Quantitative Aptitude PYQs

Shortcut Trick

Total interest percentage = 3% × 11 = 33%.

Directly calculate 33% of the principal to get the simple interest.

Simple Interest = 33% of 1200 = (33 × 1200) ÷ 100

Simple Interest = 33 × 12 = 396

∴ The correct answer is ₹396.

Alternate Method

Start
P = ₹1200
11 Months
SI = ?
Rate = 3% per month
Total Rate = 11 × 3% = 33%

Given:

Principal (P) = ₹1200

Rate of Interest (R) = 3% per month

Time (T) = 11 months

Formula Used:

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

Calculation:

Since the rate is given per month and the time is given in months, we do not need to convert the time into years.

⇒ SI = (1200 × 3 × 11) ÷ 100

⇒ SI = 12 × 3 × 11

⇒ SI = 36 × 11

⇒ SI = 396

∴ The correct answer is ₹396.

Additional Information

Simple vs Compound Interest

Simple Interest is calculated only on the initial principal amount, making it uniform. Compound Interest is calculated on both the principal and the accumulated interest of previous periods.

Rate and Time Consistency

Always ensure the time unit matches the rate unit. If the rate is per annum (p.a.) but time is in months, convert time to years using T = (months ÷ 12).

Total Amount Formula

The total amount at the end of the time period is the sum of Principal and Interest: A = P + SI = P[1 + (R × T) ÷ 100].

Shortcut Trick

The sum of squares of the two consecutive odd numbers is 394, so their average square is 394 ÷ 2 = 197.

The perfect squares of odd numbers closest to 197 are 169 (which is 13²) and 225 (which is 15²).

Verifying their sum: 169 + 225 = 394.

Thus, the two consecutive odd natural numbers are 13 and 15.

Required sum = 13 + 15 = 28.

∴ The correct answer is 28.

Alternate Method

Given:

The sum of the squares of two consecutive odd natural numbers is 394.

Formula Used:

(a − b)² + (a + b)² = 2(a² + b²)

Calculation:

Let the two consecutive odd natural numbers be represented as (2n − 1) and (2n + 1).

⇒ (2n − 1)² + (2n + 1)² = 394

⇒ 2[(2n)² + 1²] = 394

⇒ 4n² + 1 = 197

⇒ 4n² = 196

⇒ n² = 49

⇒ n = 7 (Since the numbers are natural numbers, n must be positive)

The sum of the two numbers is: (2n − 1) + (2n + 1) = 4n

⇒ Sum = 4 × 7 = 28

∴ The correct answer is 28.

Additional Information

Consecutive Odd and Even Numbers

Consecutive odd or even numbers always differ by 2. Symmetrically, they can be represented as (2n − 1) and (2n + 1) for odds, and (2n − 2), 2n, (2n + 2) for evens to simplify calculations.

Important Algebraic Identity

The algebraic identity (a + b)² + (a − b)² = 2(a² + b²) is highly useful for rapidly simplifying problems that involve the sum of squares of terms differing by a constant value.

Quadratic Roots Nature

When a word problem on natural numbers reduces to a quadratic equation ax² + bx + c = 0, the discriminant D = b² − 4ac must be a perfect square, and negative roots are rejected.

Shortcut Trick

Time to fill tank: P = 30 × 7/5 = 42 hours, Q = 28 × 5/2 = 70 hours.

Total capacity = LCM(42, 70) = 210 units ⇒ P's efficiency = 5 units/hr, Q's efficiency = 3 units/hr.

Volume filled by P and Q in 7 hours = (5 + 3) × 7 = 56 units.

R empties 56 units in 8 hours ⇒ R's efficiency = −56 ÷ 8 = −7 units/hr.

Time taken by P, Q, and R together = 210 ÷ (5 + 3 − 7) = 210 ÷ 1 = 210 hours.

∴ The correct answer is 210 hours.

Alternate Method

Given:

Pipe P can fill 5/7 part of a tank in 30 hours.

Pipe Q can fill 2/5 part of the same tank in 28 hours.

Both P and Q were kept open for 7 hours, then closed.

Pipe R empties the filled water in 8 hours.

Calculations:

⇒ Total time taken by P to fill the complete tank = 30 × (7/5) = 42 hours.

⇒ Total time taken by Q to fill the complete tank = 28 × (5/2) = 70 hours.

⇒ Work done by Pipe P and Pipe Q together in 7 hours = (5 + 3) × 7 = 8 × 7 = 56 units.

