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Chapter 3: Problem 20

Simplify. $$ \frac{\frac{3}{8}}{12} $$

Short Answer

Expert verified
The simplified form is \(\frac{1}{32}\).

Step by step solution

01

Understand the Problem

First, observe that the problem is to simplify the compound fraction. This involves simplifying the fraction \(\frac{\frac{3}{8}}{12}\).
02

Write the Fraction as Multiplication

Recognize that dividing by a number is the same as multiplying by its reciprocal. So, rewrite the fraction \(\frac{\frac{3}{8}}{12}\) as \(\frac{3}{8} \times \frac{1}{12}\).
03

Multiply the Numerators and Denominators

Multiply the numerators and the denominators: \(\frac{3 \times 1}{8 \times 12}\). This becomes \(\frac{3}{96}\).
04

Simplify the Fraction

Simplify the fraction \(\frac{3}{96}\) by finding the greatest common divisor (GCD) of 3 and 96, which is 3. Divide both the numerator and the denominator by their GCD: \(\frac{3 \div 3}{96 \div 3} = \frac{1}{32}\).

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Key Concepts

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

Understanding Fractions
Fractions represent a part of a whole. They are written as \(\frac{a}{b}\) where 'a' is the numerator and 'b' is the denominator. The numerator tells us how many parts we have, and the denominator tells us how many parts make up a whole. For example, \(\frac{3}{8}\) means we have 3 out of 8 parts.
This concept also applies to compound fractions, which are fractions within fractions. Simplifying compound fractions like \(\frac{\frac{3}{8}}{12}\) can seem tricky, but they follow the same basic rules. First, understand that dividing by a number is the same as multiplying by its reciprocal.
Multiplying Fractions
Multiplication of fractions is straightforward once broken down. When multiplying fractions, you multiply the numerators together and the denominators together. In our example, we converted \(\frac{\frac{3}{8}}{12}\) to \(\frac{3}{8} \times \frac{1}{12}\). This becomes \(\frac{3 \times 1}{8 \times 12}\), which simplifies to \(\frac{3}{96}\).
Keep these steps in mind when multiplying fractions:
  • Write down both fractions.
  • Multiply the numerators together.
  • Multiply the denominators together.
  • Simplify the result if possible.
This method works for any fraction multiplication, making it easier to handle even compound fractions.
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) helps simplify fractions. It's the largest number that divides both the numerator and the denominator without leaving a remainder. In our example, we arrived at \(\frac{3}{96}\).
To simplify, find the GCD of 3 and 96. The GCD is 3 because it is the largest number that divides both 3 and 96. We then divide both the numerator and the denominator by their GCD: \(\frac{3 \div 3}{96 \div 3} = \frac{1}{32}\).
Steps for finding and using the GCD:
  • List the factors of each number.
  • Identify the highest common factor.
  • Divide both numerator and denominator by the GCD.
Simplifying makes fractions easier to understand and work with in further calculations.

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Most popular questions from this chapter

Estimate each of the following by estimating each mixed numeral as a whole number or as a mixed numeral where the fraction part is \(\frac{1}{2}\) and by estimating each fraction as \(0, \frac{1}{2},\) or 1 . $$ 1 \frac{5}{8}+1 \frac{27}{28} \cdot 6 \frac{35}{74} $$

Add and simplify. $$ \frac{1}{10}+\frac{2}{100}+\frac{3}{1000} $$

Estimate each of the following as \(0, \frac{1}{2},\) or 1 . $$ \frac{7}{15} $$

Add and simplify. $$ \frac{3}{8}+\frac{1}{8}+\frac{2}{8} $$

Estimate each of the following as \(0, \frac{1}{2},\) or 1 . $$ \frac{4}{5} $$
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