Search Previous Year Questions

Shortcut Trick

Each exterior angle of a regular polygon = 360° ÷ n

Given n = 24 sides.

⇒ One exterior angle = 360° ÷ 24 = 15°

⇒ Sum of 8 exterior angles = 8 × 15° = 120°

∴ The correct answer is 120°.

Alternate Method

Given:

Number of sides of the regular polygon (n) = 24

Number of angles to be summed = 8

Formula Used:

1. Sum of all exterior angles of any convex polygon = 360°
2. Measure of each exterior angle = 360° ÷ n

Calculations:

⇒ Total sum of all exterior angles = 360°

⇒ Since the polygon is regular, all 24 exterior angles are equal.

⇒ Measure of one exterior angle = 360° ÷ 24

⇒ One exterior angle = 15°

⇒ Sum of 8 such exterior angles = 8 × 15°

⇒ Sum = 120°

∴ The correct answer is 120°.

Additional Information

Sum of Interior Angles

The sum of all interior angles of an n-sided polygon is given by the formula (n − 2) × 180°.

Relation between Angles

At any vertex of a polygon, the sum of the interior angle and the exterior angle is always 180° (linear pair).

Regular Polygon Properties

In a regular polygon, all sides are equal and all interior/exterior angles are equal. The measure of each interior angle is [(n − 2) × 180°] ÷ n.

Logic followed here is.

Logic: The original number is determined by formulating and solving a linear equation based on the given conditions, followed by calculating the sum of its digits.

Step 1: Finding the original number

Let the original number be x.

According to the statement, one-third of the number (x / 3) increased by 55 results in 83.

(x / 3) + 55 = 83

Subtract 55 from both sides of the equation:

x / 3 = 83 − 55

x / 3 = 28

Multiply both sides by 3 to isolate x:

x = 28 × 3

x = 84

The original number is 84.

 

Step 2: Finding the sum of the digits

The digits of the number 84 are 8 and 4.

Sum of the digits = 8 + 4

Sum of the digits = 12

 

Therefore, the sum of the digits of the original number is 12.

Shortcut Trick

Present sum of ages = 24 years.

Sum after 2 years = 24 + (2 persons × 2 years) = 28 years.

After 2 years, ratio of Lokesh to Taresh = 3 : 1 (Total = 4 units).

4 units = 28 years ⇒ 1 unit = 7 years.

Lokesh's age after 2 years = 3 units = 3 × 7 = 21 years.

Lokesh's present age = 21 − 2 = 19 years.

∴ The correct answer is 19 years.

Alternate Method

Given: Sum of present ages = 24 years | Future condition (after 2 years) = Lokesh is 3 times Taresh.

Formula: Total Sum Change = Present Sum + (Number of Persons × Number of Years)

Let Lokesh's present age be L and Taresh's present age be T.

⇒ L + T = 24 --- (Eq. 1)

⇒ T = 24 − L

According to the condition after 2 years:

⇒ L + 2 = 3 × (T + 2)

⇒ L + 2 = 3 × ((24 − L) + 2)

⇒ L + 2 = 3 × (26 − L)

⇒ L + 2 = 78 − 3L

⇒ L + 3L = 78 − 2

⇒ 4L = 76

⇒ L = 76 ÷ 4 = 19

∴ The correct answer is 19 years.

Additional Information

Sum of Ages Property

If the sum of ages of 'n' people is S, then after 'x' years, the new sum will be S + (n × x).

Constant Age Difference

The difference between the ages of two individuals remains constant throughout their lives, regardless of the years passed.

Ratio of Ages

When a ratio is given for a future or past time, always adjust the total sum or individual ages to that specific point in time before applying the ratio.

Shortcut Trick

Calculate the Effective Interest Rate: 6% × 2 years = 12%.

Since 12% of the Principal equals ₹720:

12% → 720

1% → 720 ÷ 12 = 60

100% (Principal) → 60 × 100 = 6000

∴ The correct answer is ₹6000.

Alternate Method

Given:

Simple Interest (SI) = ₹720

Rate (R) = 6% per annum

Time (T) = 2 years

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Or, Principal (P) = (SI × 100) ÷ (R × T)

Calculations:

⇒ P = (720 × 100) ÷ (6 × 2)

⇒ P = 72000 ÷ 12

⇒ P = 6000

∴ The correct answer is ₹6000.

Additional Information

Total Amount Formula

The total amount (A) at the end of the period is the sum of the Principal and Simple Interest: A = P + SI.

Effective Rate of Interest

In Simple Interest, the effective rate for the entire duration is simply the product of the annual rate and time (R × T).