⇒ The tank has 56 units of water when P and Q are closed.

⇒ Pipe R alone empties these 56 units in 8 hours.

⇒ Efficiency of Pipe R = −56 ÷ 8 = −7 units/hour.

⇒ Net efficiency when Pipes P, Q, and R are opened together = 5 + 3 − 7 = 1 unit/hour.

⇒ Time taken by P, Q, and R together to fill the empty tank = Total Work ÷ Net efficiency.

⇒ Time taken = 210 ÷ 1 = 210 hours.

∴ The correct answer is 210 hours.

Additional Information

Pipes and Cisterns Efficiency Rule

When a pipe fills a tank, its efficiency is taken as positive. When a pipe (or leak) empties a tank, its efficiency is taken as negative.

LCM Method

The total capacity of the tank is assumed to be the LCM of the time taken by individual pipes to fill or empty the tank completely. Efficiency = Total Capacity ÷ Time Taken.

Combined Work

If multiple pipes are operating together, their combined net efficiency is the algebraic sum of their individual efficiencies: Combined Efficiency = E₁ + E₂ − E₃.

Shortcut Trick

Total work = LCM(10, 20) = 20 units.

Combined efficiency of A and B = 2 + 1 = 3 units/day.

Work done in 6 days = 6 × 3 = 18 units.

Fraction of work done = 18 ÷ 20 = 9/10.

Remaining work = 1 − 9/10 = 1/10.

∴ The correct answer is 1/10.

Alternate Method

Given:

Time taken by A = 10 days

Time taken by B = 20 days

Number of days worked together = 6 days

Calculations:

⇒ Total efficiency of A and B = 2 + 1 = 3 units/day.

⇒ Work done by A and B together in 6 days = 6 × 3 = 18 units.

⇒ Remaining work = Total work − Work done = 20 − 18 = 2 units.

⇒ Fraction of work left = Remaining work ÷ Total work

⇒ Fraction of work left = 2 ÷ 20 = 1/10.

∴ The correct answer is 1/10.

Additional Information

Pipes & Cisterns

This concept follows the same LCM efficiency logic as Time & Work, where an inlet pipe does positive work (+ efficiency) and an outlet pipe does negative work (− efficiency).

Work & Wages

Wages are always distributed in the ratio of the actual work done by each individual. If they work for the same duration, wages are distributed in the ratio of their efficiencies.

Alternate Day Work

When individuals work on alternate days, find the work done in a complete cycle (e.g., 2 days) and then multiply to get as close to total work as possible before adding the remaining work individually.

Shortcut Trick

Since the investment time is equal for all, Profit Ratio = Investment Ratio.

Sum of profit ratio = 34 + 75 + 6 = 115 units.

Value of 115 units = ₹3,600 ⇒ 1 unit = 3600 ÷ 115 = 720 ÷ 23.

Difference between Q and R's shares = 75 − 6 = 69 units.

Required difference = 69 × (720 ÷ 23) = 3 × 720 = ₹2,160.

∴ The correct answer is ₹2,160.

Alternate Method

Profit Ratio = 34 : 75 : 6 (Total 115 units = ₹3600)

P
Q
R

34 units

Difference = 69 units

6 units

Given:

Investment ratio of P, Q, and R = 34 : 75 : 6

Total Profit at the end of the year = ₹3,600

Formula Used:

Profit Ratio = Investment Ratio (when the time period is identical for all partners)

Calculations:

⇒ Profit ratio of P : Q : R = 34 : 75 : 6

⇒ Total profit units = 34 + 75 + 6 = 115 units

⇒ 115 units = ₹3,600

⇒ 1 unit = 3600 ÷ 115 = 720 ÷ 23

⇒ Difference in ratio between Q and R = 75 − 6 = 69 units

⇒ Difference in their profit shares = 69 units × (720 ÷ 23)

⇒ Difference = 3 × 720

⇒ Difference = ₹2,160

∴ The correct answer is ₹2,160.

Additional Information

Simple Partnership

When all the partners invest their capitals for the same time period, it is called a simple partnership. The profit is distributed exactly in the ratio of their invested capitals.

Compound Partnership

When partners invest for different time periods, the profit ratio is calculated by multiplying the capital invested by the time period of investment for each partner (Profit ∝ Capital × Time).

Working vs Sleeping Partner

A working partner receives a fixed salary or a specific percentage of the total profit for managing the business. The remaining profit is then divided among all partners based on their investments.

