Search Previous Year Questions

Shortcut Trick

Convert all animals to equivalent cows based on eating capacity: 1 horse = 2 cows, 1 sheep = 0.5 cows.

A's grazing = 18 horses × 4 months = (18 × 2) × 4 = 144 cow-months.

B's grazing = (24 cows × 6 months) + (36 sheep × 4 months) = 144 + (36 × 0.5) × 4 = 216 cow-months.

A's rent share = A's grazing ÷ Total grazing = 144 ÷ (144 + 216) = 144 ÷ 360 = 2/5.

∴ The correct answer is 2/5.

Alternate Method

Given:

A grazes 18 horses for 4 months.

B grazes 24 cows for 6 months and 36 sheep for 4 months.

Equivalence: 3 horses = 6 cows ⇒ 1 horse = 2 cows.

Equivalence: 4 cows = 8 sheep ⇒ 1 sheep = 0.5 cows.

Formula Used:

Rent Share = Individual Grazing Units ÷ Total Grazing Units

Grazing Units = Equivalent Number of Animals × Grazing Time

A: 144 units B: 216 units Total Grazing Equivalent = 360 Units

A's Rent Share = 144 ÷ 360 = 2/5

Calculations:

⇒ Convert A's animals to equivalent cows: 18 horses = 36 cows.

⇒ A's grazing units = 36 cows × 4 months = 144 cow-months.

⇒ Convert B's animals to equivalent cows: 36 sheep = 18 cows.

⇒ B's grazing units = (24 cows × 6 months) + (18 cows × 4 months).

⇒ B's grazing units = 144 + 72 = 216 cow-months.

⇒ Total grazing units = A's units + B's units = 144 + 216 = 360 cow-months.

⇒ A's fraction of the rent = 144 ÷ 360 = 2/5.

∴ The correct answer is 2/5.

Additional Information

Partnership with Proportional Usage

When resources are shared, rent or profit is divided based on the product of the quantity of usage and time: Profit Ratio = (Quantity₁ × Time₁) : (Quantity₂ × Time₂).

Base Conversion in Ratios

To accurately compare different types of assets (like varying animals), they must be converted into a common base unit using an equivalence ratio before summing up the values.

Compound Proportions

When a share depends on multiple factors (such as number of animals and duration of months), the compound ratio is the product of the simple ratios of those respective factors.

Shortcut Trick

Let the initial investments of X, Y, and Z be 2, 3, and 5 units respectively.

Profit ratio is directly proportional to the product of Investment × Time.

X's share = 2 × 18 = 36

Y's share = (3 × 6) + (1.5 × 12) = 18 + 18 = 36

Z's share = (5 × 3) + (3 × 15) = 15 + 45 = 60

Ratio of profit shares = 36 : 36 : 60 ⇒ 3 : 3 : 5.

∴ The correct answer is 3:3:5.

Alternate Method

Given:

Ratio of initial capitals of X, Y, and Z = 2 : 3 : 5

Total duration of the business = 18 months

After 3 months, Z withdrew 40% of his capital.

After another 3 months (i.e., after 6 months total), Y withdrew half of his capital.

Total profit = ₹3,24,000 (not required to find the ratio)

Formula Used:

Profit Ratio = (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃)

2 units × 18 months

3 × 6m 1.5 × 12m

5×3m 3 × 15m

X Y Z

= 36 = 36 = 60

Calculations:

⇒ Let the initial capitals of X, Y, and Z be 2x, 3x, and 5x respectively.

⇒ X invested 2x for the entire duration of 18 months.

⇒ X's equivalent capital = 2x × 18 = 36x

⇒ Y invested 3x for 6 months. He then withdrew half (1.5x) for the remaining 12 months.

⇒ Y's equivalent capital = (3x × 6) + (1.5x × 12) = 18x + 18x = 36x

⇒ Z invested 5x for 3 months. He then withdrew 40% (2x) leaving 3x for the remaining 15 months.

⇒ Z's equivalent capital = (5x × 3) + (3x × 15) = 15x + 45x = 60x

⇒ Ratio of profit shares of X : Y : Z = 36x : 36x : 60x

⇒ Dividing by the common factor of 12x, we get the ratio as 3 : 3 : 5.

