Percentage All Competence Aptitude

The statement "p% of p is 36" can be expressed as the equation:

\[ \dfrac{p}{100} \times p = 36 \]

To find the value of \( p \), you can solve this equation:

\[ p^2 = 3600 \]

Taking the square root of both sides:

\[ p = \sqrt{3600} \]

\[ p = 60 \]

So, \( p \) is equal to 60. Therefore, the correct option is 60.

To find 20% of 25% of 300, you can multiply the two percentages and then multiply the result by 300:

\[ 20\% \text{ of } (25\% \text{ of } 300) \]

\[ = 0.20 \times (0.25 \times 300) \]

\[ = 0.20 \times 75 \]

\[ = 15 \]

So, 20% of 25% of 300 is 15. Therefore, the correct option is 15.

To express half of 1 per cent as a decimal, you can follow these steps:

1. Convert 1 per cent to a decimal: \(1\% = 0.01\).
2. Find half of 0.01: \(0.5 \times 0.01\).

\[0.5 \times 0.01 = 0.005\]

So, half of 1 per cent, written as a decimal, is \(0.005\). Therefore, the correct option is \(0.005\).

Let's denote A's income as \( A \) and B's income as \( B \).

The given statement is: 60% of A's income is equal to 75% of B's income.

This can be expressed as the equation:

\[ 0.6 \times A = 0.75 \times B \]

Now, we need to find \( x \), which represents B’s income as a percentage of A’s income:

\[ B = \dfrac{x}{100} \times A \]

Now, substitute the first equation into the second equation:

\[ 0.6 \times A = 0.75 \times \left( \dfrac{x}{100} \times A \right) \]

Now, solve for \( x \):

\[ 0.6 = 0.75 \times \dfrac{x}{100} \]

Multiply both sides by \( \dfrac{100}{0.75} \):

\[ x = \dfrac{0.6 \times 100}{0.75} \]

\[ x = 80 \]

Therefore, \( B \)’s income is equal to 80% of \( A \)’s income. The value of \( x \) is 80. So, the correct option is 80.

First, let's calculate the average income (\( A \)) per person:

\[ A = \dfrac{T}{16} \]

where \( T \) is the total monthly income of 16 persons, given as $80,800.

\[ A = \dfrac{80,800}{16} = 5,050 \]

Now, it's given that one person's income (\( I \)) is 120% of the average income:

\[ I = 1.2 \times A \]

Substitute the value of \( A \) into the equation:

\[ I = 1.2 \times 5,050 = 6,060 \]

Therefore, the income of one person is $6,060. So, the correct option is 6,060.

1:1 is equivalent to 100%
This implies 1/1 = 100%
1 →, 100%
 

5:4 is equivalent is x 
5/4 = 1.25

$1.25 \to \frac{100*1.25}{1}=125\%$

Let's say the initial price of kerosene oil is $P$, and the person initially consumes $Q$ units of kerosene oil. Therefore, the initial expenditure ($E_1$) is given by:

$$E_1=P\times Q$$

After the price increase of 25%, the new price of kerosene oil is $1.25P$. The person would need to reduce their consumption ($Q^{\prime }$) to maintain the same expenditure. The new expenditure ($E_2$) is given by:

$$E_2=1.25P\times Q^{\prime }$$

The person needs to find $Q^{\prime }$ such that $E_2=E_1$, so:

$$P\times Q=1.25P\times Q^{\prime }$$

Now, solve for $Q^{\prime }$:

$$Q^{\prime }=\frac{Q}{1.25}$$

Calculate the percentage reduction in consumption:

$$\text{Percentage Reduction}=\left(1-\frac{Q^{\prime }}{Q}\right)\times 100$$

Substitute the value of $Q^{\prime }$:

$$\text{Percentage Reduction}=\left(1-\frac{Q}{1.25Q}\right)\times 100$$

Simplify:

$$\text{Percentage Reduction}=\left(1-\frac{1}{1.25}\right)\times 100$$

$$\text{Percentage Reduction}=\left(1-0.8\right)\times 100$$

$$\text{Percentage Reduction}=0.2\times 100$$

$$\text{Percentage Reduction}=20\mathrm{\%}$$

Therefore, a person must reduce their consumption of kerosene oil by 20% to ensure that their expenditure does not increase. Hence, the correct option is 20%.

