Use the 3 methods to solve this problem and draw a graph. 5 T-shirts cost R120. How much will 9 cost?

Question

UpStudy AI · Accepted Answer

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.
9 T-shirts will cost R216.

Use the 3 methods to solve this problem and draw a graph. 5 T-shirts cost R120. How much will 9 cost?

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