Three distinct varieties of coffee beans with unit prices of ₹50/kg, ₹70/kg, and an unstated price are blended together in the ratio of 2 : 3 : 2. If the blended mixture is retailed at ₹64/kg to yield a profit of 20%, what is the cost price per kilogram of the third variety?

Correct Option: ₹31.6/kg

Let's solve the problem step-by-step to find the cost price per kilogram of the third variety of coffee beans.

Step 1: Calculate the Cost Price (CP) per kg of the blended mixture.

We are given that the blended mixture is sold (Selling Price, SP) at ₹64/kg to yield a profit of 20%.

The formula relating Selling Price, Cost Price, and Profit Percentage is:

$\text{SP}=\text{CP}\times \left(1+\frac{\text{Profit \%}}{100}\right)$

Substituting the given values into the equation:

$64=\text{CP}\times \left(1+\frac{20}{100}\right)$

$64=\text{CP}\times \frac{120}{100}$

$64=\text{CP}\times 1.2$

$\text{CP}=\frac{64}{1.2}=\frac{640}{12}=\text{₹}\frac{160}{3}\text{per kg}$

Step 2: Set up the equation for the total cost of the mixture.

The three varieties of coffee beans are mixed in the ratio 2 : 3 : 2.

Let the quantities of the three varieties mixed be 2 kg, 3 kg, and 2 kg respectively. Then, the total weight of the blended mixture is:

$\text{Total Weight}=2+3+2=7\text{kg}$

Let the cost price per kg of the third variety be ₹x/kg.

The unit prices of the three varieties are:

• Variety 1: ₹50/kg

• Variety 2: ₹70/kg

• Variety 3: ₹x/kg

The total cost of the 7 kg mixture can be calculated in two ways:

1. Sum of individual costs of each variety:

$\text{Total Cost}=\left(2\times 50\right)+\left(3\times 70\right)+\left(2\times x\right)$

$\text{Total Cost}=100+210+2x=310+2x$

2. Using the average cost price per kg of the mixture:

$\text{Total Cost}=7\times \frac{160}{3}=\frac{1120}{3}$

Step 3: Solve for x.

Equating the two expressions for Total Cost:

$310+2\mathrm{x<mo>=</mo><mfrac><mn>1120</mn><mn>3</mn></mfrac>}$

Multiply both sides by 3 to eliminate the denominator:

$3\times \left(310+2x\right)=1120$

$930+6x=1120$

$6x=1120-930$

$6x=190$

$x=\frac{190}{6}=\frac{95}{3}\approx \mathrm{31.666...}\text{≈ ₹31.6/kg}$

Thus, the cost price per kilogram of the third variety is ₹31.6/kg.