∴ The correct answer is 3:3:5.

Additional Information

Partnership Profit Ratio

The profit is distributed among partners in the ratio of the product of their capital and the time period for which it is invested: Profit Ratio = (C₁ × T₁) : (C₂ × T₂).

Variable Investments

When a partner adds or withdraws capital mid-way, their total effective capital is calculated as the sum of the varying capital amounts multiplied by their respective time durations.

Sleeping vs Active Partner

An active partner may receive a fixed salary or a certain percentage of the total profit for managing the business before the remaining profit is distributed as per the capital-time ratio.

Shortcut Trick

Age of new student = Old Average + (New Total Members × Increase in Average)

Old Average = 16 years

New Total Members = 40 + 1 = 41 students

Increase in Average = 0.2 years

Age of new student = 16 + (41 × 0.2) = 16 + 8.2 = 24.2 years

∴ The correct answer is 24.2 years.

Alternate Method

Given:

⇒ Average age of 40 students = 16 years

⇒ Increase in average age = 0.2 years

Formula Used:

⇒ Total Age = Number of Students × Average Age

⇒ Age of New Student = Total age of new group − Total age of old group

40 Students Average = 16

1 New Student Age = x

41 Students (Total) New Average = 16.2 (+0.2)

Calculations:

⇒ Total age of 40 students = 40 × 16 = 640 years

⇒ Total students after admission = 40 + 1 = 41

⇒ New average age = 16 + 0.2 = 16.2 years

⇒ Total age of 41 students = 41 × 16.2 = 664.2 years

⇒ Age of the new student = 664.2 − 640 = 24.2 years

∴ The correct answer is 24.2 years.

Additional Information

1. Weighted Average

When combining two different groups, the overall combined average is calculated as A = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

2. Replacement in Average

If an existing person is replaced by a new person and the average changes, the value of the new person = Value of removed person + (Total members × Change in average).

3. Person Leaving a Group

When a person leaves the group, their age/value can be found directly by: Value = Old Average − (Remaining Members × Change in average).

Shortcut Trick

Supervisor's salary is ₹10,000 more than the group average (₹25,000 − ₹15,000).

When excluded, this ₹10,000 deficit is shared equally among the remaining 8 employees.

Decrease in their average = 10,000 ÷ 8 = ₹1,250

New average of 8 employees = 15,000 − 1,250 = ₹13,750

∴ The correct answer is ₹13,750.

Alternate Method

Given:

Total number of people = 8 employees + 1 supervisor = 9

Monthly average salary of 9 people = ₹15,000

Supervisor's salary = ₹25,000

Formula Used:

Total Value = Average × Number of Items

8 Employees 1 Sup.

Total = ₹1,10,000 ₹25,000

Total Salary of 9 = 9 × ₹15,000 = ₹1,35,000

Calculations:

⇒ Total salary of all 9 people = 9 × 15,000 = ₹1,35,000

⇒ Total salary of 8 employees = Total salary of 9 people − Supervisor's salary

⇒ Total salary of 8 employees = 1,35,000 − 25,000 = ₹1,10,000

⇒ Average salary of 8 employees = 1,10,000 ÷ 8

⇒ Average salary of 8 employees = ₹13,750

∴ The correct answer is ₹13,750.

Additional Information

Exclusion in Averages

When a value is removed from a group, the new average is calculated directly as: New Average = Old Average − [ (Value of Excluded Item − Old Average) ÷ Remaining Number of Items ].

Weighted Average

Used when combining two groups of varying sizes and averages: Combined Average = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂).

Net Deviation Method

The sum of all deviations from the true average is always zero. Adding an item greater than the average increases the overall average, while removing it decreases the average.

Shortcut Trick

Ratio of women to men = 35% : 65% = 7 : 13

Let the base average be the men's average = 68 (in thousands).

Deviation of women's average from base = 72 − 68 = +4

Net change in overall average = (+4 × 7) ÷ (7 + 13) = +28 ÷ 20 = +1.4

Overall average = 68 + 1.4 = 69.4

∴ The correct answer is 69.4 (in ₹1000).

