A jeweller 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 is 24-carat).

Let us assume the quantity of 18 carat gold as x gm and 12 carat gold as y gm.

Given, total weight of new bar = 120 gm.

x + y = 120 ……(i)

Pure gold is 24 carat.

So, the purity of 18 carat gold = $\dfrac{18}{24} \times 100\% = \dfrac{3}{4} \times 100\% = 75\%.$

Purity of 12 carat gold = $\dfrac{12}{24} \times 100\% = 50\%.$

Purity of 16 carat gold = $\dfrac{16}{24} \times 100\% = \dfrac{200}{3}\%$

According to condition,

$75\% \text{ of } x + 55\% \text{ of } y = \dfrac{200}{3}\% \text{ of } 120 \\[1em] \Rightarrow \dfrac{75}{100}x + \dfrac{50}{100}y = \dfrac{200}{3 \times 100} \times 120 \\[1em] \dfrac{3}{4}x + \dfrac{1}{2}y = 80 \\[1em] \dfrac{3x + 2y}{4} = 80 \\[1em] 3x + 2y = 320 \space ……(ii)$

Multiplying (i) by 2 we get,

⇒ 2x + 2y = 240 …….(iii)

Subtracting eq (iii) from eq (ii)

⇒ (3x + 2y) - (2x + 2y) = 320 - 240

⇒ 3x - 2x + 2y - 2y = 80

⇒ x = 80.

Substituting value of x in (i) we get,

⇒ 80 + y = 120

⇒ y = 40.

Hence, jeweller requires 80 gm of 18 carat gold and 40 gm of 12 carat gold to obtain a bar of 16 carat gold weighing 120 gm.