Arithmetic

Mixture & Alligation

Easy

The strength of a salt solution is $p\%$ if $100 \mathrm{ml}$ 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

$1: 4$

$3: 10$

$1:3$

$2: 5$

✅ Correct Option: 3

When we say a solution has p% strength, it means 100 ml of that solution contains p grams of salt. This is a standard way to measure concentration.

When different solutions are mixed, the strength of the resulting mixture is the weighted average of the individual strengths, where the weights are the proportions in which they're mixed.

Let the strengths of solutions A, B, and C be a%, b%, and c% respectively.

When A, B, C are mixed in ratio 1:2:3:

Total parts = 1 + 2 + 3 = 6

Resulting strength = $\tfrac{1 \times a + 2 \times b + 3 \times c}{6} = 20$ %

$a + 2b + 3c = 120$ ... (1)

When A, B, C are mixed in ratio 3:2:1:

Total parts = 3 + 2 + 1 = 6

Resulting strength = $\tfrac{3 \times a + 2 \times b + 1 \times c}{6} = 30$ %

$3a + 2b + c = 180$ ... (2)

We have two equations with three unknowns, so we'll express two variables in terms of the third.

$(3a + 2b + c) - (a + 2b + 3c) = 180 - 120$

$2a - 2c = 60$

$c = a - 30$ ... (3)

Substituting (3) into equation (1):

$a + 2b + 3(a - 30) = 120$

$a + 2b + 3a - 90 = 120$

$4a + 2b = 210$

$b = 105 - 2a$ ... (4)

Solution D is made by mixing B and C in ratio 2:7.

Using the weighted average formula:

Total parts = 2 + 7 = 9

Strength of D = $\tfrac{2 \times b + 7 \times c}{9}$

Substituting our expressions for b and c:

$D = \tfrac{2 \times (105 - 2a) + 7 \times (a - 30)}{9}$

$D = \tfrac{210 - 4a + 7a - 210}{9}$

$D = \tfrac{3a}{9}$

$D = \tfrac{a}{3}$

The ratio of strength of D to A is:

$D : A = \tfrac{a}{3} : a = 1 : 3$

Therefore, the ratio is 1:3

We observe how the middle terms (210 and -210) cancelled out perfectly! This often happens in mixture problems and is a sign that our algebra is on the right track. The beauty of this problem is that even though we have three unknowns, the specific ratios given allow us to find the relationship we need without determining the individual values.

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