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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)