Shortcut Trick

The difference between two people's ages remains constant over time.

From the future condition: 6(Q + 8) = 6(P + 8) − 24

Dividing by 6 gives: Q + 8 = P + 8 − 4 ⇒ P − Q = 4

We are given the present age condition: 7P − Q = 46

Subtracting (P − Q = 4) from (7P − Q = 46) gives: 6P = 42 ⇒ P = 7

Substituting P gives: Q = 7 − 4 = 3 years.

∴ The correct answer is 3 years.

Alternate Method

Given:

Seven times the present age of P exceeds Q's present age by 46 years.

After 8 years, 6 times Q's age is 24 years less than 6 times P's age.

Present AgeP = P yearsQ = Q yearsAfter 8 YearsP' = P + 8Q' = Q + 8+ 8 yearsRelations: 7P - Q = 46 | 6(Q+8) = 6(P+8) - 24

Calculations:

Let the present ages of P and Q be P and Q respectively.

⇒ 7P − Q = 46   --- (Equation 1)

According to the condition after 8 years:

⇒ 6(Q + 8) = 6(P + 8) − 24

Dividing the entire equation by 6:

⇒ Q + 8 = P + 8 − 4

⇒ Q = P − 4

⇒ P − Q = 4   --- (Equation 2)

Now, substitute Q = P − 4 in Equation 1:

⇒ 7P − (P − 4) = 46

⇒ 6P + 4 = 46

⇒ 6P = 42

⇒ P = 7 years

Substitute the value of P back in Equation 2:

⇒ 7 − Q = 4

⇒ Q = 3 years

∴ The correct answer is 3 years.

Additional Information

Constant Age Difference

The difference in ages between any two individuals always remains constant over time. If P is 4 years older than Q today, P will always be 4 years older than Q regardless of the years passed.

Ratio of Ages

While the absolute difference remains constant, the ratio of their ages changes continuously. As they grow older, the ratio of their ages approaches 1:1.

Average Age Concept

If the average age of a group is A today, then after N years, their new average age will simply be A + N years, assuming no new members are added or removed.

Shortcut Trick

The ratio of the sides is 5 : 12 : 13, which forms a Pythagorean triplet indicating it is a right-angled triangle.

Area = ½ × base × height = ½ × 5x × 12x = 30x²

Given area = 1080 ⇒ 30x² = 1080 ⇒ x² = 36 ⇒ x = 6

Perimeter = 5x + 12x + 13x = 30x = 30 × 6 = 180 cm.

∴ The correct answer is 180 cm.

Alternate Method

Given:

Ratio of sides = 5 : 12 : 13

Area of the triangle = 1080 cm²

Formula Used:

For a triangle with sides a, b, c, semi-perimeter s = (a + b + c) ÷ 2

Area by Heron's formula = √[s(s − a)(s − b)(s − c)]

5x (30 cm)
12x (72 cm)
13x (78 cm)
Area = 1080 cm²

Calculations:

⇒ Let the sides of the triangle be 5x, 12x, and 13x.

⇒ Semi-perimeter, s = (5x + 12x + 13x) ÷ 2 = 30x ÷ 2 = 15x

⇒ Area = √[15x(15x − 5x)(15x − 12x)(15x − 13x)]

⇒ 1080 = √[15x × 10x × 3x × 2x]

⇒ 1080 = √[900x⁴]

⇒ 1080 = 30x²

⇒ x² = 1080 ÷ 30

⇒ x² = 36

⇒ x = 6

⇒ Perimeter of the triangle = 5x + 12x + 13x = 30x

⇒ Perimeter = 30 × 6 = 180 cm

∴ The correct answer is 180 cm.

Additional Information

Pythagorean Triplets

A set of three positive integers a, b, c that satisfy a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Identifying these can instantly simplify right-angled triangle problems and skip complex formulas.

Heron's Formula

Used to calculate the area of any triangle when all three side lengths are known: Area = √[s(s − a)(s − b)(s − c)], where s is the semi-perimeter.

Right-Angled Triangle Area

For a right-angled triangle, the area can be directly calculated as ½ × base × perpendicular height, bypassing the need for semi-perimeters entirely.

Shortcut Trick

Total time = 7 days (Feb) + 31 days (Mar) + 23 days (Apr) = 61 days.

Since 2024 is a leap year (366 days), Time (T) = 61 ÷ 366 = 1/6 years.

