Quantitative Aptitude PYQs

Shortcut Trick

Ratio of Profit = (Investment × Time)

Riya : Kabir : Sana = (13500 × 8) : (21000 × 6) : (18000 × 12)

Riya : Kabir : Sana = 108 : 126 : 216 = 6 : 7 : 12

Total units = 6 + 7 + 12 = 25 units → ₹12,600

Kabir's share (7 units) = (12600 ÷ 25) × 7 = 504 × 7 = ₹3,528

∴ The correct answer is ₹3,528.

Alternate Method

Given:

Riya's investment (C_R) = ₹13,500, Time (T_R) = 8 months

Kabir's investment (C_K) = ₹21,000, Time (T_K) = 6 months

Sana's investment (C_S) = ₹18,000, Time (T_S) = 12 months

Total Profit = ₹12,600

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

Calculations:

⇒ Riya's Profit Share : Kabir's Profit Share : Sana's Profit Share

⇒ (13500 × 8) : (21000 × 6) : (18000 × 12)

⇒ 108000 : 126000 : 216000

⇒ 108 : 126 : 216

⇒ 6 : 7 : 12 (Dividing by 18)

⇒ Total Ratio = 6 + 7 + 12 = 25 units

⇒ 25 units = ₹12,600

⇒ 1 unit = 12600 ÷ 25 = ₹504

⇒ Kabir's share = 7 units = 7 × 504 = ₹3,528

∴ The correct answer is ₹3,528.

Additional Information

Partnership Profit Sharing Rule

Profit is distributed in the ratio of the product of Capital and Time. P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Active vs Sleeping Partner

An Active partner manages the business and may receive a salary or extra profit %, while a Sleeping partner only provides capital.

Variable Time Investment

If a partner changes investment mid-way, calculate total investment as: (C₁ × T₁) + (C₂ × T₂) for that partner.

Shortcut Trick

For a dishonest vendor selling at Cost Price (CP):

Profit % = (Error in weight ÷ Actual weight given) × 100

Error = 60 g − 50 g = 10 g

Profit % = (10 ÷ 50) × 100 = 20%

∴ The correct answer is 20%.

Alternate Method

Given:

Stated Weight (what the customer pays for) = 60 g

Actual Weight (what the vendor gives) = 50 g

Formula Used:

Profit % = [(Stated Weight − Actual Weight) ÷ Actual Weight] × 100

Let the Cost Price of 1 g of salt be ₹1.

⇒ Total Cost Price (CP) of salt actually given (50 g) = ₹50

⇒ Total Selling Price (SP) (Price of 60 g at CP) = ₹60

⇒ Profit earned = SP − CP = ₹60 − ₹50 = ₹10

⇒ Profit Percentage = (Profit ÷ CP) × 100

⇒ Profit % = (10 ÷ 50) × 100

⇒ Profit % = 20%

∴ The correct answer is 20%.

Additional Information

Dishonest Dealer Weight Gain

Profit % = [Gain ÷ (True Weight − Gain)] × 100. This applies when the dealer sells at CP but uses a faulty less weight.

Cheating While Buying and Selling

If a dealer cheats x% in weight while buying and y% while selling, Net Profit % = [((100 + x)(100 + y) ÷ 100) − 100].

Markup and Weight Cheating

If a dealer marks up price by P% and uses a weight of W₁ instead of W₂, then Net Profit % = [((100 + P) × W₂) ÷ (100 × W₁) − 1] × 100.

Shortcut Trick

For a number to be divisible by 9, the sum of its digits must be divisible by 9.

Sum of existing digits in 1_36 = 1 + 3 + 6 = 10.

The next multiple of 9 greater than 10 is 18.

Required digit = 18 − 10 = 8.

∴ The correct answer is 8.

Alternate Method

Given: Number = 1x36 (where x is the missing digit), Divisor = 9.

Formula Used: Divisibility Rule of 9 → Sum of digits ÷ 9 = Integer.

Calculations:

⇒ Sum of digits = 1 + x + 3 + 6

⇒ Sum = 10 + x

⇒ For the number to be divisible by 9, (10 + x) must be a multiple of 9 (9, 18, 27...).

⇒ Since x is a single digit (0−9), 10 + x = 18 is the only valid case.

⇒ x = 18 − 10

⇒ x = 8

∴ The correct answer is 8.

Additional Information

Divisibility by 3

Similar to the rule for 9, a number is divisible by 3 if the sum of its digits is a multiple of 3.

