$3 \frac{2}{3}+3 \frac{2}{5}$
To calculate the sum \(3 \frac{2}{3} + 3 \frac{2}{5}\), we first analyze the structure of the expression. It is a sum of two mixed numbers. A mixed number, represented as \(A \frac{B}{C}\), is defined as \(A + \frac{B}{C}\). Therefore, we can approach this by either summing the whole numbers and fractions separately or by converting each mixed number into an improper fraction. Converting to improper fractions is often a more systematic method for addition.
First, we convert \(3 \frac{2}{3}\) to an improper fraction using the formula \(A \frac{B}{C} = \frac{A \cdot C + B}{C}\):
\[3 \frac{2}{3} = \frac{3 \cdot 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}.\]
Next, we convert \(3 \frac{2}{5}\) to an improper fraction:
\[3 \frac{2}{5} = \frac{3 \cdot 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}.\]
To add the two fractions \(\frac{11}{3}\) and \(\frac{17}{5}\), we need a common denominator. The least common multiple of 3 and 5 is 15. We rewrite each fraction with 15 as the denominator:
\[\frac{11}{3} = \frac{11 \cdot 5}{3 \cdot 5} = \frac{55}{15},\]
\[\frac{17}{5} = \frac{17 \cdot 3}{5 \cdot 3} = \frac{51}{15}.\]
Now, we add the two fractions:
\[\frac{55}{15} + \frac{51}{15} = \frac{55 + 51}{15} = \frac{106}{15}.\]
Finally, we convert the improper fraction \(\frac{106}{15}\) back to a mixed number. We divide 106 by 15:
\[106 \div 15 = 7 \text{ with a remainder of } 1,\]
because \(15 \cdot 7 = 105\). Thus, \(\frac{106}{15} = 7 \frac{1}{15}\). The sum of \(3 \frac{2}{3}\) and \(3 \frac{2}{5}\) is \(7 \frac{1}{15}\).
Suppose you are baking and need to measure ingredients. If a recipe calls for \(3 \frac{2}{3}\) cups of flour for the cake layers and \(3 \frac{2}{5}\) cups of flour for the frosting, you need to know the total amount of flour required. By adding these mixed numbers, you can determine that you need a total of \(7 \frac{1}{15}\) cups of flour to complete your recipe accurately. Correctly adding these amounts helps ensure the perfect consistency for your baked goods.