Problem 96 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \frac{1}{3}\left(-\frac{5}{16}\right) $$

Step-by-Step Solution

Answer

\( \frac{-5}{48} \)

Step 1: Identify the Operation

We need to multiply two fractions: \( \frac{1}{3} \) and \( -\frac{5}{16} \). Multiplying fractions involves multiplying the numerators together and the denominators together.

Step 2: Multiply the Numerators

Multiply the numerators: \( 1 \times (-5) = -5 \). This gives us the numerator of the resulting fraction.

Step 3: Multiply the Denominators

Multiply the denominators: \( 3 \times 16 = 48 \). This gives us the denominator of the resulting fraction.

Step 4: Combine the Results

Combine the results from Step 2 and Step 3 to form the new fraction: \( \frac{-5}{48} \).

Key Concepts

Understanding Numerators

In every fraction, the top number is called the numerator. It expresses how many parts of the whole we are dealing with. When you multiply two fractions together, you multiply their numerators directly. Take for example the fractions \( \frac{1}{3} \) and \( -\frac{5}{16} \). Their numerators are 1 and -5, respectively. During multiplication, you perform the following operation: \( 1 \times (-5) \), which results in -5, forming the numerator of the new fraction.

It's important to remember that the sign of the numbers should be considered at this stage. If you have a negative numerator, like in \( -\frac{5}{16} \), it directly affects the result of your multiplication, potentially resulting in a negative fraction.

Deciphering Denominators

A denominator is the bottom number of a fraction indicating into how many parts the whole is divided. When multiplying fractions, both denominators are multiplied together to form the new fraction's denominator. For the fractions \( \frac{1}{3} \) and \( -\frac{5}{16} \), their denominators are 3 and 16, respectively. Therefore, you multiply them together: \( 3 \times 16 \), which equals 48. This gives the denominator of the resultant fraction.

• Denominators determine how big each piece of the fraction is.
• The larger the denominator, the smaller each part is.

By multiplying the denominators, you determine how many total parts exist in relation to the whole.

Explaining Negative Fractions

A negative fraction has a negative sign in front of either its numerator or its denominator, or the entire fraction. In multiplication, when one fraction is negative, it influences the sign of the result.

For example, multiplying \( \frac{1}{3} \) by \( -\frac{5}{16} \), the numerator becomes -5 and the denominator is 48, resulting in \( \frac{-5}{48} \).

• If both fractions were negative, the result would be positive:
• \( -a/b \times -c/d = ac/bd \)
• If one fraction is negative, the product stays negative:
• \( a/b \times -c/d = -ac/bd \)

Always remember that a negative sign in a fraction means the value is less than zero, so its position (numerator or denominator) doesn't affect its magnitude but only its sign.