A dishonest dealer sells the goods at 10% profit on cost price but uses false meter. So his over all profit is 30% find the cm. In 1 m that he used.

Answer

To solve the problem step by step, we need to find out how much the dishonest dealer measures in centimeters when he claims to sell 1 meter of goods.

### Step 1: Understand the given information
- The dealer sells goods at a 10% profit on the cost price.
- His overall profit is 30%.
- We need to find out how many centimeters he actually uses when he claims to sell 1 meter (100 cm).

### Step 2: Define the cost price
Let’s assume the cost price (CP) of 1 cm of the goods is 1 rupee. Therefore, the cost price of 100 cm (1 meter) is:
\[ \text{CP of 1 meter} = 100 \text{ cm} \times 1 \text{ rupee/cm} = 100 \text{ rupees} \]

### Step 3: Calculate the selling price with 10% profit
If the dealer sells at a 10% profit, the selling price (SP) for 100 cm is:
\[ \text{SP} = \text{CP} + 10\% \text{ of CP} \]
\[ \text{SP} = 100 + 0.10 \times 100 = 100 + 10 = 110 \text{ rupees} \]

### Step 4: Calculate the actual profit based on the overall profit
The overall profit is given as 30%. Therefore, the actual selling price (SP) based on the overall profit is:
\[ \text{Actual SP} = \text{CP} + 30\% \text{ of CP} \]
\[ \text{Actual SP} = 100 + 0.30 \times 100 = 100 + 30 = 130 \text{ rupees} \]

### Step 5: Set up the relationship between SP and the actual measured length
Now, we have two selling prices:
- Selling price based on 10% profit: 110 rupees
- Actual selling price based on 30% profit: 130 rupees

To find the relationship, we can set up a proportion:
\[ \frac{110 \text{ rupees}}{130 \text{ rupees}} = \frac{100 \text{ cm}}{x \text{ cm}} \]
Where \( x \) is the actual length in centimeters that the dealer measures when he claims to sell 1 meter.

### Step 6: Cross-multiply to solve for \( x \)
Cross-multiplying gives us:
\[ 110x = 130 \times 100 \]
\[ 110x = 13000 \]

### Step 7: Solve for \( x \)
Now, divide both sides by 110:
\[ x = \frac{13000}{110} \]
\[ x = 118.18 \text{ cm} \]

### Step 8: Calculate the actual length in centimeters
Since the dealer claims to sell 1 meter (100 cm), we need to find out how much he actually measures:
\[ \text{Actual length} = 100 \text{ cm} - x \]
\[ \text{Actual length} = 100 - 118.18 = -18.18 \text{ cm} \]

This indicates that the dealer is using a false meter that is shorter than the actual meter.

### Final Answer
The dishonest dealer measures approximately **846.15 cm** when he claims to sell 1 meter.