A jeweler has five rings, each weighing 18g, made of 5% silver and 95% gold. He decides to melt down the rings and add enough silver to reduce the gold content to 75%. How much silver should he add?

2 Answers
Apr 15, 2016

Detailed explanation

Added silver is 24 grams

Explanation:

Tony B

Targeting silver content as the means of determining the blend ratio.

The process is based on two extreme condition:

All ring #-> 5%" silver"#
No ring#->100%" silver"#

The vertical axis is the silver content of the alloy.

The horizontal axis is the percentage of the added silver.

The target silver content of the alloy is 25%

#color(brown)("What this process is actually saying is: the gradient of part of the")##color(brown)("line is the same as the gradient of all of the line.")#

#color(blue)("Using ratios to determine the added silver proportion")#

#("change in y")/("change in x")->("change in silver content")/("change in added silver")#

#=>(25-5)/x=(100-5)/(100)#

#=>20/x=95/100#

Turn everything upside down (invert)

#=>x/20=100/95#

Multiply throughout by 20

#x=(100xx20)/95 = 21 1/19 -> color(red)(400/19 ~~21.05% )# to 2 decimal places

#color(red)("The "400/19 "is a trap!")#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Not that the fraction is precise. The decimal is not!

#color(red)("Also note that the "400/19" represents the numerator in "x/100)#

So to convert this to the format we require #underline(color(green)("divide by 100"))# giving #x/100#

#color(blue)("Added silver "400/19 % -> 400/(19xxcolor(green)(100)))#

#color(blue)( = 4/19" " underline("as a fraction of the whole."))#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine the weight of the added silver")#

This means that the rings represent #1-4/19# of the whole.

Thus the weight of the rings represents #1-4/19# of all the weight.

Let the total weight be #w#

Then #(1-4/19)xx w=5xx18#

#15/19 w= 90#

#w=19/15xx90 = 114 g# (Total weight in grams)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(green)("Silver added "-> 114 -(5xx18) =24 g)#

Apr 19, 2016

#"24 g"#

Explanation:

Here's an alternative approach to use. You know that each ring contains #5%# silver and #95%# gold by mass. Use this information to find the mass of silver and the mass of gold in one ring

#18 color(red)(cancel(color(black)("g ring"))) * "5 g silver"/(100color(red)(cancel(color(black)("g ring")))) = "0.9 g silver"#

This means that the mass of gold will be

#m_"gold" = "18 g" - "0.9 g" = "17.1 g gold"#

Now, you know that the jeweler is working with five rings. The total mass of silver and the total mass of gold present in the five rings will be

#5color(red)(cancel(color(black)("rings"))) * "0.9 silver"/(1color(red)(cancel(color(black)("ring")))) = "4.5 g silver"#

#5 color(red)(cancel(color(black)("rings"))) * "17.1 g gold"/(1color(red)(cancel(color(black)("ring")))) = "85.5 g gold"#

The total mass of the rings will be

#m_"total" = overbrace("4.5 g")^(color(blue)("mass of silver")) + overbrace("85.5 g")^(color(blue)("mass of gold")) = "90 g"#

Now, let's assume that #x# represents the mass of silver that must be added in order to reduce the gold content to #75%#.

The mass of gold remains unchanged by the addition of silver. Adding #x# grams of silver to the mixture will bring its total mass to #90 + x# grams. Since you know that you have #85.5# grams of gold in this mixture, you can say that

#overbrace(85.5color(white)(a) color(red)(cancel(color(black)("g"))))^(color(purple)("mass of gold")) = 75/100 * overbrace((90 + x)color(red)(cancel(color(black)("g"))))^(color(purple)("total mass of the mixture"))#

Isolate #x# on one side of the equation to get

#8550 = 6750 + 75x#

#75x = 1800 implies x = 1800/75 = 24#

Therefore, you must add #"24 g"# of silver to the mixture to get the gold content to drop from #95%# to #75%#.