- Perform multiplication first: \(\frac{1}{4} \times \frac{3}{5} = \frac{3}{20}\).
- Find a common denominator to add: \(\frac{2}{5} = \frac{8}{20}\).
- Add the fractions together: \(\frac{3}{20} + \frac{8}{20} = \frac{11}{20}\).
- Final result: \(\boxed{\frac{11}{20}}\).
Explanation
- Understanding the Problem
To evaluate the expression \(\frac{1}{4} \times \frac{3}{5} + \frac{2}{5}\), we follow the standard order of operations, which dictates that multiplication must be performed before addition. The problem requires us to multiply two fractions, \(\frac{1}{4}\) and \(\frac{3}{5}\), and then add the result to \(\frac{2}{5}\). We will ensure that our calculations are precise by using a common denominator for the addition step.
- Performing Multiplication
We first perform the multiplication \(\frac{1}{4} \times \frac{3}{5}\). To multiply two fractions, we multiply their numerators together to get the new numerator and their denominators together to get the new denominator:
\[\frac{1}{4} \times \frac{3}{5} = \frac{1 \times 3}{4 \times 5} = \frac{3}{20}\]
- Performing Addition
Now we need to add the result to \(\frac{2}{5}\). The expression becomes \(\frac{3}{20} + \frac{2}{5}\). To add these, we must have a common denominator. Since 20 is a multiple of 5, we can convert \(\frac{2}{5}\) to a fraction with a denominator of 20 by multiplying both the numerator and the denominator by 4:
\[\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}\]
Now we add the two fractions:
\[\frac{3}{20} + \frac{8}{20} = \frac{3 + 8}{20} = \frac{11}{20}\]
- Conclusion
After calculating the product and performing the addition, we have obtained the final value \(\frac{11}{20}\). This fraction is already in its simplest form as 11 is a prime number and not a factor of 20. Thus, the final value is \(\boxed{\frac{11}{20}}\).
Examples
Imagine you are cooking and a recipe requires you to take a quarter of a bag of sugar that is already partially filled to three-fifths capacity. You then add another two-fifths of a bag. By understanding how to calculate and combine these fractional portions, you can accurately determine the total amount of sugar you have added to your mixture, ensuring your recipe measurements are precise. Calculations like this are essential in everyday life for everything from cooking to home improvement projects.