Search Previous Year Questions

Shortcut Trick

Income : Expenditure ratio is 8 : 5.

Expenditure 5 units = ₹10,000 ⇒ 1 unit = ₹2,000 ∴ Initial Income = 8 × 2,000 = ₹16,000.

New Income = 16,000 ± 40% = 16,000 × 1.4 = ₹22,400.

New Expenditure = 10,000 − 10% = 10,000 × 0.9 = ₹9,000.

New Saving = 22,400 − 9,000 = ₹13,400.

∴ The correct answer is ₹13,400.

Alternate Method

Given: Ratio of Income (I₁) to Expenditure (E₁) = 8 : 5. Initial Expenditure = ₹10,000. Income rise = 40%. Expenditure fall = 10%.

Formula: Saving = Income − Expenditure

Calculations:

⇒ Let Initial Income = 8x and Initial Expenditure = 5x.

⇒ 5x = 10,000 ⇒ x = 2,000.

⇒ Initial Income = 8 × 2,000 = ₹16,000.

⇒ New Income = 16,000 + (40/100 × 16,000) = 16,000 + 6,400 = ₹22,400.

⇒ New Expenditure = 10,000 − (10/100 × 10,000) = 10,000 − 1,000 = ₹9,000.

⇒ New Saving = New Income − New Expenditure = 22,400 − 9,000 = ₹13,400.

∴ The correct answer is ₹13,400.

Additional Information

Income, Expenditure, and Savings Relationship

Income = Expenditure + Savings. Any change in two variables directly affects the third.

Percentage Change Formula

New Value = Original Value × (100 ± Percentage Change)/100.

Ratio Unitary Method

If a : b is the ratio and the value of 'b' is V, then the value of 1 unit = V/b and value of 'a' = (V/b) × a.

Shortcut Trick

Net percentage change = 5% (gain) − (−10% loss) = 15%

This 15% difference corresponds to the price increase of ₹120.

15% → ₹120

1% → ₹8

Initial Selling Price (at 10% loss) = 90% → 90 × 8 = ₹720

∴ The correct answer is ₹720.

Alternate Method

Given: Initial loss = 10%, New gain = 5%, Increase in SP = ₹120.

Formula Used: Selling Price (SP) = Cost Price (CP) × (100 ± Profit/Loss)% / 100

⇒ Let the Cost Price (CP) of the watch be 100 units.

⇒ Initial Selling Price (SP₁) = 100 − 10 = 90 units.

⇒ New Selling Price (SP₂) = 100 + 5 = 105 units.

⇒ Difference between SP₂ and SP₁ = 105 − 90 = 15 units.

⇒ According to the question, 15 units = ₹120.

⇒ 1 unit = 120 ÷ 15 = ₹8.

⇒ Initial Selling Price (SP₁) = 90 × 8 = ₹720.

∴ The correct answer is ₹720.

Additional Information

Profit and Loss Percentage

Profit % = (Profit ÷ CP) × 100 and Loss % = (Loss ÷ CP) × 100. Profit and loss are always calculated on the Cost Price unless otherwise mentioned.

Selling Price Calculation

When there is a gain of G%, SP = CP × (100 + G) ÷ 100. When there is a loss of L%, SP = CP × (100 − L) ÷ 100.

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount. MP = CP + Markup.

Shortcut Trick

Test the options to satisfy the continued proportion condition: (13 − x) / (25 − x) = (25 − x) / 49.

Try x = 4: (13 − 4) / (25 − 4) = 9 / 21 = 3 / 7.

Check the second ratio: (25 − 4) / 49 = 21 / 49 = 3 / 7.

Since the ratios are equal, x = 4 is the required natural number.

∴ The correct answer is 4.

Alternate Method

Given: Original numbers are 13 and 25. The fourth proportional of the resulting numbers after subtracting x is 49.

Formula Used: If three numbers a, b, and c are in continued proportion, then a/b = b/c, where c is the third proportional (or the fourth term in the sequence a, b, b, c).

