Quantitative Aptitude PYQs

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.

Shortcut Trick

Let Cost Price (CP) = 100%.

Marked Price (MP) = 100% + 20% = 120%.

Selling Price (SP) = 120% − (15% of 120%) = 120% − 18% = 102%.

Profit = SP − CP = 102% − 100% = 2%.

Given, 2% = ₹300 ⇒ 100% = ₹15,000.

∴ The correct answer is ₹15,000.

Alternate Method

Given:

Markup = 20%

Discount = 15%

Profit = ₹300

Formula Used:

MP = CP × (100 + Markup%) ÷ 100

SP = MP × (100 − Discount%) ÷ 100

Profit = SP − CP

CP = 100xMP = 120xSP = 102x+20% Markup-15% DiscountProfit = 2x = ₹300

Calculations:

⇒ Let the Cost Price (CP) be 100x.

⇒ Marked Price (MP) = 100x × (100 + 20) ÷ 100 = 120x

⇒ Selling Price (SP) = 120x × (100 − 15) ÷ 100 = 120x × 85 ÷ 100 = 102x

⇒ Profit = 102x − 100x = 2x

⇒ 2x = 300

⇒ x = 150

⇒ CP = 100x = 100 × 150 = 15000

∴ The correct answer is ₹15,000.

Additional Information

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). SP = MP × (100 − D%) ÷ 100.

Successive Discounts

If two successive discounts of a% and b% are given, the single equivalent discount is (a + b − a×b ÷ 100)%.

Relation between CP and MP

CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%). This is a direct shortcut formula to find CP or MP without having to calculate SP.

Shortcut Trick Let Cost Price (CP) = 100 units ⇒ Marked Price (MP) = 100 + 25% = 125 units. Net discount = 10% + 20% − (10 × 20) ÷ 100 = 28%. Selling Price (SP) = 125 × (1 − 0.28) = 125 × 0.72 = 90 units. Loss = CP − SP = 100 − 90 = 10 units. 10 units = ₹400 ⇒ 1 unit = ₹40. MP = 125 units × 40 = ₹5000. ∴ The correct answer is ₹5000. Alternate Method Given: Markup = 25% above CP Consecutive discounts = 10% and 20% Loss = ₹400

CP = 100x

MP = 125x

SP = 90x

+25%

-10%, -20%

Loss = 100x − 90x = 10x 10x = ₹400 ⇒ x = 40

Calculations: Let the Cost Price (CP) of the article be 100x. ⇒ Marked Price (MP) = 100x + (25/100) × 100x = 125x Two consecutive discounts are given on the Marked Price: 10% and 20%. ⇒ Selling Price (SP) = MP × (100 − D1)/100 × (100 − D2)/100 ⇒ SP = 125x × (100 − 10)/100 × (100 − 20)/100 ⇒ SP = 125x × 0.90 × 0.80 ⇒ SP = 125x × 0.72 = 90x Loss is incurred when CP > SP. ⇒ Loss = CP − SP ⇒ Loss = 100x − 90x = 10x According to the given data, Loss = ₹400. ⇒ 10x = 400 ⇒ x = 40 Now, substitute the value of x to find the Marked Price: ⇒ MP = 125x = 125 × 40 = ₹5000 ∴ The correct answer is ₹5000. Additional Information Successive Discounts When two discounts d1% and d2% are given consecutively, the effective discount is: Net Discount = d1 + d2 − (d1 × d2) ÷ 100. Marked Price & Profit/Loss Marked price is the price printed on an article. Discount is always calculated on Marked Price (MP), while Profit or Loss is always calculated on Cost Price (CP). Single Equivalent Discount for 3 Discounts For three successive discounts a%, b% and c%, final SP = MP × (100 − a) ÷ 100 × (100 − b) ÷ 100 × (100 − c) ÷ 100.

Shortcut Trick

Let the remaining mixture (after removing 30 L) be 10x (since ratio is 3:5:2).

Quantities of P and Q in the remaining mixture are 3x and 5x respectively.

After adding 12 L of P and 8 L of Q, the difference is given as 20 L: (5x + 8) − (3x + 12) = 20.

⇒ 2x − 4 = 20 ⇒ x = 12.

Remaining mixture = 10 × 12 = 120 L. Initial total quantity = 120 + 30 = 150 L.

∴ The correct answer is 150 litres.