Constant Interest

In Simple Interest, the interest remains the same for every year because it is always calculated on the original Principal.

Shortcut Trick

Assume total goods = 100 units.

P1 (60 units) → Profit = 60 × 21 = +1260

P2 (40% of 40 = 16 units) → Profit = 16 × 46 = +736

P3 (Remaining 24 units) → Loss = 24 × (−29) = −696

Overall profit = (1260 + 736 − 696) ÷ 100 = 1300 ÷ 100 = 13%

∴ The correct answer is 13.

Alternate Method

Given:

Part 1 = 60% of goods sold at 21% profit.

Part 2 = 40% of (100 − 60)% = 16% of goods sold at 46% profit.

Part 3 = (100 − 60 − 16)% = 24% of goods sold at 29% loss.

Formula Used:

Overall Profit/Loss % = (w₁p₁ + w₂p₂ + w₃p₃) ÷ (w₁ + w₂ + w₃)

Calculations:

⇒ Let total quantity be 100 units.

⇒ Profit from Part 1 = 60 × (21/100) = 12.6 units

⇒ Profit from Part 2 = 16 × (46/100) = 7.36 units

⇒ Loss from Part 3 = 24 × (29/100) = 6.96 units

⇒ Net Profit = 12.6 + 7.36 − 6.96

⇒ Net Profit = 19.96 − 6.96 = 13 units

⇒ Overall Profit % (x) = (13 ÷ 100) × 100 = 13%

⇒ x = 13

∴ The correct answer is 13.

Additional Information

Weighted Average Profit

When items are sold in different lots, the overall profit is the weighted sum of profits divided by the total quantity: Net % = Σ(n_i × p_i) ÷ Σn_i.

Profit and Loss Basics

Profit % = (Profit ÷ CP) × 100 and Loss % = (Loss ÷ CP) × 100. Loss is always treated as a negative value in weighted average calculations.

Successive Percentage Change

If a value changes by a% then b%, the effective change is [a + b + (a × b)/100]%. This applies to successive discounts or markup followed by discount.

Shortcut Trick

Apply the Chain Rule formula: (M₁ × T₁) ÷ W₁ = (M₂ × T₂) ÷ W₂

Substitute values: (6 × 8) ÷ 120 = (9 × 4) ÷ W₂

⇒ 48 ÷ 120 = 36 ÷ W₂

⇒ 1 ÷ 2.5 = 36 ÷ W₂

⇒ W₂ = 36 × 2.5 = 90

∴ The correct answer is 90 items.

Alternate Method

Given: M₁ = 6 machines, W₁ = 120 items, T₁ = 8 hours.

Find: W₂ (items) for M₂ = 9 machines and T₂ = 4 hours.

Formula: (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂

Step-by-step calculation:

⇒ Work done per machine hour = 120 ÷ (6 × 8)

⇒ Rate = 120 ÷ 48 = 2.5 items per machine hour

⇒ Total machine hours in Case 2 = 9 machines × 4 hours = 36 machine hours

⇒ Total items (W₂) = 36 × 2.5

⇒ W₂ = 90 items

∴ The correct answer is 90 items.

Additional Information

The General Chain Rule Formula

For work problems involving multiple entities, time, and efficiency: (M₁ × D₁ × H₁ × E₁) ÷ W₁ = (M₂ × D₂ × H₂ × E₂) ÷ W₂, where M is number of men/machines, D is days, H is hours, E is efficiency, and W is work.

Direct and Inverse Proportion

Work is directly proportional to the number of machines and time. Conversely, for a fixed amount of work, the number of machines is inversely proportional to the time taken.

Unitary Method

Always find the output of 1 unit of the entity in 1 unit of time (e.g., 1 machine in 1 hour) to simplify complex variations.

Shortcut Trick

Total Work (LCM of 12 and 8) = 24 units.

Efficiency: A = 2, B = 3. Combined efficiency = 5.

Work done by A and B in 2 days = 5 × 2 = 10 units.

Remaining work = 24 − 10 = 14 units.

Time taken by A to finish remaining work = 14 ÷ 2 = 7 days.

Total time to complete the work = 2 + 7 = 9 days.

∴ The correct answer is 9 days.

Alternate Method

Given: Time taken by A = 12 days, Time taken by B = 8 days. Work started together for 2 days.

Formula Used: Total Work = Efficiency × Time taken.

⇒ Combined efficiency of A and B = 2 + 3 = 5 units/day.

⇒ Work completed in first 2 days = 5 × 2 = 10 units.

