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Chapter 1: Problem 27

Multiply or divide as indicated. See Examples 4 and 5 . $$ 3\left(-\frac{1}{18}\right) $$

Short Answer

Expert verified
The result is \(-\frac{1}{6}\).

Step by step solution

01

Identify the Operation

The operation indicated is multiplication, as signified by the expression \( 3\left(-\frac{1}{18}\right) \).
02

Understand Sign Rules

Multiplying a positive number by a negative number results in a negative product.
03

Multiply the Numbers

Multiply the absolute values: \( 3 \times \frac{1}{18} = \frac{3}{18} \).
04

Simplify the Fraction

Simplify \( \frac{3}{18} \) by finding the greatest common divisor (GCD) of 3 and 18. The GCD is 3, so divide both the numerator and the denominator by 3: \( \frac{3}{18} = \frac{1}{6} \).
05

Apply the Sign

Since the original expression involved a negative number, the final answer is negative: \( -\frac{1}{6} \).

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Key Concepts

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

Sign Rules in Multiplication
When multiplying numbers, it's crucial to understand how the signs of the numbers affect the result. Here's a simple rule to follow:
  • When you multiply two numbers with the same sign, the result is positive.
  • When you multiply two numbers with different signs, the result is negative.
In our example, we have a positive number, 3, and a negative number, \(-\frac{1}{18}\). Since their signs are different, the result of multiplying them is negative. Remembering these sign rules will always give you the correct sign of your product.
Simplifying Fractions
Simplifying fractions means expressing the fraction in its simplest form. This makes it easier to understand and compare with other fractions.
  • To simplify a fraction, find a number that divides both the numerator and the denominator without a remainder. This number is known as a factor.
  • Keep dividing both numbers by common factors until there are no more common factors other than 1.
In our exercise, we simplified \(\frac{3}{18}\) to \(\frac{1}{6}\). Both 3 and 18 have a factor of 3, so we divided the numerator and the denominator by 3 to achieve the simplest form. Simplification is an important step, as it makes fractions more manageable and easier to interpret.
Greatest Common Divisor
The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. To simplify a fraction, you often need to find the GCD of the numerator and the denominator.
  • List the factors of each number.
  • Identify the greatest number that appears in both lists.
For example, the GCD of 3 and 18 is 3, because:
  • The factors of 3 are: 1, 3
  • The factors of 18 are: 1, 2, 3, 6, 9, 18
The largest factor common to both is 3, making it the GCD. Using the GCD to simplify fractions guarantees that you reach the simplest form efficiently.

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