Percentage All Competence Aptitude

Let's denote the initial strength of the school in 2000 as \( S \).

- In 2001, there is a 10% increase, so the strength becomes \( 1.1S \).
- In 2002, there is a 10% decrease, so the strength becomes \( 0.9 \times 1.1S = 0.99S \).
- In 2003, there is a 10% increase, so the strength becomes \( 1.1 \times 0.99S = 1.089S \).

Now, let's find the percentage change in the strength from 2000 to 2003:

\[ \text{Percentage Change} = \left( \dfrac{\text{Final Strength} - \text{Initial Strength}}{\text{Initial Strength}} \right) \times 100 \]

\[ \text{Percentage Change} = \left( \dfrac{1.089S - S}{S} \right) \times 100 \]

\[ \text{Percentage Change} = \left( \dfrac{0.089S}{S} \right) \times 100 \]

\[ \text{Percentage Change} = 8.9\% \]

Therefore, the strength of the school in 2003 as compared to 2000 increased by 8.9%. Hence, the correct option is "increased by 8.9%".

Let's denote the total number of staff as \( S \), the number of females as \( F \), and the number of males as \( M \).

1. \( 40\% \) of the staff is female, so the number of females (\( F \)) is \( 0.4 \times S \).
   \[ F = 0.4S \]

2. \( 60\% \) of the staff is male, so the number of males (\( M \)) is \( 0.6 \times S \).
   \[ M = 0.6S \]

3. \( 40\% \) of the females voted for you, so the number of female votes is \( 0.4 \times F \).
   \[ \text{Female votes} = 0.4 \times 0.4S \]

4. \( 60\% \) of the males voted for you, so the number of male votes is \( 0.6 \times M \).
   \[ \text{Male votes} = 0.6 \times 0.6S \]

Now, let's find the total number of votes (\( \text{Total votes} \)):
   \[ \text{Total votes} = \text{Female votes} + \text{Male votes} \]    \[ \text{Total votes} = 0.4 \times 0.4S + 0.6 \times 0.6S \]

   Simplify the expression:
   \[ \text{Total votes} = 0.16S + 0.36S \]    \[ \text{Total votes} = 0.52S \]

Now, find the percentage of votes you got:
   \[ \text{Percentage of votes} = \left( \dfrac{\text{Total votes}}{\text{Total staff}} \right) \times 100 \]    \[ \text{Percentage of votes} = \left( \dfrac{0.52S}{S} \right) \times 100 \]    \[ \text{Percentage of votes} = 52\% \]

Therefore, the percentage of votes you got was \( 52\% \). Hence, the correct option is "52%".

Let's denote the capital as C. According to the given information:

1. One-fourth of the capital receives 3% interest, which is (1/4) * C * 3/100.
2. Two-thirds of the capital receives 5% interest, which is (2/3) * C * 5/100.
3. The remainder (1 - 1/4 - 2/3) receives 11% interest, which is (1/12) * C * 11/100.

Now, let's add these interest amounts to find the total interest:

\[ \text{Total Interest} = \dfrac{1}{4} \cdot C \cdot \dfrac{3}{100} + \dfrac{2}{3} \cdot C \cdot \dfrac{5}{100} + \dfrac{1}{12} \cdot C \cdot \dfrac{11}{100} \]

Combine the terms:

\[ \text{Total Interest} = \dfrac{3C}{400} + \dfrac{10C}{300} + \dfrac{11C}{1200} \]

Now, find a common denominator:

\[ \text{Total Interest} = \dfrac{9C + 40C + 11C}{1200} \]

\[ \text{Total Interest} = \dfrac{60C}{1200} \]

Simplify:

\[ \text{Total Interest} = \dfrac{1}{20}C \]

To find the percentage, divide the total interest by the capital and multiply by 100:

\[ \text{Percentage} = \dfrac{\dfrac{1}{20}C}{C} \times 100 \]

\[ \text{Percentage} = \dfrac{1}{20} \times 100 \]

\[ \text{Percentage} = 5 \]

So, the percentage he receives on the whole is 5%. Therefore, the correct answer is 5.

Let's denote the original price of eggs as \(P\) per dozen. 

Due to a 50% increase, the new price (\(P_{\text{new}}\)) is given by:

\[ P_{\text{new}} = P + 0.5 \times P = 1.5P \]

Now, the number of eggs available for the original price is \(12\) (a dozen). Due to a 50% increase in price, 4 eggs less are available for the new price.

So, the number of eggs available for the new price is \(12 - 4 = 8\).

Now, we can set up the equation:

\[ P_{\text{new}} \times \left( \dfrac{8}{12} \right) = 24 \]

Substitute the expression for \(P_{\text{new}}\):

\[ 1.5P \times \left( \dfrac{8}{12} \right) = 24 \]

Solving for \(P\):

\[ 1.5P \times \dfrac{2}{3} = 24 \]

\[ P = \dfrac{24 \times 3}{1.5 \times 2} \]

\[ P = 36 \]

So, the original rate of eggs per dozen (\(P\)) is 36. Therefore, the correct answer is 36.

