Quantitative Aptitude PYQs

Shortcut Trick

CP of first pulse = 72 × 100/120 = ₹60/kg

CP of second pulse = 60 × 100/125 = ₹48/kg

Net CP of mixture = (60 × 2 + 48 × 1) ÷ 3 = 168/3 = ₹56/kg

Overall Profit % = ((70 − 56) ÷ 56) × 100 = (14/56) × 100 = 25%

∴ The correct answer is Profit of 25%.

Alternate Method

Given:

⇒ SP of first variety = ₹72/kg, Profit = 20%

⇒ SP of second variety = ₹60/kg, Profit = 25%

⇒ Ratio of mixture (first : second) = 2 : 1

⇒ SP of mixture = ₹70/kg

Formula Used:

⇒ Cost Price (CP) = SP × 100 / (100 + Profit%)

⇒ Profit % = [(SP − CP) / CP] × 100

Type 1: 2 kg CP = ₹60/kg

Type 2: 1 kg CP = ₹48/kg

Mix CP: 3 kg Avg = ₹56/kg

Mix SP ₹70/kg

Profit = ₹14/kg (+25%)

Calculations:

⇒ Let the quantities of the two varieties be 2 kg and 1 kg respectively.

⇒ CP of 1^st variety = 72 × 100 / (100 + 20) = 7200 / 120 = ₹60 per kg

⇒ Total CP for 2 kg of 1^st variety = 60 × 2 = ₹120

⇒ CP of 2^nd variety = 60 × 100 / (100 + 25) = 6000 / 125 = ₹48 per kg

⇒ Total CP for 1 kg of 2^nd variety = 48 × 1 = ₹48

⇒ Total CP of 3 kg mixture = 120 + 48 = ₹168

⇒ Average CP of mixture = 168 ÷ 3 = ₹56 per kg

⇒ The SP of the mixture is given as ₹70 per kg

⇒ Overall Profit = SP − CP = 70 − 56 = ₹14 per kg

⇒ Profit Percentage = (Profit / CP) × 100 = (14 / 56) × 100 = (1 / 4) × 100 = 25%

∴ The correct answer is Profit of 25%.

Additional Information

Cost Price (CP)

The base price at which goods are purchased. It is the reference value for calculating profit or loss percentages. Formula: CP = SP × 100 / (100 + Profit%).

Mixtures & Alligation

A mathematical rule used to find the weighted average price of a mixture when two components with different prices are mixed in a specific ratio.

Profit & Loss Percentage

Profit and loss are always calculated over the overall Cost Price unless explicitly stated otherwise. Profit% = [(SP − CP) / CP] × 100.

Shortcut Trick

Let SP of dresses D₁, D₂, and D₃ be 5, 6, and 7 units respectively. (Total SP = 18)

Since profit is 20%, loss is 10%, and profit is 30% respectively, their multipliers are 1.2, 0.9, and 1.3.

Total CP = 5/1.2 + 6/0.9 + 7/1.3 = 25/6 + 20/3 + 70/13 = 65/6 + 70/13 = 1265/78

Total Profit = Total SP − Total CP = 18 − 1265/78 = 139/78

Profit % = (Total Profit ÷ Total CP) × 100 = (139/78 ÷ 1265/78) × 100 = 13900/1265 ≈ 10.98%

∴ The correct answer is profit of 10.98%.

Alternate Method

Given:

Selling Price (SP) ratio of D₁ : D₂ : D₃ = 5 : 6 : 7

Profit on D₁ = 20%, Loss on D₂ = 10%, Profit on D₃ = 30%

Formula Used:

Cost Price (CP) = [SP ÷ (100 ± Profit/Loss %)] × 100

Overall Profit % = [(Total SP − Total CP) ÷ Total CP] × 100

Calculations:

Let the Selling Prices of the three dresses be 500, 600, and 700 units respectively.

