Of three positive numbers, the ratio of first and second is 3 : 4, that of second and third is 5 : 6 and the product of second and third is 4320. What is the sum of the three numbers?

Options

A. 405
B. 531
C. 177
D. 117

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the ratios
Let the three positive numbers be represented as:
- First number = \( a \)
- Second number = \( b \)
- Third number = \( c \)

According to the problem, we have the following ratios:
1. The ratio of the first and second numbers is \( \frac{a}{b} = \frac{3}{4} \).
2. The ratio of the second and third numbers is \( \frac{b}{c} = \frac{5}{6} \).

### Step 2: Express the numbers in terms of a variable
From the ratios, we can express \( a \) and \( c \) in terms of \( b \):
- From \( \frac{a}{b} = \frac{3}{4} \), we can write:
\[ a = \frac{3}{4}b \]

- From \( \frac{b}{c} = \frac{5}{6} \), we can write:
\[ c = \frac{6}{5}b \]

### Step 3: Use the product of the second and third numbers
We know that the product of the second and third numbers is given as:
\[ b \cdot c = 4320 \]
Substituting \( c \) from the previous step:
\[ b \cdot \left(\frac{6}{5}b\right) = 4320 \]
This simplifies to:
\[ \frac{6}{5}b^2 = 4320 \]

### Step 4: Solve for \( b \)
To eliminate the fraction, multiply both sides by 5:
\[ 6b^2 = 21600 \]
Now, divide both sides by 6:
\[ b^2 = 3600 \]
Taking the square root of both sides gives:
\[ b = 60 \]

### Step 5: Find \( a \) and \( c \)
Now that we have \( b \), we can find \( a \) and \( c \):
- For \( a \):
\[ a = \frac{3}{4} \cdot 60 = 45 \]

- For \( c \):
\[ c = \frac{6}{5} \cdot 60 = 72 \]

### Step 6: Calculate the sum of the three numbers
Now we can find the sum of the three numbers:
\[ a + b + c = 45 + 60 + 72 = 177 \]

### Final Answer
The sum of the three numbers is \( \boxed{177} \).

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