Multiply. Write your answer as a fraction.
$\frac{5}{9} \times 6$
To multiply the fraction \(\frac{5}{9}\) by the integer \(6\), we follow the standard rule for multiplying a fraction by a whole number. The whole number \(6\) can be thought of as a fraction with a denominator of \(1\), so \(6 = \frac{6}{1}\). Multiplication then becomes \(\frac{5}{9} \times \frac{6}{1} = \frac{5 \times 6}{9 \times 1}\). Performing the multiplication in the numerator, we get \(5 \times 6 = 30\), and in the denominator, \(9 \times 1 = 9\). Thus, the product is \(\frac{30}{9}\).
To simplify the fraction \(\frac{30}{9}\), we look for the greatest common divisor (GCD) of both the numerator (\(30\)) and the denominator (\(9\)). The divisors of \(30\) are \(1, 2, 3, 5, 6, 10, 15, 30\), and the divisors of \(9\) are \(1, 3, 9\). The largest common divisor is \(3\). We divide both the numerator and the denominator by \(3\):
\[\frac{30 \div 3}{9 \div 3} = \frac{10}{3}\]
Since \(10\) and \(3\) have no common factors other than \(1\), the fraction is now in its simplest form.
Imagine you are following a recipe that calls for \(\frac{5}{9}\) of a cup of sugar for one serving. If you decide to make 6 servings, you need to multiply the quantity by 6 to determine the total amount of sugar required. By calculating \(\frac{5}{9} \times 6\), you find that you need \(\frac{10}{3}\) cups, or \(3\frac{1}{3}\) cups of sugar, ensuring your recipe remains perfectly balanced and delicious!