⇒ Total SP = 500 + 600 + 700 = 1800 units

Calculating the Cost Price of each dress:

⇒ CP of D₁ = (500 ÷ 120) × 100 = 5000/12 = 1250/3 units

⇒ CP of D₂ = (600 ÷ 90) × 100 = 6000/9 = 2000/3 units

⇒ CP of D₃ = (700 ÷ 130) × 100 = 7000/13 units

⇒ Total CP = 1250/3 + 2000/3 + 7000/13

⇒ Total CP = 3250/3 + 7000/13 = (42250 + 21000) ÷ 39 = 63250/39 units

Now, finding the overall profit margin:

⇒ Total SP = 1800 = 70200/39 units

⇒ Total Profit = Total SP − Total CP = (70200 − 63250) ÷ 39 = 6950/39 units

⇒ Profit % = [(6950/39) ÷ (63250/39)] × 100

⇒ Profit % = (6950 ÷ 63250) × 100

⇒ Profit % = (695 ÷ 6325) × 100 = (139 ÷ 1265) × 100 ≈ 10.988%

∴ The correct answer is profit of 10.98%.

Additional Information

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount. The ratio SP / MP gives the multiplying factor for the exact discount applied.

Successive Percentage Changes

When applying successive profit, loss, or discounts, the net percentage change is evaluated by: A + B + (AB / 100)%, where profit is treated as positive and loss/discount as negative.

Dishonest Dealer Concept

When a dealer uses false weights, the actual cost is strictly based on the weight given, while the selling price is based on the claimed weight. Profit % = (Error ÷ True Value − Error) × 100.

Shortcut Trick

In successive chain transactions, the equivalent multiplying factor method is the fastest.

Net multiplying factor = 1.60 (wholesaler markup) × 0.85 (wholesaler discount) × 1.25 (retailer markup) × 0.90 (retailer discount)

Net factor = 1.60 × 0.85 × 1.25 × 0.90 = 1.53

14490 = Original CP × 1.53 ⇒ CP = 14490 ÷ 1.53 ≈ 9470.588

∴ The correct answer is ₹9,471.

Alternate Method

Wholesaler CP = 100x

Wholesaler MP = 160x

Retailer CP = 136x

Retailer MP = 170x

Final SP = 153x

+60% −15% +25% −10%

Final SP = 153x = ₹14,490

Given:

Final Selling Price (paid by customer) = ₹14,490

Let the original cost price of the microwave to the wholesaler be 100x.

Calculations:

⇒ Wholesaler's Marked Price (MP) = 100x + 60% of 100x = 160x

⇒ Trade discount given by wholesaler = 15% of 160x = 24x

⇒ Retailer's Cost Price (CP) = 160x − 24x = 136x

⇒ Retailer's Marked Price (MP) = 136x + 25% of 136x = 136x + 34x = 170x

⇒ Festival discount given by retailer = 10% of 170x = 17x

⇒ Final Selling Price (SP) = 170x − 17x = 153x

⇒ According to the given data, Final SP = 14490

⇒ 153x = 14490

⇒ x = 14490 ÷ 153 ≈ 94.7058

⇒ Wholesaler's Cost Price = 100x = 100 × 94.7058 = 9470.58

⇒ Approximate original cost price = ₹9,471

∴ The correct answer is ₹9,471.

Additional Information

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). The formula used is: SP = MP × (1 − Discount / 100).

Markup Percentage

Markup is always calculated on the Cost Price (CP) before any discount is offered. The formula used is: MP = CP × (1 + Markup / 100).

Successive Transactions

When an article passes through multiple hands where percentage increases or decreases occur consecutively, the Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100) × ...

Shortcut Trick

Since only alcohol is added, the total quantity of water remains constant throughout the process.

Initial Water % × Initial Volume = Final Water % × Final Volume

20% × 60 = 15% × Final Volume

Final Volume = (20 × 60) ÷ 15 = 1200 ÷ 15 = 80 litres

Amount of alcohol added = Final Volume − Initial Volume = 80 − 60 = 20 litres.

∴ The correct answer is 20 litres.

Alternate Method

Given:

Initial volume of solution = 60 litres

Percentage of water initially = 20%

Target percentage of water finally = 15%

Formula Used:

We can solve this using the rule of Alligation on the alcohol percentage.

