Unit 2 - Notes

PEA307 6 min read

Unit 2: Advanced percentage and Income based questions

Part 1: Advanced Percentage

Percentage (per-cent) means "out of 100." Advanced percentage problems require moving beyond basic calculations to apply logical deductions, algebraic equations, and fraction conversions to solve complex real-world scenarios efficiently.

1. Percentage to Fraction Conversion

Memorizing percentage-to-fraction conversions is the foundation of solving advanced analytical problems quickly. It eliminates the need for complex decimal multiplication and division.

Concept:
To convert a percentage to a fraction, divide by 100. To convert a fraction to a percentage, multiply by 100.
$x\% = \frac{x}{100}$

Standard Conversion Table:

$1 = 100\%$
$1/2 = 50\%$
$1/3 = 33.33\%$ or $33\frac{1}{3}\%$
$1/4 = 25\%$
$1/5 = 20\%$
$1/6 = 16.66\%$ or $16\frac{2}{3}\%$
$1/7 = 14.28\%$ or $14\frac{2}{7}\%$
$1/8 = 12.5\%$ or $12\frac{1}{2}\%$
$1/9 = 11.11\%$ or $11\frac{1}{9}\%$
$1/10 = 10\%$
$1/11 = 9.09\%$ or $9\frac{1}{11}\%$
$1/12 = 8.33\%$ or $8\frac{1}{3}\%$
$1/16 = 6.25\%$ or $6\frac{1}{4}\%$
$1/20 = 5\%$

Derived Fractions:
You can derive complex percentages using base fractions.

$3/8 = 3 \times (1/8) = 3 \times 12.5\% = 37.5\%$
$5/6 = 5 \times (1/6) = 5 \times 16.66\% = 83.33\%$

2. Successive Percentage Change

When a value undergoes a percentage change, and then the new value undergoes another percentage change, it is called successive percentage change.

Formula Method:
If a value changes by $x\%$ and then by $y\%$ , the net effective percentage change is given by:
$\text{Net Change} = x + y + \frac{x \times y}{100}$

Use a positive sign ( $+$ ) for an increase.
Use a negative sign ( $-$ ) for a decrease.

Ratio / Fraction Method (Preferred for complex percentages):
Convert percentages to fractions.

Example: A population increases by $16.66\%$ (which is $1/6$ ) and then decreases by $12.5\%$ (which is $1/8$ ).
First change: Initial = 6, Final = $6 + 1 = 7$ .
Second change: Initial = 8, Final = $8 - 1 = 7$ .
Overall Initial = $6 \times 8 = 48$ . Overall Final = $7 \times 7 = 49$ .
Net Change = $\frac{49 - 48}{48} \times 100 = \frac{1}{48} \times 100 = 2.08\%$ increase.

3. Problems Related to Examination

Examination problems usually revolve around passing marks, maximum marks, or Venn diagrams involving sets of students.

Type A: Pass/Fail Score Comparisons

Scenario: Student A gets $x\%$ and fails by $a$ marks. Student B gets $y\%$ and passes by $b$ marks above the minimum.
Logic: The difference in their percentage equals the sum of their absolute mark differences from the passing line.
Equation: $(y\% - x\%) \text{ of Total Marks} = a + b$

Type B: Venn Diagram Problems

Scenario: $x\%$ failed in Subject 1, $y\%$ failed in Subject 2, and $z\%$ failed in both.
Logic: Use Set Theory.
- $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
- Total Failed in at least one subject = $x + y - z$
- Total Passed in both = $100\% - (\text{Total Failed in at least one})$

4. Problems Related to Election

Election problems involve total voters, invalid votes, and the margin of victory.

Key Terminology:

Total Enrolled Voters: The voting list (100%).
Votes Cast: Voters who actually voted.
Valid Votes: Votes cast minus invalid votes.
Winning Margin: Difference between the winner's valid votes and the loser's valid votes.

Step-by-Step Approach:

Let total voters = $100x$ .
Subtract non-voters to get votes cast.
Subtract invalid votes to get valid votes.
Distribute valid votes between Winner ( $W$ ) and Loser ( $L$ ).
Equate $W - L$ to the given majority/margin to find $x$ .

5. Partial Percentage

These problems involve dividing a total quantity into parts, where each part undergoes a different percentage change.

Example Scenario:
A man invests a part of ₹10,000 at 5% and the rest at 10%.

Method 1 (Equation): Let Part 1 be $x$ , Part 2 be $10,000 - x$ . Apply percentages respectively.
Method 2 (Alligation): If the overall percentage is known, use the rule of alligation to find the ratio of the parts.

6. Percentage Error

Percentage error occurs when a mistake is made in calculation, such as multiplying by the wrong number.