⇒ (13 − x) × 49 = (25 − x) × (25 − x)

⇒ 637 − 49x = 625 − 50x + x²

⇒ x² − 50x + 49x + 625 − 637 = 0

⇒ x² − x − 12 = 0

⇒ x² − 4x + 3x − 12 = 0

⇒ x(x − 4) + 3(x − 4) = 0

⇒ (x − 4)(x + 3) = 0

⇒ x = 4 or x = −3

⇒ Since x must be a natural number, x = 4.

∴ The correct answer is 4.

Additional Information

Mean Proportional

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

Third Proportional

If a : b :: b : c, then c is 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

To maximize the product of two numbers with a fixed sum, the numbers should be as close to each other as possible.

Given a + b = 41, the closest integers are 20 and 21.

Check if 20 and 21 are coprime: HCF(20, 21) = 1 (Condition satisfied).

Maximum product = 20 × 21 = 420.

∴ The correct answer is 420.

Alternate Method

Given:

Sum of two coprime numbers, a + b = 41.

Calculations:

⇒ Let the numbers be a and b. We know a + b = 41.

⇒ According to the AM-GM inequality, for a constant sum, the product is maximized when numbers are equal: a = b = 41 ÷ 2 = 20.5.

⇒ Since a and b must be integers, we test integers closest to 20.5, which are 20 and 21.

⇒ Verify coprime condition: Factors of 20 = {1, 2, 4, 5, 10, 20} and Factors of 21 = {1, 3, 7, 21}.

⇒ HCF(20, 21) = 1, therefore they are coprime.

⇒ Maximum Product = 20 × 21 = 420.

⇒ Testing other pairs: 19 × 22 = 418; 18 × 23 = 414. The product decreases as the difference between numbers increases.

∴ The correct answer is 420.

Additional Information

Coprime Numbers

Two numbers are coprime (or relatively prime) if their Greatest Common Divisor (GCD) or HCF is 1. Any two consecutive integers are always coprime.

Sum and Product Relationship

If the sum of two positive numbers is constant, their product is maximum when the numbers are equal, and their product is minimum when the difference between them is maximum.

Prime Sum Property

If the sum S = a + b is a prime number, then every pair of positive integers (a, b) that sums to S is coprime, because HCF(a, S−a) = HCF(a, S), and a prime S has no divisors other than 1 and itself.

Shortcut Trick

Let the Original Cost Price (CP) be 100 units. Initial SP = 132 units.

New CP = 100 units + 20 and New SP = 132 units − 42.

New Profit is 27.5%, so New SP = 127.5% of New CP.

⇒ 132 units − 42 = 1.275 × (100 units + 20) = 127.5 units + 25.5.

⇒ (132 − 127.5) units = 25.5 + 42 ⇒ 4.5 units = 67.5.

⇒ 1 unit = 67.5 ÷ 4.5 = 15. ∴ 100 units = 100 × 15 = ₹1,500.

∴ The correct answer is ₹1,500.

Alternate Method

Given: Original Profit = 32%, Increase in CP = ₹20, Decrease in SP = ₹42, New Profit = 27.5%.

Formula Used: Selling Price (SP) = Cost Price (CP) × (100 + Profit %)/100

⇒ Let the Original Cost Price be x.

⇒ Original Selling Price = 1.32x.

⇒ New Cost Price = x + 20.

⇒ New Selling Price = 1.32x − 42.

⇒ According to the problem: 1.32x − 42 = (x + 20) × (127.5/100)

⇒ 1.32x − 42 = (x + 20) × 1.275

⇒ 1.32x − 42 = 1.275x + 25.5

⇒ 1.32x − 1.275x = 25.5 + 42

⇒ 0.045x = 67.5

⇒ x = 67.5 ÷ 0.045 = 1500.

∴ The correct answer is ₹1,500.