Let's denote the income before tax as \( I \). The individual pays 30% income tax on this amount, so the tax paid (\( T \)) is given by:

\[ T = 0.30 \times I \]

Now, there is a surcharge of 10% on this tax, so the total tax paid (\( \text{Total Tax} \)) is:

\[ \text{Total Tax} = T + 0.10 \times T \]

\[ \text{Total Tax} = 1.10 \times T \]

Substitute the expression for \( T \):

\[ \text{Total Tax} = 1.10 \times (0.30 \times I) \]

\[ \text{Total Tax} = 0.33 \times I \]

Now, to find the net tax rate, we need to calculate the percentage of the total tax with respect to the original income:

\[ \text{Net Tax Rate} = \left( \dfrac{\text{Total Tax}}{I} \right) \times 100 \]

\[ \text{Net Tax Rate} = \left( \dfrac{0.33 \times I}{I} \right) \times 100 \]

\[ \text{Net Tax Rate} = 33\% \]

Therefore, the net tax rate the individual has to pay is 33%. The closest option is 33%, so the correct answer is:

\[ \text{Net Tax Rate} = 33\% \]

Let's denote the number of eggs the man bought as \( E \).

1. Initially, 10% of the eggs are rotten, so the remaining good eggs are \( 90\% \) of the total:
   \[ \text{Good eggs after first step} = 0.9 \times E \]

2. The man gives 80% of the remaining eggs to his neighbors, so the eggs he has left are \( 20\% \) of the good eggs after the first step:
   \[ \text{Eggs left after second step} = 0.2 \times (0.9 \times E) \]

According to the given information, he is left with 36 eggs:
   \[ 0.2 \times (0.9 \times E) = 36 \]

Now, solve for \( E \):
   \[ E = \dfrac{36}{0.2 \times 0.9} \]

   \[ E = \dfrac{36}{0.18} \]

   \[ E = 200 \]

Therefore, the man initially bought 200 eggs. So, the correct answer is 200.

Let's denote the average of the original set A as \( \bar{A} \).

The average of a set is calculated by summing up all the values in the set and then dividing by the number of values.

\[ \bar{A} = \dfrac{27 + 28 + 30 + 32 + 33}{5} \]

Now, if we include integer \( k \) and the average increases by 30%, the new average (\( \bar{A'} \)) is given by:

\[ \bar{A'} = 1.3 \times \bar{A} \]

Now, we can set up the equation:

\[ \dfrac{27 + 28 + 30 + 32 + 33 + k}{6} = 1.3 \times \bar{A} \]

Plug in the expression for \( \bar{A} \):

\[ \dfrac{150 + k}{6} = 1.3 \times \left( \dfrac{150}{5} \right) \]

Solve for \( k \):

\[ \dfrac{150 + k}{6} = 1.3 \times 30 \]

\[ 150 + k = 6 \times 39 \]

\[ 150 + k = 234 \]

\[ k = 84 \]

Therefore, the value of integer \( k \) is 84. So, the correct answer is 84.

Let the annual salary in the year 2006 be \(S\). According to the given information, the annual salary in 2007 is 10% more than that in 2006, so:

\[ \text{Salary in 2007} = S + 0.10 \times S \]

Given that the salary in 2007 is \(8,80,000\), we can set up the equation:

\[ 8,80,000 = S + 0.10 \times S \]

Combine like terms:

\[ 8,80,000 = 1.10 \times S \]

Now, solve for \(S\):

\[ S = \dfrac{8,80,000}{1.10} \]

\[ S = 8,00,000 \]

Therefore, the annual salary in the year 2006 was \(8,00,000\). So, the correct answer is \(8,00,000\).

Let \(S\) be the original side length of the cube, and \(V\) be the original volume of the cube.

The original volume \(V\) is given by \(V = S^3\).

If each side of the cube is increased by 10%, the new side length (\(S_{\text{new}}\)) will be \(1.10 \times S\). The new volume (\(V_{\text{new}}\)) is given by \((1.10 \times S)^3\).

Now, let's find the percentage increase in volume:

\[ \text{Percentage Increase} = \left( \dfrac{V_{\text{new}} - V}{V} \right) \times 100 \]

\[ \text{Percentage Increase} = \left( \dfrac{(1.10 \times S)^3 - S^3}{S^3} \right) \times 100 \]

\[ \text{Percentage Increase} = \left( \dfrac{1.331 \times S^3 - S^3}{S^3} \right) \times 100 \]

\[ \text{Percentage Increase} = \left( \dfrac{0.331 \times S^3}{S^3} \right) \times 100 \]

\[ \text{Percentage Increase} = 33.1\% \]

Therefore, if each side of a cube is increased by 10%, the volume of the cube will increase by 33.1%. So, the correct answer is 33.1%.

Let's denote the marks obtained by D as \(D\), by C as \(C\), by B as \(B\), and by A as \(A\).

Given information:
1. \(C\) got 25% more than \(D\), so \(C = D + 0.25D = 1.25D\).
2. \(B\) got 10% less than \(C\), so \(B = C - 0.10C = 0.90C\).
3. \(A\) got 25% more than \(B\), so \(A = B + 0.25B = 1.25B\).
4. \(D\) got 320 marks out of 500, so \(D = 320\).

Now, let's calculate \(C\):\[ C = 1.25D = 1.25 \times 320 \]

\[ C = 400 \]

Now, let's calculate \(B\):\[ B = 0.90C = 0.90 \times 400 \]

\[ B = 360 \]

Finally, let's calculate \(A\):\[ A = 1.25B = 1.25 \times 360 \]

\[ A = 450 \]

Therefore, the marks obtained by A were 450. So, the correct answer is 450.