Search Previous Year Questions

Shortcut Trick

Since it is an internal transfer, the total number of employees remains constant.

Initial Ratio A : B = 5 : 7 (Total 12 units) | Final Ratio = 1 : 1 = 6 : 6 (Total 12 units).

Shift from Dept B to Dept A = 7 − 6 = 1 unit (or 6 − 5 = 1 unit).

Given, 1 unit = 28 employees.

Original Dept A = 5 × 28 = 140 | Original Dept B = 7 × 28 = 196.

∴ The correct answer is 140 in A, 196 in B.

Alternate Method

Given:

Original ratio of employees in A and B = 5 : 7

Number of employees moved from B to A = 28

New ratio after transfer = 1 : 1

Calculations:

Let the original number of employees in Department A be 5x and in Department B be 7x.

⇒ New count in A = 5x + 28

⇒ New count in B = 7x − 28

⇒ (5x + 28) / (7x − 28) = 1 / 1

⇒ 5x + 28 = 7x − 28

⇒ 7x − 5x = 28 + 28

⇒ 2x = 56

⇒ x = 28

⇒ Original Dept A = 5 × 28 = 140

⇒ Original Dept B = 7 × 28 = 196

∴ The correct answer is 140 in A, 196 in B.

Additional Information

Constant Sum Property

When items are shifted between two groups, the total sum of items across both groups remains unchanged (Initial Sum = Final Sum).

Ratio Balancing

To solve ratio changes involving constant sums, equate the sum of ratio parts by multiplying with suitable constants.

Cross-Multiplication Method

For problems with additions/subtractions in ratios: (a × d) − (b × c) logic can be used to find the value of one ratio unit directly.

Shortcut Trick

The Simple Interest (SI) is calculated as a constant percentage of the Principal amount for every time period.

By definition, Simple Interest = Principal × Rate (%) × Time.

Converting the percentage Rate into a fraction gives: SI = (Principal × Rate × Time) ÷ 100.

∴ The correct answer is Simple Interest = (Principal × Rate × Time) / 100.

Alternate Method

Given: Principal (P), Rate of Interest per annum (R), and Time in years (T).

Formula Used: Simple Interest (SI) = (P × R × T) / 100.

⇒ Simple Interest is defined as the cost of borrowing money, where the interest is calculated only on the original Principal amount.

⇒ Interest for one year = P × (R / 100)

⇒ Interest for T years = (P × R / 100) × T

⇒ Simple Interest (SI) = (Principal × Rate × Time) / 100

∴ The correct answer is Simple Interest = (Principal × Rate × Time) / 100.

Additional Information

Total Amount (A)

The final value including both Principal and Interest is calculated as A = P + SI, which can also be written as A = P(1 + (R × T)/100).

Time Unit Consistency

The Time (T) must be in the same unit as the Rate (R). If the rate is per annum, time must be in years. (Divide months by 12 or days by 365 if necessary).

Simple vs. Compound Interest

Unlike Simple Interest, Compound Interest is calculated on the principal plus any accumulated interest from previous periods.

Shortcut Trick

The markup percentage is +20% and the discount percentage is (2 ÷ 30) × 100 = 20/3%.

Using the successive percentage formula: Net Change = Markup − Discount − [(Markup × Discount) ÷ 100]

Net Gain % = 20 − 20/3 − [20 × (20/3) ÷ 100] = 20 − 20/3 − 4/3

Net Gain % = 20 − 24/3 = 20 − 8 = 12%.

∴ The correct answer is 12% gain.

Alternate Method

Given:

The markup on the article is 20% above the cost price.

A discount of ₹2 is offered on a purchase of ₹30.

Formula Used:

Marked Price (MP) = CP × [1 + (Markup% ÷ 100)]

Selling Price (SP) = MP − Discount

Profit % = [(SP − CP) ÷ CP] × 100

⇒ Let the cost price (CP) of the article be ₹100.

