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A trader bought maize for Ksh 20 per kilogram and beans for Ksh 60 per kilogram. She mixed the maize and beans and sold the mixture at Ksh 48 per kilogram. If she made a 60% profit, determine the ratio of maize to beans per kilogram in the mixture.

(4 marks)

Sagot :

Certainly! Let's solve this step-by-step:

### Step 1: Calculate the Cost Price per Kilogram
We know the trader sells the mixture for Ksh 48 per kilogram and makes a 60% profit. Let's determine the cost price per kilogram of the mixture.

Given:
- Selling price per kg = Ksh 48
- Profit percentage = 60%

We can calculate the cost price per kilogram using the formula:
[tex]\[ \text{Cost Price (CP) per kg} = \frac{\text{Selling Price}}{1 + \text{Profit Percentage}} \][/tex]

Substituting the values, we get:
[tex]\[ \text{CP per kg} = \frac{48}{1 + 0.60} = \frac{48}{1.6} = 30 \text{ Ksh} \][/tex]

### Step 2: Set up the Mixture Equation
Let [tex]\( x \)[/tex] be the proportion of maize in the mixture, and [tex]\( 1 - x \)[/tex] be the proportion of beans in the mixture. The cost price per kg of the mixture can be determined by the weighted average of the costs of maize and beans.

Given:
- Cost of maize per kg = Ksh 20
- Cost of beans per kg = Ksh 60

The equation for the cost price of the mixture is:
[tex]\[ \text{Cost Price per kg} = 20x + 60(1 - x) \][/tex]

Substituting the calculated cost price:
[tex]\[ 30 = 20x + 60(1 - x) \][/tex]

### Step 3: Solve for [tex]\( x \)[/tex]
Simplify the equation:
[tex]\[ 30 = 20x + 60 - 60x \][/tex]
[tex]\[ 30 = 60 - 40x \][/tex]
[tex]\[ 40x = 60 - 30 \][/tex]
[tex]\[ 40x = 30 \][/tex]
[tex]\[ x = \frac{30}{40} = \frac{3}{4} = 0.75 \][/tex]

So, the proportion of maize in the mixture is [tex]\( 0.75 \)[/tex] (or 75%).

### Step 4: Determine the Proportion of Beans
The proportion of beans is:
[tex]\[ 1 - x = 1 - 0.75 = 0.25 \][/tex]

So, the proportion of beans in the mixture is [tex]\( 0.25 \)[/tex] (or 25%).

### Step 5: Calculate the Ratio of Maize to Beans
The ratio of maize to beans per kilogram is:
[tex]\[ \text{Ratio (Maize:Beans)} = \frac{\text{Proportion of Maize}}{\text{Proportion of Beans}} = \frac{0.75}{0.25} = 3 \][/tex]

### Conclusion:
The ratio of maize to beans per kilogram in the mixture is [tex]\( 3:1 \)[/tex].
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