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].