Additional Information

Profit and Loss Fundamentals

Profit % is always calculated on the Cost Price (CP) unless otherwise stated. Profit = SP − CP.

Relationship between CP and SP

SP = CP × (100 + P)/100 for profit and SP = CP × (100 − L)/100 for loss.

Successive Changes

When CP and SP both change, the New Profit Percentage is applied to the Modified Cost Price to find the Modified Selling Price.

Shortcut Trick

Let the Assumed Mean (A) be ₹1200.

Calculate deviations (d = x − A): −300, −200, −100, 0, 100, 200, 300.

Net deviation (∑fd) = 12(−300) + 12(−200) + 14(−100) + 13(0) + 14(100) + 11(200) + 14(300)

∑fd = −3600 − 2400 − 1400 + 0 + 1400 + 2200 + 4200 = 400.

Total employees (∑f) = 90. Corrected Mean = 1200 + (400 ÷ 90) = 1200 + 4.44 = 1204.44.

∴ The correct answer is ₹1204.44.

Alternate Method

Given: Weekly wages (x) and Number of employees (f).

Formula Used: Arithmetic Mean (μ) = ∑fx ÷ ∑f

⇒ Total employees (∑f) = 12 + 12 + 14 + 13 + 14 + 11 + 14 = 90

⇒ Total wages (∑fx) = (900 × 12) + (1000 × 12) + (1100 × 14) + (1200 × 13) + (1300 × 14) + (1400 × 11) + (1500 × 14)

⇒ ∑fx = 10800 + 12000 + 15400 + 15600 + 18200 + 15400 + 21000 = 108400

⇒ Arithmetic Mean = 108400 ÷ 90

⇒ Arithmetic Mean = 1204.444...

∴ The correct answer is ₹1204.44.

Additional Information

Arithmetic Mean of Discrete Series

It is the sum of the products of observations and their respective frequencies divided by the total frequency: Mean = ∑f_ix_i ÷ ∑f_i.

Assumed Mean Method

Used to simplify calculations: Mean = A + (∑fd ÷ N), where A is the assumed mean and d = x − A is the deviation.

Properties of Mean

The sum of deviations of all observations from their arithmetic mean is always zero (∑(x − μ) = 0).

Given:

6^x − 6^x−1 = 180

Formula used:

a^m−n = a^m ÷ aⁿ

a^m = aⁿ ⇒ m = n (when bases are same)

Calculations:

6^x − 6^x−1 = 180

⇒ 6^x − (6^x ÷ 6¹) = 180

⇒ 6^x (1 − 1/6) = 180

⇒ 6^x (5/6) = 180

⇒ 6^x = 180 × (6 ÷ 5)

⇒ 6^x = 36 × 6

⇒ 6^x = 216

⇒ 6^x = 6³

⇒ x = 3

Now, substitute the value of x in the expression (2x − 1) / (2x + 3):

⇒ (2 × 3 − 1) / (2 × 3 + 3)

⇒ (6 − 1) / (6 + 3)

⇒ 5/9

∴ The correct answer is option (4).

Shortcut Trick

Rate 10% = 1/10, so the Multiplier for 1 year is 11/10.

For 2 years, the ratio of Principal to Amount = 10² : 11² = 100 : 121.

⇒ 121 units = ₹9,075

⇒ 1 unit = 9075 ÷ 121 = 75

⇒ Principal (100 units) = 75 × 100 = ₹7,500.

∴ The correct answer is ₹7,500.

Alternate Method

Given:

Total Amount (A) = ₹9,075

Rate of Interest (R) = 10% per annum

Time period (n) = 2 years

Formula Used:

A = P(1 + R/100)ⁿ

Calculations:

⇒ 9075 = P(1 + 10/100)²

⇒ 9075 = P(110/100)²

⇒ 9075 = P(11/10)²

⇒ 9075 = P × (121/100)

⇒ P = (9075 × 100) ÷ 121

⇒ P = 75 × 100

⇒ P = ₹7,500

∴ The correct answer is ₹7,500.

