Question

A goldsmith has two alloys of gold, the first having a purity of 53% and the second having a purity of 83%. If x grams of the first alloy are mixed with y grams of the second, obtaining 100 grams of an alloy which contains 72.5% gold, find x to the nearest gram.

Answer 1

Correct answer: x = 35 grams and y = 65 grams

Explanation:

Given:

53% purity of the first alloy

83% purity of the second alloy

x = ? mass of the first alloy

y = ? mass of the second alloy

x + y = 100 grams the mass of the mixture

72.5% purity of the mixture

We can solve the problem by setting up a two system of equations:

First: Second:

53 x + 83 y = 100 · 72.5 x + y = 100

x + y = 100 x : y = (83 - 72.5) : (72.5 - 53)

First system:

y = 100 - x

From the second, we express y as a function of x and replace in the first

and get:

53 x + 83 (100 - x) = 7,250

53 x + 8,300 - 83 x = 7,250

30 x = 1050 ⇒ x = 1050 / 30 = 35 grams

x = 35 grams

y = 100 - 35 = 65 grams

y = 65 grams

Second system:

x + y = 100

x : y = (83 - 72.5) : (72.5 - 53) = 10.5 : 19.5 = 105 : 195 = 7 : 13

x : y = 7 : 13 ⇒ x = (7/13) y

From the second, we express x as a function of y and replace in the first

and get:

(7/13) y + y = 100 / ·(13) ⇒ 7 y + 13 y = 1300 ⇒ 20 y = 1300

y = 1300 / 20 = 65 grams

y = 65 grams

x + 65 = 100 ⇒ x = 100 - 65 = 35 grams

x = 35 grams

God is with you!!!