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Shortcut Trick

Total difference in the daily wage bill = Excess + Shortfall = 4000 + 2000 = ₹6000.

Difference in daily pay per worker = 500 − 450 = ₹50.

Total temporary workers = Total Difference ÷ Difference per Worker = 6000 ÷ 50 = 120.

∴ The correct answer is 120.

Alternate Method

Given: Pay Rate₁ = ₹500, Pay Rate₂ = ₹450, Budget Excess = ₹4,000, Budget Shortfall = ₹2,000.

Formula Used: Total Wage Bill = Number of Workers × Daily Pay Rate

⇒ Let n be the number of workers and B be the company’s budget.

⇒ Case 1: 500 × n = B + 4000

⇒ Case 2: 450 × n = B − 2000

⇒ Subtracting Case 2 from Case 1: (500n − 450n) = (B + 4000) − (B − 2000)

⇒ 50n = 4000 + 2000

⇒ 50n = 6000

⇒ n = 6000 ÷ 50 = 120.

∴ The correct answer is 120.

Additional Information

Excess and Deficit Rule

When comparing two scenarios where one results in an excess (E) and the other in a shortfall or deficit (D), the total difference between the two conditions is calculated as (E + D).

Linear Equations in One Variable

These problems follow the standard form ax + b = c, where x represents the unknown quantity (like the number of workers) and can be solved by isolating the variable.

Rate-Quantity Relationship

The change in total value is equal to the quantity multiplied by the change in rate: ΔTotal = Quantity × ΔRate.

Given:

(32 ÷ 0.8 × x) − 72 = 720 ÷ (4 × 5 + 10)

Formula used:

Solve using the BODMAS rule (Bracket, Of, Division, Multiplication, Addition, Subtraction).

Calculation:

Let the missing value be x.

(32 ÷ 0.8 × x) − 72 = 720 ÷ (4 × 5 + 10)

⇒ (40 × x) − 72 = 720 ÷ (20 + 10)

⇒ 40 × x − 72 = 720 ÷ 30

⇒ 40 × x − 72 = 24

⇒ 40 × x = 24 + 72

⇒ 40 × x = 96

⇒ x = 96 ÷ 40

⇒ x = 2.4

∴ The correct answer is option (3).

Shortcut Trick

The Third Proportional of two numbers a and b is given by the formula b² ÷ a.

⇒ Third Proportional = (16 × 16) ÷ 8

⇒ 256 ÷ 8 = 32

∴ The correct answer is 32.

Alternate Method

Given:

First number (a) = 8

Second number (b) = 16

Formula Used:

If three numbers a, b, and c are in continued proportion, then:

a : b :: b : c

⇒ a ÷ b = b ÷ c

Calculations:

⇒ Let the third proportional be x.

⇒ 8 : 16 :: 16 : x

⇒ 8 ÷ 16 = 16 ÷ x

⇒ 1 ÷ 2 = 16 ÷ x

⇒ x = 16 × 2

⇒ x = 32

∴ The correct answer is 32.

Additional Information

Mean Proportional

The Mean Proportional between two numbers a and b is calculated as √(a × b).

Fourth Proportional

For three numbers a, b, and c, the fourth proportional (x) satisfies a : b = c : x, which means x = (b × c) ÷ a.

Continued Proportion

Three quantities are in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third.

Shortcut Trick

List integers that leave remainder 2 when divided by 5: 2, 7, 12, 17, 22, ...

List integers that leave remainder 3 when divided by 7: 3, 10, 17, 24, ...

Identify the first common number that appears in both sequences.

The smallest positive integer satisfying both conditions is 17.

∴ The correct answer is 17.

Alternate Method

Given:

Number (N) ÷ 5 → Remainder = 2

Number (N) ÷ 7 → Remainder = 3

Step-by-step Calculation:

⇒ Let N = 5k + 2 and N = 7m + 3

⇒ 5k + 2 = 7m + 3

⇒ 5k = 7m + 1

⇒ If m = 1, 5k = 8 (Not an integer)

⇒ If m = 2, 5k = 7(2) + 1 = 15

⇒ k = 15 ÷ 5 = 3

⇒ Smallest N = 5(3) + 2 = 17

∴ The correct answer is 17.

Additional Information

Chinese Remainder Theorem (CRT)

CRT provides a systematic way to solve systems of linear congruences with pairwise coprime moduli like 5 and 7.

General Form of Solution

The general solution for such problems is N = (LCM(d₁, d₂) × n) + x, where x is the smallest positive integer and n = 0, 1, 2...

