Answer
- Oil costs R34.95 per litre.
- Petrol costs R12.67 per litre.
- The LCM of \( abc, a^2b, bc^2 \) is \( a^2bc^2 \).
- The LCM of \( 2a^2 + 2a - 4, 3a^2 - 3, 6a + 12 \) is \( 6(a - 1)(a + 2)(a + 1) \).
Solution
Let's break down the problem step by step.
### Problem 6.5: Cost of Oil and Petrol
**Known Conditions:**
1. The total cost of 2 litres of oil and 30 litres of petrol is R450.
2. The cost of petrol alone is R380.
**Let:**
- \( x \) = cost of 1 litre of oil
- \( y \) = cost of 1 litre of petrol
**From the information given, we can set up the following equations:**
1. \( 2x + 30y = 450 \) (Equation 1)
2. \( 30y = 380 \) (Equation 2)
**Step 1: Solve Equation 2 for \( y \):**
\[
y = \frac{380}{30}
\]
**Step 2: Calculate \( y \):**
\[
y = \frac{380}{30} = 12.67
\]
**Step 3: Substitute \( y \) back into Equation 1 to find \( x \):**
\[
2x + 30(12.67) = 450
\]
\[
2x + 380.1 = 450
\]
\[
2x = 450 - 380.1
\]
\[
2x = 69.9
\]
\[
x = \frac{69.9}{2} = 34.95
\]
**Final Results:**
- Cost of 1 litre of oil \( x \) = R34.95
- Cost of 1 litre of petrol \( y \) = R12.67
---
### Problem 7: Calculate the LCM
#### 7.1.1 LCM of \( abc, a^2b, bc^2 \)
**Step 1: Identify the highest powers of each variable:**
- For \( a \): highest power is \( a^2 \)
- For \( b \): highest power is \( b^1 \) (from \( abc \))
- For \( c \): highest power is \( c^2 \)
**Step 2: Write the LCM:**
\[
\text{LCM} = a^2 b c^2
\]
#### 7.1.2 LCM of \( 2a^2 + 2a - 4, 3a^2 - 3, 6a + 12 \)
**Step 1: Factor each expression:**
1. \( 2a^2 + 2a - 4 = 2(a^2 + a - 2) = 2(a - 1)(a + 2) \)
2. \( 3a^2 - 3 = 3(a^2 - 1) = 3(a - 1)(a + 1) \)
3. \( 6a + 12 = 6(a + 2) \)
**Step 2: Identify the unique factors:**
- From \( 2(a - 1)(a + 2) \): factors are \( 2, (a - 1), (a + 2) \)
- From \( 3(a - 1)(a + 1) \): factors are \( 3, (a - 1), (a + 1) \)
- From \( 6(a + 2) \): factors are \( 6, (a + 2) \)
**Step 3: Take the highest power of each factor:**
- \( 2 \) from \( 2 \)
- \( 3 \) from \( 3 \)
- \( (a - 1) \) from \( 3(a - 1)(a + 1) \)
- \( (a + 2) \) from \( 6(a + 2) \)
- \( (a + 1) \) from \( 3(a - 1)(a + 1) \)
**Step 4: Write the LCM:**
\[
\text{LCM} = 6(a - 1)(a + 2)(a + 1)
\]
### Summary of Results:
- Cost of oil: R34.95
- Cost of petrol: R12.67
- LCM of \( abc, a^2b, bc^2 \): \( a^2bc^2 \)
- LCM of \( 2a^2 + 2a - 4, 3a^2 - 3, 6a + 12 \): \( 6(a - 1)(a + 2)(a + 1) \)
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