Formula:
$\text{Percentage Error} = \left( \frac{\text{Error}}{\text{True Value}} \right) \times 100$
$\text{Where Error} = |\text{True Value} - \text{Measured/Calculated Value}|$

Classic Example:
A student multiplied a number by $3/5$ instead of $5/3$ .

Assume the number is the LCM of denominators (3 and 5) = 15.
True Value = $15 \times (5/3) = 25$ .
Calculated Value = $15 \times (3/5) = 9$ .
Error = $25 - 9 = 16$ .
Percentage Error = $(16 / 25) \times 100 = 64\%$ .

Part 2: Income Based Questions (Profit, Loss, and Discount)

This section translates percentage concepts into commercial contexts. "Income" here relates to revenue generated through transactions involving costs, sales, and markups.

1. Concept of Cost Price, Selling Price, and Marked Price

Cost Price (CP): The price at which an article is purchased or manufactured. Profit and Loss are ALWAYS calculated on CP (unless explicitly stated otherwise).
Selling Price (SP): The price at which an article is sold to the customer.
Marked Price (MP) / Maximum Retail Price (MRP): The price printed on the article. Discount is ALWAYS calculated on MP.

Core Formulas:

$\text{Profit} = SP - CP$ (when $SP > CP$ )
$\text{Loss} = CP - SP$ (when $CP > SP$ )
$\text{Profit \%} = (\text{Profit} / CP) \times 100$
$\text{Loss \%} = (\text{Loss} / CP) \times 100$
$\text{Discount} = MP - SP$
$\text{Discount \%} = (\text{Discount} / MP) \times 100$

Inter-relationships:

$SP = CP \times \frac{100 + \text{Profit\%}}{100}$
$SP = MP \times \frac{100 - \text{Discount\%}}{100}$
Therefore, $\frac{CP}{MP} = \frac{100 - \text{Discount\%}}{100 + \text{Profit\%}}$

2. Successive Discount

When a discount is offered on an already discounted price, it is called a successive discount.

Equivalent Discount Formula:
If two successive discounts of $d_1\%$ and $d_2\%$ are given, the single equivalent discount is:
$\text{Equivalent Discount} = d_1 + d_2 - \frac{d_1 \times d_2}{100}$
(Note: This is derived from the successive percentage change formula where $x$ and $y$ are negative).

Ratio Method for Multiple Discounts:
For discounts of $a\%$ , $b\%$ , and $c\%$ :

Convert to fractions.
Calculate the ratio of SP to MP for each step.
Multiply the ratios. Total SP / Total MP will give the overall discount fraction.

3. Problems Based on Number of Articles

These problems manipulate the quantities bought and sold rather than just the prices.

Type A: "CP of $x$ articles = SP of $y$ articles"

Method: Let $CP \times x = SP \times y \implies \frac{SP}{CP} = \frac{x}{y}$
If $x > y$ , there is a profit. If $x < y$ , there is a loss.
$\text{Profit/Loss \%} = \frac{|x - y|}{y} \times 100$

Type B: "Buy $x$ , Get $y$ Free"
In this promotional scheme, the customer gets $(x + y)$ articles but only pays for $x$ articles.

The Marked Price (MP) applies to all $(x + y)$ articles.
The Selling Price (SP) corresponds to the price of $x$ articles.
$\text{Discount} = \text{Price of } y \text{ articles}$ .
$\text{Discount \%} = \left( \frac{y}{x + y} \right) \times 100$

4. Dishonest Seller

A dishonest seller claims to sell goods at Cost Price (or at a stated profit/loss) but uses false weights to cheat the customer, thereby increasing his actual profit margin.

Core Concept:
The seller's actual cost is based on the actual weight given, while his revenue is based on the claimed weight charged to the customer.

Formula for Dishonest Dealer (Claiming to sell at CP):
$\text{Profit \%} = \left( \frac{\text{Error}}{\text{True Weight} - \text{Error}} \right) \times 100$
Or:
$\text{Profit \%} = \left( \frac{\text{Claimed Weight} - \text{Actual Weight}}{\text{Actual Weight}} \right) \times 100$

Advanced Dishonest Seller (Markup + False Weight):
If a seller marks up goods by $x\%$ and uses a false weight (gives $W_{actual}$ instead of $W_{claimed}$ ):

Assume CP of 1 gram = ₹1.
Total CP for the seller = $W_{actual}$ (since he only parts with this much).
Total SP for the seller = $W_{claimed} \times \left(1 + \frac{x}{100}\right)$ (customer pays for claimed weight at marked up price).
$\text{Overall Profit \%} = \left( \frac{SP - CP}{CP} \right) \times 100$ .

Unit 1

Unit 3