Verified Answer :

  • Decompose the mixed numbers \(1 \frac{3}{4}\) and \(2 \frac{2}{2}\) into their integer and fractional parts.
  • Simplify \(2 \frac{2}{2}\) to 3, as the fraction \(\frac{2}{2}\) is equal to 1.
  • Add the integer parts and the fractional part together: \(1 + 3 + \frac{3}{4} = 4 \frac{3}{4}\).
  • The final result is \(\boxed{4 \frac{3}{4}}\).

Explanation

  1. Analysis of the Problem

We are given the expression \(1 \frac{3}{4} + 2 \frac{2}{2}\). To simplify this, we first recall that a mixed number \(a \frac{b}{c}\) is equivalent to the sum \(a + \frac{b}{c}\). This allows us to break down each part of the expression into its integer and fractional components. Understanding this representation is essential for performing arithmetic with mixed numbers.

  1. Simplifying the Terms

We process each term separately:

  1. The first term is \(1 \frac{3}{4}\), which can be written as \(1 + \frac{3}{4}\).
  2. The second term is \(2 \frac{2}{2}\). We observe that the fractional part is \(\frac{2}{2}\), which equals \(1\). Therefore, \(2 \frac{2}{2} = 2 + \frac{2}{2} = 2 + 1 = 3\).

By simplifying these terms, the original expression simplifies to \(1 + \frac{3}{4} + 3\).

  1. Final Calculation and Conclusion

Now, we add the results from the previous step:

  • Combine the integer parts: \(1 + 3 = 4\).
  • Keep the fractional part: \(\frac{3}{4}\).

Combining these, we get \(4 + \frac{3}{4} = 4 \frac{3}{4}\).
Alternatively, as an improper fraction, this is \(\frac{16}{4} + \frac{3}{4} = \frac{19}{4}\). Both representations are valid, but \(4 \frac{3}{4}\) is the most standard form for a mixed number. The final answer is \(\boxed{4 \frac{3}{4}}\).

Examples

Mixed numbers are frequently used in everyday life, especially in cooking. For instance, if a recipe calls for \(1 \frac{3}{4}\) cups of flour and you need to prepare a larger batch requiring an additional \(2 \frac{2}{2}\) cups (which simplifies to 3 cups), knowing how to combine these amounts helps you measure ingredients accurately. By simplifying each part first, you can determine that you need a total of \(4 \frac{3}{4}\) cups of flour for your recipe.

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