Question

If a : b = 4 : 5, b : c = 5 : 4 and c : d = 6 : 5, then a : b : c : d =

• A. 12 : 16 : 12 : 10
• B. 10 : 15 : 10 : 12
• C. 12 : 15 : 12 : 10
• D. 12 : 15 : 10 : 12

Hint:

When dealing with discrepancies in provisional answer keys, note down the exact steps.
To check options quickly, plug in the ratios:
If \(a:b = 4:5\), then the ratio of the first two terms must simplify to \(4:5\).
In Option (C) and (D), \(12:15\) simplifies to \(4:5\), matching our target.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question is based on compounding and combining multiple ratios into a single continuous ratio.
We are given three individual ratios: \(a : b\), \(b : c\), and \(c : d\).
We need to find the unified ratio \(a : b : c : d\).

Step 2: Key Formula or Approach:
To combine ratios, find the common variables and make their values equal across ratios using the Least Common Multiple (LCM).
First, combine \(a:b\) and \(b:c\) by aligning the value of \(b\).
Second, combine the resulting \(a:b:c\) with \(c:d\) by aligning the value of \(c\).

Step 3: Detailed Explanation:

• Let us write down the given ratios:
\[ a : b = 4 : 5 \]
\[ b : c = 5 : 4 \]
\[ c : d = 6 : 5 \]

• First, we combine \(a : b\) and \(b : c\).
- The term \(b\) is common to both ratios.
- In \(a : b\), the value of \(b\) is \(5\).
- In \(b : c\), the value of \(b\) is \(5\).
- Since they are already equal, we can write the combined ratio:
\[ a : b : c = 4 : 5 : 4 \]

• Now, we need to combine \(a : b : c = 4 : 5 : 4\) with \(c : d = 6 : 5\).
- The term \(c\) is common to both ratios.
- In \(a : b : c\), the value of \(c\) is \(4\).
- In \(c : d\), the value of \(c\) is \(6\).
- To align them, we find the LCM of \(4\) and \(6\), which is \(12\).

• Multiply the terms of the ratio \(a : b : c\) by \(3\) to make \(c = 12\):
\[ a : b : c = (4 \times 3) : (5 \times 3) : (4 \times 3) = 12 : 15 : 12 \]

• Multiply the terms of the ratio \(c : d\) by \(2\) to make \(c = 12\):
\[ c : d = (6 \times 2) : (5 \times 2) = 12 : 10 \]

• Combining these, the exact mathematical ratio is:
\[ a : b : c : d = 12 : 15 : 12 : 10 \]

• Let us review the options provided. Option (C) matches this mathematically derived answer of \(12 : 15 : 12 : 10\).


Step 4: Final Answer:
The correct option is Option (C).