M : N = 2 : 7 and M : O = 4 : 3, then what is (M + N) : (N + O)?

Options

A. `18 : 17`
B. `17 :19`
C. `17 : 7`
D. `18 : 3`

Answer

To solve the problem, we need to find the ratio \( (M + N) : (N + O) \) given the ratios \( M : N = 2 : 7 \) and \( M : O = 4 : 3 \).

### Step-by-Step Solution:

1. **Express N in terms of M**:
From the ratio \( M : N = 2 : 7 \), we can express \( N \) in terms of \( M \):
\[ \frac{M}{N} = \frac{2}{7} \implies N = \frac{7M}{2} \]

**Hint**: Use cross-multiplication to isolate \( N \).

2. **Express O in terms of M**:
From the ratio \( M : O = 4 : 3 \), we can express \( O \) in terms of \( M \):
\[ \frac{M}{O} = \frac{4}{3} \implies O = \frac{3M}{4} \]

**Hint**: Again, use cross-multiplication to isolate \( O \).

3. **Calculate \( M + N \)**:
Now, we substitute the value of \( N \) into \( M + N \):
\[ M + N = M + \frac{7M}{2} = \frac{2M}{2} + \frac{7M}{2} = \frac{9M}{2} \]

**Hint**: Find a common denominator to combine the terms.

4. **Calculate \( N + O \)**:
Next, we substitute the values of \( N \) and \( O \) into \( N + O \):
\[ N + O = \frac{7M}{2} + \frac{3M}{4} \]
To add these, we need a common denominator. The least common multiple of 2 and 4 is 4:
\[ N + O = \frac{14M}{4} + \frac{3M}{4} = \frac{17M}{4} \]

**Hint**: Convert both fractions to have the same denominator before adding.

5. **Form the ratio \( (M + N) : (N + O) \)**:
Now we can form the ratio:
\[ (M + N) : (N + O) = \frac{9M}{2} : \frac{17M}{4} \]
To simplify this ratio, we can multiply both sides by 4 to eliminate the fractions:
\[ = \frac{9M \cdot 4}{2} : 17M = \frac{36M}{2} : 17M = 18 : 17 \]

**Hint**: When simplifying ratios, you can multiply both sides by the same number to eliminate fractions.

### Final Answer:
The ratio \( (M + N) : (N + O) \) is \( 18 : 17 \).