Multiply $\frac{4}{5}$ by $7 \frac{1}{3}$. Convert the product to a mixed number.
To multiply a fraction by a mixed number, we first need to convert the mixed number into an improper fraction. The mixed number given is \(7 \frac{1}{3}\). An improper fraction is formed by multiplying the whole number part (7) by the denominator (3) and adding the numerator (1), all over the original denominator. Thus, \(7 \frac{1}{3} = \frac{7 \times 3 + 1}{3} = \frac{21 + 1}{3} = \frac{22}{3}\). Now, the problem reduces to multiplying the fraction \(\frac{4}{5}\) by the improper fraction \(\frac{22}{3}\).
To multiply two fractions, we multiply their numerators together to get the new numerator and their denominators together to get the new denominator. The calculation is as follows: \[\frac{4}{5} \times \frac{22}{3} = \frac{4 \times 22}{5 \times 3} = \frac{88}{15}\] This gives us the improper fraction \(\frac{88}{15}\).
To convert the improper fraction \(\frac{88}{15}\) back into a mixed number, we divide the numerator (88) by the denominator (15). Performing the division, we find that \(88 \div 15 = 5\) with a remainder of \(88 - (15 \times 5) = 88 - 75 = 13\). The whole number part of the mixed number is 5, and the fractional part is the remainder over the original denominator, which is \(\frac{13}{15}\). Thus, the product expressed as a mixed number is \(5 \frac{13}{15}\).
Understanding how to multiply fractions and mixed numbers is essential for tasks like adjusting recipes. For example, if a recipe calls for \(7 \frac{1}{3}\) cups of flour and you decide to make \(\frac{4}{5}\) of the original batch to fit your smaller baking pan, you would calculate \(7 \frac{1}{3} \times \frac{4}{5} = 5 \frac{13}{15}\) cups of flour. Mastering these calculations ensures that your proportions remain correct, helping you achieve perfect results every time you cook or bake.