⇒ The marked price (MP) is 20% above CP: MP = 100 + 20% of 100 = ₹120.

⇒ The discount is ₹2 for a purchase of ₹30. This means the discount rate = 2 ÷ 30 = 1/15.

⇒ Total discount on the MP of ₹120 = 120 × (1 ÷ 15) = ₹8.

⇒ The selling price (SP) = MP − Discount = 120 − 8 = ₹112.

⇒ Gain = SP − CP = 112 − 100 = ₹12.

⇒ Gain percentage = (12 ÷ 100) × 100 = 12%.

∴ The correct answer is 12% gain.

Additional Information

Relationship between CP and MP

The ratio of MP to CP can be expressed as MP/CP = (100 + Profit%) ÷ (100 − Discount%).

Successive Discount Formula

If two successive discounts of a% and b% are given, the equivalent single discount is [a + b − (ab/100)]%.

Markup Percentage

Markup % = [(MP − CP) ÷ CP] × 100. It represents how much the retailer has increased the price above the original cost.

Shortcut Trick

The smallest number which leaves a remainder r when divided by a, b, and c is given by: LCM(a, b, c) + r

⇒ Required number = LCM(9, 12, 15) + 3

⇒ Required number = 180 + 3 = 183

∴ The correct answer is 183.

Alternate Method

Given:

Divisors = 9, 12, and 15

Common remainder in each case = 3

Formula Used:

Required Number = LCM(divisors) × k + Remainder

For the smallest possible number, we take k = 1.

Calculations:

Find the prime factorization of each divisor to calculate the LCM:

⇒ 9 = 3²

⇒ 12 = 2² × 3

⇒ 15 = 3 × 5

The Least Common Multiple (LCM) is the product of the highest powers of all prime factors involved:

⇒ LCM(9, 12, 15) = 2² × 3² × 5

⇒ LCM(9, 12, 15) = 4 × 9 × 5

⇒ LCM(9, 12, 15) = 180

Now, add the common remainder to obtain the required smallest number:

⇒ Required number = LCM(9, 12, 15) + 3

⇒ Required number = 180 + 3 = 183

∴ The correct answer is 183.

Additional Information

General Form of Such Numbers

Any number that leaves a remainder r when divided by divisors a, b, and c is written in the form: (LCM(a, b, c) × k) + r, where k ≥ 1 is an integer.

LCM with Different Remainders (Constant Difference)

If a number leaves remainders p, q, and r when divided by a, b, and c respectively, such that (a − p) = (b − q) = (c − r) = k (constant difference), then the smallest such number is: LCM(a, b, c) − k.

HCF with Remainders

To find the greatest number that divides x, y, and z leaving remainders p, q, and r respectively, calculate: HCF(x − p, y − q, z − r).

Shortcut Trick

Mode is the value that appears most frequently in a given data set.

Frequency of 46 → 3 times (46, 46, 46).

Frequency of 41, 43, and 45 → 2 times each.

Frequency of all other values (40, 42, 44, 48, 50, 51, 52, 53, 55) → 1 time each.

Since 46 has the highest frequency of 3, it is the mode.

∴ The correct answer is 46.

Alternate Method

Given data: 43, 40, 41, 53, 50, 41, 44, 45, 43, 46, 48, 46, 51, 55, 52, 42, 46, 45

Formula: Mode = Observation with the maximum frequency.

To find the Mode, we construct a frequency distribution table:

⇒ 40: 1 time

⇒ 41: 2 times

⇒ 42: 1 time

⇒ 43: 2 times

⇒ 44: 1 time

⇒ 45: 2 times

⇒ 46: 3 times

⇒ 48: 1 time

⇒ 50, 51, 52, 53, 55: 1 time each

⇒ The observation with the maximum frequency is 46.

∴ The correct answer is 46.

Additional Information

Empirical Relation

For a moderately skewed distribution, the relationship is: Mode = 3 × Median − 2 × Mean.

