Sonia was asked to find the sum of all the prime numbers between 10 and 40. She found the sum to be 180. Then which of the following statements is true?
(i) She missed one prime number between 10 and 20
(ii) She added one extra prime number between 10 and 20
(iii) She missed one prime number between 20 and 30
(iv) Her total is correct

Options

A. (i)
B. (ii)
C. (iii)
D. (iv)

Answer

To solve the problem, we need to find the sum of all the prime numbers between 10 and 40 and verify Sonia's result of 180.

### Step-by-Step Solution:

1. **Identify the Prime Numbers between 10 and 40:**
- The prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
- The prime numbers between 10 and 40 are:
- 11, 13, 17, 19, 23, 29, 31, 37.

2. **List the Prime Numbers:**
- Let's write down the identified prime numbers:
- 11, 13, 17, 19, 23, 29, 31, 37.

3. **Calculate the Sum of the Prime Numbers:**
- Now, we will add these prime numbers together:
- \( 11 + 13 = 24 \)
- \( 24 + 17 = 41 \)
- \( 41 + 19 = 60 \)
- \( 60 + 23 = 83 \)
- \( 83 + 29 = 112 \)
- \( 112 + 31 = 143 \)
- \( 143 + 37 = 180 \)

4. **Verify Sonia's Result:**
- After calculating, we find that the sum of all the prime numbers between 10 and 40 is indeed 180.

5. **Evaluate the Statements:**
- (i) She missed one prime number between 10 and 20: **False** (She included all).
- (ii) She added one extra prime number between 10 and 20: **False** (She did not add any extra).
- (iii) She missed one prime number between 20 and 30: **False** (She included all).
- (iv) Her total is correct: **True** (The sum is indeed 180).

### Conclusion:
The correct statement is (iv) Her total is correct.