Percentage All Competence Aptitude

Let's denote the total number of candidates who appeared in the examination as "T."

According to the given information:

1. 60% of the candidates passed in English, which means 40% failed in English.
2. 70% of the candidates passed in Mathematics, which means 30% failed in Mathematics.
3. 20% failed in both subjects.

Now, let's express the number of candidates who failed in English, failed in Mathematics, and failed in both subjects in terms of "T."

1. Number of candidates who failed in English = 40% of T
2. Number of candidates who failed in Mathematics = 30% of T
3. Number of candidates who failed in both subjects = 20% of T

Since 20% failed in both subjects, we can write an equation:

\[ \text{Number of candidates who failed in English} + \text{Number of candidates who failed in Mathematics} - \text{Number of candidates who failed in both subjects} = T \]

\[ (40\%T + 30\%T - 20\%T) = T \]

Now, simplify the equation:

\[ 50\%T = T \]

Divide both sides by 50% (or 0.5):

\[ T = \dfrac{T}{0.5} = 2T \]

So, the total number of candidates who appeared in the examination (\(T\)) is twice the number of candidates who failed in both subjects. Given that 2500 candidates passed in both subjects:

\[ T = 2 \times 2500 = 5000 \]

Therefore, the correct answer is 5000 candidates who appeared at the examination.

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

Given information:
1. \(A\) got 10% less than \(B\): \(A = B - 0.10B = 0.90B\)
2. \(B\) got 25% more than \(C\): \(B = C + 0.25C = 1.25C\)
3. \(C\) got 20% less than \(D\): \(C = D - 0.20D = 0.80D\)

Now, we are given that \(A\) got 360 marks. Substitute the expressions from above:

\[0.90B = 360\]

Since \(B = 1.25C\), substitute this in the equation:

\[0.90(1.25C) = 360\]

Solve for \(C\):

\[1.125C = 360\]

\[C = \dfrac{360}{1.125} \approx 320\]

Now, substitute \(C\) back into the expression for \(B\) to find \(B\):

\[B = 1.25 \times 320 = 400\]

Now, substitute \(C\) into the expression for \(D\) to find \(D\):

\[C = 0.80D\]

\[320 = 0.80D\]

\[D = \dfrac{320}{0.80} = 400\]

Now, we know \(D\) obtained 400 marks out of 500. To find the percentage obtained by \(D\), use the formula:

\[ \text{Percentage} = \left( \dfrac{\text{Marks obtained}}{\text{Total marks}} \right) \times 100 \]

\[ \text{Percentage obtained by } D = \left( \dfrac{400}{500} \right) \times 100 = 80\% \]

Therefore, D obtained 80% of the full marks. The correct answer is 80%.

Let's denote the maximum marks as \(M\) and the passing percentage as 40%.

According to the given information, the student gets 90 marks and fails by 10 marks. This means the passing marks are \(90 + 10 = 100\).

Now, we can set up an equation using the passing percentage:

\[ \dfrac{100}{M} \times 100 = 40\% \]

Solve for \(M\):

\[ \dfrac{100}{M} \times 100 = 40 \]

Multiply both sides by \(M\):

\[ 100 \times 100 = 40M \]

\[ 10000 = 40M \]

Divide both sides by 40:

\[ M = \dfrac{10000}{40} \]

\[ M = 250 \]

Therefore, the maximum marks are 250. Hence, the correct answer is 250.

Let's denote the maximum marks as \(M\) and the passing percentage as 40%.

According to the given information, the student gets 90 marks and fails by 10 marks. This means the passing marks are \(90 + 10 = 100\).

Now, we can set up an equation using the passing percentage:

\[ \dfrac{100}{M} \times 100 = 40\% \]

Solve for \(M\):

\[ \dfrac{100}{M} \times 100 = 40 \]

Multiply both sides by \(M\):

\[ 100 \times 100 = 40M \]

\[ 10000 = 40M \]

Divide both sides by 40:

\[ M = \dfrac{10000}{40} \]

\[ M = 250 \]

Therefore, the maximum marks are 250. Hence, the correct answer is 250.

Let's denote the original number as \(x\).

The value of the number increased by 20% is \(1.20x\) (since an increase of 20% is equivalent to multiplying by 1.20).

The value of the number decreased by 25% is \(0.75x\) (since a decrease of 25% is equivalent to multiplying by 0.75).

According to the given information, the difference between these two values is 36:

\[ 1.20x - 0.75x = 36 \]

Combine like terms:

\[ 0.45x = 36 \]

Now, solve for \(x\):

\[ x = \dfrac{36}{0.45} \]

\[ x = 80 \]

Therefore, the original number is 80. Hence, the correct answer is 80.

Let's denote the original rate of income tax as \(x\). 

If the income tax is increased by 19%, the new rate of income tax (\(x_{\text{new}}\)) can be expressed as:

\[ x_{\text{new}} = x + 0.19x \]

Similarly, if the net income is reduced by 1%, the new net income is \(100\% - 1\% = 99\%\) of the original net income.

