Question

# Find the percentage of pure gold in 22-carat gold, if 24-carat gold is 100% pure.

Hint: The constant is based on the concept of proportionality. Let the purity in carat scale be k carat and the corresponding to k the purity in % scale be p%. So, p is proportional to k. Take the constant of proportionality to be c and remove the proportionality sign. Now find c using the statement that 24 carat is 100% pure. Put c and 22 carat in the equation to get % purity.

To start with the question, we let the purity in carat scale be k carat and the corresponding to k the purity in % scale be p%.
Let us first try to interpret the question in mathematical terms. We know that the purity in carat scale is directly proportional to the % purity.
$k\propto p$
Now according to the definition of proportionality, we can represent the above statement as:
$k=cp$
In the above equation, c is the constant of proportionality.
Now according to the question 24 carat represents 100% purity. Representing it mathematically using the above equation, we get
$k=cp$
$\Rightarrow 24=100c$
$\Rightarrow c=\dfrac{24}{100}\text{ }..........\text{(i)}$
Therefore, the value of the constant of proportionality is $\dfrac{24}{100}$ .
We will now find the purity corresponding to 22 carats.
$k=cp$
On substituting the value of c from equation (i), we get
$k=\dfrac{24}{100}\times p$
$\Rightarrow 22=\dfrac{24}{100}\times p$
$\Rightarrow 22\times \dfrac{100}{24}=p$
$\Rightarrow p=91.66%$
Therefore, the answer to the above question is 91.66%.

Note: In the question related to proportionality, the key thing is to find the constant of proportionality. Also, the other important thing is the calculation part, which might be a bit complex due to decimal calculations, so be careful while calculating.