Problem 66 - Introductory Algebra for College Students

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$1 \frac{3}{4} \div 2 \frac{5}{8}$$

Step-by-Step Solution

Answer

The answer is \(\frac{2}{3}\).

Step 1: Converting Mixed Numbers to Improper Fractions

Convert the mixed numbers \(1 \frac{3}{4}\) and \(2 \frac{5}{8}\) into improper fractions. The formula to do this is \(whole \ number \times denominator + numerator = new \ numerator\). So, \(1 \frac{3}{4}\) becomes \(\frac{4 \times 1 + 3}{4}=\frac{7}{4}\) and \(2 \frac{5}{8}\) becomes \(\frac{8 \times 2 + 5}{8}=\frac{21}{8}\).

Step 2: Dividing the Fractions

Apply the rule that the division of fractions is equivalent to the multiplication of the first fraction by the reciprocal of the second. So, \(\frac{7}{4} ÷ \frac{21}{8}\) becomes \(\frac{7}{4} \times \frac{8}{21}\).

Step 3: Multiplying the Fractions

Multiply the fractions \(\frac{7}{4} \times \frac{8}{21}\). Multiply the numerators together to get the numerator of the result, and multiply the denominators together to get the denominator of the result. So, \(\frac{7 \times 8}{4 \times 21}=\frac{56}{84}\).

Step 4: Simplifying the Resulting Fraction

Simplify \(\frac{56}{84}\) to lowest terms by dividing the numerator and the denominator by their greatest common divisor, which is 28. So, \(\frac{56}{84}\) simplifies to \(\frac{2}{3}\).