If `a: b = 2/9 : 1/3, b: c = 2/7 : 5/14` and `d : c = 7/10 : 3/5` then `a : b :c : d` is:

Options

A. `4:6 : 7 : 9`
B. `16 : 24 : 30 : 35`
C. `8 : 12 : 15 : 7`
D. `30 : 35 : 24 : 16`

Answer

To solve the problem of finding the ratio \( a : b : c : d \) given the ratios \( a : b = \frac{2}{9} : \frac{1}{3} \), \( b : c = \frac{2}{7} : \frac{5}{14} \), and \( d : c = \frac{7}{10} : \frac{3}{5} \), we will follow these steps:

### Step 1: Simplify the ratios

1. **Convert the ratios to a common format**:
- For \( a : b = \frac{2}{9} : \frac{1}{3} \):
\[ a : b = \frac{2}{9} : \frac{1}{3} = \frac{2}{9} : \frac{3}{9} = 2 : 3 \]
- For \( b : c = \frac{2}{7} : \frac{5}{14} \):
\[ b : c = \frac{2}{7} : \frac{5}{14} = \frac{2}{7} : \frac{5 \times 2}{14 \times 2} = 2 : 5 \]
- For \( d : c = \frac{7}{10} : \frac{3}{5} \):
\[ d : c = \frac{7}{10} : \frac{3}{5} = \frac{7}{10} : \frac{6}{10} = 7 : 6 \]

### Step 2: Express all ratios in terms of a common variable

2. **Let’s express \( a, b, c, d \) using a common variable**:
- From \( a : b = 2 : 3 \), we can write:
\[ a = 2k \quad \text{and} \quad b = 3k \]
- From \( b : c = 2 : 5 \), we can express \( c \) in terms of \( b \):
\[ b = 3k \implies 3k : c = 2 : 5 \implies c = \frac{5}{2} \times 3k = \frac{15k}{2} \]
- From \( d : c = 7 : 6 \), we can express \( d \) in terms of \( c \):
\[ c = \frac{15k}{2} \implies d : \frac{15k}{2} = 7 : 6 \implies d = \frac{7}{6} \times \frac{15k}{2} = \frac{105k}{12} = \frac{35k}{4} \]

### Step 3: Find a common multiple to express all ratios

3. **Finding a common multiple**:
- The ratios are now:
\[ a = 2k, \quad b = 3k, \quad c = \frac{15k}{2}, \quad d = \frac{35k}{4} \]
- To eliminate fractions, we can multiply each term by 4 (the least common multiple of the denominators):
\[ a = 8k, \quad b = 12k, \quad c = 30k, \quad d = 35k \]

### Step 4: Write the final ratio

4. **Final ratio**:
- Thus, the final ratio \( a : b : c : d \) is:
\[ a : b : c : d = 8k : 12k : 30k : 35k = 8 : 12 : 30 : 35 \]
- To simplify this, we can divide each term by their greatest common divisor (1 in this case):
\[ 8 : 12 : 30 : 35 \]

### Final Answer
The final ratio \( a : b : c : d \) is:
\[ 8 : 12 : 30 : 35 \]