`2a=3b=5c=3d`, find `a:b:c:d=?`

Options

A. A) 15:10:6:10
B. B) 1;1:1:1
C. C) 2:3:4:5
D. D) 5:6:7:8

Answer

To solve the equation \(2a = 3b = 5c = 3d\) and find the ratio \(a:b:c:d\), we can follow these steps:

### Step 1: Set a common variable
Let \(K\) be the common value for \(2a\), \(3b\), \(5c\), and \(3d\). Therefore, we can write:
\[ 2a = K, \quad 3b = K, \quad 5c = K, \quad 3d = K \]

### Step 2: Express each variable in terms of \(K\)
Now, we can express \(a\), \(b\), \(c\), and \(d\) in terms of \(K\):
\[ a = \frac{K}{2}, \quad b = \frac{K}{3}, \quad c = \frac{K}{5}, \quad d = \frac{K}{3} \]

### Step 3: Find a common denominator
To find the ratio \(a:b:c:d\), we need to express all variables with a common denominator. The least common multiple (LCM) of the denominators \(2\), \(3\), and \(5\) is \(30\).

### Step 4: Rewrite each variable with the common denominator
Now, we can rewrite each variable:
\[ a = \frac{K}{2} = \frac{15K}{30}, \quad b = \frac{K}{3} = \frac{10K}{30}, \quad c = \frac{K}{5} = \frac{6K}{30}, \quad d = \frac{K}{3} = \frac{10K}{30} \]

### Step 5: Form the ratio
Now we can express the ratio \(a:b:c:d\):
\[ a:b:c:d = \frac{15K}{30} : \frac{10K}{30} : \frac{6K}{30} : \frac{10K}{30} \]
This simplifies to:
\[ a:b:c:d = 15 : 10 : 6 : 10 \]

### Step 6: Simplify the ratio
To simplify the ratio, we can divide each term by the greatest common divisor (GCD). The GCD of \(15\), \(10\), \(6\), and \(10\) is \(1\), so the ratio remains:
\[ a:b:c:d = 15 : 10 : 6 : 10 \]

### Final Ratio
Thus, the final ratio is:
\[ \boxed{15 : 10 : 6 : 10} \]