Initial alcohol % = 100% − 20% = 80%

Pure alcohol added % = 100%

Final mixture alcohol % = 100% − 15% = 85%

80% Initial 60 Litres

100% Added x Litres

85% Mixture

15

5

Ratio = 15 : 5 = 3 : 1

Calculations:

From the Alligation diagram, the ratio of the initial volume to the pure added alcohol is:

⇒ Initial Volume : Added Alcohol = 15 : 5

⇒ Initial Volume : Added Alcohol = 3 : 1

⇒ 3 ratio units = 60 litres

⇒ 1 ratio unit = 60 ÷ 3 = 20 litres

⇒ Added alcohol = 20 litres

∴ The correct answer is 20 litres.

Additional Information

Constant Component Principle

When one ingredient is added or removed from a mixture, the total quantity of the other ingredient remains constant. Equating the amount of the constant element (like water in this problem) gives a direct one-step equation.

Rule of Alligation

Used to find the ratio in which two mixtures or ingredients of given concentrations must be mixed to produce a mixture of a desired concentration. The cross-difference of concentrations directly provides the ratio of quantities.

Pure Substance Concentration

When applying alligation, pure substances like pure alcohol or pure milk are always taken as 100% concentration for that element, while pure water is taken as 0% of that active ingredient.

Shortcut Trick

Total Work = LCM(8, 12) = 24 units.

Efficiency of A = 24 ÷ 8 = 3 units/day, and Efficiency of B = 24 ÷ 12 = 2 units/day.

Combined efficiency = 3 + 2 = 5 units/day.

Work done in 3 days = 5 × 3 = 15 units.

Remaining work = 24 − 15 = 9 units, which is 9 ÷ 24 = 3/8 of the total work.

∴ The correct answer is 3/8.

Alternate Method

Given:

Time taken by A = 8 days

Time taken by B = 12 days

Time they worked together = 3 days

Formula / Method Used:

LCM Method: Total Work = LCM of given days, Efficiency = Total Work ÷ Days

Calculations:

⇒ Total efficiency of A and B = 3 + 2 = 5 units/day

⇒ Work done by A and B together in 3 days = 5 × 3 = 15 units

⇒ Remaining work = Total Work − Work Done

⇒ Remaining work = 24 − 15 = 9 units

⇒ Part of the work left = Remaining Work ÷ Total Work

⇒ Part of the work left = 9 ÷ 24 = 3/8

∴ The correct answer is 3/8.

Additional Information

Pipes & Cisterns

This topic follows the same logic as Time & Work. The only difference is that an emptying pipe has a negative efficiency. Net work in 1 hour for a filling pipe (x hours) and emptying pipe (y hours) = (1 ÷ x) − (1 ÷ y).

Work & Wages

Total wages are always distributed in proportion to the total work completed by each individual. If workers operate for the same number of days, their wages are directly proportional to their efficiencies.

Alternate Days Work

When individuals work consecutively on alternate days, group their efforts by calculating the combined work done in one complete cycle (e.g., 2 days) to easily track total completion time.

Shortcut Trick

Mean CP of the mixture = 75 ÷ 1.1 = 750/11 ₹/kg.

Applying alligation on CP: (80 − 750/11) : (750/11 − 65)

Ratio of Type 1 to Type 2 = 130/11 : 35/11 = 26 : 7

Given 26 parts = 40 kg ⇒ 1 part = 40 ÷ 26 = 20/13 kg.

Quantity of Type 2 = 7 parts = 7 × (20/13) = 140/13 ≈ 10.77 kg.

∴ The correct answer is 10.77 kg.

Alternate Method

Given:

Quantity of Type 1 lentils (Q₁) = 40 kg, Cost Price (CP₁) = ₹65/kg

Cost Price of Type 2 lentils (CP₂) = ₹80/kg

Selling Price of mixture (SP) = ₹75/kg, Profit = 10%

Formula Used:

Cost Price = (Selling Price × 100) ÷ (100 + Profit%)

Calculations:

⇒ Mean CP of the mixture = (75 × 100) ÷ 110 = 750/11 ₹/kg

⇒ Let the quantity of Type 2 lentils be x kg.