Successive Division Method

For larger numbers, use N = d₁k + r₁ and substitute into the second congruence to solve for k iteratively.

Shortcut Trick

Ratio of the two complementary angles is 3 : 2 (since 2 × Angle₁ = 3 × Angle₂).

Total units = 3 + 2 = 5 units.

Given, total sum = 90º (complementary property).

5 units = 90º ⇒ 1 unit = 18º.

Larger angle = 3 units = 3 × 18º = 54º.

∴ The correct answer is 54º.

Alternate Method

Given:

Sum of two angles = 90º

Relationship: 2 × (One angle) = 3 × (Other angle)

Formula Used:

For Complementary Angles, θ₁ + θ₂ = 90º.

Calculations:

Let the two angles be x and y.

⇒ x + y = 90º    ---(i)

⇒ 2x = 3y    ---(ii)

⇒ x = 3y / 2

Substituting the value of x in equation (i):

⇒ (3y / 2) + y = 90º

⇒ (3y + 2y) / 2 = 90º

⇒ 5y / 2 = 90º

⇒ 5y = 180º

⇒ y = 36º

Now, finding the value of x:

⇒ x = 90º − 36º

⇒ x = 54º

Comparing 54º and 36º, the larger angle is 54º.

∴ The correct answer is 54º.

Additional Information

Complementary Angles

Two angles are called complementary if their sum is exactly 90º. Each angle is the complement of the other.

Supplementary Angles

Two angles are called supplementary if their sum is exactly 180º. They form a linear pair when placed adjacently on a straight line.

Adjacent Angles

Angles that have a common vertex and a common side but do not overlap are known as adjacent angles.

Shortcut Trick

Convert decimals into their simplest fractions: 5 = 5/1, 8.3 = 83/10, and 0.19 = 19/100.

Formula: LCM of fractions = LCM of Numerators ÷ HCF of Denominators.

LCM(5, 83, 19) = 5 × 83 × 19 = 7885 (since all are co-prime).

HCF(1, 10, 100) = 1.

Result = 7885 ÷ 1 = 7885.

∴ The correct answer is 7885.

Alternate Method

Given: Numbers are 5, 8.3, and 0.19.

Formula Used: To find the LCM of decimals, equalize the number of decimal places and find the LCM of the resulting integers.

⇒ Step 1: Make decimal places equal: 5.00, 8.30, 0.19.

⇒ Step 2: Treat them as integers: 500, 830, and 19.

⇒ Step 3: Find prime factors:

⇒ 500 = 2² × 5³

⇒ 830 = 2 × 5 × 83

⇒ 19 = 19 (Prime)

⇒ Step 4: LCM = Highest powers of all prime factors = 2² × 5³ × 83 × 19.

⇒ LCM = 4 × 125 × 83 × 19 = 500 × 1577 = 788500.

⇒ Step 5: Adjust decimal: 788500 ÷ 100 = 7885.

∴ The correct answer is 7885.

Additional Information

LCM of Fractions

The Least Common Multiple (LCM) of a set of fractions is given by: (LCM of Numerators) ÷ (HCF of Denominators).

HCF of Fractions

The Highest Common Factor (HCF) of a set of fractions is given by: (HCF of Numerators) ÷ (LCM of Denominators).

Co-prime Numbers

Numbers that have only 1 as a common factor are co-prime. The LCM of co-prime numbers is always equal to their product.

Shortcut Trick

When an object covers equal distances at different speeds, the Average Speed is the Harmonic Mean of the speeds.

Average Speed = 3 ÷ (1/v₁ + 1/v₂ + 1/v₃)

⇒ Average Speed = 3 ÷ (1/40 + 1/60 + 1/120)

⇒ Average Speed = 3 ÷ (6/120) = 3 × 20 = 60 km/hr.

∴ The correct answer is 60 km/hr.

Alternate Method

Given: Three segments of 40 km each. Speeds = 40 km/hr, 60 km/hr, and 120 km/hr.

Formula Used: Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Total Distance = 40 + 40 + 40 = 120 km

⇒ Time for 1^st segment (t₁) = 40 ÷ 40 = 1 hour

⇒ Time for 2^nd segment (t₂) = 40 ÷ 60 = 2/3 hour

⇒ Time for 3^rd segment (t₃) = 40 ÷ 120 = 1/3 hour

⇒ Total Time = 1 + 2/3 + 1/3 = 1 + 1 = 2 hours

⇒ Average Speed = 120 ÷ 2 = 60 km/hr

∴ The correct answer is 60 km/hr.

Additional Information

Two Equal Distance Segments

If a journey is divided into two equal halves with speeds x and y, the average speed is 2xy ÷ (x + y).

