If M : N = 3 : 5 and M : O = 3 : 7, then what is (M + N) : (N + O)?

Options

A. `5 : 4`
B. `2 : 3`
C. `6 : 5`
D. `4 :5`

Answer

To solve the problem, we need to find the ratio (M + N) : (N + O) given the ratios M : N = 3 : 5 and M : O = 3 : 7.

### Step-by-Step Solution:

1. **Express N in terms of M using the ratio M : N = 3 : 5.**
\[ \frac{M}{N} = \frac{3}{5} \implies N = \frac{5}{3}M \]

**Hint:** Use cross multiplication to express one variable in terms of another.

2. **Express O in terms of M using the ratio M : O = 3 : 7.**
\[ \frac{M}{O} = \frac{3}{7} \implies O = \frac{7}{3}M \]

**Hint:** Again, use cross multiplication to express O in terms of M.

3. **Substitute N and O into the expression (M + N) : (N + O).**
\[ M + N = M + \frac{5}{3}M = \frac{3}{3}M + \frac{5}{3}M = \frac{8}{3}M \]
\[ N + O = \frac{5}{3}M + \frac{7}{3}M = \frac{5 + 7}{3}M = \frac{12}{3}M = 4M \]

**Hint:** Combine like terms to simplify the expressions for M + N and N + O.

4. **Now, form the ratio (M + N) : (N + O).**
\[ (M + N) : (N + O) = \frac{8}{3}M : 4M \]

5. **Simplify the ratio.**
\[ = \frac{8}{3} : 4 = \frac{8}{3} : \frac{12}{3} = \frac{8}{12} = \frac{2}{3} \]

**Hint:** To simplify a ratio, divide both parts by the common factor.

### Final Answer:
The ratio (M + N) : (N + O) is \( 2 : 3 \).