Problem 68 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ 30\left(\frac{5}{6}\right) $$

Step-by-Step Solution

Answer

The simplified result of the expression is 25.

Step 1: Understand the Expression

The expression given is: \( 30 \left( \frac{5}{6} \right) \). This means we need to multiply the whole number 30 by the fraction \( \frac{5}{6} \).

Step 2: Replace Multiplication with Division

Since multiplying by a fraction is equivalent to multiplying by the numerator and dividing by the denominator, we can express the multiplication as a division problem: \( 30 \cdot \frac{5}{6} = \frac{30 \cdot 5}{6} \).

Step 3: Multiply the Numerators

Multiply the numerator 30 by 5: \( 30 \cdot 5 = 150 \). The expression now becomes \( \frac{150}{6} \).

Step 4: Simplify the Fraction

Divide 150 by 6 to simplify the fraction: \( 150 \div 6 = 25 \). Thus, the simplified result of the expression is 25.

Key Concepts

Simplifying Fractions

Simplifying fractions is all about reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. A simpler fraction is easier to understand and work with. When you're presented with a fraction like \( \frac{150}{6} \), your goal is to divide both the numerator and the denominator by their greatest common divisor (GCD).

In the case of \( \frac{150}{6} \), both 150 and 6 can be divided evenly by 6, which is their GCD. Performing this division, we get:

• \( 150 \div 6 = 25 \)
• \( 6 \div 6 = 1 \)

This simplifies the fraction to \( \frac{25}{1} \), which simply equals 25. Simplifying makes our problem-solving process more straightforward and less cluttered with large numbers.

Division in Multiplication

When you're multiplying fractions, you're also engaging with division. The key trick is understanding that multiplying by a fraction is the same as multiplying by its numerator and dividing by its denominator.

Take our problem: multiplying 30 by \( \frac{5}{6} \). This translates to first multiplying the number by 5 and then dividing by 6, or:

• First, compute \( 30 \times 5 = 150 \)
• Next, divide 150 by 6

This process is a core principle in fraction multiplication. Whenever you see a fraction multiplication problem, remember: multiply then divide. This understanding will streamline your operations in fraction work.

Numerators and Denominators

Numerators and denominators are the essential parts of any fraction. The numerator is the top number and tells us how many parts we have, while the denominator, at the bottom, tells us into how many parts the whole is divided.

For example, in \( \frac{5}{6} \), 5 is the numerator showing that we have 5 parts, and 6 is the denominator showing that each part is one-sixth of the whole. These two components work together to give a complete picture of the fraction or ratio.

While performing operations, whether it's multiplication, division, or simplifying fractions, recognizing the role of numerators and denominators can guide you. You multiply or divide numerators with each other, and denominators with each other, but remember, the actual division (to simplify or finalize computations) occurs with the final resultant values. Understanding this organization will help decode not just simple problems, but more complex ones as well in fraction operations.