Problem 75 - Algebra 1: Concepts and Skills

Question

Simplify. $$ \frac{5}{6} \cdot \frac{9}{4} \cdot \frac{1}{3} \div \frac{1}{2} $$

Step-by-Step Solution

Answer

The simplification of the given complex fraction is \( \frac{5}{4} \)

Step 1: Convert the division into multiplication

Convert the division operation to multiplication by taking the reciprocal of the fraction: \( \frac{5}{6} \times \frac{9}{4} \times \frac{1}{3} \times 2 \)

Step 2: Multiply the fractions

Multiply the fractions one-by-one. Multiply the numerators and denominators separately to get a single fraction: \( \frac{5 \times 9 \times 1 \times 2}{6 \times 4 \times 3} = \frac{90}{72}\)

Step 3: Simplify the fraction

Divide the numerator and denominator by the greatest common divisor (18) to simplify the fraction: \( \frac{90}{72} = \frac{5}{4}\)

Key Concepts

Multiplication of Fractions

When multiplying fractions, you multiply the numerators together and the denominators together. This will give you a single fraction. For example, in our exercise, we have fractions such as \( \frac{5}{6} \) and \( \frac{9}{4} \). Multiply:

• Numerators: \( 5 \times 9 = 45 \)
• Denominators: \( 6 \times 4 = 24 \)

So, \( \frac{5}{6} \times \frac{9}{4} = \frac{45}{24} \). This same process is carried out when multiplying multiple fractions. Simply line them up, multiply the numerators and denominators, and write down the answer as a single fraction. However, it's good to simplify along the way to keep the numbers manageable.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a crucial concept when simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder.
In our exercise, after multiplying the fractions, we were left with \( \frac{90}{72} \). To simplify this fraction, we determined the GCD of 90 and 72, which is 18.

• Divide the numerator by 18: \( 90 \div 18 = 5 \)
• Divide the denominator by 18: \( 72 \div 18 = 4 \)

This gives us the simplified fraction of \( \frac{5}{4} \). Finding the GCD can sometimes involve listing the factors of each number and choosing the largest factor common to both.

Reciprocal of Fractions

The reciprocal of a fraction is a number that when multiplied with the original fraction results in 1. It is found by swapping the numerator and the denominator.
In this exercise, we had to handle division by a fraction \( \frac{1}{2} \). To turn it into multiplication, we took its reciprocal, which is 2. Therefore, \( \frac{1}{2} \) becomes 2 in the expression.

• For a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).
• Multiplying \( \frac{a}{b} \) by \( \frac{b}{a} \) gives 1, as \( \frac{a \times b}{b \times a} \), simplifies to \( \frac{ab}{ab} = 1 \).

Using reciprocals is a handy technique to simplify division into a multiplication problem, making it easier to follow through with the calculations.