A sum was split into three parts. The first part was lent at 10% per annum for 4 years. The second part was lent at 20% per annum for 6 years. The third part was lent at 30% per annum for 5 years. Each part was lent at simple interest and the same amount of simple interest was realised from each. Find the ratio of the first, second and third parts.

Options

A. `15:5:2`
B. `20:7:2`
C. `15:5:4`
D. `20:9:4`

Answer

To solve the problem step by step, we will find the ratio of the three parts lent at different interest rates and durations, given that the simple interest earned from each part is the same.

### Step 1: Define the parts
Let the three parts of the sum be denoted as \( X_1 \), \( X_2 \), and \( X_3 \).

### Step 2: Write the formula for Simple Interest
The formula for Simple Interest (SI) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest per annum, and \( T \) is the time in years.

### Step 3: Set up the equations for each part
1. For the first part \( X_1 \) lent at 10% for 4 years:
\[ SI_1 = \frac{X_1 \times 10 \times 4}{100} = \frac{40X_1}{100} = 0.4X_1 \]

2. For the second part \( X_2 \) lent at 20% for 6 years:
\[ SI_2 = \frac{X_2 \times 20 \times 6}{100} = \frac{120X_2}{100} = 1.2X_2 \]

3. For the third part \( X_3 \) lent at 30% for 5 years:
\[ SI_3 = \frac{X_3 \times 30 \times 5}{100} = \frac{150X_3}{100} = 1.5X_3 \]

### Step 4: Set the simple interests equal
Since the simple interests from each part are equal, we can set them equal to each other:
\[ 0.4X_1 = 1.2X_2 = 1.5X_3 \]

### Step 5: Express each part in terms of a common variable
Let \( SI = k \) (a common simple interest amount). Then we can express each part as:
1. From \( 0.4X_1 = k \):
\[ X_1 = \frac{k}{0.4} = \frac{10k}{4} = \frac{5k}{2} \]

2. From \( 1.2X_2 = k \):
\[ X_2 = \frac{k}{1.2} = \frac{10k}{12} = \frac{5k}{6} \]

3. From \( 1.5X_3 = k \):
\[ X_3 = \frac{k}{1.5} = \frac{10k}{15} = \frac{2k}{3} \]

### Step 6: Find the ratio of \( X_1 : X_2 : X_3 \)
Now we have:
\[ X_1 = \frac{5k}{2}, \quad X_2 = \frac{5k}{6}, \quad X_3 = \frac{2k}{3} \]
To find the ratio, we can express them with a common denominator. The least common multiple (LCM) of 2, 6, and 3 is 6.

1. Convert \( X_1 \):
\[ X_1 = \frac{5k}{2} = \frac{15k}{6} \]

2. \( X_2 \) is already:
\[ X_2 = \frac{5k}{6} \]

3. Convert \( X_3 \):
\[ X_3 = \frac{2k}{3} = \frac{4k}{6} \]

### Step 7: Write the ratio
Now we can write the ratio:
\[ X_1 : X_2 : X_3 = \frac{15k}{6} : \frac{5k}{6} : \frac{4k}{6} \]
Cancelling \( k \) and \( 6 \) gives:
\[ 15 : 5 : 4 \]

### Final Answer
Thus, the ratio of the first, second, and third parts is:
\[ \boxed{15 : 5 : 4} \]