Question

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?

Answer 1

Detailed explanation

Added silver is 24 grams

Explanation:

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)`

Answer 2

`"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%`.