Median

The Median is the middle value when the data is arranged in ascending or descending order.

Bimodal Distribution

If a data set contains two values that share the highest frequency, the distribution is called bimodal.

Shortcut Trick

Subtract the extra cost of tables from the total to equalize all items to chair prices.

Extra cost of 5 tables = 5 × 400 = ₹2000.

Remaining total for 12 items (5 + 7) = 7040 − 2000 = ₹5040.

Price of 1 chair = 5040 ÷ 12 = ₹420.

Price of 1 table = 420 + 400 = ₹820.

∴ The correct answer is ₹820.

Alternate Method

Given: Total cost of 5 tables and 7 chairs = ₹7040. A table costs ₹400 more than a chair.

Let the cost of one chair = ₹x

Then, the cost of one table = ₹(x + 400)

⇒ 5 × (x + 400) + 7 × x = 7040

⇒ 5x + 2000 + 7x = 7040

⇒ 12x + 2000 = 7040

⇒ 12x = 7040 − 2000

⇒ 12x = 5040

⇒ x = 5040 ÷ 12

⇒ x = 420

Cost of one table = x + 400 = 420 + 400 = 820.

∴ The correct answer is ₹820.

Additional Information

Linear Equations in One Variable

A linear equation is an algebraic equation in which each term has an exponent of 1. It is solved by isolating the variable on one side.

Unitary Method

This method involves finding the value of a single unit (like one chair) and then calculating the required value for other units.

Substitution Method

When one variable (Table) is expressed in terms of another (Chair), we use substitution to simplify the equation into a single variable.

Shortcut Trick

Since the Median is 28.5, the Median Class is 20−30.

Total frequency N = 52 + x ⇒ N/2 = 26 + 0.5x.

Cumulative frequency before median class cf = 5 + x and frequency f = 20.

Applying the simplified relation: 28.5 − 20 = [(26 + 0.5x − 5 − x) ÷ 20] × 10.

8.5 = (21 − 0.5x) ÷ 2 ⇒ 17 = 21 − 0.5x ⇒ 0.5x = 4 ⇒ x = 8.

∴ The correct answer is 8.

Alternate Method

Given: Median = 28.5, Class Intervals and Frequencies as provided.

Formula Used: Median = L + [((N/2) − cf) ÷ f] × h

Calculations:

⇒ Total frequency N = 5 + x + 20 + 15 + 7 + 5 = 52 + x

⇒ Median 28.5 lies in class 20−30, so L = 20, f = 20, h = 10

⇒ Cumulative frequency of preceding class cf = 5 + x

⇒ 28.5 = 20 + [((52 + x) ÷ 2 − (5 + x)) ÷ 20] × 10

⇒ 8.5 = [((52 + x) − 2(5 + x)) ÷ 2] × (1 ÷ 2)

⇒ 8.5 = (52 + x − 10 − 2x) ÷ 4

⇒ 8.5 × 4 = 42 − x

⇒ 34 = 42 − x

⇒ x = 42 − 34 = 8

∴ The correct answer is 8.

Additional Information

Median for Grouped Data

It is the value that divides the data into two equal halves. Formula: L + [((N/2) − cf) ÷ f] × h, where cf is the cumulative frequency of the class preceding the median class.

Empirical Formula

The relationship between Mean, Median, and Mode is: 3 × Median = Mode + 2 × Mean.

Mode for Grouped Data

The value with the highest frequency. Formula: L + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h.

Shortcut Trick

Effective Compound Interest (CI) rate for 2 years at 10% = 10 + 10 + (10×10)/100 = 21%.

Effective Simple Interest (SI) rate for 2 years at 10% = 10 × 2 = 20%.

Ratio of CI to SI for 2 years = 21 : 20.

Given 21 units = ₹136.50 ⇒ 1 unit = 136.50 ÷ 21 = ₹6.50.