Alternate Method

Initial MixP:Q:R = 3:5:2− 30 LRemainingP=3x, Q=5x+12L P+8L QFinalP = 3x + 12Q = 5x + 8Given: Final Q − Final P = 20 L

Given:

Initial ratio of chemicals P, Q, and R = 3:5:2

Quantity of mixture removed = 30 litres

Quantity of P added = 12 litres

Quantity of Q added = 8 litres

Final amount of Q − Final amount of P = 20 litres

Calculations:

⇒ Let the remaining quantity of the mixture be 10x (since 3 + 5 + 2 = 10).

⇒ In the remaining mixture, quantity of P = 3x and quantity of Q = 5x.

⇒ After addition, resultant P = 3x + 12.

⇒ After addition, resultant Q = 5x + 8.

⇒ According to the given condition: Resultant Q − Resultant P = 20

⇒ (5x + 8) − (3x + 12) = 20

⇒ 2x − 4 = 20

⇒ 2x = 24 ⇒ x = 12

⇒ Remaining quantity of the mixture = 10x = 10 × 12 = 120 litres.

⇒ Initial total quantity = Remaining quantity + Removed quantity

⇒ Initial total quantity = 120 + 30 = 150 litres.

∴ The correct answer is 150 litres.

Additional Information

Mixture Replacement Rule

When a certain volume of a mixture is withdrawn, the ratio of the constituent elements in the remaining mixture remains exactly the same as in the original mixture.

Ratio & Proportion

If elements are present in the ratio a:b:c, their respective quantities can be assumed to be ax, bx, and cx, where the common multiplier 'x' represents a unit part of the total mixture.

Alligation Method

It is a general rule used to find the ratio in which two components of different concentrations are mixed to yield a desired concentration. The ratio of components is inversely proportional to the differences in their concentrations.

Shortcut Trick

Total effective interest rate = (3 × 5%) + (5 × 7%) + (2 × 9%)

⇒ Total rate = 15% + 35% + 18% = 68%

We are given that 68% of the total Principal = ₹6800

⇒ Principal (100%) = (6800 ÷ 68) × 100 = ₹10000

∴ The correct answer is ₹10000.

Alternate Method

Total Time = 10 Years

3 yrs @ 5% 15%

5 yrs @ 7% 35%

2 yrs @ 9% 18%

Total Interest = 15% + 35% + 18% = 68% 68% = ₹6800 → 100% = ₹10000

Given:

Rate for the first 3 years (R₁) = 5% per annum

Rate for the next 5 years (R₂) = 7% per annum

Total time given = 10 years, so remaining time (T₃) = 10 − (3 + 5) = 2 years

Rate for the remaining 2 years (R₃) = 9% per annum

Total Simple Interest (SI) paid at the end = ₹6800

Formula Used:

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

Calculations:

⇒ Total SI = SI₁ + SI₂ + SI₃

⇒ 6800 = (P × 5 × 3) ÷ 100 + (P × 7 × 5) ÷ 100 + (P × 9 × 2) ÷ 100

⇒ 6800 = (15P ÷ 100) + (35P ÷ 100) + (18P ÷ 100)

⇒ 6800 = (15P + 35P + 18P) ÷ 100

⇒ 6800 = 68P ÷ 100

⇒ P = (6800 × 100) ÷ 68

⇒ P = 10000

∴ The correct answer is ₹10000.

Additional Information

Effective Rate of Simple Interest

When money is borrowed at different rates for different periods, the total effective percentage interest is simply the sum of the products of each rate and its respective time period: R_effective = R₁T₁ + R₂T₂ + ...

Amount in Simple Interest

The total amount to be repaid at the end of the term is the sum of the Principal and the Simple Interest generated: A = P + SI = P(1 + RT ÷ 100).

Difference Between SI and CI

Unlike Compound Interest where interest is calculated on the accumulated interest at the end of each cycle, Simple Interest is always calculated solely on the initial Principal amount for the entire duration.

Shortcut Trick Base area of a cone = πr2 Lateral surface area = πrl Equating both as per the given condition: πr2 = πrl Canceling π and r from both sides directly gives: l = r ∴ The correct answer is r.

Alternate Method

r l h

Given: Radius of the cone = r Slant height of the cone = l Area of the base = Lateral surface area Formula Used: Area of base of a cone = πr2 Lateral surface area of a cone = πrl Calculations: ⇒ According to the question, Area of base = Lateral surface area ⇒ πr2 = πrl ⇒ Dividing both sides by πr (since r > 0) ⇒ r = l ∴ The correct answer is r.