Divisibility by 11

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

Divisibility by 8

A number is divisible by 8 if the last three digits of the number are divisible by 8.

Shortcut Trick

Distance covered in first 3 hr = 80 × 3 = 240 km.

Remaining distance = 540 − 240 = 300 km.

Remaining time = 300 ÷ 60 = 5 hr.

Total time taken = 3 + 5 = 8 hr.

∴ The correct answer is 8 hr.

Alternate Method

Given:

Total distance = 540 km

Initial speed (s₁) = 80 km/h | Initial time (t₁) = 3 hr

Reduced speed (s₂) = 60 km/h

Formula Used:

Distance = Speed × Time | Time = Distance ÷ Speed

Calculations:

⇒ Distance covered in first part = 80 km/h × 3 hr = 240 km

⇒ Remaining distance = Total distance − Distance covered in Part 1

⇒ Remaining distance = 540 − 240 = 300 km

⇒ Time taken for remaining distance = 300 km ÷ 60 km/h = 5 hr

⇒ Total time = Initial time + Remaining time

⇒ Total time = 3 hr + 5 hr = 8 hr

∴ The correct answer is 8 hr.

Additional Information

Average Speed

Average Speed = Total Distance ÷ Total Time. In this case, Average Speed = 540 ÷ 8 = 67.5 km/h.

Unit Conversion

To convert Speed from km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

Relation between S, T, and D

Speed is inversely proportional to Time (S ∝ 1/T) when Distance is constant, and Speed is directly proportional to Distance (S ∝ D) when Time is constant.

Shortcut Trick

Let the Cost Price (CP) be 100.

Marked Price (MP) = 100 + 165 = 265.

Required Selling Price (SP) for 112% gain = 100 + 112 = 212.

Discount amount = 265 − 212 = 53.

Discount % = (53 ÷ 265) × 100 = 1/5 × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given:

Markup percentage = 165% above CP

Profit percentage = 112%

Formula Used:

1. Marked Price (MP) = CP × (100 + Markup %)/100
2. Selling Price (SP) = CP × (100 + Gain %)/100
3. Discount % = [(MP − SP) ÷ MP] × 100

Calculations:

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

⇒ Since the Marked Price (MP) is 165% above CP:

⇒ MP = 100 + 165% of 100 = 100 + 165 = 265 units.

⇒ To gain 112%, the Selling Price (SP) must be:

⇒ SP = 100 + 112% of 100 = 100 + 112 = 212 units.

⇒ Discount = MP − SP = 265 − 212 = 53 units.

⇒ Discount % = (Discount ÷ MP) × 100

⇒ Discount % = (53 ÷ 265) × 100

⇒ Discount % = 1/5 × 100 = 20%.

∴ The correct answer is 20%.

Additional Information

Markup Percentage

It is the percentage by which the cost price is increased to set the marked price: Markup % = [(MP − CP) ÷ CP] × 100.

Effective Formula

The relation between Profit, Markup, and Discount is given by: (100 + P%) = (100 + M%) × (100 − D%) ÷ 100.

Discount Calculation

Discount is always calculated on the Marked Price (MP), whereas Profit or Loss is generally calculated on the Cost Price (CP).

Shortcut Trick

Identify the ratio of investment for P, Q, and R as 2 : 3 : 5.

Calculate total ratio parts: 2 + 3 + 5 = 10 units.

Find the difference in parts between R and P: 5 − 2 = 3 units.

Apply unitary method: 10 units → ₹10,000, so 1 unit → ₹1,000.

Required difference = 3 units × ₹1,000 = ₹3,000.

∴ The correct answer is ₹3,000.

Alternate Method

Given: Investment ratio of P : Q : R = 2 : 3 : 5. Total Profit = ₹10,000.

Formula Used: Profit share of a partner = (Partner's ratio / Total ratio) × Total Profit

⇒ Total ratio units = 2 + 3 + 5 = 10 units

⇒ Value of 10 units = ₹10,000

⇒ Value of 1 unit = 10,000 ÷ 10 = ₹1,000

⇒ Profit share of P = 2 × 1,000 = ₹2,000

⇒ Profit share of R = 5 × 1,000 = ₹5,000

⇒ Difference between R and P = ₹5,000 − ₹2,000 = ₹3,000

∴ The correct answer is ₹3,000.

Additional Information

Basic Partnership Rule

When the time period of investment is the same for all partners, the profit is distributed directly in the ratio of their capitals.

Compound Partnership

When partners invest for different time periods, Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂).