⇒ Total Cost Price = (65 × 40) + (80 × x) = 2600 + 80x

⇒ Total Selling Price = 75 × (40 + x) = 3000 + 75x

⇒ We know, SP = CP × 1.1 (for 10% profit)

⇒ 3000 + 75x = 1.1 × (2600 + 80x)

⇒ 3000 + 75x = 2860 + 88x

⇒ 88x − 75x = 3000 − 2860

⇒ 13x = 140

⇒ x = 140 ÷ 13 ≈ 10.77 kg

∴ The correct answer is 10.77 kg.

Additional Information

Alligation Rule

When two ingredients are mixed, their ratio is given by: (Quantity of Cheaper) / (Quantity of Dearer) = (Price of Dearer − Mean Price) / (Mean Price − Price of Cheaper).

Mean Price (Cost Price)

Alligation must always be applied on the Cost Price (CP) of the mixture. If the Selling Price (SP) and Profit/Loss percentages are given, first find the CP using: CP = (SP × 100) / (100 + Profit%).

Direct Algebraic Method

Alternatively, mixture problems can be solved directly using the total cost balance equation: Q₁ × CP₁ + Q₂ × CP₂ = (Q₁ + Q₂) × Mean CP.

Shortcut Trick

Total Work = LCM(12, 16) = 48 units. Efficiency: (A + B) = 4, (B + C) = 3.

Given: A works 5 days, B works 7 days, C works 13 days = Total Work.

Rewrite as: 5(A + B) + 2(B + C) + 11C = 48 units.

⇒ 5(4) + 2(3) + 11C = 48 ⇒ 26 + 11C = 48 ⇒ C = 2 units/day.

Time taken by C alone = Total Work ÷ Efficiency = 48 ÷ 2 = 24 days.

∴ The correct answer is 24 days.

Alternate Method

Given:

Time taken by (A + B) to complete the work = 12 days

Time taken by (B + C) to complete the work = 16 days

A works for 5 days, B works for 7 days, and C works for 13 days to complete the task.

Calculations:

Let the total work be 48 units (LCM of 12 and 16).

⇒ Efficiency of (A + B) = 48 ÷ 12 = 4 units/day

⇒ Efficiency of (B + C) = 48 ÷ 16 = 3 units/day

Let efficiency of A, B, and C be a, b, and c respectively.

⇒ a + b = 4

⇒ b + c = 3

According to the given condition, A works for 5 days, B for 7 days, and C for 13 days:

⇒ 5a + 7b + 13c = 48

We can group the terms to use the known combined efficiencies:

⇒ 5a + 5b + 2b + 2c + 11c = 48

⇒ 5(a + b) + 2(b + c) + 11c = 48

Substitute the values of (a + b) and (b + c):

⇒ 5 × 4 + 2 × 3 + 11c = 48

⇒ 20 + 6 + 11c = 48

⇒ 26 + 11c = 48

⇒ 11c = 48 − 26

⇒ 11c = 22

⇒ c = 2 units/day

Total time taken by C alone to complete the work = Total Work ÷ Efficiency of C

⇒ Time taken by C = 48 ÷ 2 = 24 days

∴ The correct answer is 24 days.

Additional Information

Time and Work Relation

Time taken to complete a work is inversely proportional to the efficiency. Work = Efficiency × Time.

Combined Efficiency Rule

When multiple people work together, their combined efficiency is the sum of their individual efficiencies: E_Total = E₁ + E₂ + ... + E_n.

Pipes and Cisterns

The concept is identical to Time & Work, but an emptying pipe has negative efficiency, decreasing the total work done per unit time.

MDH Formula

If M₁ men can do W₁ work in D₁ days working H₁ hours/day, then (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂.

Shortcut Trick

When distance is constant, speed and time are inversely proportional: S₁ × T₁ = S₂ × T₂.

⇒ 20 × 4 = S₂ × 2.5

⇒ S₂ = 80 ÷ 2.5 = 32 km/h

Increase in speed = S₂ − S₁ = 32 − 20 = 12 km/h

∴ The correct answer is 12 km/h.

