Evaluate: $2 \frac{7}{10} \times \frac{5}{6}$
We are asked to compute the product of a mixed number and a fraction: \(2 \frac{7}{10} \times \frac{5}{6}\). To solve this, we first convert the mixed number into an improper fraction. A mixed number \(a \frac{b}{c}\) is equivalent to \(\frac{a \times c + b}{c}\). Here, \(a = 2, b = 7,\) and \(c = 10\), so \(2 \frac{7}{10} = \frac{2 \times 10 + 7}{10} = \frac{27}{10}\).
Now, we set up the multiplication:
\[ \frac{27}{10} \times \frac{5}{6} \]
To simplify the calculation, we can express the product as:
\[ \frac{27 \times 5}{10 \times 6} \]
We can simplify by canceling common factors. Notice that \(5\) and \(10\) share a common factor of \(5\), and \(27\) and \(6\) share a common factor of \(3\):
\[ \frac{27 \times 5}{10 \times 6} = \frac{27 \times 1}{2 \times 6} = \frac{9 \times 3 \times 1}{2 \times 2 \times 3} = \frac{9}{4} \]
So, the result of the multiplication is \(\frac{9}{4}\).
Finally, we convert the improper fraction \(\frac{9}{4}\) back into a mixed number or decimal form if needed. Since \(9 \div 4 = 2\) with a remainder of \(1\), we have:
\[ \frac{9}{4} = 2 \frac{1}{4} = 2.25 \]
Thus, the final answer is \(\boxed{2.25}\).
Suppose you are baking cookies and the recipe requires 2 7/10 cups of sugar, but you only want to make 5/6 of the full batch. To find the amount of sugar you need, you would perform the calculation \(2 \frac{7}{10} \times \frac{5}{6} = 2.25\) cups. This helps in adjusting quantities for smaller or larger portions, ensuring your baking proportions remain correct.