Question

A jeweler has bars of $18$ carat gold and $12$ carat gold. How much of each must be melted together to obtain a bar of $16$ carat gold weighing $120$ grams? (pure gold $24$ carat)

Answer 1

Hint: First we have to define what the terms we need to solve the problem are.
Since a pure gold is $24$ carat, the jeweler has bars two different carats of gold which are eighteen and twelve respectively. He also needs to melt that two golds to get the carat of sixteen at the weight of one twenty grams.

Complete step-by-step solution:
Since as per the given information, we assume that $x$ is the $18$ carat gold and $y$ is the $12$ carat gold.
Also, the overall grams after melting both the carat golds needs to be $120$ grams; hence by adding the x and y we will need that one twenty grams’,
we convert that into the equation form we get; $x + y = 120$ ………….. $(1)$
Let $18$ carat gold and $12$ carat gold can also write in the equation form $18x + 12y = 16 \times 120$ ……...$(2)$
(since each must be melted together to obtain a bar of $16$ carat gold and we multiple to given one twenty grams), now we have two equations and we need to simplify using the elimination method, which help as to eliminate one variable after that we can find the second variable using the first variable
So, in both equations at least one variable need to same and hence multiply equation $(1)$ with $12$ we get;
$12x + 12y = 12 \times 120$ now cancel this with comparing the equation 2.
Hence, we get $x = 80$ (by elimination method), as we said we have found the first variable so let us substitute in the equation 1 to get the second variable thus put $x = 80$ in $x + y = 120$ we get; $y = 40$
Since x is the $18$ carat gold and y is the $12$ carat gold; now we find the quantity of $18$ carat gold is $80$
And the quantity of $12$ carat gold is $40$ to obtain the bar of $16$ carat gold weighing $120$ grams.

Note: The purity of the gold is $24$ carat, we are also able to solve this problem using this $24$ carat.
Since $120$ grams are the weight that overall, we need so $120 - x$ (x is unknown grams)
Hence $x\dfrac{{18}}{{24}} + (120 - x)\dfrac{{12}}{{24}} = 120 \times \dfrac{{16}}{{24}}$ if we simplify this, we get same result as above.