SI = (P × R × T) ÷ 100

SI = 4000 × (6.75 ÷ 100) × (1 ÷ 6) = 45

∴ The correct answer is ₹45.

Alternate Method

Given:

Principal (P) = ₹4000

Rate of Interest (R) = 6.75% per annum

Time period: Deposited on 22 February 2024, Withdrawn on 23 April 2024

Formula Used:

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

Principal = ₹4000 | Rate = 6.75% p.a.

22 Feb
29 Feb
31 Mar
23 Apr

7 days

31 days

23 days

Total = 61 days

Simple Interest = (P × R × T) ÷ 100

Calculations:

⇒ 2024 is a leap year, so February has 29 days.

⇒ Number of days in Feb = 29 − 22 = 7 days (excluding the deposit day)

⇒ Number of days in Mar = 31 days

⇒ Number of days in Apr = 23 days (including the withdrawal day)

⇒ Total duration (T) = 7 + 31 + 23 = 61 days

⇒ Converting days to years: T = 61 ÷ 366 = 1/6 year

⇒ SI = (4000 × 6.75 × 1) ÷ (100 × 6)

⇒ SI = (40 × 6.75) ÷ 6

⇒ SI = 270 ÷ 6 = 45

∴ The correct answer is ₹45.

Additional Information

Day Counting Rule

When calculating interest for a specific timeline, it is an international banking standard to count either the day of deposit or the day of withdrawal, but not both.

Leap Year Conversion

A leap year has 366 days and occurs in years divisible by 4. To convert days into years during a leap year, divide the total calculated days by 366 instead of 365.

Simple Interest vs Compound Interest

Simple Interest is charged only on the original principal, whereas Compound Interest accrues on both the principal and the accumulated interest from previous periods.

Shortcut Trick

Directly expand the brackets and combine the like terms together.

⇒ 3x² − 5x + 5x² − 15 + 16

⇒ (3 + 5)x² − 5x + (16 − 15)

⇒ 8x² − 5x + 1

∴ The correct answer is 8x² − 5x + 1.

Alternate Method

Given:

Expression: x(3x − 5) + 5(x² − 3) + 16

Calculation:

⇒ Expand the first term: x × 3x − x × 5 = 3x² − 5x

⇒ Expand the second term: 5 × x² − 5 × 3 = 5x² − 15

⇒ Substitute the expanded values back into the expression:

⇒ 3x² − 5x + 5x² − 15 + 16

⇒ Group the like terms (terms with x², terms with x, and constant terms):

⇒ (3x² + 5x²) − 5x + (−15 + 16)

⇒ 8x² − 5x + 1

∴ The correct answer is 8x² − 5x + 1.

Additional Information

Distributive Property

The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together: a(b + c) = a × b + a × c.

Like Terms in Algebra

Like terms are terms that have the exact same variables and exponents.

Only like terms can be added or subtracted to simplify algebraic expressions. For example, 3x² and 5x² are like terms and can be added to get 8x².

BODMAS Rule

BODMAS stands for Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction. It determines the correct sequence of operations when simplifying a mathematical expression.

Shortcut Trick

A number divisible by 88 must be divisible by both 8 and 11.

By adding the options to 91673, only adding 23 yields a number that satisfies the divisibility rule for 8.

91673 + 23 = 91696

The last 3 digits (696) are divisible by 8 (696 ÷ 8 = 87). Also, difference of alternate sums (9+6+6) − (1+9) = 11.

∴ The correct answer is 23.

Alternate Method

Given:

Number = 91673

Divisor = 88

Formula Used:

Smallest number to be added = Divisor − Remainder

Calculations:

⇒ Divide 91673 by 88 to find the remainder.

⇒ 91673 = (88 × 1041) + 65

⇒ The remainder is 65.

⇒ Number to be added = 88 − 65

⇒ Number to be added = 23

∴ The correct answer is 23.

Additional Information

Divisibility Rule for 88

A number is divisible by 88 if it is completely divisible by both of its co-prime factors, 8 and 11.

Divisibility Rule for 8

A number is divisible by 8 if the number formed by its last three digits is completely divisible by 8.

Divisibility Rule for 11

A number is divisible by 11 if the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or a multiple of 11.

Finding the Number to be Added

The smallest number to be added to a given number to make it completely divisible by a divisor is given by: Divisor − Remainder.

Shortcut Trick

For any regular polygon with N sides, the exterior angle is 360° ÷ N.

