$5 \frac{3}{4} + 2 \frac{3}{5}$

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05-11-2026
Mathematics

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$5 \frac{3}{4} + 2 \frac{3}{5}$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T18:55:26-05:00

Separate the expression into integer and fractional parts: $(5 + 2) + (\frac{3}{4} + \frac{3}{5})$.
Add the integer parts to get 7.
Find a common denominator of 20 to add the fractions: $\frac{15}{20} + \frac{12}{20} = \frac{27}{20} = 1 \frac{7}{20}$.
Combine the results: $7 + 1 \frac{7}{20} = 8 \frac{7}{20}$.
The final result is $\boxed{8 \frac{7}{20}}$.

Explanation

Analysis of the problem

We need to calculate the sum of two mixed numbers, $5 \frac{3}{4}$ and $2 \frac{3}{5}$. A mixed number is a sum of an integer and a proper fraction. The expression can be written as $(5 + \frac{3}{4}) + (2 + \frac{3}{5})$. To solve this, we can add the integer parts and the fractional parts separately.

Grouping and Common Denominators

First, group the integer parts: $5 + 2 = 7$. Next, group the fractional parts: $\frac{3}{4} + \frac{3}{5}$. To add these two fractions, we must find a common denominator. The least common multiple of 4 and 5 is 20. We convert each fraction:
\[\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}\]
\[\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}\]
Summing these gives: $\frac{15}{20} + \frac{12}{20} = \frac{27}{20}$.

Final Calculation and Result

Now, convert the improper fraction $\frac{27}{20}$ back to a mixed number. Since 20 goes into 27 one time with a remainder of 7, we have $\frac{27}{20} = 1 \frac{7}{20}$. Finally, add this to the sum of the integer parts: $7 + 1 \frac{7}{20} = 8 \frac{7}{20}$. Alternatively, as an improper fraction, this is $8 \times \frac{20}{20} + \frac{7}{20} = \frac{160+7}{20} = \frac{167}{20}$. The result is $\boxed{8 \frac{7}{20}}$.

Examples

Understanding how to work with mixed numbers is essential for real-world scenarios such as construction or cooking. For example, if you are building a shelf and need two boards of length $5 \frac{3}{4}$ feet and $2 \frac{3}{5}$ feet, adding these values helps you calculate the total length of wood required for your project: $5 \frac{3}{4} + 2 \frac{3}{5} = 8 \frac{7}{20}$ feet. This skill ensures accuracy in measurements and material procurement.