Let's denote the initial price of rice as \( P \), the initial consumption as \( Q \), and the initial expenditure as \( E_1 \).

\[ E_1 = P \times Q \]

After the price increase of 25%, the new price of rice is \( 1.25P \). To maintain the same expenditure (\( E_1 \)), the person would need to reduce their consumption (\( Q' \)). The new expenditure (\( E_2 \)) is given by:

\[ E_2 = 1.25P \times Q' \]

The person needs to find \( Q' \) such that \( E_2 = E_1 \), so:

\[ P \times Q = 1.25P \times Q' \]

Now, solve for \( Q' \):

\[ Q' = \dfrac{Q}{1.25} \]

Calculate the percentage reduction in consumption:

\[ \text{Percentage Reduction} = \left(1 - \dfrac{Q'}{Q}\right) \times 100 \]

Substitute the value of \( Q' \):

\[ \text{Percentage Reduction} = \left(1 - \dfrac{Q}{1.25Q}\right) \times 100 \]

Simplify:

\[ \text{Percentage Reduction} = \left(1 - \dfrac{1}{1.25}\right) \times 100 \]

\[ \text{Percentage Reduction} = \left(1 - 0.8\right) \times 100 \]

\[ \text{Percentage Reduction} = 0.2 \times 100 \]

\[ \text{Percentage Reduction} = 20\% \]

Therefore, a householder must reduce their consumption of rice by 20% to ensure that their expenditure does not increase. Hence, the correct option is 20%.

Let's calculate the number of successful boys and girls first:

1. For boys:
   \[ \text{Successful boys} = 60\% \text{ of } 640 \]    \[ \text{Successful boys} = 0.6 \times 640 = 384 \]

2. For girls:
   \[ \text{Successful girls} = 80\% \text{ of } 360 \]    \[ \text{Successful girls} = 0.8 \times 360 = 288 \]

Now, calculate the total number of students who were successful:\[ \text{Total successful students} = \text{Successful boys} + \text{Successful girls} \]\[ \text{Total successful students} = 384 + 288 = 672 \]

The total number of students who appeared for the examination:\[ \text{Total students} = \text{Boys} + \text{Girls} \]\[ \text{Total students} = 640 + 360 = 1000 \]

Now, calculate the percentage of failure:\[ \text{Percentage of failure} = \left(1 - \dfrac{\text{Total successful students}}{\text{Total students}}\right) \times 100 \]\[ \text{Percentage of failure} = \left(1 - \dfrac{672}{1000}\right) \times 100 \]\[ \text{Percentage of failure} = \left(1 - 0.672\right) \times 100 \]\[ \text{Percentage of failure} = 0.328 \times 100 \]

Therefore, the percentage of failure is \(32.8\%\). Hence, the correct option is \(32.8\%\).

Let's calculate the number of successful boys and girls first:

1. For boys:
   \[ \text{Successful boys} = 60\% \text{ of } 640 \]    \[ \text{Successful boys} = 0.6 \times 640 = 384 \]

2. For girls:
   \[ \text{Successful girls} = 80\% \text{ of } 360 \]    \[ \text{Successful girls} = 0.8 \times 360 = 288 \]

Now, calculate the total number of students who were successful:\[ \text{Total successful students} = \text{Successful boys} + \text{Successful girls} \]\[ \text{Total successful students} = 384 + 288 = 672 \]

The total number of students who appeared for the examination:\[ \text{Total students} = \text{Boys} + \text{Girls} \]\[ \text{Total students} = 640 + 360 = 1000 \]

Now, calculate the percentage of failure:\[ \text{Percentage of failure} = \left(1 - \dfrac{\text{Total successful students}}{\text{Total students}}\right) \times 100 \]\[ \text{Percentage of failure} = \left(1 - \dfrac{672}{1000}\right) \times 100 \]\[ \text{Percentage of failure} = \left(1 - 0.672\right) \times 100 \]\[ \text{Percentage of failure} = 0.328 \times 100 \]

Therefore, the percentage of failure is \(32.8\%\). Hence, the correct option is \(32.8\%\).