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Chapter 4: Problem 106

Multiply, and write the answer in simplified form. \(-\frac{3}{8} \cdot \frac{4}{15}\)

Short Answer

Expert verified
-\frac{1}{10}

Step by step solution

01

Multiply the Numerators

To multiply two fractions, first multiply the numerators together. In this case, multiply \(-3\) and 4. \(-3 \times 4 = -12\).
02

Multiply the Denominators

Next, multiply the denominators together. Multiply 8 and 15. \(8 \times 15 = 120\).
03

Form the New Fraction

Combine the results from steps 1 and 2 to form the new fraction. \(-\frac{12}{120}\).
04

Simplify the Fraction

Simplify the fraction \(-\frac{12}{120}\) by finding the greatest common divisor (GCD) of 12 and 120, which is 12. Divide both the numerator and the denominator by 12. \(-\frac{12 \div 12}{120 \div 12} = -\frac{1}{10}\).

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Key Concepts

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

Simplifying Fractions
Simplifying fractions means reducing them to their simplest form.
This makes fractions easier to understand and use in calculations.
To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD).
For instance, consider a fraction \(\frac{12}{120}\).
The GCD of 12 and 120 is 12.
So, we divide both 12 and 120 by 12 to get \(\frac{1}{10}\).
We will discuss the GCD in more detail later.
Simplified fractions are more straightforward to compare, add, and subtract.
Numerator and Denominator
Understanding numerators and denominators is essential when working with fractions.
The numerator is the top part of a fraction and tells how many parts we have.
The denominator is the bottom part and tells how many parts make up a whole.
For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator.
When we multiply fractions, we multiply the numerators together and the denominators together.
In the given exercise, we multiplied the numerators \-3\ and 4 to get \-12\, and the denominators 8 and 15 to get 120.
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) is the largest number that can divide both the numerator and the denominator without leaving a remainder.
Finding the GCD is crucial for simplifying fractions.
To find the GCD of two numbers, like 12 and 120, you can use several methods:
  • **Prime Factorization**: Break down both numbers into their prime factors and find the common factors. The GCD is the product of these common factors.
  • **Euclidean Algorithm**: A more advanced method involving division and remainders.
For 12 and 120, the GCD is 12.
Dividing the numerator and denominator of \(\frac{12}{120}\) by 12, we get the simplified fraction \(\frac{1}{10}\).
Simplifying using the GCD ensures the fraction is in its simplest form, making it easier to work with.

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Most popular questions from this chapter

In the following exercises, perform the indicated operations and simplify. $$\frac{11}{12}-\frac{2}{3}$$

In the following exercises, perform the indicated operation. $$ 9 \frac{13}{20}-4 \frac{11}{20} $$

In the following exercises, subtract. $$-\frac{31}{30}-\frac{7}{30}$$

Convert the mixed number to an improper fraction. $$ 3 \frac{2}{7} $$

In the following exercises, translate and solve. The quotient of \(p\) and \(-4\) is \(-8 .\)
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