Question

Alpha and Beta are two chemical fertilizers. Alpha consists of \(N, P, K\) and Beta consists of \(N\) and \(P\). A mixture of Alpha and Beta is prepared in which the ratio of \(N, P\) and \(K\) is \(26%\), \(68%\) and \(6%\). The ratio of \(N, P, K\) in Alpha is \(20%\), \(70%\) and \(10%\). What is the ratio of \(N\) and \(P\) in Beta?

• A. \(27%\) and \(73%\)
• B. \(33%\) and \(67%\)
• C. \(35%\) and \(65%\)
• D. \(30%\) and \(70%\)

Hint:

Using the component that is missing in one part (like \(K\) in Beta) is the fastest way to find the mixing ratio in mixture problems involving three or more components.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
By using the Alligation method on one of the common components (like Potassium \(K\)), we can find the mixing ratio of Alpha and Beta. Then we can determine the composition of the other elements.
Step 2: Key Formula or Approach:
Ratio of mix \(A:B = (m - b) / (a - m)\) where \(m\) is mix concentration, \(a\) is concentration in Alpha, and \(b\) is concentration in Beta.
Step 3: Detailed Explanation:
1. Find mixing ratio using \(K\):
Potassium (\(K\)) in Alpha = \(10%\).
Potassium (\(K\)) in Beta = \(0%\) (since Beta only has \(N\) and \(P\)).
Potassium (\(K\)) in the Mixture = \(6%\).
By alligation:
\( \frac{\text{Alpha}}{\text{Beta}} = \frac{|6 - 0|}{|10 - 6|} = \frac{6}{4} = \frac{3}{2} \).
2. Calculate Nitrogen (\(N\)) in Beta:
Let Nitrogen in Beta be \(x%\). Nitrogen in Alpha is \(20%\). Nitrogen in the mixture is \(26%\).
\[ \frac{3 \times 20 + 2 \times x}{3 + 2} = 26 \]
\[ \frac{60 + 2x}{5} = 26 \]
\[ 60 + 2x = 130 \implies 2x = 70 \implies x = 35% \text{ (Nitrogen)} \]
3. Calculate Phosphorus (\(P\)) in Beta:
Since Beta only contains \(N\) and \(P\):
\(P = 100% - 35% = 65%\).
Step 4: Final Answer:
The ratio of \(N\) and \(P\) in Beta is \(35%\) and \(65%\).