Step 1: Understanding the Question:
We are given two separate ratios involving three variables A, B, and C.
The goal is to find the direct ratio between the first variable (A) and the last variable (C).
Step 2: Key Formula or Approach:
To find the ratio \(A : C\), we can use the method of multiplying the fractions:
\[\frac{A}{C} = \frac{A}{B} \times \frac{B}{C}\]
Alternatively, we can equate the common term 'B' in both ratios.
Step 3: Detailed Explanation:
Method 1: Fraction Multiplication
- Given \(\frac{A}{B} = \frac{2}{3}\)
- Given \(\frac{B}{C} = \frac{4}{5}\)
- Multiplying them: \[\frac{A}{B} \times \frac{B}{C} = \frac{2}{3} \times \frac{4}{5}\]
- The 'B' terms cancel out on the left side, leaving \(\frac{A}{C}\).
- Calculation: \[\frac{2 \times 4}{3 \times 5} = \frac{8}{15}\]
- Therefore, \(A : C = 8 : 15\).
Method 2: Equating the Common Term (B)
- In Ratio 1 (\(A:B\)), \(B = 3\).
- In Ratio 2 (\(B:C\)), \(B = 4\).
- To combine them, we find the LCM of 3 and 4, which is 12.
- Modify Ratio 1: Multiply by 4 \(\rightarrow A:B = (2 \times 4) : (3 \times 4) = 8 : 12\).
- Modify Ratio 2: Multiply by 3 \(\rightarrow B:C = (4 \times 3) : (5 \times 3) = 12 : 15\).
- Now that \(B\) is the same, we can write the combined ratio \(A:B:C = 8:12:15\).
- From this, the ratio \(A:C = 8:15\).
Step 4: Final Answer:
The ratio \(A : C\) is \(8 : 15\).