The interior angle (angle between adjacent sides) is 180° − Exterior angle.

Given N = 2n, Exterior angle = 360° ÷ 2n = 180° ÷ n.

Interior angle = 180° − (180° ÷ n) = 180° × (1 − 1/n).

∴ The correct answer is (1 − (1/n)) × 180°.

Alternate Method

Given: Number of sides (N) = 2n.

Formula Used: Each interior angle of a regular polygon = [(N − 2) × 180°] ÷ N.

⇒ N = 2n

⇒ Interior angle = [(2n − 2) × 180°] ÷ 2n

⇒ Interior angle = [2(n − 1) × 180°] ÷ 2n

⇒ Interior angle = [(n − 1) × 180°] ÷ n

⇒ Interior angle = (n/n − 1/n) × 180°

⇒ Interior angle = (1 − 1/n) × 180°

∴ The correct answer is (1 − (1/n)) × 180°.

Additional Information

Shortcut Trick

Required percentage = (Marks of B ÷ Marks of A) × 100

Required percentage = (50 ÷ 75) × 100

Required percentage = (2 ÷ 3) × 100 ≈ 66.67%

∴ The correct answer is 66.67%.

Alternate Method

Given:

Marks obtained by B = 50

Marks obtained by A = 75

Formula Used:

Percentage = (Value ÷ Base Value) × 100

Calculations:

⇒ Required % = (50 ÷ 75) × 100

⇒ Required % = 0.66666... × 100

⇒ Required % = 66.666...

⇒ Required % ≈ 66.67%

∴ The correct answer is 66.67%.

Additional Information

Percentage Base Rule

The value following the word 'than' or 'of' in a comparison is usually taken as the denominator (base value).

Fraction to Percentage Conversion

Standard fractions like 1/3 = 33.33% and 2/3 = 66.67% are frequently used in competitive exams for faster calculation.

Shortcut Trick

For two numbers P and Q, the Mean Proportional is √(P × Q).

Given Mean Proportional = 120 and one number = 150.

⇒ 120² = P × 150

⇒ 14400 = P × 150

⇒ P = 1440 ÷ 15 = 96.

∴ The correct answer is 96.

Alternate Method

Given:

First number = P

Second number = 150

Mean Proportional (M.P.) = 120

Formula Used:

Mean Proportional = √(First number × Second number)

Calculations:

⇒ 120 = √(P × 150)

Squaring both sides:

⇒ (120)² = P × 150

⇒ 14400 = P × 150

⇒ P = 14400 ÷ 150

⇒ P = 96

∴ The correct answer is 96.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is x if a/x = x/b, which gives x = √(ab).

Third Proportional

If a : b :: b : c, then c is called the third proportional to a and b, calculated as c = b²/a.

Fourth Proportional

If a : b :: c : d, then d is the fourth proportional to a, b, and c, calculated as d = (b × c)/a.

Shortcut Trick

Effective discount = a + b − (a × b) ÷ 100 = 35 + 30 − (35 × 30) ÷ 100 = 65 − 10.5 = 54.5%.

Ratio of Marked Price (MP) to Cost Price (CP): MP ÷ CP = (100 + Gain%) ÷ (100 − Discount%).

MP ÷ 7,000 = (100 + 17) ÷ (100 − 54.5) = 117 ÷ 45.5.

MP = (7,000 × 117) ÷ 45.5 = (7,000 × 1170) ÷ 455.

MP = 18.02 × 1,000 ≈ ₹18,000.

∴ The correct answer is ₹18,000.

Alternate Method

Given: CP = ₹7,000, Gain = 17%, Successive Discounts = 35% and 30%.
Formulas: SP = CP × (100 + P%) ÷ 100 | MP = SP × 100 ÷ (100 − D%) | Successive Discount = a + b − ab ÷ 100.

Steps:
⇒ Calculate Selling Price (SP): SP = 7,000 × (117 ÷ 100) = 70 × 117 = ₹8,190.
⇒ Calculate Single Equivalent Discount: 35 + 30 − (35 × 30 ÷ 100) = 65 − 10.5 = 54.5%.
⇒ Use SP and Discount to find MP: MP × (100 − 54.5) ÷ 100 = 8,190.
⇒ MP × 45.5 ÷ 100 = 8,190 ⇒ MP = (8,190 × 100) ÷ 45.5.
⇒ MP = 819,000 ÷ 45.5 = ₹18,000.

