A merchant bought a batch of goods. He fixed the price at 30% above his cost price. He managed to sell half of the stock at this price, one quarter of the stock at a discount of 20% on the marked price and the remainder at a discount of 40% on the marked price. Find his percentage profit on the batch of goods.
My working:
Let $T$ be the total profit on the batch of goods sold at the marked price. Let $mp$ be the marked price and $cp$ be the cost price.
Then
$\frac{1}{2}T=mp-cp$
$\frac{1}{4}T=80\%mp-cp$
$\frac{1}{4}T=60\%mp-cp$
But $mp=130\%cp$
So
$\frac{1}{2}T=130\%cp-cp$
$\frac{1}{4}T=80\%(130\%cp)-cp$
$\frac{1}{4}T=60\%(130\%cp)-cp$
which implies $T=\frac{13}{10}cp-cp+\frac{26}{25}cp-cp+\frac{39}{50}cp-cp$, then $T=\frac{3}{25}cp$. What we got is a loss! The total profit is less than the cost price?
The correct answer is 10.5%.
Could anybody tell me where did I do wrong and how could I fix it?
Is there any easier approach to the problem?
Thanks.
Take total CP as $x$
For half stocks,
CP$=\frac {x}{2}$
SP$=\frac {x}{2}.\frac {13}{10}$
For 1st quater stocks,
CP $=\frac {x}{4}$
SP$=\frac {x}{4}.\frac {13}{10}.\frac {100-20}{100}$
Same goes for 2nd quarter stocks, just with different discount.
You will get total SP = $1.105×x $
which is in agreement with the given answer.