Question:medium

A grocer mixes coffee powder of 2 types, one of which is priced at Rs. 60 and the other at Rs. 90. What should be the ratio of combining the two, to sell the blended mix coffee powder of the two types at Rs. 80?

Updated On: Jul 15, 2026
  • 2:1
  • 2:3
  • 1:2
  • 3:2
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The Correct Option is C

Approach Solution - 1

Step 1: Understanding the Question.
Two types of coffee, priced Rs. 60 and Rs. 90, are blended to sell at Rs. 80. We need the ratio in which the two types are mixed.

Step 2: Key Formula or Approach.
The rule of alligation says that when two items are mixed, the ratio of their quantities is inverse to how far each price is from the mean price.
\[ \frac{\text{Quantity of cheaper}}{\text{Quantity of dearer}} = \frac{\text{Dearer price} - \text{Mean price}}{\text{Mean price} - \text{Cheaper price}} \]

Step 3: Detailed Explanation.
Here the cheaper coffee costs Rs. 60, the dearer coffee costs Rs. 90, and the mean (blended) price is Rs. 80.
\[ \text{Dearer price} - \text{Mean price} = 90 - 80 = 10 \]
\[ \text{Mean price} - \text{Cheaper price} = 80 - 60 = 20 \]
So the ratio of cheaper to dearer coffee is:
\[ \frac{60\text{-type}}{90\text{-type}} = \frac{10}{20} = \frac{1}{2} \]

Step 4: Final Answer.
The two coffee powders should be mixed in the ratio 1:2 (Rs. 60 type to Rs. 90 type). \[ \boxed{1:2} \]
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Approach Solution -2

Think of the blend as one whole quantity split between the Rs. 60 coffee and the Rs. 90 coffee. Let \( f \) be the fraction of the total that is the Rs. 60 coffee, so \( (1-f) \) is the fraction that is the Rs. 90 coffee. The blended price is the weighted average of the two prices: \[ 60f + 90(1-f) = 80. \] Expanding, \( 60f + 90 - 90f = 80 \), so \( -30f = -10 \) and \( f = \frac{1}{3} \). This means one third of the total is the Rs. 60 coffee and two thirds is the Rs. 90 coffee, a ratio of \( \frac{1}{3} : \frac{2}{3} \), which simplifies to \( 1:2 \). Compare each option to this fraction split.

  1. Option (A): 2:1: This puts two thirds of the mixture as the Rs. 60 coffee, so \( f = \frac{2}{3} \), the opposite of the \( f = \frac{1}{3} \) required. Substituting back, \( 60(\frac{2}{3}) + 90(\frac{1}{3}) = 40+30=70 \), giving Rs. 70, not Rs. 80.
  2. Option (B): 2:3: Here \( f = \frac{2}{5} \), a bit more of the cheaper coffee than the \( \frac{1}{3} \) the equation calls for. Checking, \( 60(\frac{2}{5}) + 90(\frac{3}{5}) = 24+54=78 \), giving Rs. 78.
  3. Option (C): 1:2: Here \( f = \frac{1}{3} \), exactly the fraction found by solving the weighted-average equation. Checking, \( 60(\frac{1}{3}) + 90(\frac{2}{3}) = 20+60=80 \), giving exactly Rs. 80.
  4. Option (D): 3:2: Here \( f = \frac{3}{5} \), well above the required \( \frac{1}{3} \), meaning far too much of the cheaper coffee. Checking, \( 60(\frac{3}{5}) + 90(\frac{2}{5}) = 36+36=72 \), giving Rs. 72.

Solving for the fraction of cheaper coffee needed in the blend gives exactly \( 1:2 \), matching option (C) alone.

Therefore, the correct answer is 1:2.

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