Question

In one alloy there is $60\% $ gold in its total mass, while in another alloy it is $35\% $. $12kg$ of the first alloy was melted together with $8kg$ of the second one to form a third alloy. Find the percentage of gold in the new alloy.

Answer 1

Hint: Note down all the values. Find the amount of gold in first alloy in terms of weight. Then find the amount of gold in the second alloy in terms of weight. When they are melted together, the amount of gold in the third alloy is the sum of the gold present in the previous two alloys. Now find its percentage in the total weight.

Complete step-by-step answer:
Let us note down the given data firstly,
Percentage of gold in the first alloy is $60\% $.
Percentage of gold in the second alloy is $35\% $.
Total weight of the first alloy used is $12kg$.
Total weight of the second alloy used is $8kg$.
It is given that the both alloys were melted together and prepared a third alloy.
So, the weight of the third alloy is the sum of the weights of the previous two alloys that are melted.
Hence, weight of the third alloy $ = 12 + 8$
                    $ = 20kg$
Now, let us find out the amount of gold present in the first alloy.
It is given that $60\% $ of gold is present in $12kg$ alloy.
So, Weight of the gold in the first alloy $ = $ $60\% $ of $12kg$
                                                      $\
   = \dfrac{{60}}{{100}} \times 12 \\
   = \dfrac{3}{5} \times 12 \\
   = 7.2kg \\
\ $
So, the first alloy has $7.2kg$ of gold in it.
Similarly,
Now, let us find out the amount of gold present in the second alloy.
It is given that $35\% $ of gold is present in $8kg$ alloy.
So, Weight of the gold in second alloy$ = $ $35\% $ of $8kg$
                                                    $\
   = \dfrac{{35}}{{100}} \times 8 \\
   = \dfrac{7}{{20}} \times 8 \\
   = \dfrac{{7 \times 4}}{{10}} \\
   = 2.8kg \\
\ $
So, first alloy has $2.8kg$ of gold in it.
So, the weight of gold in the third alloy is the sum of the weights of the previous two alloys that are melted.
Weight of gold in the third alloy $ = 7.2 + 2.8$
                    $ = 10kg$
But we are asked to find out the percentage value of the gold.
So, the percentage of gold in the third alloy can be calculated by dividing the amount of gold in the third alloy with the total amount and then multiplying with hundred.
So, the percentage of gold in third alloy $ = \dfrac{{10}}{{20}} \times 100$
                         $\
   = \dfrac{1}{2} \times 100 \\
   = 50\% \\
\ $
Hence there is $50\% $ of gold in the third alloy.

Note: This type of percentage related problems seems to be easy but we must take care while calculating. This problem can be also asked with fractions by replacing the percentages. The way we solve will be a little bit different but the concept behind will be the same. To convert a fraction into percentage, we must multiply the fraction by 100 and perform the mathematical operation.