Question

If 1 g of gold 10 carats of fine, 1 g of gold 11 carats of fine, 2 g of gold 12 carats of fine, and 5 g of gold 13 carats of fine be mixed together, find the fineness of the resulting compound.
(a) 12.5
(b) $12\dfrac{2}{9}$
(c) 12
(d) 15

Answer 1

Hint: First, we will find the total quantity of gold given in question. Then, to find the fineness of the compound we will use the formula $\text{carats of fine}\times \dfrac{\text{quantity of gold}}{\text{total grams of gold}}$ . By putting the values and solving them we will get the answer which will not be directly present in the option. So, we will then take option value and check one by one. We will also convert option (b) from mixed fraction into normal fraction by using the formula $\left( divisor\times quotient \right)+remainder$ . Thus, we will get the correct answer.

Complete step-by-step answer:
Here, we are given the fineness of gold whose values are in carats along with the quantity of gold. So, we will first find the total gram of gold from the question.
So, it is given that 1 g of gold 10 carats of fine, 1 g of gold 11 carats of fine, 2 g of gold 12 carats of fine, and 5 g of gold 13 carats of fine. We can write total grams of gold as
Total grams of gold $=1+1+2+5=9g$
 Now, we have to find the fineness of compound i.e. mixing gold along with-it fineness. So, we can write it as
$\text{carats of fine}\times \dfrac{\text{quantity of gold}}{\text{total grams of gold}}$
Thus, using this formula and adding all the quantity given to us we will get as
Fineness \[=10\times \dfrac{1}{9}+11\times \dfrac{1}{9}+12\times \dfrac{2}{9}+13\times \dfrac{5}{9}\]
On solving this we will get as,
\[=\dfrac{10}{9}+\dfrac{11}{9}+\dfrac{24}{9}+\dfrac{65}{9}\]
On adding the numerator part as denominator is same for all we get as,
\[=\dfrac{10+11+24+65}{9}=\dfrac{110}{9}\]
Thus, we can say that the fineness of compounds is \[\dfrac{110}{9}\] .
Now, this value is not an option directly. So, we will check all the options which match with the answer.
On taking option (a), (c), (d) we can directly know that is not the correct answer. Now, if we convert option (b) from mixed fraction into normal fraction. Basically, mixed fraction is in the form of $quotient\dfrac{remainder}{divisor}$ . So, to convert it into fraction form we have to multiply $\left( divisor\times quotient \right)+remainder$ . By this we will get a fraction in the form of $\dfrac{dividend}{divisor}$ .
So, we can write $12\dfrac{2}{9}$ as \[\dfrac{\left( 12\times 9 \right)+2}{9}=\dfrac{110}{9}\] .
Hence, option (b) is the correct answer.

Note: To find fineness of compound, do not simply multiply quantity of gold with number of carats of fine. This will lead to incorrect answers. On doing this directly, we will get as \[\left( 1\times 10 \right)+\left( 1\times 11 \right)+\left( 2\times 12 \right)+\left( 5\times 13 \right)\] which will given the answer \[10+11+24+65=110\] which is not answer given in option. So, please understand the question carefully and then try to solve it.