A drink of chocolate and milk contains 8% pure chocolate by volume. If 10 litres of pure milk are added to 50 litres of this drink, the percentage of chocolate in the new drink is:
This problem involves calculating the percentage of a component in a mixture after adding another component. We need to determine the initial amount of chocolate, find the new total volume after adding milk, and then calculate the new percentage.
Understanding the Initial Chocolate Drink Mixture
We start with 50 litres of a drink that contains 8% pure chocolate by volume. This means that a certain portion of the 50 litres is chocolate, and the rest is likely milk or other components. Since pure milk is added later, we can assume the original drink was a mixture of chocolate and milk, with 8% being the chocolate part.
- Initial volume of the drink = 50 litres
- Percentage of chocolate in the initial drink = 8%
Calculating the Volume of Pure Chocolate
To find the actual volume of pure chocolate in the initial 50 litres, we calculate 8% of 50 litres.
\(\text{Volume of chocolate} = 8\% \text{ of } 50 \text{ litres}\)
\(\text{Volume of chocolate} = \frac{8}{100} \times 50\)
\(\text{Volume of chocolate} = \frac{400}{100}\)
\(\text{Volume of chocolate} = 4 \text{ litres}\)
So, there are 4 litres of pure chocolate in the original 50 litres of the drink.
Calculating the New Total Volume
Pure milk is added to the original drink. The volume of pure chocolate remains unchanged because only milk is added, not more chocolate. The total volume of the drink increases.
- Volume of pure milk added = 10 litres
- Initial volume of the drink = 50 litres
The new total volume is the sum of the initial volume and the volume of added milk.
\(\text{New total volume} = \text{Initial volume} + \text{Volume of added milk}\)
\(\text{New total volume} = 50 \text{ litres} + 10 \text{ litres}\)
\(\text{New total volume} = 60 \text{ litres}\)
The new total volume of the drink is 60 litres.
Calculating the New Percentage of Chocolate
The amount of pure chocolate in the new mixture is still 4 litres. The total volume of the mixture is now 60 litres. To find the new percentage of chocolate, we use the formula:
\(\text{Percentage of chocolate} = \left( \frac{\text{Volume of chocolate}}{\text{New total volume}} \right) \times 100\)
\(\text{Percentage of chocolate} = \left( \frac{4 \text{ litres}}{60 \text{ litres}} \right) \times 100\)
\(\text{Percentage of chocolate} = \frac{4}{60} \times 100\)
\(\text{Percentage of chocolate} = \frac{1}{15} \times 100\)
\(\text{Percentage of chocolate} = \frac{100}{15}\)
Now, we simplify the fraction $\frac{100}{15}$ and convert it to a mixed number.
\(\frac{100}{15} = \frac{20 \times 5}{3 \times 5} = \frac{20}{3}\)
To convert $\frac{20}{3}$ to a mixed number, we divide 20 by 3:
\(20 \div 3 = 6 \text{ with a remainder of } 2\)
So, $\frac{20}{3}$ as a mixed number is \(6 \frac{2}{3}\).
The new percentage of chocolate in the drink is \(6 \frac{2}{3}\%\).
Let's summarize the values:
| Component | Initial Volume (litres) | Added Volume (litres) | Final Volume (litres) |
|---|---|---|---|
| Pure Chocolate | 4 | 0 | 4 |
| Milk (or other) | 46 (50-4) | 10 | 56 |
| Total Drink | 50 | 10 | 60 |
New percentage of chocolate = \(\frac{4}{60} \times 100 = \frac{20}{3}\% = 6 \frac{2}{3}\%\).
Mixture Problem Solution Summary
We started with a drink containing a specific amount of chocolate. We calculated the pure volume of chocolate. Then, we added milk, which increased the total volume but not the volume of chocolate. Finally, we recalculated the percentage of chocolate based on the new total volume. The key is to track the constant amount of the substance (chocolate) whose percentage is being measured, and the changing total volume.
Revision Table: Chocolate Drink Mixture
| Concept | Details | Formula/Calculation |
|---|---|---|
| Initial Volume | Volume of the original drink | 50 litres |
| Initial Chocolate % | Percentage of chocolate in the original drink | 8% |
| Initial Chocolate Volume | Actual volume of chocolate in the initial drink | \(8\% \text{ of } 50 = 4 \text{ litres}\) |
| Added Milk Volume | Volume of pure milk added | 10 litres |
| New Total Volume | Total volume after adding milk | \(50 + 10 = 60 \text{ litres}\) |
| New Chocolate % | Percentage of chocolate in the new drink | \(\left( \frac{\text{Chocolate Volume}}{\text{New Total Volume}} \right) \times 100\) |
| Final Calculation | Calculation of the new percentage | \(\left( \frac{4}{60} \right) \times 100 = \frac{20}{3}\% = 6 \frac{2}{3}\%\) |
Additional Information: Percentage in Mixtures
Mixture problems often involve calculating the concentration (usually as a percentage) of a specific substance within a total volume or weight. When a pure substance is added to a mixture, the amount of the original components remains unchanged, but the total amount of the mixture changes. When a pure solvent (like milk in this case) is added, it dilutes the solute (chocolate), reducing its percentage concentration.
- Concentration: Amount of solute per unit amount of mixture (e.g., grams per litre, volume percentage).
- Dilution: Process of reducing the concentration of a solute in a solution by adding more solvent.
- Formula for Percentage Concentration: \(\left( \frac{\text{Amount of Solute}}{\text{Total Amount of Mixture}} \right) \times 100\).
In dilution problems, the amount of the solute stays constant while the total amount of the mixture increases, leading to a decrease in percentage concentration.
Important Questions from To Make a Mixture from Two Mixtures
One cup has juice and water in the ratio 5 ∶ 2, while another cup of the same capacity has them in the ratio 7 ∶ 4, respectively. If contents of both the cups (when full) are poured in a vessel, then what will be the final ratio of water to juice in the vessel?
A and B are solutions of acid and water. The ratios of water and acid in A and B are 4 : 5 and 1 : 2 respectively. If x liters of A is mixed with y liters of B, then the ratio of water and acid in the mixture becomes 8 : 13 What is x : y?
Mixture A contains chocolate and milk in the ratio 4 ∶ 3 and mixture B contains chocolate and milk in the ratio 5 ∶ 2. A and B are taken in the ratio 5 ∶ 6 and mixed to form a new mixture. The percentage of chocolate in the new mixture is closest to:
If 80 litres of milk solution has 60% milk in it, then how much milk should be added to make milk 80% in the solution?
A 40 - litre mixture contains 25% alcohol and 75% water. If 10 litres of water are added to the mixture, the percentage of alcohol in the new mixture is:
