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Chapter 4: Problem 5

Simplify. $$-\frac{2}{5} \div \frac{7}{6}$$

Short Answer

Expert verified
-\frac{12}{35}

Step by step solution

01

- Identify the Division of Fractions

The given expression is \(-\frac{2}{5} \div \frac{7}{6}\). In order to simplify this expression, the first step is to understand that dividing by a fraction is the same as multiplying by its reciprocal.
02

- Find the Reciprocal

The reciprocal of \(\frac{7}{6}\) is \(\frac{6}{7}\). This means we replace the division operation with multiplication by the reciprocal: \(-\frac{2}{5} \times \frac{6}{7}\).
03

- Multiply the Fractions

Multiply the numerators (top parts) together and the denominators (bottom parts) together: \(\text{Numerator:} -2 \times 6 = -12\) and \(\text{Denominator:} 5 \times 7 = 35\). Thus, \(-\frac{2}{5} \times \frac{6}{7} = -\frac{12}{35}\).
04

- Simplify if Necessary

In this case, \(-\frac{12}{35}\) cannot be simplified further since 12 and 35 have no common factors other than 1.

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Key Concepts

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

Reciprocal
When dividing fractions, one essential concept is the reciprocal. A reciprocal of a number or fraction is what you multiply it by to get 1. In other words, if the product of two numbers is 1, those two numbers are reciprocals of each other. For a fraction like \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). For example, the reciprocal of \(\frac{7}{6}\) is \(\frac{6}{7}\).
To divide by a fraction, you multiply by its reciprocal. This is critical to understand as it transforms the division problem into a multiplication problem, making your calculations easier.
Multiplication of Fractions
Multiplying fractions involves straightforward steps. Each fraction has a numerator and a denominator. When multiplying two fractions, you multiply the numerators together and the denominators together.
For instance, in the problem \(-\frac{2}{5} \times \frac{6}{7}\), you do the following calculations:
  • The numerators: -2 and 6 are multiplied to get -12.
  • The denominators: 5 and 7 are multiplied to get 35.
So, the product of these two fractions is \(-\frac{12}{35}\).
Always remember to multiply across the top and the bottom.
Simplifying Fractions
Once you have multiplied the fractions, the next step is simplification. This means reducing the fraction to its simplest form. A fraction is simplified when the numerator and the denominator have no common factor other than 1.
To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator and divide both by that number. However, in cases where the GCD is 1, like \(-\frac{12}{35}\), the fraction is already in its simplest form because 12 and 35 have no common factors other than 1.
Simplifying fractions helps in getting the most concise and understandable form of your answer.

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Most popular questions from this chapter

Convert the mixed number to an improper fraction. $$10 \frac{3}{5}$$

Simplify the expression. $$\frac{5 x^{4}}{15 x^{3}}$$

Convert the improper fraction to a mixed number. $$\frac{51}{10}$$

Plot the fraction on the number line. $$-\frac{3}{2}$$

Divide. Write the quotient as a mixed number. $$5281 \div 5$$
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