# Mrs.Sharma sold a jewellery at a loss of 5%. If he had sold it at Rs.5,200 more, she would have gained 8%. Find the cost price of the jewellery ?

Jul 14, 2018

$\text{cost price} = R s 40000$

A lot of detail given so that you can see where everything comes from.

#### Explanation:

$\textcolor{b l u e}{\text{Modelling the given conditions}}$

Set the original selling price as $x$
Let the cost price be $y$

$\textcolor{b r o w n}{\text{Consider the 5% loss condition. }}$

To get get just the cost price back we have $\frac{100}{100} \times y$

But what she got back was: $\frac{100 - 5}{100} \times y$

And we know this is the original selling price so we have:

95/100ycolor(white)("d")=color(white)("d")xcolor(white)("dddd") =>color(white)("dddd") ycolor(white)("d")=color(white)("d")100/95 x" "..Equation(1)

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$\textcolor{b r o w n}{\text{Consider the 8% profit}}$

To obtain this profit she sold it for Rs 5200 more than the original selling price of $x$. So we have:

y+8%y=x+5200

y(1+8%)=x+5200

$y \left(\frac{108}{100}\right) = x + 5200$

$y = \left[\left(\frac{100}{108}\right) x\right] + \left[\frac{100}{108} \times 5200\right] \text{ } \ldots E q u a t i o n \left(2\right)$
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$\textcolor{b l u e}{\text{Putting it all together}}$

Substitute for $y$ in $E q u a t i o n \left(2\right)$ using $E q u a t i o n \left(1\right)$

$\frac{100}{95} x = \frac{100}{108} x + \left[\frac{100}{108} \times 5200\right]$

Subtract $\frac{100}{108} x$ from both sides

$\frac{100 x}{95} - \frac{100 x}{108} = \frac{100 \times 5200}{108}$

Lets make all the denominators the same so we need to change the 95 into 108.

Note that $95 \times \frac{108}{95} = 108$ so by applying this we have:

$\frac{100 \times \frac{108}{95} \times x}{108} - \frac{100 x}{108} = \frac{100 \times 5200}{108}$

Multiply all of both sides by 108

$\left(100 \times \frac{108}{95} \times x\right) - 100 x = 100 \times 5200$

$\frac{10800}{95} x - 100 x = 520000 \leftarrow \text{ Using decimals will introduce}$
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color(white)("dddddddddddddddddddddddd")" with fractions"

$\frac{260}{19} x = 520000$

$x = 520000 \times \frac{19}{260} = 38000$
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$\textcolor{b l u e}{\text{Determine the cost price.}}$

Using $E q u a t i o n \left(1\right)$

$y = \frac{100}{95} x \textcolor{w h i t e}{\text{d")->color(white)("d}} y = \frac{100}{95} \times 38000$

$\text{cost price} = y = R s 40000$

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