# 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?

Apr 15, 2016

Detailed explanation

#### 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%

$\textcolor{b r o w n}{\text{What this process is actually saying is: the gradient of part of the}}$$\textcolor{b r o w n}{\text{line is the same as the gradient of all of the line.}}$

$\textcolor{b l u e}{\text{Using ratios to determine the added silver proportion}}$

$\left(\text{change in y")/("change in x")->("change in silver content")/("change in added silver}\right)$

$\implies \frac{25 - 5}{x} = \frac{100 - 5}{100}$

$\implies \frac{20}{x} = \frac{95}{100}$

Turn everything upside down (invert)

$\implies \frac{x}{20} = \frac{100}{95}$

Multiply throughout by 20

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

$\textcolor{red}{\text{The "400/19 "is a trap!}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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

$\textcolor{red}{\text{Also note that the "400/19" represents the numerator in } \frac{x}{100}}$

So to convert this to the format we require $\underline{\textcolor{g r e e n}{\text{divide by 100}}}$ giving $\frac{x}{100}$

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

color(blue)( = 4/19" " underline("as a fraction of the whole."))
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the weight of the added silver}}$

This means that the rings represent $1 - \frac{4}{19}$ of the whole.

Thus the weight of the rings represents $1 - \frac{4}{19}$ of all the weight.

Let the total weight be $w$

Then $\left(1 - \frac{4}{19}\right) \times w = 5 \times 18$

$\frac{15}{19} w = 90$

$w = \frac{19}{15} \times 90 = 114 g$ (Total weight in grams)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{g r e e n}{\text{Silver added } \to 114 - \left(5 \times 18\right) = 24 g}$

Apr 19, 2016

$\text{24 g}$

#### 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

${m}_{\text{gold" = "18 g" - "0.9 g" = "17.1 g gold}}$

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

${m}_{\text{total" = overbrace("4.5 g")^(color(blue)("mass of silver")) + overbrace("85.5 g")^(color(blue)("mass of gold")) = "90 g}}$

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

$8550 = 6750 + 75 x$

$75 x = 1800 \implies x = \frac{1800}{75} = 24$

Therefore, you must add $\text{24 g}$ of silver to the mixture to get the gold content to drop from 95% to 75%.