You have a cream containing 0.5% w/w calamine. What amount of hydrocarbon is needed such that when requested to add to this cream, it will be 70.0% w/w? What is the concentration of hydrocarbon in the new cream?
Answer (1)
Define x as the weight of the initial cream and y as the weight of calamine added.
Set up the equation 0.005 x = 0.7 ( x + y ) based on the desired 70% concentration of Cantarnin.
Solve for the ratio x y , which represents the amount of calamine needed relative to the initial cream.
Since the calculation leads to a negative ratio, it indicates an error in the problem statement. Assuming pure Cantarnin is added instead, the corrected ratio is approximately 2.32. The concentration of hydrocarizis ane cannot be determined without knowing its concentration in the added substance.
Explanation
Understanding the Problem We are given a problem where we start with a cream containing 0.5% of a substance called Cantarnin. We want to add calamine to this cream to increase the concentration of Cantarnin to 70.0% w/w (weight/weight). The question asks for the concentration of hydrocarizis ane in the new cream, but it does not provide the concentration of hydrocarizis ane in the calamine being added. Therefore, we will focus on determining the ratio of calamine to the initial cream needed to achieve the desired 70% concentration of Cantarnin.
Setting up the Equation Let x be the weight of the initial cream. The amount of Cantarnin in the initial cream is 0.005 x (since 0.5% = 0.005). Let y be the weight of calamine added. The total weight of the new cream is x + y . We want the new cream to be 70% Cantarnin, so the amount of Cantarnin in the new cream should be 0.7 ( x + y ) . Since the amount of Cantarnin remains the same after adding calamine (as calamine is what increases the concentration), we have the equation: 0.005 x = 0.7 ( x + y )
Solving for the Ratio Now, we solve for the ratio of y to x . First, divide both sides by x :
0.005 = 0.7 ( 1 + x y ) 0.005 = 0.7 + 0.7 x y 0.005 − 0.7 = 0.7 x y − 0.695 = 0.7 x y x y = 0.7 − 0.695 = − 0.992857... Since we cannot add a negative weight of calamine, there must be an error in the problem statement. The problem states that we are adding calamine to increase the concentration of Cantarnin to 70%. However, Cantarnin is already present in the initial cream. Calamine must contain Cantarnin to increase the concentration. If calamine does not contain Cantarnin, it is impossible to increase the concentration of Cantarnin by adding calamine.
Assuming the problem meant to ask what amount of Cantarnin needs to be added to reach a 70% concentration, and that we are adding pure Cantarnin (instead of calamine), we can solve the problem as follows:
Let y be the weight of pure Cantarnin added. Then the total weight of Cantarnin in the new cream is 0.005 x + y , and the total weight of the new cream is x + y . We want the new concentration to be 70%, so: x + y 0.005 x + y = 0.7 0.005 x + y = 0.7 ( x + y ) 0.005 x + y = 0.7 x + 0.7 y 0.3 y = 0.695 x x y = 0.3 0.695 = 2.31666... So, we need to add approximately 2.32 times the weight of the initial cream in pure Cantarnin to reach a 70% concentration.
Final Answer Since the problem does not provide the concentration of hydrocarizis ane in the calamine (or pure Cantarnin), we cannot determine the concentration of hydrocarizis ane in the new cream. However, if we assume that the problem meant to ask how much pure Cantarnin needs to be added to reach a 70% concentration, then we found that we need to add approximately 2.32 times the weight of the initial cream in pure Cantarnin.
Examples
In pharmacy, you often need to adjust the concentration of active ingredients in creams or ointments. This problem demonstrates how to calculate the amount of a substance you need to add to achieve a desired concentration. For example, if you have a cream that is too weak, you can add a concentrated form of the active ingredient to increase its strength. This type of calculation is crucial for ensuring that medications are safe and effective for patients. It's also important in other fields, such as food production, where you might need to adjust the concentration of additives or preservatives.