Alternate Method

Given:

Total employees = 80

Percentage of women = 35%

Average monthly salary of women = 72 (in ₹1000)

Average monthly salary of men = 68 (in ₹1000)

Formula Used:

Overall Average = (Total sum of salaries) ÷ (Total number of employees)

Men (68)Women (72)Avg (69.4)7 parts13 partsWeight: 13Weight: 7

Calculations:

⇒ Number of women = 35% of 80 = 28

⇒ Number of men = 80 − 28 = 52

⇒ Total salary of women = 28 × 72 = 2016 (in ₹1000)

⇒ Total salary of men = 52 × 68 = 3536 (in ₹1000)

⇒ Total salary of all employees = 2016 + 3536 = 5552 (in ₹1000)

⇒ Overall average salary = 5552 ÷ 80 = 69.4

∴ The correct answer is 69.4 (in ₹1000).

Additional Information

Weighted Average

When two groups of sizes n₁ and n₂ have individual averages x₁ and x₂, the combined group average is computed using the formula: (n₁x₁ + n₂x₂) ÷ (n₁ + n₂).

Deviation Method for Averages

To simplify calculations without multiplying large numbers, assume a base average. Determine the total surplus or deficit relative to this base, divide it evenly among all items, and adjust the base average accordingly.

Rule of Alligation

It establishes the ratio in which two components of different values must be combined to yield a specific average value. The ratio is derived as (Dearer Value − Mean Value) : (Mean Value − Cheaper Value).

Shortcut Trick

Difference in discount percentages = 15% − 10% = 5%

Extra discount amount is simply 5% of the Marked Price.

Extra discount = (5 ÷ 100) × 3500 = 175

∴ The correct answer is ₹175.

Alternate Method

Given:

Marked Price (MP) = ₹3,500

Initial Discount = 10%

New Discount = 15%

MP = ₹3500

10% = ₹350

15% = ₹525

Extra Discount = 5% = ₹175

Formula Used:

Discount Value = (Discount % ÷ 100) × MP

Extra Discount = New Discount Value − Initial Discount Value

Calculations:

⇒ Initial Discount Value = (10 ÷ 100) × 3500

⇒ Initial Discount Value = ₹350

⇒ New Discount Value = (15 ÷ 100) × 3500

⇒ New Discount Value = ₹525

⇒ Extra Discount = 525 − 350

⇒ Extra Discount = ₹175

∴ The correct answer is ₹175.

Additional Information

Marked Price vs Selling Price

Selling Price (SP) is obtained after applying the discount on the Marked Price. Formula: SP = MP − Discount.

Successive Discounts

When two discounts d₁% and d₂% are given consecutively, the single effective discount percentage is given by: d₁ + d₂ − (d₁ × d₂) ÷ 100.

Markup Percentage

The percentage by which the Marked Price is increased over the Cost Price. Markup % = ((MP − CP) ÷ CP) × 100.

Shortcut Trick

Let ordinary coffee price = 1 unit, premium = 3 units, and original ordinary quantity = x kg.

Original bill = (5 × 3) + (x × 1) = 15 + x

Swapped bill = (x × 3) + (5 × 1) = 3x + 5

Bill increased by 40%, meaning New Bill ÷ Old Bill = 140% = 7 / 5

⇒ (3x + 5) / (15 + x) = 7 / 5 ⇒ 15x + 25 = 105 + 7x ⇒ 8x = 80 ⇒ x = 10 kg

∴ The correct answer is 1 : 2.

Alternate Method

Original OrderPremium: 5 kg × 3POrdinary: x kg × 1PTotal: P(15 + x)+40% BillSwapped OrderPremium: x kg × 3POrdinary: 5 kg × 1PTotal: P(3x + 5)New Bill = 1.4 × Original Bill

Given:

Original quantity of premium coffee = 5 kg

Price of premium coffee = 3 × Price of ordinary coffee

Bill increased by 40% after swapping quantities.