SI (20 units) = 20 × 6.50 = ₹130.

∴ The correct answer is ₹130.

Alternate Method

Given:

Rate (R) = 10% per annum

Time (T) = 2 years

Compound Interest (CI) = ₹136.50

Formula Used:

CI = P[(1 + R/100)^T − 1]

SI = (P × R × T) ÷ 100

Calculations:

⇒ 136.50 = P[(1 + 10/100)² − 1]

⇒ 136.50 = P[(1.1)² − 1]

⇒ 136.50 = P[1.21 − 1]

⇒ 136.50 = 0.21 × P

⇒ P = 136.50 ÷ 0.21 = ₹650

Now, calculate Simple Interest:

⇒ SI = (650 × 10 × 2) ÷ 100

⇒ SI = 13000 ÷ 100 = ₹130

∴ The correct answer is ₹130.

Additional Information

Difference between CI and SI for 2 years

The difference (D) between Compound Interest and Simple Interest for 2 years is given by D = P(R/100)².

Successive Percentage Formula

For a two-year period, the effective compound interest rate can be calculated using the successive change formula: x + y + (xy/100).

CI and SI for the First Year

When interest is compounded annually, the Simple Interest and Compound Interest for the first year are always equal.

Shortcut Trick

To find the lowest price, calculate the product of the remaining price factors (100% − discount%).

Shop I: 0.13 × 0.63 = 0.0819

Shop II: 0.19 × 0.63 = 0.1197

Shop III: 0.43 × 0.46 = 0.1978

Shop IV: 0.09 × 0.75 × 0.41 = 0.0277

The shop with the minimum factor (Shop IV) will have the lowest price.

∴ The correct answer is Shop IV.

Alternate Method

Given: Marked Price (MP) is the same for all shops. Successive discounts are provided for four shops (I, II, III, and IV).

Formula Used: Selling Price (SP) = MP × (1 − d₁/100) × (1 − d₂/100) × ... where d₁, d₂ are successive discount percentages.

⇒ Let the Marked Price (MP) be 100 units.

⇒ Shop I SP = 100 × (100 − 87)/100 × (100 − 37)/100 = 100 × 0.13 × 0.63 = 8.19 units.

⇒ Shop II SP = 100 × (100 − 81)/100 × (100 − 37)/100 = 100 × 0.19 × 0.63 = 11.97 units.

⇒ Shop III SP = 100 × (100 − 57)/100 × (100 − 54)/100 = 100 × 0.43 × 0.46 = 19.78 units.

⇒ Shop IV SP = 100 × (100 − 91)/100 × (100 − 25)/100 × (100 − 59)/100 = 100 × 0.09 × 0.75 × 0.41 = 2.7675 units.

⇒ Comparing all Selling Prices: 2.7675 < 8.19 < 11.97 < 19.78.

∴ The correct answer is Shop IV.

Additional Information

Single Equivalent Discount

For two successive discounts a% and b%, the effective discount is (a + b − ab/100)%.

Discount and Selling Price

Selling Price = Marked Price × (100 − Discount%)/100. A higher total discount results in a lower selling price.

Successive Multiplication

When multiple discounts are applied, the order of discounts does not change the final selling price.

Given:

4^x − 4^x−1 = 48

Formula used:

a^m−n = a^m / aⁿ

Calculation:

4^x − (4^x / 4¹) = 48

⇒ 4^x (1 − 1/4) = 48

⇒ 4^x (3/4) = 48

⇒ 4^x = (48 × 4) / 3

⇒ 4^x = 16 × 4

⇒ 4^x = 64

⇒ 4^x = 4³

⇒ x = 3

Now, substitute x = 3 into the expression (2x − 1) / (2x + 3):

⇒ (2 × 3 − 1) / (2 × 3 + 3)

⇒ (6 − 1) / (6 + 3)

⇒ 5/9

∴ The correct answer is option (2).