Additional Information

Effective Rate of Interest

For 2 years, the effective interest rate can be calculated as x + y + (xy/100)%. For 10%, it is 10 + 10 + 1 = 21%.

Difference between CI and SI

The difference between Compound Interest and Simple Interest for 2 years is given by the formula: Diff = P × (R/100)².

Compound Interest Formula

Total Compound Interest = Amount − Principal = P[(1 + R/100)ⁿ − 1].

Shortcut Trick

Assume distance d = 240 km (LCM of 5, 4, 15, 20, 16, 24).

Onward time = (96 ÷ 15) + (144 ÷ 20) = 6.4 + 7.2 = 13.6 hrs.

Return time = (60 ÷ 16) + (180 ÷ 24) = 3.75 + 7.5 = 11.25 hrs.

Total Average Speed = Total Distance ÷ Total Time = 480 ÷ 24.85 = 19.32 km/hr.

∴ The correct answer is 19.32 km/hr.

Alternate Method

Given: Onward journey speeds are 15 km/hr (2/5^th) and 20 km/hr (3/5^th). Return journey speeds are 16 km/hr (1/4^th) and 24 km/hr (3/4^th).

Formula: Average Speed = (Total Distance) ÷ (Total Time taken)

Let the distance from A to B be d.

⇒ Total distance for the entire trip = d + d = 2d

⇒ Time for onward journey (T₁) = (2d/5 ÷ 15) + (3d/5 ÷ 20)

⇒ T₁ = 2d/75 + 3d/100 = (8d + 9d)/300 = 17d/300

⇒ Time for return journey (T₂) = (d/4 ÷ 16) + (3d/4 ÷ 24)

⇒ T₂ = d/64 + 3d/96 = d/64 + d/32 = (d + 2d)/64 = 3d/64

⇒ Total Time (T) = 17d/300 + 3d/64 = (544d + 450d)/9600 = 994d/9600

⇒ Average Speed = 2d ÷ (994d/9600) = (2 × 9600) ÷ 994 = 19200 ÷ 994

⇒ Average Speed ≈ 19.3158... km/hr

∴ The correct answer is 19.32 km/hr.

Additional Information

Average Speed for Equal Distances

If two equal distances are covered at speeds v₁ and v₂, average speed = 2v₁v₂ ÷ (v₁ + v₂).

Weighted Average Speed

When different parts of a journey (d₁, d₂, ...) are covered at different speeds (v₁, v₂, ...), Average Speed = (d₁ + d₂ + ...) ÷ (d₁/v₁ + d₂/v₂ + ...).

Unit Conversions

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

Given: ABCD is a trapezium with AB || DC. AB = 14 cm, DC = 72 cm, BC = 64 cm, and AD = 47 cm. E and F are the midpoints of AC and BD respectively.

Formula Used: In a trapezium with bases a and b, the distance between the midpoints of the diagonals is |a − b| / 2.

Calculations:

⇒ Length of the parallel side AB = 14 cm

⇒ Length of the parallel side DC = 72 cm

⇒ According to the geometry property of a trapezium, the distance EF joining the midpoints of the diagonals is:

⇒ EF = (DC − AB) ÷ 2

⇒ EF = (72 − 14) ÷ 2

⇒ EF = 58 ÷ 2

⇒ EF = 29 cm

The calculated length is 29 cm, which corresponds to Option (2).

∴ The correct answer is 29 cm.

Additional Information

Median of a Trapezium

The line segment joining the midpoints of the non-parallel sides is called the median. Its length is equal to half the sum of the parallel sides: Median = (a + b) ÷ 2.

Midpoint of Diagonals Theorem

The segment connecting the midpoints of the diagonals of a trapezium is always parallel to the bases and its length is half the absolute difference of the bases.

Area of a Trapezium

The area of a trapezium is calculated as ½ × (sum of parallel sides) × height, where the height is the perpendicular distance between the bases.