Formula Used:

Total Value = (Quantity₁ × Price₁) + (Quantity₂ × Price₂)

Calculation:

⇒ Let the original quantity of ordinary coffee = x kg

⇒ Let the price of ordinary coffee = ₹P per kg

⇒ Then, the price of premium coffee = ₹3P per kg

⇒ Original Bill = (5 × 3P) + (x × P) = P(15 + x)

⇒ After swapping, quantity of premium coffee = x kg

⇒ Quantity of ordinary coffee = 5 kg

⇒ New Bill = (x × 3P) + (5 × P) = P(3x + 5)

⇒ The new bill is 40% more than the original bill.

⇒ New Bill = 1.4 × Original Bill

⇒ P(3x + 5) = 1.4 × P(15 + x)

⇒ 3x + 5 = 21 + 1.4x

⇒ 3x − 1.4x = 21 − 5

⇒ 1.6x = 16

⇒ x = 10 kg

⇒ Ratio of premium to ordinary coffee = 5 : 10 = 1 : 2

∴ The correct answer is 1 : 2.

Additional Information

Percentage Increase

If a value increases by R%, the new value becomes (1 + R/100) × Original Value. A 40% increase means the multiplier is 1.4 or 140/100 = 7/5.

Ratio & Proportion

A ratio indicates how many times one number contains another. The ratio a : b can also be expressed as the fraction a/b. Simplifying ratios involves dividing both terms by their highest common factor (HCF).

Mixtures & Total Value

When combining multiple items, the total value is the sum of the product of quantity and unit price for each item. Total Value = (Quantity₁ × Price₁) + (Quantity₂ × Price₂).

Shortcut Trick

For Compound Interest, the amount at the end of a year acts as the principal for the very next year.

Interest for the 3^rd year = A₃ − A₂ = 7040 − 6400 = ₹640

Rate % = (Interest ÷ Principal) × 100

Rate % = (640 ÷ 6400) × 100 = 10%

∴ The correct answer is 10%.

Alternate Method

Given:

Amount after 2 years (A₂) = ₹6400

Amount after 3 years (A₃) = ₹7040

(Note: Based on the given options, the interest is compounded annually)

Formula Used:

A = P(1 + R/100)ⁿ

 Year 0 Year 2 Year 3 Principal ₹6400 ₹7040     + ₹640 (1 Year Interest) Rate = (640 ÷ 6400) × 100 = 10%

Calculations:

⇒ For 2 years: 6400 = P(1 + R/100)² --- (Equation 1)

⇒ For 3 years: 7040 = P(1 + R/100)³ --- (Equation 2)

⇒ Dividing Equation 2 by Equation 1:

⇒ 7040 ÷ 6400 = [P(1 + R/100)³] ÷ [P(1 + R/100)²]

⇒ 704 ÷ 640 = 1 + R/100

⇒ 1.1 = 1 + R/100

⇒ R/100 = 0.1

⇒ R = 10%

∴ The correct answer is 10%.

Additional Information

Compound Interest Successive Years

The amount at the end of any year acts as the principal for the very next year. Rate = [(A_n − A_n-1) ÷ A_n-1] × 100.

Simple Interest vs Compound Interest

For the first year, SI and CI are identical. From the second year onwards, CI earns interest on the accumulated interest, whereas SI is always calculated on the initial principal only.

Difference Between CI and SI

For 2 years, Difference = P(R/100)². For 3 years, Difference = P(R/100)² × (3 + R/100).

Shortcut Trick

The actual Selling Price (₹300) is not required to find the Loss %.

Let the Cost Price (CP) of 1 pen = 1 unit.

Total CP of 10 pens = 10 units.

Loss = CP of 4 pens = 4 units.

Loss % = (Loss ÷ Total CP) × 100 = (4 ÷ 10) × 100 = 40%.

∴ The correct answer is 40%.

Alternate Method

Given:

Selling Price (SP) of 10 pens = ₹300

Loss = Cost Price (CP) of 4 pens

Formula Used:

Loss = Total CP − Total SP

Loss % = (Loss ÷ Total CP) × 100

Total Cost Price (10 Pens) = ₹500

SP (10 Pens) = ₹300

Loss = ₹200 (Selling Amount) (CP of 4 pens) Loss % = (200 ÷ 500) × 100 = 40%

⇒ Let the Cost Price (CP) of 1 pen be ₹x.

