To solve this problem, we need to calculate the price of the initial coffee mixture and then determine the ratio in which this mixture should be combined with coffee that costs sh. 40 per kilogram to achieve the desired price of sh. 52 per kilogram.
### Step 1: Calculate the Price of the Initial Mixture
We are given:
- Coffee 1 costs sh. 50 per kilogram.
- Coffee 2 costs sh. 60 per kilogram.
- The mixing ratio of Coffee 1 to Coffee 2 is 2:3.
The price of the initial mixture ([tex]\(P_{\text{mixture}}\)[/tex]) can be calculated using the given ratio:
[tex]\[
P_{\text{mixture}} = \frac{(\text{price of Coffee 1} \times \text{ratio1}) + (\text{price of Coffee 2} \times \text{ratio2})}{\text{ratio1} + \text{ratio2}}
\][/tex]
Substituting the values:
[tex]\[
P_{\text{mixture}} = \frac{(50 \times 2) + (60 \times 3)}{2 + 3}
\][/tex]
Calculate the numerator:
[tex]\[
P_{\text{mixture}} = \frac{100 + 180}{5} = \frac{280}{5} = 56 \text{ sh}
\][/tex]
### Step 2: Determine the Ratio for Mixing with Coffee at sh. 40 per Kilogram
We need to find the ratio [tex]\( x:1 \)[/tex] (where [tex]\( x \)[/tex] is the amount of the initial mixture) to achieve a final price of sh. 52 per kilogram by mixing the initial mixture with coffee costing sh. 40 per kilogram.
Let [tex]\( x \)[/tex] be the ratio of the initial mixture, then:
[tex]\[
\frac{P_{\text{mixture}} \times x + 40 \times 1}{x + 1} = 52
\][/tex]
Substituting the value of [tex]\( P_{\text{mixture}} \)[/tex]:
[tex]\[
\frac{56x + 40}{x + 1} = 52
\][/tex]
### Step 3: Solving for [tex]\( x \)[/tex]
Multiply both sides by [tex]\( x + 1 \)[/tex]:
[tex]\[
56x + 40 = 52(x + 1)
\][/tex]
Distribute 52 on the right-hand side:
[tex]\[
56x + 40 = 52x + 52
\][/tex]
Subtract [tex]\( 52x \)[/tex] from both sides:
[tex]\[
4x + 40 = 52
\][/tex]
Subtract 40 from both sides:
[tex]\[
4x = 12
\][/tex]
Divide by 4:
[tex]\[
x = 3
\][/tex]
So, the ratio of the initial mixture to the coffee costing sh. 40 should be 3:1.
### Conclusion
The desired price of sh. 52 per kilogram can be achieved by mixing the initial coffee mixture (priced at sh. 56 per kilogram) with coffee costing sh. 40 per kilogram in a ratio of 3:1.
[tex]\[
\boxed{3:1}
\][/tex]