Additional Information Total Surface Area of a Cone The total surface area includes the area of the circular base and the curved lateral surface: TSA = πr(l + r). Volume of a Cone The volume represents the 3D space enclosed by the cone: V = (1÷3)πr2h, where h is the perpendicular height. Slant Height Relationship The slant height (l), base radius (r), and perpendicular height (h) form a right-angled triangle. Applying Pythagoras theorem: l = √(r2 + h2).

Shortcut Trick

Area of remaining field = Area of Square − Area of Pond

Area of remaining field = (Side)² − 616

Area of remaining field = 28² − 616

Area of remaining field = 784 − 616 = 168 m²

∴ The correct answer is 168 m².

Alternate Method

Pond Area = 616 m2

28 m

28 m Field Field

Given:

Side of the square field (a) = 28 m

Area of the circular pond = 616 m²

Formula Used:

Area of a square = a²

Area of remaining part = Area of square − Area of internal shape

Calculations:

⇒ Area of the square field = 28 × 28

⇒ Area of the square field = 784 m²

⇒ Area of the remaining field = Area of square field − Area of pond

⇒ Area of the remaining field = 784 − 616 = 168 m²

∴ The correct answer is 168 m².

Additional Information

Square Properties

Area = a² | Perimeter = 4a | Diagonal = a√2

Circle Properties

Area = πr² | Circumference = 2πr (where r is the radius)

Largest Circle in a Square

When a circle perfectly fits inside a square of side a, its radius is a/2. It occupies exactly π/4 (approx. 78.5%) of the square's total area.

Shortcut Trick

Volume of any right prism = Base Area × Height.

Since the base area remains constant, Volume ∝ Height.

Therefore, the percentage change in volume will be exactly equal to the percentage change in height.

Given percentage increase in height = 50%.

⇒ Percentage increase in volume = 50%.

∴ The correct answer is 50%.

Alternate Method

Given:

Increase in height of the right prism = 50%

Base area remains constant.

Formula Used:

Volume of a Right Prism = Base Area × Height

h V = Ah Area = A

1.5h V' = 1.5Ah Area = A

+50% Height

Calculations:

⇒ Let the initial height of the prism = h

⇒ Let the constant base area of the prism = A

⇒ Initial Volume (V₁) = A × h = Ah

⇒ New height of the prism = h + (50 ÷ 100) × h = 1.5h

⇒ New Volume (V₂) = A × 1.5h = 1.5Ah

⇒ Increase in Volume = V₂ − V₁ = 1.5Ah − Ah = 0.5Ah

⇒ Percentage Increase in Volume = (Increase in Volume ÷ Initial Volume) × 100

⇒ Percentage Increase = (0.5Ah ÷ Ah) × 100 = 50%

∴ The correct answer is 50%.

Additional Information

Volume of a Prism

Volume = Base Area × Height. If the base area is kept constant, the volume is directly proportional to the height.

Volume of a Pyramid or Cone

Volume = 1/3 × Base Area × Height. It is exactly one-third the volume of a right prism having the same base area and height.

Successive Percentage Change

If the height changes by a% and the base area changes by b%, the overall percentage change in volume can be evaluated using the successive formula: a + b + (a × b)/100.

Shortcut Trick

The percentage area of a minor segment formed by an equilateral triangle at the center is independent of the circle's radius.

Required % = [(Area of Sector − Area of Triangle) ÷ Total Area] × 100

Required % = (1 ÷ 6 − √3 ÷ 4π) × 100 = (0.1667 − 1.732 ÷ 12.56) × 100

Required % = (0.1667 − 0.1379) × 100 = 0.0288 × 100 = 2.88%

∴ The correct answer is 2.88%.

Alternate Method

Given:

Radius of the clock (r) = 20 cm

Central angle (θ) = 60° (since the chord and two radii form an equilateral triangle)

π = 3.14, √3 = 1.732

Formula Used:

Total Area of Clock = πr²

Area of Sector = (θ ÷ 360) × πr²

Area of Equilateral Triangle = (√3 ÷ 4) × r²

O

60° r = 20 r = 20 Chord A B

Painted Minor Segment

Calculations:

⇒ Total Area = 3.14 × 20² = 3.14 × 400 = 1256 cm²

⇒ Area of Sector = (60 ÷ 360) × 1256 = 1256 ÷ 6 ≈ 209.33 cm²

⇒ Area of Triangle = (1.732 ÷ 4) × 400 = 1.732 × 100 = 173.20 cm²

⇒ Area of Painted Segment = Area of Sector − Area of Triangle

⇒ Area of Painted Segment = 209.33 − 173.20 = 36.13 cm²

⇒ Required Percentage = (36.13 ÷ 1256) × 100 = 2.876% ≈ 2.88%

∴ The correct answer is 2.88%.