⇒ Total CP of 10 pens = ₹10x

⇒ Loss = ₹4x

⇒ Since Loss = Total CP − Total SP

⇒ 4x = 10x − 300

⇒ 10x − 4x = 300

⇒ 6x = 300

⇒ x = 300 ÷ 6 = 50

⇒ Total CP of 10 pens = 10 × 50 = ₹500

⇒ Loss = 4 × 50 = ₹200

⇒ Loss % = (Loss ÷ Total CP) × 100

⇒ Loss % = (200 ÷ 500) × 100 = 40%

∴ The correct answer is 40%.

Additional Information

Profit and Loss Basics

Profit = SP − CP, and Loss = CP − SP. Both Profit % and Loss % are always calculated on the Cost Price (CP) unless specifically mentioned otherwise.

Direct Short-Trick Concept

If the loss on selling 'a' items equals the Cost Price of 'b' items, the Loss % = (b ÷ a) × 100 directly without needing the actual SP amount.

Marked Price & Discount

Discount is always offered on the Marked Price (MP). SP = MP − Discount. The equivalent discount of two successive discounts x% and y% is (x + y − x × y ÷ 100)%.

Shortcut Trick

Assume quantities identical to the ratio: 2, 3, and 1 (Total = 6 items).

Total CP = 2 × (10 ÷ 4) + 3 × (12 ÷ 5) + 1 × (5 ÷ 2) = 5 + 7.2 + 2.5 = ₹14.7

Total SP of 6 items = 6 × (7 ÷ 3) = ₹14.0

Loss = Total CP − Total SP = 14.7 − 14.0 = ₹0.7

Loss % = (0.7 ÷ 14.7) × 100 = (1 ÷ 21) × 100 ≈ 4.76%

∴ The correct answer is Loss of 4.76%.

Alternate Method

Given:

Buying price of 1^st variety = 4 for ₹10

Buying price of 2^nd variety = 5 for ₹12

Buying price of 3^rd variety = 2 for ₹5

Ratio of the three varieties mixed = 2 : 3 : 1

Selling price of the mixture = 3 for ₹7

Formula Used:

Loss = Total CP − Total SP

Loss % = (Loss ÷ Total CP) × 100

Comparison of Cost and Selling Price

Total Cost Price (CP) = ₹14.7

Total Selling Price (SP) = ₹14.0

Loss = ₹0.7

₹14.7

Calculations:

⇒ Let the number of mangoes bought be 2, 3, and 1 respectively, making the total number of mangoes 6.

⇒ CP of 1^st variety = 10 ÷ 4 = ₹2.5 per mango

⇒ CP of 2^nd variety = 12 ÷ 5 = ₹2.4 per mango

⇒ CP of 3^rd variety = 5 ÷ 2 = ₹2.5 per mango

⇒ Total CP = (2 × 2.5) + (3 × 2.4) + (1 × 2.5)

⇒ Total CP = 5 + 7.2 + 2.5 = ₹14.7

⇒ SP of 1 mixture mango = ₹(7 ÷ 3)

⇒ Total SP for all 6 mangoes = 6 × (7 ÷ 3) = ₹14.0

⇒ Since the Cost Price is greater than the Selling Price, there is an overall loss.

⇒ Loss = 14.7 − 14.0 = ₹0.7

⇒ Loss % = (0.7 ÷ 14.7) × 100

⇒ Loss % = (1 ÷ 21) × 100 ≈ 4.76%

∴ The correct answer is Loss of 4.76%.

Additional Information

Profit and Loss Fundamentals

A profit is realized when the Selling Price (SP) is greater than the Cost Price (CP). A loss occurs when the Cost Price is greater than the Selling Price.

Percentage Calculation Rule

Unless explicitly mentioned otherwise, both profit percentage and loss percentage are strictly computed relative to the Cost Price base: Percentage = (Value ÷ CP) × 100.

Weighted Mixtures Concept

When combining different quantities at varying rates, treating the given mixing ratio units as the actual physical quantity simplifies the arithmetic needed to evaluate the overall expenditure and returns.