Answer
9 T-shirts will cost R216.
Solution
Certainly! Let's solve the problem **"5 T-shirts cost R120. How much will 9 cost?"** using **three different methods** and then illustrate the relationship with a **graph**.
---
## **Method 1: Proportion (Ratio) Method**
**Step 1:** Set up the proportion based on the given information.
\[
\frac{\text{Number of T-shirts}_1}{\text{Cost}_1} = \frac{\text{Number of T-shirts}_2}{\text{Cost}_2}
\]
\[
\frac{5}{120} = \frac{9}{x}
\]
**Step 2:** Cross-multiply to solve for \( x \).
\[
5x = 120 \times 9
\]
\[
5x = 1080
\]
\[
x = \frac{1080}{5} = 216
\]
**Answer:**
\[
\boxed{R216}
\]
---
## **Method 2: Unit Price Method**
**Step 1:** Determine the cost per T-shirt.
\[
\text{Unit Price} = \frac{\text{Total Cost}}{\text{Number of T-shirts}} = \frac{120}{5} = R24 \text{ per T-shirt}
\]
**Step 2:** Multiply the unit price by the desired number of T-shirts.
\[
\text{Total Cost for 9 T-shirts} = 9 \times 24 = R216
\]
**Answer:**
\[
\boxed{R216}
\]
---
## **Method 3: Algebraic (Equation) Method**
**Step 1:** Let \( C \) be the total cost and \( T \) be the number of T-shirts. Since the cost is directly proportional to the number of T-shirts, we can write:
\[
C = k \times T
\]
**Step 2:** Find the constant of proportionality \( k \) using the given data.
\[
120 = k \times 5
\]
\[
k = \frac{120}{5} = 24
\]
**Step 3:** Use the value of \( k \) to find the cost for 9 T-shirts.
\[
C = 24 \times 9 = 216
\]
**Answer:**
\[
\boxed{R216}
\]
---
## **Graphical Representation**
To visualize the relationship between the number of T-shirts and the total cost, let's plot a graph.
### **Graph Details:**
- **X-axis (Horizontal):** Number of T-shirts
- **Y-axis (Vertical):** Total Cost (R)
- **Slope:** Represents the unit price per T-shirt (R24)
### **Plot Points:**
1. **Point A:** (0, 0) - No T-shirts cost R0.
2. **Point B:** (5, 120) - 5 T-shirts cost R120.
3. **Point C:** (9, 216) - 9 T-shirts cost R216.
### **Graph Sketch:**
```
^
|
| C (9,216)
|
|
| B (5,120)
|
|
|_______________________________>
0 5 9 Number of T-shirts
```
**Description:**
- The graph is a straight line, indicating a **linear relationship** between the number of T-shirts and the total cost.
- The **slope** of the line is R24 per T-shirt, showing a constant rate of increase.
- As you move from left to right, the cost increases proportionally with the number of T-shirts.
**Conclusion:**
All three methods consistently determine that **9 T-shirts will cost R216**. The graph visually confirms this linear relationship, ensuring the solution's accuracy.
Reviewed and approved by the UpStudy tutoring team
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