If a= 2b/3, b= 2c/3, and c=2d/3, then find the ratio of b and d:

Options

A. A. 8/9
B. B. 4/9
C. C. 4/3
D. D. 5/27

Answer

To solve the problem step by step, we will use the given relationships between the variables a, b, c, and d.

### Step 1: Write down the given equations
We have the following relationships:
1. \( a = \frac{2b}{3} \)
2. \( b = \frac{2c}{3} \)
3. \( c = \frac{2d}{3} \)

### Step 2: Eliminate variable a
Since we need to find the ratio of b and d, we can ignore the equation involving a, as it does not directly relate to b and d.

### Step 3: Substitute for c in terms of d
From the third equation, we can express c in terms of d:
\[ c = \frac{2d}{3} \]

### Step 4: Substitute c into the equation for b
Now, substitute the expression for c into the equation for b:
\[ b = \frac{2c}{3} \]
Substituting \( c = \frac{2d}{3} \):
\[ b = \frac{2 \left( \frac{2d}{3} \right)}{3} \]

### Step 5: Simplify the expression for b
Now simplify the expression:
\[ b = \frac{4d}{9} \]

### Step 6: Find the ratio of b to d
Now we need to find the ratio of b to d:
\[ \text{Ratio} = \frac{b}{d} = \frac{\frac{4d}{9}}{d} \]

### Step 7: Simplify the ratio
This simplifies to:
\[ \frac{b}{d} = \frac{4}{9} \]

### Conclusion
Thus, the ratio of b to d is:
\[ b : d = 4 : 9 \]