Ratio and Proportion

To find a specific share from a total, multiply the total by (Individual Ratio ÷ Sum of all Ratios).

Shortcut Trick

The effective rate of interest for 3 years at 5% per annum is:

Effective Interest Rate = 3 × 5% = 15%

Simple Interest (SI) = 15% of ₹920

⇒ SI = (15 ÷ 100) × 920 = ₹138

∴ The correct answer is ₹138.

Alternate Method

Given:

⇒ Principal (P) = ₹920

⇒ Rate of Interest (R) = 5% per annum

⇒ Time (T) = 3 years

Formula Used:

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

Calculations:

⇒ SI = (920 × 5 × 3) ÷ 100

⇒ SI = (920 × 15) ÷ 100

⇒ SI = 13800 ÷ 100

⇒ SI = ₹138

∴ The correct answer is ₹138.

Additional Information

Simple Interest Formula

Interest is computed only on the original principal. SI = (P × R × T) ÷ 100, where P is the principal, R is the rate per annum, and T is time in years.

Total Amount Formula

The total accumulated amount is the sum of the principal and interest: A = P + SI = P × (1 + (R × T) ÷ 100).

Constant Interest Property

In simple interest, the interest amount remains exactly the same every year as long as the principal and rate of interest do not change.

Shortcut Trick

Check options for Sum = 88 and divisibility by HCF = 8.

For 40 and 48: 40 + 48 = 88 (Matches given Sum).

Check LCM: 40 = 8 × 5 and 48 = 8 × 6.

LCM = 8 × 5 × 6 = 240 (Matches given LCM).

∴ The correct answer is 40 and 48.

Alternate Method

Given: HCF = 8, LCM = 240, and Sum = 88.

Formula Used: Product of two numbers = HCF × LCM; Numbers = HCF × x and HCF × y (where x, y are co-prime).

⇒ Let the numbers be 8x and 8y.

⇒ 8x + 8y = 88 ⇒ x + y = 11

⇒ 8 × x × y = 240 (since LCM = HCF × x × y)

⇒ xy = 240 ÷ 8 = 30

⇒ We need two numbers whose sum is 11 and product is 30. By inspection or solving t² − 11t + 30 = 0, we get t = 5 and t = 6.

⇒ x = 5 and y = 6.

⇒ Number 1 = 8 × 5 = 40

⇒ Number 2 = 8 × 6 = 48

∴ The correct answer is 40 and 48.

Additional Information

Product Property

For any two numbers a and b, the Product of numbers = HCF(a, b) × LCM(a, b).

Divisibility Rule

The HCF of two or more numbers is always a factor of their LCM and also a factor of their Sum and Difference.

Co-prime Factorization

If HCF is h, numbers can be written as hx and hy, where LCM = h × x × y and gcd(x, y) = 1.

Shortcut Trick

In an equilateral triangle with side a, the length of the median AD (which is also the altitude) is given by:

AD = (√3 ÷ 2) × a

The centroid G divides the median AD in the ratio 2 : 1. Thus, the segment GD is exactly 1/3 of the median AD:

GD = a ÷ (2√3)

Substituting a = 24 cm:

GD = 24 ÷ (2√3) = 12 ÷ √3 = 4√3 cm

∴ The correct answer is 4√3 cm.

Alternate Method

Given:

Side of the equilateral triangle ABC (a) = 24 cm

AD is the median and G is the centroid.

Formula Used:

Height/Median of an equilateral triangle of side a = (√3 ÷ 2) × a

The centroid G divides the median AD from vertex to base in the ratio 2 : 1 (AG : GD = 2 : 1).

Therefore, GD = (1 ÷ 3) × AD

Shortcut Trick

The difference between ratio terms remains constant over time.

Present ratio of P : Q = 4 : 5 (difference = 1 unit)

8 years ago ratio of P : Q = 2 : 3 (difference = 1 unit)

Since the difference is already balanced, the change in ratio units = 4 − 2 = 2 units → 8 years.

⇒ 1 unit = 4 years → Present ages: P = 16 years, Q = 20 years.

Harmonic Mean (HM) = (2 × 16 × 20) ÷ (16 + 20) = 640 ÷ 36 = 160 ÷ 9 = 17(7)/(9) years.

∴ The correct answer is 17(7)/(9) years.

Alternate Method

Given:

Present ratio of ages of P and Q = 4 : 5

Ratio of their ages 8 years ago = 2 : 3

Formula Used:

Harmonic Mean (HM) of two values a and b = 2ab ÷ (a + b)

Present Ages Representation P (4x)

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.