∴ The correct answer is ₹18,000.

Additional Information

Relationship between MP and CP

When both discount percentage and profit percentage are known, the direct relation is MP/CP = (100 + P%)/(100 - D%).

Successive Discounts

Two successive discounts of x% and y% are equivalent to a single discount of [x + y − (xy/100)]%.

Shortcut Trick

Mode is the observation that appears most frequently in the data set.

Count the occurrences of each unique number:

Value 1 appears 2 times; Value 3 appears 2 times; Value 4 appears 2 times; Value 5 appears 1 time; Value 6 appears 1 time.

Value 7 appears 4 times; Value 8 appears 3 times.

Since 7 has the highest frequency, it is the mode.

∴ The correct answer is 7.

Alternate Method

Given: Observations = 7, 5, 6, 3, 3, 7, 1, 7, 8, 7, 1, 4, 8, 8, 4

Step 1: Arrange data in Ascending Order: 1, 1, 3, 3, 4, 4, 5, 6, 7, 7, 7, 7, 8, 8, 8

⇒ Frequency of 1 = 2

⇒ Frequency of 3 = 2

⇒ Frequency of 4 = 2

⇒ Frequency of 5 = 1

⇒ Frequency of 6 = 1

⇒ Frequency of 7 = 4

⇒ Frequency of 8 = 3

⇒ Highest Frequency = 4 (for the number 7)

∴ The correct answer is 7.

Additional Information

Empirical Formula

The relationship between central tendencies is given by: Mode = 3 × Median − 2 × Mean.

Bimodal and Multimodal

If two observations have the same maximum frequency, the data set is Bimodal. If more than two, it is Multimodal.

Median for Discrete Data

For n observations, if n is odd, Median = ((n+1)/2)^th term. If n is even, Median = Average of (n/2)^th and (n/2 + 1)^th terms.

Shortcut Trick

Combine the ratios: A : B = 13 : 9 and B : C = 18 : 11.

To equalize B, multiply A : B by 2 → A : B : C = 26 : 18 : 11.

Total units = 26 + 18 + 11 = 55 units.

Value of 55 units = ₹14,200 → 1 unit = 14,200 ÷ 55 = 2840 ÷ 11.

Share of C = 11 units = (2840 ÷ 11) × 11 = ₹2,840.

∴ The correct answer is ₹2,840.

Alternate Method

Given:

Total profit = ₹14,200

Ratio A : B = 13 : 9

Ratio B : C = 18 : 11

Calculations:

⇒ To find the combined ratio, we make the term for B common.

⇒ A : B = (13 × 2) : (9 × 2) = 26 : 18

⇒ B : C = 18 : 11

⇒ Combined Ratio A : B : C = 26 : 18 : 11

⇒ Sum of ratio parts = 26 + 18 + 11 = 55

⇒ Share of C = (Ratio of C ÷ Total Ratio) × Total Profit

⇒ Share of C = (11 ÷ 55) × 14,200

⇒ Share of C = (1 ÷ 5) × 14,200

⇒ Share of C = ₹2,840

∴ The correct answer is ₹2,840.

Additional Information

Ratio Balancing

When two ratios share a common variable, multiply the terms to make the common variable's value identical in both ratios before combining.

Profit Distribution in Partnership

Profit is distributed based on the Ratio of Investment × Time. If time is equal, profit ratio = capital ratio.

Compound Ratio

The compound ratio of a : b and c : d is found by multiplying corresponding terms: (ac) : (bd).

Shortcut Trick

Let Income = 100, Savings = 26, Expenditure = 74.

New Income = 100 + 20% = 120.

New Expenditure = 74 + 40% = 74 + 29.6 = 103.6.

New Savings = 120 − 103.6 = 16.4.

Change in Savings = 16.4 − 26 = −9.6 units. (Decrease)

Since 100 units = ₹43,000, 1 unit = ₹430. Change = 9.6 × 430 = ₹4,128.

∴ The correct answer is decrease by ₹4,128.

Alternate Method

Given: Initial Income = ₹43,000 | Savings % = 26%.

Formula Used: Income = Expenditure + Savings

⇒ Initial Savings = 26% of 43,000 = 0.26 × 43,000 = ₹11,180.

⇒ Initial Expenditure = 43,000 − 11,180 = ₹31,820.

⇒ New Income = 43,000 × 1.20 = ₹51,600.

