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

Options

A. `60:30:15:44:17`
B. `80:40:15:20:19`
C. `90:45:30:40:16`
D. `77:15:19:20:16`

Answer

To find the ratio \( a : b : c : d : e \) given the ratios \( a:b::2:1 \), \( b:c::3:2 \), \( c:d::3:4 \), and \( d:e::5:2 \), we will follow these steps:

### Step 1: Express the ratios in terms of a common variable
1. From \( a : b = 2 : 1 \), we can express:
\[ a = 2x \quad \text{and} \quad b = x \]

### Step 2: Use the second ratio to express \( c \) in terms of \( b \)
2. From \( b : c = 3 : 2 \), we can express:
\[ b = 3y \quad \text{and} \quad c = 2y \]
Since \( b = x \), we equate:
\[ x = 3y \implies y = \frac{x}{3} \]
Substituting \( y \) into \( c \):
\[ c = 2y = 2 \left(\frac{x}{3}\right) = \frac{2x}{3} \]

### Step 3: Use the third ratio to express \( d \) in terms of \( c \)
3. From \( c : d = 3 : 4 \), we can express:
\[ c = 3z \quad \text{and} \quad d = 4z \]
Since \( c = \frac{2x}{3} \), we equate:
\[ \frac{2x}{3} = 3z \implies z = \frac{2x}{9} \]
Substituting \( z \) into \( d \):
\[ d = 4z = 4 \left(\frac{2x}{9}\right) = \frac{8x}{9} \]

### Step 4: Use the fourth ratio to express \( e \) in terms of \( d \)
4. From \( d : e = 5 : 2 \), we can express:
\[ d = 5w \quad \text{and} \quad e = 2w \]
Since \( d = \frac{8x}{9} \), we equate:
\[ \frac{8x}{9} = 5w \implies w = \frac{8x}{45} \]
Substituting \( w \) into \( e \):
\[ e = 2w = 2 \left(\frac{8x}{45}\right) = \frac{16x}{45} \]

### Step 5: Summarize all variables
Now we have:
- \( a = 2x \)
- \( b = x \)
- \( c = \frac{2x}{3} \)
- \( d = \frac{8x}{9} \)
- \( e = \frac{16x}{45} \)

### Step 6: Find a common denominator to express the ratios
To express \( a : b : c : d : e \) in the simplest form, we need a common denominator. The least common multiple of the denominators (1, 1, 3, 9, 45) is 45.

- Convert each term:
- \( a = 2x = \frac{90x}{45} \)
- \( b = x = \frac{45x}{45} \)
- \( c = \frac{2x}{3} = \frac{30x}{45} \)
- \( d = \frac{8x}{9} = \frac{40x}{45} \)
- \( e = \frac{16x}{45} = \frac{16x}{45} \)

### Step 7: Write the final ratio
Thus, the ratio \( a : b : c : d : e \) is:
\[ 90 : 45 : 30 : 40 : 16 \]

### Final Answer
The final ratio is:
\[ a : b : c : d : e = 90 : 45 : 30 : 40 : 16 \]

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