Question

Evaluate each expression. $$ 3 \cdot \frac{5}{6}-\frac{1}{2} $$

Short answer

The answer is 2.

Step-by-step solution

Multiply the Fraction

First, multiply the integer 3 by the fraction \(\frac{5}{6}\). This involves multiplying the numerator of the fraction by the integer and keeping the denominator the same: \(\frac{3 \times 5}{6} = \frac{15}{6}\).

Simplify the Fraction

Simplify \(\frac{15}{6}\) to its simplest form. Both the numerator and the denominator can be divided by their greatest common divisor, which is 3: \(\frac{15 \div 3}{6 \div 3} = \frac{5}{2}\).

Subtract the Fractions

To subtract \(\frac{1}{2}\) from \(\frac{5}{2}\), make sure that the denominators are the same. Since both fractions already have the same denominator, simply subtract the numerators: \(\frac{5 - 1}{2} = \frac{4}{2}\).

Simplify the Result

Simplify \(\frac{4}{2}\) to its simplest form by dividing the numerator and the denominator by their greatest common divisor, which is 2: \(\frac{4 \div 2}{2 \div 2} = 2\).

Key concepts

Multiplying Fractions

Multiplying fractions involves multiplying the numerators (top numbers) together and the denominators (bottom numbers) together. In this problem, we start by multiplying the integer 3 by the fraction \(\frac{5}{6}\). Treat the integer as a fraction by placing it over 1: \(\frac{3}{1}\). Then multiply the numerators and denominators: \(\frac{3 \times 5}{1 \times 6} = \frac{15}{6}\). This gives us a new fraction that can be simplified in the next step.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. To simplify \(\frac{15}{6}\), you find the greatest common divisor (GCD) of the numerator and denominator. The GCD of 15 and 6 is 3. Divide both the numerator and the denominator by 3: \(\frac{15 \div 3}{6 \div 3} = \frac{5}{2}\). The fraction \(\frac{5}{2}\) is now in its simplest form.

Subtracting Fractions

When subtracting fractions, it is essential for the fractions to have the same denominator. Here, we need to subtract \(\frac{1}{2}\) from \(\frac{5}{2}\). Both fractions already have the same denominator of 2, so we can directly subtract their numerators: \(\frac{5-1}{2} = \frac{4}{2}\). This leaves us with a simplified fraction ready for the final step.

Greatest Common Divisor (GCD)

The Greatest Common Divisor, or GCD, is the largest number that divides both the numerator and the denominator without leaving a remainder. To simplify \(\frac{4}{2}\), we find the GCD of 4 and 2, which is 2. Dividing the numerator and the denominator by 2 gives us the simplest form: \(\frac{4 \div 2}{2 \div 2} = 2\). This is our final answer.