Unequal Distance Segments

For unequal distances d₁, d₂, ... with speeds v₁, v₂, ..., Average Speed = (d₁ + d₂ + ...) ÷ (d₁/v₁ + d₂/v₂ + ...).

Unit Conversion

To convert speed from km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

Shortcut Trick

Use the Weighted Average method for calculation.

Ratio of goods quantity = 50% : (40% of 50%) : (Remaining 30%) = 50 : 20 : 30 = 5 : 2 : 3

Overall Profit x% = (5 × 36 + 2 × 14 + 3 × (−21)) ÷ (5 + 2 + 3)

x = (180 + 28 − 63) ÷ 10 = 145 ÷ 10 = 14.5%

∴ The correct answer is 14.5%.

Alternate Method

Given: 50% goods at 36% profit, 40% of remaining (50%) at 14% profit, and still remaining (30%) at 21% loss.

Formula Used: Overall Profit% = (Total Profit ÷ Total CP) × 100

Let the Total Cost Price (CP) of 100 units of goods be ₹100.

⇒ Case 1: CP = ₹50 | Profit = 36% of 50 = 50 × 0.36 = ₹18

⇒ Remaining goods = 100 − 50 = 50 units

⇒ Case 2: CP = 40% of 50 = 20 units | Profit = 14% of 20 = 20 × 0.14 = ₹2.8

⇒ Still Remaining goods = 50 − 20 = 30 units

⇒ Case 3: CP = 30 units | Loss = 21% of 30 = 30 × 0.21 = ₹6.3 (Loss is −6.3)

⇒ Total Profit = 18 + 2.8 − 6.3 = 14.5

⇒ x% = (14.5 ÷ 100) × 100 = 14.5%

∴ The correct answer is 14.5%.

Additional Information

Weighted Average Profit/Loss

Overall Profit % = (n₁p₁ + n₂p₂ + ... + n_kp_k) ÷ (n₁ + n₂ + ... + n_k), where n is quantity and p is profit percentage (negative for loss).

Successive Changes vs Parts

When goods are divided into parts, we use weighted averages. When one value changes multiple times, we use successive percentage formula: a + b + (ab/100).

Profit and Loss Basics

Profit % = (Profit ÷ CP) × 100 | SP = CP × (100 + P%) ÷ 100. Always calculate profit and loss on the Cost Price (CP) unless stated otherwise.

Shortcut Trick

The Total Sum is simply the product of the Average and the Total count of numbers.

Total Sum = Average × Number of items

Total Sum = 7 × 3 = 21

∴ The correct answer is 21.

Alternate Method

Given:

Average of three numbers = 7

Number of quantities (n) = 3

Formula Used:

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

Calculations:

⇒ 7 = (Sum of three numbers) ÷ 3

⇒ Sum of three numbers = 7 × 3

⇒ Sum of three numbers = 21

∴ The correct answer is 21.

Additional Information

Arithmetic Mean

The most common type of average, calculated by adding all numbers in a set and dividing by the count of numbers.

Weighted Average

Used when different values have different levels of importance or frequency: (w₁x₁ + w₂x₂) ÷ (w₁ + w₂).

Average of Consecutive Numbers

For a series in Arithmetic Progression (AP), the average is simply (First Term + Last Term) ÷ 2.

Shortcut Trick

Use the algebraic identity: (Difference)² = (Sum)² − 4 × (Product)

⇒ (Difference)² = 27² − 4 × 176

⇒ (Difference)² = 729 − 704 = 25

⇒ Difference = √25 = 5

∴ The correct answer is 5.

Alternate Method

Given: Sum of two numbers = 27, Product = 176.

Formula Used: Factorisation method to find the numbers x and y such that x + y = 27 and xy = 176.

⇒ Let the two positive numbers be x and y.

⇒ x + y = 27 and x × y = 176.

⇒ Factorizing 176 to find pairs whose sum is 27: 176 = 1 × 176, 2 × 88, 4 × 44, 8 × 22, 11 × 16.

⇒ Since 11 + 16 = 27, the numbers are 16 and 11.

⇒ Positive difference = 16 − 11 = 5.

∴ The correct answer is 5.

Additional Information

Algebraic Relationship

The core relationship between sum, product, and difference is given by (x + y)² − (x − y)² = 4xy.

Quadratic Equation Formulation

If sum (S) and product (P) of two numbers are known, the numbers are the roots of the quadratic equation t² − St + P = 0.

Nature of Roots

For the difference to be a rational number, the value (Sum² − 4 × Product) must be a perfect square.