Percentages and averages are among the most common aptitude topics. They are used in marks, profit and loss, discounts, ratio comparisons, mixtures, and data interpretation. A strong understanding of these basics helps in many other chapters as well.

What questions occur in this topic?

You may be asked to find a percentage of a number, compute an increase or decrease, compare two values, find the average of a set of quantities, or solve word problems involving marks, profit, loss, discount, and population or salary changes.

In most aptitude questions, the focus is on converting the problem into a simple percentage or average relationship and then calculating quickly.

Meaning of percentage

A percentage means “per hundred.” So, 25% means 25 out of 100, or 25/100 = 0.25.

• x% of y = (x/100) × y
• 1% of a number = number / 100
• 50% = 1/2, 25% = 1/4, 20% = 1/5, 10% = 1/10

Basic percentage formulas

• Percentage = (Part / Whole) × 100
• Part = (Percentage / 100) × Whole
• Whole = (Part × 100) / Percentage
• Percentage increase = [(New value - Original value) / Original value] × 100
• Percentage decrease = [(Original value - New value) / Original value] × 100

Important percentage concepts

• To increase a value by x%: multiply by (1 + x/100).
• To decrease a value by x%: multiply by (1 - x/100).
• To find the original value after percentage change: divide by the final multiplier.
• Successive percentage change: use multiplication of factors, not simple addition.

Successive percentage changes

If a quantity increases by a% and then by b%, the net multiplier is (1 + a/100) × (1 + b/100) . This is the key idea behind many exam questions on salary, population, sales, and price changes.

For example, if a price rises by 20% and then falls by 10%, the overall change is not 10% but 1.20 × 0.90 = 1.08, which means a net increase of 8%.

Profit, loss, and discount

• Profit = Selling Price - Cost Price
• Loss = Cost Price - Selling Price
• Profit % = (Profit / Cost Price) × 100
• Loss % = (Loss / Cost Price) × 100
• Discount % = (Discount / Marked Price) × 100
• Selling Price = Marked Price × (1 - Discount %/100)

What is an average?

An average is a representative value of a group of numbers. It tells us the central value of the data set.

Average = (Sum of all observations) / (Number of observations)

Basic average formulas

• Average of n numbers = Sum of numbers / n
• Sum of numbers = Average × n
• Number of terms = Sum / Average
• Weighted average = (Σ value × weight) / (Σ weight)

Common average short-cuts

• If each number is increased by k, the average also increases by k.
• If each number is decreased by k, the average also decreases by k.
• If each number is multiplied by k, the average is also multiplied by k.
• Average of consecutive numbers is the middle value when there are odd numbers.

Useful number patterns

• Average of first n natural numbers = (n + 1)/2
• Sum of first n natural numbers = n(n + 1)/2
• Average of first n even numbers = n + 1
• Average of first n odd numbers = n

How to solve percentage and average questions

1. Read the problem carefully and identify the base value.
2. Convert the statement into a percentage or average formula.
3. Keep the unit and reference value consistent.
4. Use direct formula shortcuts whenever they are faster.
5. Check whether the question asks for increase, decrease, original value, or average.

Quick revision summary

• Percentage means per hundred.
• Use x% = x/100 to convert quickly.
• Always ask: what is the base value?
• For changes, use multipliers instead of adding percentages directly.
• Average is total sum divided by total count.
• Weighted average is useful when quantities have different importance.