Now, let's set up an equation using the relationship between the rate of income tax and the net income:

\[ (100\% - x_{\text{new}}) = 0.99 \times (100\% - x) \]

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

\[ 100\% - (x + 0.19x) = 0.99 \times (100\% - x) \]

Combine like terms:

\[ 100\% - 1.19x = 0.99 \times (100\% - x) \]

Expand and simplify:

\[ 100\% - 1.19x = 99\% - 0.99x \]

Isolate \(x\):

\[ -1.19x + 0.99x = 99\% - 100\% \]

\[ -0.20x = -1\% \]

Now, solve for \(x\):

\[ x = \dfrac{-1\%}{-0.20} \]

\[ x = 5\% \]

Therefore, the original rate of income tax is \(5\%\). Hence, the correct answer is 5%.

Let's denote the original number as \( x \).

According to the given information, when the number is reduced by 25%, it becomes 225:

\[ 0.75x = 225 \]

Now, solve for \( x \):

\[ x = \dfrac{225}{0.75} \]

\[ x = 300 \]

Now, the question is asking by what percentage should the number be increased to become 375:

\[ 300 \times (1 + \text{increase percentage}) = 375 \]

Simplify this equation:

\[ 1 + \text{increase percentage} = \dfrac{375}{300} \]

\[ 1 + \text{increase percentage} = \dfrac{5}{4} \]

Now, find the increase percentage:

\[ \text{increase percentage} = \dfrac{5}{4} - 1 \]

\[ \text{increase percentage} = \dfrac{1}{4} \]

Finally, convert this fraction to a percentage:

\[ \text{increase percentage} = \dfrac{1}{4} \times 100\% \]

\[ \text{increase percentage} = 25\% \]

Therefore, the number should be increased by 25% to become 375. Hence, the correct answer is 25%.

Let's denote the original price of the article as \( P \).

First, the price is decreased by 20%, which means the new price (\( P_1 \)) is given by:\[ P_1 = P - 0.20P = 0.80P \]

Then, the resulting price is increased by 30%, which means the final price (\( P_2 \)) is given by:\[ P_2 = P_1 + 0.30P_1 = 1.30 \times 0.80P = 1.04P \]

According to the given information, the resulting price is $416:\[ 1.04P = 416 \]

Now, solve for \( P \):\[ P = \dfrac{416}{1.04} \]

\[ P = 400 \]

Therefore, the original price of the article is $400. Hence, the correct answer is 400.

Let's assume the original price of cloth is \( P \) and the original quantity sold is \( Q \).

The original gross receipt (\( \text{Original Receipt} \)) is given by the product of the price and the quantity:

\[ \text{Original Receipt} = P \times Q \]

When the price is reduced by 25%, the new price (\( P' \)) becomes \( 0.75P \) (since a reduction of 25% is equivalent to multiplying by 0.75). The quantity sold increases by 20%, so the new quantity (\( Q' \)) becomes \( 1.20Q \) (since an increase of 20% is equivalent to multiplying by 1.20).

The new gross receipt (\( \text{New Receipt} \)) is given by the product of the new price and the new quantity:

\[ \text{New Receipt} = 0.75P \times 1.20Q \]

Now, let's find the effect on gross receipt:

\[ \text{Effect on Receipt} = \dfrac{\text{New Receipt} - \text{Original Receipt}}{\text{Original Receipt}} \times 100 \]

Substitute the expressions for the new and original receipts:

\[ \text{Effect on Receipt} = \dfrac{(0.75P \times 1.20Q) - (P \times Q)}{P \times Q} \times 100 \]

Simplify this expression:

\[ \text{Effect on Receipt} = \dfrac{0.9PQ - PQ}{PQ} \times 100 \]

\[ \text{Effect on Receipt} = \dfrac{-0.1PQ}{PQ} \times 100 \]

\[ \text{Effect on Receipt} = -10\% \]

Therefore, the effect on the gross receipt of the shop is a 10% decrease. Hence, the correct answer is a 10% decrease.

Let's denote the original number as \(x\).

First, the number is increased by 10%, which means the new number (\(x_1\)) is given by:\[ x_1 = x + 0.10x = 1.10x \]

Then, the resulting number is decreased by 10%, which means the final number (\(x_2\)) is given by:\[ x_2 = x_1 - 0.10x_1 = 0.90 \times 1.10x = 0.99x \]

Now, let's calculate the net change:\[ \text{Net Change} = \dfrac{x_2 - x}{x} \times 100 \]

Substitute the values:\[ \text{Net Change} = \dfrac{0.99x - x}{x} \times 100 \]

Simplify the expression:\[ \text{Net Change} = \dfrac{-0.01x}{x} \times 100 \]

Cancel out \(x\) from the numerator and denominator:\[ \text{Net Change} = -0.01 \times 100 \]

\[ \text{Net Change} = -1\% \]

Therefore, the net change in the number is a 1% decrease. Hence, the correct answer is a 1% decrease.