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Information for a period:
- Sales: $1,500,000
- Purchases: $1,000,000
- Closing Inventory: $50,000
Inventory turnover was 12 times and the gross profit margin was 40%. There were also returns and carriage inwards. Calculate opening inventory.
1. Gross Profit = $1,500,000 $\times$ 40% = $600,000.
2. Cost of Sales (COS) = $1,500,000 - $600,000 = $900,000.
3. Average Inventory = COS / Turnover = $900,000 / 12 = $75,000.
4. Average Inv = (Opening + Closing) / 2 $\Rightarrow$ $75,000 = (Opening + $50,000) / 2.
5. Opening Inventory = $150,000 - $50,000 = $100,000.
In the US, the power of Judicial Review was established in:
The 1803 Supreme Court case Marbury v. Madison established the principle of judicial review in the United States.
Three objects of masses $m_1$, $m_2$, and $m_3$ fall from rest along three different frictionless paths from the same height. Their speeds on reaching the ground will be in the ratio:
By conservation of energy: $mgh = \dfrac{1}{2}mv^2 \Rightarrow v = \sqrt{2gh}$
The final speed depends only on height $h$ and $g$, not on mass or the shape of the frictionless path. Since all start from the same height, all reach the ground with the same speed.
Ratio $= \mathbf{1:1:1}$
Find the value of $\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{\sin^2 x}{1+2^x}\,dx$
Let $I=\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{\sin^2x}{1+2^x}dx$. Use the property $\int_{-a}^a f(x)dx$: replace $x\to-x$:
$I=\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{\sin^2x}{1+2^{-x}}dx=\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{2^x\sin^2x}{1+2^x}dx$
Adding: $2I=\displaystyle\int_{-\pi/2}^{\pi/2}\sin^2x\,dx=\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{1-\cos2x}{2}dx=\dfrac{\pi}{2}$
$I=\dfrac{\pi}{4}$
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