`a:b::2:3,b:c::4:1,c:d::2:5` find `a:b:c:d.`

Options

A. `14:8:9:7`
B. `17:24:7:14`
C. `16:24:6:15`
D. `19:25:8:17`

Answer

To solve the problem of finding the ratio \( a:b:c:d \) given the ratios \( a:b::2:3 \), \( b:c::4:1 \), and \( c:d::2:5 \), we can follow these steps:

### Step 1: Write the ratios in fraction form
From the given ratios, we can express them as:
- \( \frac{a}{b} = \frac{2}{3} \) (1)
- \( \frac{b}{c} = \frac{4}{1} \) (2)
- \( \frac{c}{d} = \frac{2}{5} \) (3)

### Step 2: Express \( b \) in terms of \( a \)
From equation (1), we can express \( a \) in terms of \( b \):
\[ a = \frac{2}{3}b \]

### Step 3: Express \( c \) in terms of \( b \)
From equation (2), we can express \( c \) in terms of \( b \):
\[ c = \frac{b}{4} \]

### Step 4: Express \( d \) in terms of \( c \)
From equation (3), we can express \( d \) in terms of \( c \):
\[ d = \frac{5}{2}c \]

### Step 5: Substitute \( c \) in terms of \( b \) into the equation for \( d \)
Substituting \( c = \frac{b}{4} \) into the equation for \( d \):
\[ d = \frac{5}{2} \left(\frac{b}{4}\right) = \frac{5b}{8} \]

### Step 6: Now we have expressions for \( a \), \( b \), \( c \), and \( d \) in terms of \( b \)
- \( a = \frac{2}{3}b \)
- \( b = b \)
- \( c = \frac{b}{4} \)
- \( d = \frac{5b}{8} \)

### Step 7: Find a common denominator to express all ratios in terms of \( b \)
The common denominator for \( 3, 1, 4, \) and \( 8 \) is \( 24 \).
- For \( a \):
\[ a = \frac{2}{3}b = \frac{16}{24}b \]
- For \( b \):
\[ b = \frac{24}{24}b \]
- For \( c \):
\[ c = \frac{b}{4} = \frac{6}{24}b \]
- For \( d \):
\[ d = \frac{5b}{8} = \frac{15}{24}b \]

### Step 8: Write the final ratio \( a:b:c:d \)
Now we can express the ratio \( a:b:c:d \) as:
\[ a:b:c:d = 16:24:6:15 \]

### Step 9: Simplify the ratio
To simplify the ratio, we can divide each term by the greatest common divisor (GCD) of \( 16, 24, 6, \) and \( 15 \), which is \( 1 \) (as they have no common factors):
\[ a:b:c:d = 16:24:6:15 \]

### Final Answer
The final ratio is:
\[ a:b:c:d = 16:24:6:15 \]