Math problem help (Solving Systems by Addition or Subtraction)?

Question

Hey, could someone help me with this math problem? For some reason I can't get the answer to work...

How much 24 karat gold should be added to 50 dwt of 14 karat gold to make 18 karat gold? How much 18 karat gold will there be?

24 karat gold is 100% gold. 14 karat gold is 58.3% gold. 18 karat gold is 75% gold.

dwt stands for pennyweights, the unit used to weigh precious metals.

Could you work through the problem step by step, by using addition or subtraction to solve the system? Maybe then I can see what I'm doing wrong...

Thanks.

Answer 1

Okay... let x = amount of gold to add.

.583(50) + x = .75 (50 + x)

which means .583(50) is the weight of gold in 14k gold (58.3% of 50dwt) + X weight of pure gold will equal 75% (18k) of the TOTAL weight (50 + x).

fractionally it would be 14/24 (50) + x = 18/24 (50 +x)

multiply the equation by 24

14(50) + 24x = 18(50 + x)

calculate

700 + 24x = 900 + 18x

combine

24x - 18x = 900-700
6x = 200
x = 33 1/3 dwt pure gold

Answer 2

Let x dwt of 24 karat or pure gold is required.Therefore according to the problem,50(58.3/100)+x=(75/100)(50+x) => 50(583/1000)+(3/4)(50+x) => 583/20 + x = 150/4 +3x/4 =>583+20x=750+15x[multiplying all terms of the equation by20] =>20x-15x=750-583 =>5x=167 =>x=167/5=33,4 dwt .Therefore 33.4 dwt of 24 karat gold would be required. A special thank to the asker for providing all the details

Answer 3

i'm not sure but i think as you have 58.3% of the 50 dwt gold then you should add 16.7 dwt 100% gold to make the the percentage 75% of 18 karat. You will eventually get a 66.7 dwt 18 karat!
75-58.3= 16.7
16.7+50=66.7

Answer 4

if 14k = 58.3% pure, then 50dwt of 14k contains 29.15dwt pure and 20.85dwt impurities.
50dwt * .583 = 29.15 dwt
50dwt - 29.15dwt = 20.85dwt
if 18k = 75% pure, then total dwt is 3/4 pure and 1/4 impure... that equals a 4 to 1 ratio... if you have 20.85dwt of impurities and that is 1/4 of total dwt then the remaining total dwt of gold must be 4 times that or 4*20.85 = 83.4dwt total weight pure gold.
83.4dwt total - 29.15dwt original = 54.25dwt pure gold to be added... result is by adding 54.25dwt pure (24k) to 50dwt (14k) yields 104.25dwt (18k)... (20.85dwt impurities/29.15dwt+54.25dwt = .25 or 25% impure/75% pure)
50dwt (14k) + 54.25dwt (24k) = 104.25dwt (18k)