Answer
**Example 3:**
a) \( \frac{1}{4} + \frac{1}{3} \times \left( \frac{4}{5} + \frac{1}{10} \right) = \frac{11}{20} \)
b) \( -\left(-\frac{1}{2}\right)^{3} - \left( -2 \frac{1}{2} \right)^{2} = -\frac{49}{8} \)
Solution
Certainly! Let's solve both parts of Example 3 step by step using the BODMAS/BIDMAS rule, which stands for:
- **B**rackets
- **O**rders (i.e., powers and roots, etc.)
- **D**ivision and **M**ultiplication
- **A**ddition and **S**ubtraction
### Example 3:
#### a) \( \frac{1}{4} + \frac{1}{3} \times \left( \frac{4}{5} + \frac{1}{10} \right) \)
**Step 1: Solve Inside the Brackets**
First, calculate the expression inside the brackets:
\[
\frac{4}{5} + \frac{1}{10}
\]
To add these fractions, find a common denominator. The least common denominator (LCD) for 5 and 10 is 10.
\[
\frac{4}{5} = \frac{8}{10} \quad \text{(since } 4 \times 2 = 8 \text{ and } 5 \times 2 = 10\text{)}
\]
Now, add the fractions:
\[
\frac{8}{10} + \frac{1}{10} = \frac{9}{10}
\]
So, the expression becomes:
\[
\frac{1}{4} + \frac{1}{3} \times \frac{9}{10}
\]
**Step 2: Perform the Multiplication**
Next, multiply \( \frac{1}{3} \) by \( \frac{9}{10} \):
\[
\frac{1}{3} \times \frac{9}{10} = \frac{9}{30} = \frac{3}{10} \quad \text{(simplified by dividing numerator and denominator by 3)}
\]
Now, the expression is:
\[
\frac{1}{4} + \frac{3}{10}
\]
**Step 3: Perform the Addition**
Add \( \frac{1}{4} \) and \( \frac{3}{10} \):
To add these fractions, find a common denominator. The LCD for 4 and 10 is 20.
\[
\frac{1}{4} = \frac{5}{20} \quad \text{and} \quad \frac{3}{10} = \frac{6}{20}
\]
Now, add them:
\[
\frac{5}{20} + \frac{6}{20} = \frac{11}{20}
\]
**Answer for part a:** \( \frac{11}{20} \)
---
#### b) \( -\left(-\frac{1}{2}\right)^{3} - \left( -2 \frac{1}{2} \right)^{2} \)
First, let's interpret the mixed number \( -2 \frac{1}{2} \):
\[
-2 \frac{1}{2} = -\frac{5}{2}
\]
So the expression becomes:
\[
-\left(-\frac{1}{2}\right)^{3} - \left(-\frac{5}{2}\right)^{2}
\]
**Step 1: Calculate Each Term Separately**
1. **First Term:** \( -\left(-\frac{1}{2}\right)^{3} \)
- Calculate the exponent:
\[
\left(-\frac{1}{2}\right)^{3} = -\frac{1}{8}
\]
- Apply the negative sign outside:
\[
-\left(-\frac{1}{8}\right) = \frac{1}{8}
\]
2. **Second Term:** \( -\left(-\frac{5}{2}\right)^{2} \)
- Calculate the exponent:
\[
\left(-\frac{5}{2}\right)^{2} = \frac{25}{4}
\]
- Apply the negative sign outside:
\[
-\frac{25}{4}
\]
**Step 2: Combine the Results**
Now, combine both terms:
\[
\frac{1}{8} - \frac{25}{4}
\]
To subtract these fractions, find a common denominator. The LCD for 8 and 4 is 8.
\[
\frac{25}{4} = \frac{50}{8}
\]
Now, subtract:
\[
\frac{1}{8} - \frac{50}{8} = -\frac{49}{8}
\]
This can also be expressed as a mixed number:
\[
-\frac{49}{8} = -6 \frac{1}{8}
\]
**Answer for part b:** \( -\frac{49}{8} \) or \( -6 \frac{1}{8} \)
Reviewed and approved by the UpStudy tutoring team
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