Question

The percentage of metals in a mine of lead ore is $60\%.$ Now the percentage of silver is $\dfrac{3}{4}\%$ of metals and the rest is lead. If the mass of ore extracted from this mine is 8000 kg, the mass $\left( \text{in kg} \right)$ of lead is:
A. 4763
B. 4764
C. 4762
D. 4761

Answer 1

Hint: To answer this question, we use the concept percentage. We first need to find the mass of metals in the mine of lead ore. Since the mass of this lead ore is given as 8000, we find $60\%$ of this value to obtain the mass of metals. Then for this mass of metals, we calculate $\dfrac{3}{4}\%$ and subtract it from the total mass of metals to get the mass of lead.

Complete step by step solution:
We simply use the concept of percentage here to obtain the answer to the given question. It is given that the mass of the lead ore is 8000 kg. This is the total mass of the ore extracted from the mine.
$\Rightarrow \text{Mass of the lead ore = 8000 kg}$
Now in this mass of lead ore, it is given that the percentage of metals contained is $60\%.$ Hence, to find the mass of the metals, we multiply the mass of the lead ore by this percentage.
$\Rightarrow \text{Mass of metal in lead ore = 60 }\!\!\%\!\!\text{ of 8000 kg}$
Percentage as we all know means out of 100. Hence for the above equation, $60\%$ means $\dfrac{60}{100}$ times the total value.
$\Rightarrow \text{Mass of metal in lead ore = }\dfrac{60}{100}\times 8000$
Cancelling the 100 with 8000 resulting in 80,
$\Rightarrow \text{Mass of metal in lead ore = }60\times 80$
Multiplying the two numbers,
$\Rightarrow \text{Mass of metal in lead ore = }4800\text{ kg}$
Now, for this mass of metal, the percentage of silver is given as $\dfrac{3}{4}\%$ of the mass of metal which is 4800 kg.
$\Rightarrow \text{Mass of silver in the metal = }\dfrac{3}{4}\%\text{ of }4800\text{ kg}$
Again, $\dfrac{3}{4}\%$ means that it is out of 100. Therefore,
$\Rightarrow \text{Mass of silver in the metal = }\dfrac{\dfrac{3}{4}}{100}\times 4800$
Taking reciprocal of 100 in the denominator and multiplying with the $\dfrac{3}{4}$ term,
$\Rightarrow \text{Mass of silver in the metal = }\dfrac{3}{400}\times 4800$
The HCF for 4800 and 400 is 400 and 400 times 12 is 4800. Simplifying this,
$\Rightarrow \text{Mass of silver in the metal = 3}\times 12$
Multiplying 3 and 12,
$\Rightarrow \text{Mass of silver in the metal = 36 kg}$
Now to find the mass of lead, we subtract this mass of silver, 36 kg, from the total mass of metals which is 4800 kg. This can be done since the metal contains only silver and lead.
$\Rightarrow \text{Mass of lead in the metal = 4800-32}$
Subtracting the two terms,
$\Rightarrow \text{Mass of lead in the metal = 4764 kg}$
Hence, the correct option is B. The mass of lead in the metal is 4764 kg.

Note: Students need to know the concept of percentage to solve this question. They need to be careful while considering the masses at each stage. This question can also be solved by calculating the mass of metal in the same way, then to find the mass of silver, we subtract the $\dfrac{3}{4}\%$ from the total $100\%$ which gives the percentage of silver in the metal. We then calculate this percentage for the mass of metal to obtain the answer.