Question

The strength of a salt solution is p% if 100 ml of the solution contains p grams of salt. If three salt solutions A, B, C are mixed in the proportion 1 : 2 : 3, then the resulting solution has strength 20%. If instead the proportion is 3 : 2 : 1, then the resulting solution has strength 30%. A fourth solution, D, is produced by mixing B and C in the ratio 2 : 7. The ratio of the strength of D to that of A is

• A. 2:5
• B. 1:3
• C. 1:4
• D. 3:10

Answer 1

The Correct Option is B

Step 1: Define Variables

Let the concentrations (in %) of salt in solutions A, B, and C be:

\[ a = \text{concentration in A}, \quad b = \text{concentration in B}, \quad c = \text{concentration in C} \]

Step 2: First Mixing Ratio (1 : 2 : 3)

When solutions are mixed in the ratio 1:2:3, the average concentration is 20%.

\[ \frac{a + 2b + 3c}{1 + 2 + 3} = 20 \quad \Rightarrow \quad \frac{a + 2b + 3c}{6} = 20 \]

\[ a + 2b + 3c = 120 \tag{1} \]

Step 3: Second Mixing Ratio (3 : 2 : 1)

When the same solutions are mixed in the ratio 3:2:1, the resulting concentration is 30%.

\[ \frac{3a + 2b + c}{3 + 2 + 1} = 30 \quad \Rightarrow \quad \frac{3a + 2b + c}{6} = 30 \]

\[ 3a + 2b + c = 180 \tag{2} \]

Step 4: Subtract Equations (1) and (2)

Subtract equation (1) from equation (2):

\[ (3a + 2b + c) - (a + 2b + 3c) = 180 - 120 \]

\[ 2a - 2c = 60 \quad \Rightarrow \quad a - c = 30 \tag{3} \]

Step 5: Eliminate a and Solve for b and c

Now eliminate \( a \) from equation (1) using equation (3):

\[ a = c + 30 \quad \text{(from equation 3)} \]

Substitute into equation (1):

\[ (c + 30) + 2b + 3c = 120 \quad \Rightarrow \quad 2b + 4c = 90 \quad \Rightarrow \quad b + 2c = 45 \tag{4} \]

Step 6: Choose Convenient Values

By observation, set \( b = 15 \), \( c = 15 \). Then from equation (3), \( a = 15 + 30 = 45 \).

Step 7: Interpret the Results

• \( a = 45\% \)
• \( b = 15\% \)
• \( c = 15\% \)

If we mix only solutions B and C (both with 15% concentration), the resulting solution will also have 15% strength.

To achieve this same 15% concentration by mixing A and the (B+C) mixture, the required strength proportion is:

\[ \text{Required Ratio} = \frac{15}{45} = \boxed{1 : 3} \]