Multiply or divide as indicated. Write answers in lowest terms as needed. $2\frac{5}{8}÷1\frac{15}{32}$

Verified step by step guidance

1. First, convert the mixed numbers into improper fractions. For the first mixed number \(2\left(\frac{5}{8}\right)\), multiply the whole number 2 by the denominator 8 and add the numerator 5: \(2 \times 8 + 5 = 16 + 5 = 21\). So, \(2\left(\frac{5}{8}\right) = \frac{21}{8}\).
2. Next, convert the second mixed number \(1\left(\frac{15}{32}\right)\) into an improper fraction. Multiply the whole number 1 by the denominator 32 and add the numerator 15: \(1 \times 32 + 15 = 32 + 15 = 47\). So, \(1\left(\frac{15}{32}\right) = \frac{47}{32}\).
3. Now, rewrite the original expression as a division of two improper fractions: \(\frac{21}{8} \div \frac{47}{32}\).
4. To divide fractions, multiply the first fraction by the reciprocal of the second fraction: \(\frac{21}{8} \times \frac{32}{47}\).
5. Multiply the numerators together and the denominators together: numerator \(21 \times 32\), denominator \(8 \times 47\). Then simplify the resulting fraction by finding the greatest common divisor (GCD) and dividing numerator and denominator by it to write the answer in lowest terms.

Key Concepts

Mixed Numbers and Improper Fractions

Mixed numbers combine a whole number and a fraction, such as 2(5/8). To perform multiplication or division, convert mixed numbers into improper fractions by multiplying the whole number by the denominator and adding the numerator. This simplifies calculations and ensures accuracy.

Division of Fractions

Dividing fractions involves multiplying the first fraction by the reciprocal of the second. The reciprocal is found by swapping the numerator and denominator of the divisor. This method transforms division into multiplication, making it easier to solve.

Simplifying Fractions

After performing multiplication or division, simplify the resulting fraction by dividing numerator and denominator by their greatest common divisor (GCD). Simplifying fractions ensures the answer is in lowest terms, making it clearer and more concise.