⇒ New Expenditure = 31,820 × 1.40 = ₹44,548.

⇒ New Savings = 51,600 − 44,548 = ₹7,052.

⇒ Change in Savings = 11,180 − 7,052 = ₹4,128 decrease.

∴ The correct answer is decrease by ₹4,128.

Additional Information

Percentage Change in Savings

Percentage change is calculated as [(New Value − Old Value) ÷ Old Value] × 100.

Base Change Property

If income and expenditure change by different percentages, savings will change based on their weighted relative proportions.

Income-Expenditure Identity

Always remember: Income = Expenditure + Savings. Any increase in Income must be balanced by the sum of increases in Expenditure and Savings.

Shortcut Trick

Assume all 20 animals are rabbits (4 legs each).

Hypothetical total legs = 20 × 4 = 80.

Difference from actual legs = 80 − 56 = 24.

Each chicken has 2 fewer legs than a rabbit (4 − 2 = 2).

Number of chickens = 24 ÷ 2 = 12.

∴ The correct answer is 12.

Alternate Method

Given:

Total animals = 20

Total legs = 56

Let the number of chickens be x and rabbits be (20 − x).

⇒ Total legs = (Number of chickens × 2) + (Number of rabbits × 4)

⇒ 56 = 2x + 4(20 − x)

⇒ 56 = 2x + 80 − 4x

⇒ 56 = 80 − 2x

⇒ 2x = 80 − 56

⇒ 2x = 24

⇒ x = 24 ÷ 2

⇒ x = 12

∴ The correct answer is 12.

Additional Information

Direct Formula for 4-legged animals

Number of 4-legged animals (Rabbits) = (Total Legs − 2 × Total Heads) ÷ 2. Here, Rabbits = (56 − 40) ÷ 2 = 8.

Alligation Method

Average legs per animal = 56 ÷ 20 = 2.8. Ratio of Chickens to Rabbits = (4 − 2.8) : (2.8 − 2) = 1.2 : 0.8 = 3 : 2.

Linear Equations in Two Variables

Solve by simultaneous equations: x + y = Total Animals and Ax + By = Total Legs, where A and B are the number of legs of each species.

Shortcut Trick

To find the largest number leaving the same remainder, find the HCF of the differences between the numbers.

Differences: (134 − 62) = 72 and (242 − 134) = 108.

HCF of 72 and 108 = 36.

∴ The correct answer is 36.

Alternate Method

Given: Numbers are 62, 134, and 242.

Formula: Required Number = HCF(|a−b|, |b−c|, |c−a|)

⇒ Difference 1 = 134 − 62 = 72

⇒ Difference 2 = 242 − 134 = 108

⇒ Difference 3 = 242 − 62 = 180

⇒ HCF(72, 108, 180):

⇒ 72 = 2³ × 3²

⇒ 108 = 2² × 3³

⇒ 180 = 2² × 3² × 5

⇒ HCF = 2² × 3² = 4 × 9 = 36

∴ The correct answer is 36.

Additional Information

HCF with Same Remainder

The greatest number which when divides x, y, and z leaving the same remainder 'r' in each case is given by HCF of (x−y), (y−z), and (z−x).

HCF with Different Remainders

The greatest number which divides x, y, and z leaving remainders a, b, and c is HCF of (x−a), (y−b), and (z−c).

LCM Concept

LCM is the smallest number which is exactly divisible by all the given numbers.

Shortcut Trick

Average Speed for equal time intervals = Arithmetic Mean of individual speeds.

Given Time = 1 hour each, so Speed 1 = 128 kmph and Speed 2 = 146 kmph.

Average Speed = (128 + 146) ÷ 2 = 274 ÷ 2 = 137 kmph.

∴ The correct answer is 137 km/hr.

Alternate Method

Given:

Distance₁ = 128 km, Time₁ = 1 hr

Distance₂ = 146 km, Time₂ = 1 hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Total Distance = 128 + 146 = 274 km

⇒ Total Time = 1 + 1 = 2 hours

⇒ Average Speed = 274 ÷ 2

⇒ Average Speed = 137 km/hr

∴ The correct answer is 137 km/hr.

Additional Information

Average Speed (Equal Distance)

If a car covers equal distances at speeds u and v, the average speed is 2uv ÷ (u + v).

Relative Speed

When two bodies move in the same direction, relative speed is the difference (s₁ − s₂). In opposite directions, it is the sum (s₁ + s₂).

Unit Conversion

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