3}{5}\) by 6.
Instead of dividing by 6, you multiply by its reciprocal, \(\frac{1}{6}\):
\[ \frac{\frac{3}{5}}{6} = \frac{3}{5} \times \frac{1}{6} \] Multiplying by the reciprocal simplifies the process and turns division into multiplication, making the calculations more straightforward.

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Most popular questions from this chapter

Simplify. $$ \frac{\frac{2}{3}}{4} $$

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{5 x}{x+3}+\frac{x+2}{x}-\frac{6}{x^{2}+3 x} $$

Add or subtract as indicated. $$ -\frac{3}{4}+\frac{1}{12} $$

Multiply or divide as indicated. $$ \frac{x^{2}-25}{x^{2}+x-20} \cdot \frac{x^{2}+7 x+12}{x^{2}-2 x-15} $$

Solve each problem mentally. Use proportions in Exercises 23 and 24. In a certain southern state, sales tax on a purchase of \(\$ 1.50\) is \(\$ 0.12 .\) What is the sales tax on a purchase of \(\mathrm{S} 9.00 ?\)
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