Additional Information

Area of a Sector

The area of a sector of a circle with radius r and central angle θ in degrees is given by the formula (θ ÷ 360) × πr².

Area of a Circular Segment

A segment is the region bounded by a chord and an arc. The area of a minor segment is calculated as the Area of the corresponding Sector − Area of the Triangle formed by the two radii and the chord.

Equilateral Triangle Area

An equilateral triangle has all sides equal and internal angles of exactly 60°. Its total area can be determined using the direct formula (√3 ÷ 4) × side².

Shortcut Trick

Since the hole is bored through the entire height, the ratio of the volume removed depends only on the square of the radii.

Volume Removed % = (r_hole ÷ R_original)² × 100

⇒ Volume Removed % = (2 ÷ 4)² × 100

⇒ (1/2)² × 100 = 25%

∴ The correct answer is 25%.

Alternate Method

Given:

Radius of original cylinder (R) = 4 cm

Height of original cylinder (h) = 10 cm

Radius of bored hole (r) = 2 cm

Height of bored hole = 10 cm (full height)

R = 4 cm r = 2 cm

h = 10 cm

Volume Removed = Inner Hole Volume

Formula Used:

Volume of cylinder = πr²h

Percentage removed = (Volume removed ÷ Original volume) × 100

Calculations:

⇒ Original volume = π × (4)² × 10

⇒ Original volume = 160π cm³

⇒ Volume removed (Hole) = π × (2)² × 10

⇒ Volume removed = 40π cm³

⇒ Percentage removed = (40π ÷ 160π) × 100

⇒ Percentage removed = (1 ÷ 4) × 100 = 25%

∴ The correct answer is 25%.

Additional Information

Volume of a Hollow Cylinder

The volume of the material remaining after boring a hole is determined by calculating the difference in volume: V = πh(R² − r²).

Curved Surface Area of a Hollow Cylinder

The total curved surface area becomes the sum of the inner hole's area and the outer cylinder's area: CSA = 2πRh + 2πrh = 2πh(R + r).

Total Surface Area of a Hollow Cylinder

The total surface area includes the inner and outer curved areas plus the area of the top and bottom solid rings: TSA = 2πh(R + r) + 2π(R² − r²).

Shortcut Trick

For sectors of the same circle, the area is directly proportional to the central angle.

Angle of the first sector = 45°

Angle of the second sector = π/4 radians = (180° ÷ 4) = 45°

Since both central angles are equal, their areas will be exactly the same.

∴ The correct answer is 1:1.

Alternate Method

Given:

Radius of the circle (r) = 15 cm

Central angle of first sector (θ₁) = 45°

Central angle of second sector (θ₂) = π/4 radians

45° r = 15 cm First Sector

π/4 r = 15 cm Second Sector

Ratio = ?

Formula Used:

Area of a sector (when θ is in degrees) = (θ ÷ 360°) × πr²

Area of a sector (when θ is in radians) = (1 ÷ 2) × r²θ

Calculations:

⇒ Area of first sector = (45° ÷ 360°) × π × (15)²

⇒ Area of first sector = (1 ÷ 8) × 225π = 28.125π cm²

⇒ Area of second sector = (1 ÷ 2) × (15)² × (π ÷ 4)

⇒ Area of second sector = (1 ÷ 2) × 225 × (π ÷ 4) = 28.125π cm²

⇒ Ratio of areas = Area of first sector : Area of second sector

⇒ Ratio = 28.125π : 28.125π = 1 : 1

∴ The correct answer is 1:1.

Additional Information

Area of a Sector

The area of a sector of a circle with radius r and central angle θ is given by (θ ÷ 360°) × πr² if θ is measured in degrees, and (1 ÷ 2)r²θ if θ is measured in radians.

Radian to Degree Conversion

To convert an angle from radians into degrees, multiply the radian value by (180° ÷ π). For example, π radians equals 180°.

Length of an Arc

The arc length of a corresponding sector is calculated as (θ ÷ 360°) × 2πr (for degrees) or simply rθ (for radians).