Verified Answer :
- For \(\frac{2}{5} + \frac{3}{10}\), convert to common denominator 10 to get \(\frac{4+3}{10} = \frac{7}{10}\).
- For \(2 \frac{1}{4} - 1 \frac{1}{2}\), convert to improper fractions \(\frac{9}{4} - \frac{6}{4} = \frac{3}{4}\).
- For \(\frac{7}{8} - \frac{1}{4}\), convert to common denominator 8 to get \(\frac{7-2}{8} = \frac{5}{8}\).
- For \(\frac{1}{6} + \frac{1}{4} + \frac{1}{8}\), convert to common denominator 24 to get \(\frac{4+6+3}{24} = \frac{13}{24}\).
- Final results: (i) \(\boxed{\frac{7}{10}}\), (ii) \(\boxed{\frac{3}{4}}\), (iii) \(\boxed{\frac{5}{8}}\), (iv) \(\boxed{\frac{13}{24}}\).
Explanation
- Problem Analysis
We are given four arithmetic expressions involving fractions and mixed numbers to compute. To add or subtract fractions, we must first find a common denominator, which is the least common multiple (LCM) of the given denominators. Once the fractions share a common denominator, we can add or subtract their numerators while keeping the denominator the same. Finally, we should ensure each result is expressed in its simplest form.
- Solving Expression (i)
For the first expression, \(\frac{2}{5} + \frac{3}{10}\), the denominators are 5 and 10. The least common multiple of 5 and 10 is 10. We convert \(\frac{2}{5}\) to a fraction with a denominator of 10 by multiplying the numerator and denominator by 2:
\[\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}.\]
Now we add the fractions:
\[\frac{4}{10} + \frac{3}{10} = \frac{4 + 3}{10} = \frac{7}{10}.\]
The result is \(\frac{7}{10}\).
- Solving Expression (ii)
For the second expression, \(2 \frac{1}{4} - 1 \frac{1}{2}\), it is helpful to convert the mixed numbers into improper fractions.
For \(2 \frac{1}{4}\): \(2 \frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4}\).
For \(1 \frac{1}{2}\): \(1 \frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2}\).
The common denominator for 4 and 2 is 4. Converting \(\frac{3}{2}\) to have a denominator of 4:
\[\frac{3}{2} = \frac{3 \times 2}{2 \times 2} = \frac{6}{4}.\]
Subtracting the fractions:
\[\frac{9}{4} - \frac{6}{4} = \frac{9 - 6}{4} = \frac{3}{4}.\]
The result is \(\frac{3}{4}\).
- Solving Expression (iii)
For the third expression, \(\frac{7}{8} - \frac{1}{4}\), the denominators are 8 and 4. The least common multiple is 8. Converting \(\frac{1}{4}\) to have a denominator of 8:
\[\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}.\]
Subtracting the fractions:
\[\frac{7}{8} - \frac{2}{8} = \frac{7 - 2}{8} = \frac{5}{8}.\]
The result is \(\frac{5}{8}\).
- Solving Expression (iv)
For the fourth expression, \(\frac{1}{6} + \frac{1}{4} + \frac{1}{8}\), the denominators are 6, 4, and 8. The least common multiple of 6, 4, and 8 is 24.
Converting each fraction:
\[\frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24}, \quad \frac{1}{4} = \frac{1 \times 6}{4 \times 6} = \frac{6}{24}, \quad \frac{1}{8} = \frac{1 \times 3}{8 \times 3} = \frac{3}{24}.\]
Adding the fractions:
\[\frac{4}{24} + \frac{6}{24} + \frac{3}{24} = \frac{4 + 6 + 3}{24} = \frac{13}{24}.\]
The result is \(\frac{13}{24}\).
- Conclusion
We have computed all four expressions:
(i) \(\frac{2}{5} + \frac{3}{10} = \frac{7}{10}\)
(ii) \(2 \frac{1}{4} - 1 \frac{1}{2} = \frac{3}{4}\)
(iii) \(\frac{7}{8} - \frac{1}{4} = \frac{5}{8}\)
(iv) \(\frac{1}{6} + \frac{1}{4} + \frac{1}{8} = \frac{13}{24}\)
These results are in their simplest form.
Examples
Learning to manipulate fractions with common denominators is essential for many real-life applications. For example, if you are following a recipe that calls for \(\frac{1}{4}\) cup of flour and then you add \(\frac{1}{3}\) cup more, you need to find a common denominator (12) to determine you have \(\frac{3}{12} + \frac{4}{12} = \frac{7}{12}\) cup in total. Similarly, if you are measuring boards for a DIY project and need to subtract \(\frac{3}{8}\) inch from a \(\frac{7}{8}\) inch piece, the common denominator allows you to quickly calculate the remaining length. Mastering these skills helps with cooking, construction, and managing finances involving fractional parts.