⇒ Remaining work to be completed by A = 24 − 10 = 14 units.

⇒ Time taken by A alone = 14 ÷ 2 = 7 days.

⇒ Total time for the whole work = Initial 2 days + A's 7 days = 9 days.

∴ The correct answer is 9 days.

Additional Information

Work and Wages

Wages are always distributed in the ratio of the efficiency of the workers (or work done) when they work for the same duration.

Pipes and Cisterns

The same LCM concept applies where inlet pipes have positive efficiency and outlet (leakage) pipes have negative efficiency.

Alternate Day Work

In alternate day problems, calculate the work done in one cycle (sum of efficiencies) and then divide the total work by the cycle value.

Shortcut Trick

Convert decimals to fractions: 0.25 = 1/4.

Solve the inner division: 3/4 ÷ 1/4 = 3.

Add the fraction in brackets: 3 + 1/2 = 7/2.

Calculate 'of' and subtract: (2/5 × 7/2) − 1/3 = 7/5 − 1/3 = 16/15.

∴ The correct answer is 16/15.

Alternate Method

Given Expression: (2/5) of (((3/4) ÷ 0.25) + 1/2) − 1/3

We follow the BODMAS rule to simplify the expression step-by-step.

⇒ (2/5) of ((0.75 ÷ 0.25) + 0.5) − 1/3

⇒ (2/5) of (3 + 0.5) − 1/3

⇒ (2/5) of (3.5) − 1/3

⇒ (2/5) × (7/2) − 1/3

⇒ (2 × 7) / (5 × 2) − 1/3

⇒ 7/5 − 1/3

⇒ (7 × 3 − 1 × 5) / 15

⇒ (21 − 5) / 15

⇒ 16/15

∴ The correct answer is 16/15.

Additional Information

BODMAS Rule

It defines the order of operations: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Of Operator

The 'of' operator is equivalent to multiplication but takes precedence over regular division and multiplication in the hierarchy.

Reciprocal in Division

Dividing by a fraction is the same as multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

Shortcut Trick

The fourth proportional to a, b, and c is calculated as (b × c) ÷ a.

Ratio of the first two terms 3.5 : 7.5 simplifies to 7 : 15 (multiplying by 2).

For the numbers to be in proportion, 7 : 15 = 6.3 : x.

Since 7 × 0.9 = 6.3, the value of x must be 15 × 0.9 = 13.5.

∴ The correct answer is 13.5.

Alternate Method

Given:

First number (a) = 3.5

Second number (b) = 7.5

Third number (c) = 6.3

Formula Used:

Let the fourth proportional be x. Then, a : b :: c : x.

According to the property of proportions, Product of Extremes = Product of Means.

a × x = b × c

Calculations:

⇒ 3.5 × x = 7.5 × 6.3

⇒ x = (7.5 × 6.3) ÷ 3.5

⇒ x = (75 × 63) ÷ 350 (removing decimals)

⇒ x = (75 × 9) ÷ 50 (dividing 63 and 350 by 7)

⇒ x = (3 × 9) ÷ 2 (dividing 75 and 50 by 25)

⇒ x = 27 ÷ 2

⇒ x = 13.5

∴ The correct answer is 13.5.

Additional Information

Mean Proportional

For two numbers a and b, the mean proportional is given by x = √(a × b).

Third Proportional

For two numbers a and b, the third proportional c satisfies a : b = b : c, which means c = b² ÷ a.

Continued Proportion

Quantities a, b, c, d... are in continued proportion if a/b = b/c = c/d and so on.

Shortcut Trick

The Average is simply the Total Sum divided by the Count of numbers.

Average = 240 ÷ 6 = 40.

∴ The correct answer is 40.

Alternate Method

Given:

Sum of 6 numbers = 240

Total count of numbers (n) = 6

Formula Used:

Average = (Sum of all observations) ÷ (Total number of observations)

Calculations:

⇒ Total Sum = 240

⇒ Total numbers = 6

⇒ Average = 240 ÷ 6

⇒ Average = 40

∴ The correct answer is 40.

Additional Information

Basic Average Concept

The average (arithmetic mean) is the central value of a data set, calculated by dividing the sum of values by their count.

Total Sum Calculation

If the average and the number of terms are known, the Total Sum can be found as: Sum = Average × Number of terms.

Average of First 'n' Natural Numbers

The average of the first n natural numbers is given by the formula: (n + 1) ÷ 2.

Average of Arithmetic Progression (AP)

For any series in AP, the average is always (First Term + Last Term) ÷ 2.