Problem 57 - Prealgebra

Question

Perform the indicated operations. $$\frac{3}{4} \div \frac{1}{2} \cdot 6$$

Step-by-Step Solution

Answer

The result is 9.

Step 1: Understand the Operations

The given expression is \( \frac{3}{4} \div \frac{1}{2} \cdot 6 \). This involves both division and multiplication of fractions.

Step 2: Division of Fractions

When dividing fractions, we multiply by the reciprocal of the second fraction. Thus, \( \frac{3}{4} \div \frac{1}{2} \) becomes \( \frac{3}{4} \times \frac{2}{1} \).

Step 3: Perform the Multiplication

Now, we multiply the fractions: \( \frac{3}{4} \times \frac{2}{1} = \frac{3 \times 2}{4 \times 1} = \frac{6}{4} \).

Step 4: Simplify the Fraction

Simplify \( \frac{6}{4} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, \( \frac{6}{4} = \frac{3}{2} \).

Step 5: Multiply by 6

Now multiply the result by 6: \( \frac{3}{2} \times 6 \). Convert 6 into a fraction: \( 6 = \frac{6}{1} \). Thus, \( \frac{3}{2} \times \frac{6}{1} = \frac{3 \times 6}{2 \times 1} = \frac{18}{2} \).

Step 6: Simplify the Final Fraction

Simplify \( \frac{18}{2} \) by dividing the numerator by the denominator: \( \frac{18}{2} = 9 \).

Key Concepts

Division of Fractions

Dividing fractions can be simpler to understand with the concept of reciprocals and multiplication. Reciprocals are numbers that, when multiplied by the original number, result in 1. For instance, the reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\). When you divide by a fraction, you're actually multiplying by its reciprocal. So, in the exercise \(\frac{3}{4} \div \frac{1}{2}\), you find the reciprocal of \(\frac{1}{2}\) to be \(\frac{2}{1}\) and then multiply:

• First, identify the reciprocal: change the divisor \(\frac{1}{2}\) to \(\frac{2}{1}\).
• Change the division into multiplication: \(\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1}\).

This method is a handy trick not only for simplifying your work but also for understanding how division operates with fractions.

Multiplication of Fractions

When multiplying fractions, the process is straightforward and involves two major steps: multiplying the numerators together and multiplying the denominators together. In the example \(\frac{3}{4} \times \frac{2}{1}\), you multiply straight across:

• Numerators: \(3 \times 2 = 6\).
• Denominators: \(4 \times 1 = 4\).

This gives you \(\frac{6}{4}\), a result that often needs simplifying. Remember, always multiply straight across and simplify afterwards if possible. To complete the exercise, you then multiply by the whole number 6, which we convert into a fraction: \(6 = \frac{6}{1}\). Following the same multiplication rules gives \(\frac{3}{2} \times \frac{6}{1} = \frac{18}{2}\).

Simplifying Fractions

After performing fractional operations, simplification is often necessary to express the fraction in its simplest form. Simplification involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing each by this number.

• For \(\frac{6}{4}\), the GCD is 2. Divide the numerator and denominator by 2: \(\frac{6 \div 2}{4 \div 2} = \frac{3}{2}\).
• For \(\frac{18}{2}\), divide both by 2: \(\frac{18 \div 2}{2 \div 2} = \frac{9}{1} = 9\).

Simplifying makes fractions easier to interpret and use in further calculations. Always check if your result can be reduced to a simpler form to ensure you have the most concise expression.