Unit 2: Percentage and Profit and Loss - Subjective Questions

Percentage of a quantity:

To calculate of a quantity , use:

1. Finding of :

Therefore, of is .

2. Finding what percentage is of :

Therefore, is of .

The two statements are not equivalent because their percentages are calculated using different base values.

Let .

Since is more than :

The amount by which is less than is:

This difference must now be compared with :

Thus:

• is more than .
• is less than .

In general, if is more than , then is less than by:

To convert a percentage into a fraction, divide it by and simplify:

Conversions:

Therefore, the required fractions are:

Let the original value be .

After an increase of , the new value is:

After a further change of , the final value is:

Expanding:

Thus, the net percentage change is:

A decrease is represented by a negative percentage. Here, and :

Therefore, the price undergoes a net increase of .

For verification, if the original price is :

Let the original number be .

After a increase:

To restore the original number, the reduction required is:

The percentage reduction is calculated on the increased number, :

Therefore, the increased number must be reduced by .

It is not because:

• The increase is calculated on the original base of .
• The decrease is calculated on the new base of .

In general, after an increase of , the required percentage decrease is:

Let the total marks be and the passing marks be .

The first student scores and fails by marks:

The second student scores and gets marks above the passing marks:

Substitute :

The passing marks are:

Therefore, the passing percentage is:

Answer:

• Total marks:
• Passing marks:
• Passing percentage:

Let the number of registered voters be .

Since did not vote, votes cast were:

Since of the votes cast were invalid, valid votes were:

The winner received of the valid votes, so the loser received .

The victory margin was:

Therefore:

Therefore, the number of registered voters was .

Initial population:

After a increase in the first year:

After a decrease in the second year:

After a increase in the third year:

The increase in population is:

Hence, the overall percentage increase is:

Answer:

• Population after three years:
• Overall change: increase

Percentage error measures the error relative to the actual value:

Given:

• Actual length cm
• Measured length cm

The absolute error is:

The percentage error is:

Since the measured value is less than the actual value, it is an underestimate.

Answer:

• Absolute error: cm
• Percentage error:
• Nature of error: Underestimation

Cost Price (CP): The amount paid by a seller to purchase or produce an article.

Selling Price (SP): The amount for which the article is sold to the customer.

Marked Price (MP): The price printed or displayed on the article before a discount is allowed.

Profit and loss relationships:

• If , the seller earns a profit:

• If , the seller suffers a loss:

Discount relationship:

A discount is generally calculated on the marked price:

Thus, the marked price determines the discount, while the cost price determines whether the final transaction results in a profit or loss.

All expenses incurred before selling the article are included in its effective cost price.

Given:

• Purchase price
• Additional expenses
• Selling price

The effective cost price is:

The profit is:

The profit percentage is calculated on the cost price:

Answer:

• Effective cost price:
• Profit:
• Profit percentage:

Let the cost price be .

1. Selling price at a profit of :

Therefore:

2. Selling price at a loss of :

Therefore:

3. Cost price when profit is :

From :

4. Cost price when loss is :

From :

These formulas show that profit and loss percentages are normally calculated on the cost price.

Given:

• Marked price
• Discount rate
• Cost price

The discount amount is:

The selling price is:

The profit is:

The profit percentage is:

Answer:

• Discount amount:
• Selling price:
• Profit:
• Profit percentage:

The first discount is calculated on the marked price, while the second discount is calculated on the reduced price.

After the first discount of :

After the second discount of :

Therefore, the total discount amount is:

The equivalent discount percentage is:

Alternatively, the equivalent discount of successive discounts and is:

Thus:

It is not because the second discount is applied to , not to the original marked price.

Answer:

• Final selling price:
• Equivalent discount:

Assume the cost price of kg of sugar is .

The shopkeeper charges , claiming to provide kg, but actually gives only g.

The cost of g is:

Therefore, his profit is:

The profit percentage is calculated on the actual cost of the quantity supplied:

Therefore, the dishonest shopkeeper earns a profit of .

In general, if a trader gives grams instead of grams while charging the cost price of kg, then:

The cost price per article is:

The selling price per article is:

Therefore, the profit per article is:

The profit percentage is:

The same result can be obtained by comparing the numbers of articles bought and sold for the same amount:

Answer:

• Cost price per article:
• Selling price per article:
• Profit percentage:

Both articles have the same selling price, but their cost prices are different.

First article: profit

Second article: loss

Total cost price:

Total selling price:

Therefore, the loss is:

The loss percentage is:

Thus, the trader suffers an overall loss of .

When two articles are sold at a common selling price, one at an gain and the other at an loss, the overall result is always a loss of:

For , the loss is .

Let the cost prices be:

The first article is sold at a profit of :

If the second article were sold at a loss of :

Since , the two selling prices cannot be equal. Therefore, the stated conditions are not possible.

Let the required change on the second article be . For equal selling prices:

Thus, the second article must be sold at an profit, not at a loss.

This also shows that no loss percentage can produce the required common selling price because the desired selling price exceeds the second article's cost price.

Let the cost price be .

The marked price is above the cost price:

A discount of means the selling price is of the marked price:

Therefore:

Thus, the selling price is of the cost price, giving a profit of:

For :

The profit is:

Answer:

• Marked price:
• Selling price:
• Profit:
• Profit percentage:

Percentage-point change is the direct arithmetic difference between two percentage values.

Percentage change compares the difference with the original percentage value.

The vote share rises from to .

The increase in percentage points is:

The percentage increase in vote share is:

Therefore:

• Increase in vote share: percentage points
• Percentage increase relative to the original vote share:

The two quantities are different because a percentage-point change compares percentages directly, while percentage change uses the original percentage as its base.