A mixture of petrol and kerosene weighing 5 kg contains 5% kerosene. How much more kerosene must be added to make it 10%?
Understand the Problem
The question is asking us to determine how much additional kerosene needs to be added to a mixture of 5 kg that currently contains 5% kerosene in order to increase the kerosene content to 10%. We will calculate the initial amount of kerosene, then set up an equation to find the required amount of additional kerosene.
Answer
The additional amount of kerosene needed is approximately \( 0.278 \, \text{kg} \).
Answer for screen readers
The additional amount of kerosene needed is approximately ( 0.278 , \text{kg} ).
Steps to Solve
- Calculate the initial amount of kerosene in the mixture
To find the initial amount of kerosene in the 5 kg mixture, we can use the percentage given. We multiply 5 kg by 5% (0.05): $$ \text{Initial kerosene} = 5 , \text{kg} \times 0.05 = 0.25 , \text{kg} $$
- Set up the equation for the final mixture
Let $x$ be the additional amount of kerosene we need to add. After adding $x$ kg of kerosene, the total mass of the mixture becomes $5 , \text{kg} + x , \text{kg}$ and the new amount of kerosene becomes $0.25 , \text{kg} + x$. The final percentage of kerosene should equal 10%, which we can set up as an equation: $$ \frac{0.25 + x}{5 + x} = 0.10 $$
- Solve the equation for $x$
To solve for $x$, we can cross-multiply and rearrange the equation: $$ 0.25 + x = 0.10(5 + x) $$ Expanding the right side, we get: $$ 0.25 + x = 0.50 + 0.10x $$ Now, we rearrange to combine like terms: $$ x - 0.10x = 0.50 - 0.25 $$ This simplifies to: $$ 0.90x = 0.25 $$ Now, divide both sides by 0.90: $$ x = \frac{0.25}{0.90} \approx 0.2778 , \text{kg} $$
- Conclusion
Therefore, the additional amount of kerosene needed to be added is approximately 0.2778 kg.
The additional amount of kerosene needed is approximately ( 0.278 , \text{kg} ).
More Information
In this problem, we used percentages and algebraic equations to find the additional kerosene required. Understanding how to set up and solve equations is fundamental in mixture problems.
Tips
- Miscalculating the initial amount of kerosene when applying the percentage.
- Forgetting to convert the final mass correctly when setting up the equation.
- Not properly isolating $x$ in the final equation.
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