`a:b:c:d=(1)/(3):(1)/(7):(1)/(2):(1)/(5)`
Find `a:b:c:d=?`

Answer

To solve the problem of finding the ratio \( a:b:c:d \) given that \( a:b:c:d = \frac{1}{3}:\frac{1}{7}:\frac{1}{2}:\frac{1}{5} \), we can follow these steps:

### Step 1: Write down the given ratios
We start with the ratios as given:
\[ a:b:c:d = \frac{1}{3}:\frac{1}{7}:\frac{1}{2}:\frac{1}{5} \]

### Step 2: Find a common denominator
To simplify these fractions, we can find a common denominator. The denominators are 3, 7, 2, and 5. The least common multiple (LCM) of these numbers is 210.

### Step 3: Convert each fraction to have the common denominator
Now, we convert each fraction to have the common denominator of 210:
- For \( \frac{1}{3} \):
\[ \frac{1}{3} = \frac{70}{210} \]
- For \( \frac{1}{7} \):
\[ \frac{1}{7} = \frac{30}{210} \]
- For \( \frac{1}{2} \):
\[ \frac{1}{2} = \frac{105}{210} \]
- For \( \frac{1}{5} \):
\[ \frac{1}{5} = \frac{42}{210} \]

### Step 4: Rewrite the ratios with the common denominator
Now we can rewrite the ratios using the common denominator:
\[ a:b:c:d = 70:30:105:42 \]

### Step 5: Simplify the ratios
To simplify the ratio \( 70:30:105:42 \), we can find the greatest common divisor (GCD) of these numbers. The GCD of 70, 30, 105, and 42 is 1 (since they do not have any common factors other than 1). Therefore, the ratio cannot be simplified further.

### Final Answer
Thus, the final ratio is:
\[ a:b:c:d = 70:30:105:42 \]