Verified Answer :

  • Convert both mixed numbers into improper fractions: \(2 \frac{1}{2} = \frac{5}{2}\) and \(2 \frac{1}{3} = \frac{7}{3}\).
  • Multiply the numerators and denominators together: \(\frac{5}{2} \times \frac{7}{3} = \frac{35}{6}\).
  • Convert the improper fraction back into a mixed number by dividing 35 by 6, which gives 5 with a remainder of 5.
  • The final result is \(\boxed{5 \frac{5}{6}}\).

Explanation

  1. Converting Mixed Numbers to Improper Fractions

To find the product of two mixed numbers, \(2 \frac{1}{2}\) and \(2 \frac{1}{3}\), we first need to express them as improper fractions. A mixed number \(A \frac{B}{C}\) is equal to \(\frac{A \times C + B}{C}\). Applying this to our two numbers:

  • For \(2 \frac{1}{2}\), the improper fraction is \(\frac{2 \times 2 + 1}{2} = \frac{5}{2}\).
  • For \(2 \frac{1}{3}\), the improper fraction is \(\frac{2 \times 3 + 1}{3} = \frac{7}{3}\).
  1. Multiplying the Fractions

Now, we multiply the two improper fractions:
\[ \frac{5}{2} \times \frac{7}{3} = \frac{5 \times 7}{2 \times 3} = \frac{35}{6} \]
The numerator is \(5 \times 7 = 35\) and the denominator is \(2 \times 3 = 6\). The fraction \(\frac{35}{6}\) is already in its simplest form, as 35 and 6 have no common factors other than 1.

  1. Converting Back to a Mixed Number

Finally, we convert the improper fraction \(\frac{35}{6}\) back into a mixed number. We perform the division \(35 \div 6\):

  • \(35 \div 6 = 5\) with a remainder of \(5\), because \(6 \times 5 = 30\) and \(35 - 30 = 5\).

Thus, \(\frac{35}{6} = 5 \frac{5}{6}\). The final answer is \(\boxed{5 \frac{5}{6}}\).

Examples

Understanding how to multiply mixed numbers is useful in many real-world situations, such as cooking or construction. For instance, if you are following a recipe that calls for \(2 \frac{1}{2}\) batches of a mixture, and each batch requires \(2 \frac{1}{3}\) cups of a specific ingredient, you would need to calculate \(2 \frac{1}{2} \times 2 \frac{1}{3}\) to determine the total amount of that ingredient to use. Mastering this operation helps ensure your measurements are accurate and your final results are consistent.

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