Read the given statements carefully and state T for true and F for false.
(i) If the simple interest on a sum is equal to the principal and number of years is equal to the rate percent per annum, then rate of interest is 10% per annum.
(ii) I a person purchases 12 items for Rs. 36 and sells 10 items for Rs. 36, then the profit % is 15%.
(iii) The percentage decrease in the priceof a shirt when it decreased from Rs. 4000 to Rs. 2500 is 15%.

Options

A. `{:(i" "ii" "iii),(T" "T" "F):}`
B. `{:(i" "ii" "iii),(T" "F" "F):}`
C. `{:(i" "ii" "iii),(F" "T" "F):}`
D. `{:(i" "ii" "iii),(F" "F" "T):}`

Explanation

To solve the problem, we need to evaluate each statement one by one and determine if they are true (T) or false (F).

### Statement (i):
**Statement:** If the simple interest on a sum is equal to the principal and the number of years is equal to the rate percent per annum, then the rate of interest is 10% per annum.

**Solution:**
1. Let the principal amount be \( P \).
2. According to the statement, the simple interest (SI) is equal to the principal, so \( SI = P \).
3. The formula for simple interest is given by:
\[ SI = \frac{P \times r \times t}{100} \]
where \( r \) is the rate of interest and \( t \) is the time in years.
4. The statement also says that the number of years \( t \) is equal to the rate \( r \). Therefore, we can set \( t = r \).
5. Substituting \( SI \) and \( t \) into the formula:
\[ P = \frac{P \times r \times r}{100} \]
6. Simplifying this, we can cancel \( P \) (assuming \( P \neq 0 \)):
\[ 1 = \frac{r^2}{100} \]
7. Rearranging gives:
\[ r^2 = 100 \]
8. Taking the square root of both sides:
\[ r = 10\% \]
9. Thus, the statement is **True (T)**.

### Statement (ii):
**Statement:** If a person purchases 12 items for Rs. 36 and sells 10 items for Rs. 36, then the profit % is 15%.

**Solution:**
1. Cost price (CP) of 12 items = Rs. 36.
2. Therefore, cost price of 1 item:
\[ CP \text{ of 1 item} = \frac{36}{12} = Rs. 3 \]
3. Cost price of 10 items:
\[ CP \text{ of 10 items} = 10 \times 3 = Rs. 30 \]
4. Selling price (SP) of 10 items = Rs. 36.
5. Profit = Selling Price - Cost Price:
\[ \text{Profit} = 36 - 30 = Rs. 6 \]
6. Profit percentage is calculated as:
\[ \text{Profit %} = \left(\frac{\text{Profit}}{\text{Cost Price}} \times 100\right) = \left(\frac{6}{30} \times 100\right) = 20\% \]
7. The statement claims the profit percentage is 15%, which is incorrect. Therefore, the statement is **False (F)**.

### Statement (iii):
**Statement:** The percentage decrease in the price of a shirt when it decreased from Rs. 4000 to Rs. 2500 is 15%.

**Solution:**
1. Initial price = Rs. 4000, Final price = Rs. 2500.
2. Decrease in price:
\[ \text{Decrease} = 4000 - 2500 = Rs. 1500 \]
3. Percentage decrease is calculated as:
\[ \text{Percentage Decrease} = \left(\frac{\text{Decrease}}{\text{Original Price}} \times 100\right) = \left(\frac{1500}{4000} \times 100\right) \]
4. Simplifying this:
\[ \text{Percentage Decrease} = \left(\frac{1500}{4000}\right) \times 100 = \frac{150}{40} = 3.75 \times 10 = 37.5\% \]
5. The statement claims the percentage decrease is 15%, which is incorrect. Therefore, the statement is **False (F)**.

### Final Answers:
1. (i) T
2. (ii) F
3. (iii) F