Question

Select all the common denominators for 1/4 and 7/10 A 10 B 20 C 24 D 32 E 40

Answer 1

To find the common denominators for the fractions \( \frac{1}{4} \) and \( \frac{7}{10} \), we first need to determine the least common multiple (LCM) of the denominators 4 and 10.

The prime factorization of 4 is:
- \( 4 = 2^2 \)

The prime factorization of 10 is:
- \( 10 = 2^1 \times 5^1 \)

To find the LCM, we take the highest power of each prime factor present in either number:
- For \( 2 \), the highest power is \( 2^2 \) (from 4).
- For \( 5 \), the highest power is \( 5^1 \) (from 10).

So, the LCM is:
\[
LCM = 2^2 \times 5^1 = 4 \times 5 = 20
\]

Next, we check the provided options to see which ones are common denominators. The options are:
- A) 10
- B) 20
- C) 24
- D) 32
- E) 40

The common denominators must be multiples of the LCM (20).

1. **10** is not a multiple of 20.
2. **20** is a multiple of 20.
3. **24** is not a multiple of 20.
4. **32** is not a multiple of 20.
5. **40** is a multiple of 20 (as \( 40 = 20 \times 2 \)).

Thus, the common denominators for \( \frac{1}{4} \) and \( \frac{7}{10} \) from the provided options are:

- **B) 20**
- **E) 40**

Therefore, the correct answers are **B and E**.