If A : B = 2 : 3 and B : C = 4 : 5, then what is (A + C) : B?

Options

A. `8:5`
B. `7:3`
C. `23:12`
D. `25:14`

Answer

To solve the problem, we need to find the ratio (A + C) : B given the ratios A : B = 2 : 3 and B : C = 4 : 5.

### Step-by-Step Solution:

1. **Express A and B in terms of a common variable**:
From the ratio A : B = 2 : 3, we can express A and B as:
\[ A = 2x \quad \text{and} \quad B = 3x \]
where \(x\) is a common multiplier.

**Hint**: Use a variable to represent the ratio components.

2. **Express B and C in terms of a common variable**:
From the ratio B : C = 4 : 5, we can express B and C as:
\[ B = 4y \quad \text{and} \quad C = 5y \]
where \(y\) is another common multiplier.

**Hint**: Again, use a variable to represent the ratio components.

3. **Equate the expressions for B**:
Since both expressions represent B, we can set them equal to each other:
\[ 3x = 4y \]

**Hint**: Find a relationship between the two variables by equating the expressions.

4. **Solve for one variable in terms of the other**:
From \(3x = 4y\), we can express \(y\) in terms of \(x\):
\[ y = \frac{3}{4}x \]

**Hint**: Isolate one variable to express it in terms of the other.

5. **Substitute y back to find C**:
Now substitute \(y\) back into the expression for C:
\[ C = 5y = 5 \left(\frac{3}{4}x\right) = \frac{15}{4}x \]

**Hint**: Substitute the value of y to find the expression for C.

6. **Calculate A + C**:
Now we can find \(A + C\):
\[ A + C = 2x + \frac{15}{4}x = \frac{8}{4}x + \frac{15}{4}x = \frac{23}{4}x \]

**Hint**: Combine the terms to find the total.

7. **Find the ratio (A + C) : B**:
Now we can find the ratio:
\[ (A + C) : B = \left(\frac{23}{4}x\right) : (3x) \]
Simplifying this gives:
\[ = \frac{23}{4} : 3 = \frac{23}{4} : \frac{12}{4} = 23 : 12 \]

**Hint**: Simplify the ratio to its simplest form.

### Final Answer:
The ratio \( (A + C) : B \) is \( 23 : 12 \).