The cost of manufacturing x articles is Rs. (50+3x). The selling price of x articles is Rs. 4x.
On a graph sheet, with the same axes, and taking suitable scale draw two graphs, first for the cost of manufacturing against no. of articles and the second for the selling price against number of articles.
Use your graph to determine
(i) No. of articles to be manufactured and sold to break even (no profit and no loss),
(ii) The profit or loss made when a. 30 b. 60 articles are manufactured and sold.

Answer

To solve the problem step by step, we will first derive the equations for the cost of manufacturing and selling price, then plot the graphs, and finally determine the break-even point and profit/loss at specified quantities.

### Step 1: Define the Equations
- The cost of manufacturing \( x \) articles is given by:
\[ \text{Cost Price (CP)} = 50 + 3x \]
- The selling price of \( x \) articles is given by:
\[ \text{Selling Price (SP)} = 4x \]

### Step 2: Create a Table of Values for CP and SP
We will calculate values for \( CP \) and \( SP \) for different values of \( x \).

| \( x \) | \( CP = 50 + 3x \) | \( SP = 4x \) |
|---------|---------------------|----------------|
| 0 | \( 50 + 3(0) = 50 \) | \( 4(0) = 0 \) |
| 10 | \( 50 + 3(10) = 80 \) | \( 4(10) = 40 \) |
| 20 | \( 50 + 3(20) = 110 \)| \( 4(20) = 80 \) |
| 30 | \( 50 + 3(30) = 140 \)| \( 4(30) = 120 \)|
| 40 | \( 50 + 3(40) = 170 \)| \( 4(40) = 160 \)|
| 50 | \( 50 + 3(50) = 200 \)| \( 4(50) = 200 \)|
| 60 | \( 50 + 3(60) = 230 \)| \( 4(60) = 240 \)|

### Step 3: Plot the Graphs
1. **Graph for Cost Price (CP)**:
- Plot the points: (0, 50), (10, 80), (20, 110), (30, 140), (40, 170), (50, 200).
- Connect these points to form the line for CP.

2. **Graph for Selling Price (SP)**:
- Plot the points: (0, 0), (10, 40), (20, 80), (30, 120), (40, 160), (50, 200), (60, 240).
- Connect these points to form the line for SP.

### Step 4: Determine the Break-even Point
- The break-even point occurs where the CP and SP lines intersect.
- From the table, we see that at \( x = 50 \), both CP and SP are equal to Rs. 200.
- Therefore, the break-even point is:
\[ \text{Number of articles to break even} = 50 \]

### Step 5: Calculate Profit or Loss at Specific Quantities
1. **For 30 articles**:
- CP at \( x = 30 \) is Rs. 140.
- SP at \( x = 30 \) is Rs. 120.
- Since CP > SP, there is a loss:
\[ \text{Loss} = CP - SP = 140 - 120 = 20 \]

2. **For 60 articles**:
- CP at \( x = 60 \) is Rs. 230.
- SP at \( x = 60 \) is Rs. 240.
- Since SP > CP, there is a profit:
\[ \text{Profit} = SP - CP = 240 - 230 = 10 \]

### Final Answers
(i) The number of articles to be manufactured and sold to break even is **50 articles**.

(ii) The profit or loss made:
- For 30 articles: **Loss of Rs. 20**.
- For 60 articles: **Profit of Rs. 10**.