Alternate Method

Given:

Initial Speed (S₁) = 20 km/h

Initial Time (T₁) = 4 hours

Required Time (T₂) = 2.5 hours

Formula Used:

Distance = Speed × Time

Speed = Distance ÷ Time

City A

City B

Total Distance = 80 km

Case 1: S = 20 km/h, T = 4 hrs

Case 2: S = 32 km/h, T = 2.5 hrs

Calculations:

⇒ Total Distance = S₁ × T₁

⇒ Distance = 20 × 4 = 80 km

⇒ To cover the same distance in 2.5 hours, required New Speed (S₂) = Distance ÷ T₂

⇒ S₂ = 80 ÷ 2.5

⇒ S₂ = 32 km/h

⇒ Required increase in speed = New Speed − Initial Speed

⇒ Required increase = 32 − 20 = 12 km/h

∴ The correct answer is 12 km/h.

Additional Information

Proportionality of Speed, Time, and Distance

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

Average Speed

Average Speed = Total Distance Covered ÷ Total Time Taken. It is not always the simple average of speeds unless time intervals are equal.

Average Speed for Equal Distances

When a person travels equal distances at speeds x and y, the average speed for the whole journey is (2xy) ÷ (x + y).

Shortcut Trick

Distance covered by the first train in 2 hours (6:00 AM to 8:00 AM) = 60 × 2 = 120 km.

Remaining distance at 8:00 AM = 540 − 120 = 420 km.

Relative speed of both trains moving towards each other = 60 + 90 = 150 km/h.

Time taken to meet after 8:00 AM = 420 ÷ 150 = 2.8 hours = 2 hours 48 minutes.

Meeting time = 8:00 AM + 2 hours 48 minutes = 10:48 AM.

∴ The correct answer is 10:48 AM.

Alternate Method

Given:

Distance between Station A and Station B = 540 km

Speed of Train 1 (from Station A) = 60 km/h

Speed of Train 2 (from Station B) = 90 km/h

Formula Used:

Distance = Speed × Time

Relative Speed (Opposite Directions) = S₁ + S₂

Time = Distance ÷ Speed

A at 8:00 AM B

6:00 AM 8:00 AM

120 km

420 km

Total: 540 km

60 km/h →

← 90 km/h

2 hours

Calculations:

⇒ Train 1 starts at 6:00 AM and Train 2 starts at 8:00 AM. Hence, Train 1 travels alone for 2 hours.

⇒ Distance covered by Train 1 in 2 hours = 60 × 2 = 120 km.

⇒ Remaining distance between the two trains at 8:00 AM = 540 − 120 = 420 km.

⇒ At 8:00 AM, both trains are moving towards each other.

⇒ Relative Speed = Speed of Train 1 + Speed of Train 2

⇒ Relative Speed = 60 + 90 = 150 km/h.

⇒ Time required to meet = Remaining distance ÷ Relative speed

⇒ Time = 420 ÷ 150 = 14 ÷ 5 hours.

⇒ Converting hours into hours and minutes: (14 ÷ 5) hours = 2 hours and (4 ÷ 5) × 60 minutes.

⇒ Time taken to meet = 2 hours 48 minutes after 8:00 AM.

⇒ Meeting Time = 8:00 AM + 2 hours 48 minutes = 10:48 AM.

∴ The correct answer is 10:48 AM.

Additional Information

Relative Speed in Opposite Directions

When two objects move towards each other, their relative speed is the sum of their individual speeds: S_R = S₁ + S₂.

Relative Speed in Same Direction

When two objects move in the same direction, their relative speed is the absolute difference of their individual speeds: S_R = |S₁ − S₂|.

Average Speed

If an object covers equal consecutive distances at speeds S₁ and S₂, the average speed over the entire journey is given by (2 × S₁ × S₂) ÷ (S₁ + S₂).

Shortcut Trick

Initial radius (r) = 10 ÷ 2 = 5 m, New radius (R) = 5 + 1 = 6 m.

Percentage increase in radius = (1 ÷ 5) × 100 = 20%.

Since area is proportional to the square of the radius, apply successive percentage change: 2x + (x² ÷ 100).

Percentage increase in area = 2 × 20 + (20² ÷ 100) = 40 + 4 = 44%.

∴ The correct answer is 44%.

Alternate Method

Given:

Diameter of the circular garden = 10 m

Width of the concrete path = 1 m

Formula Used:

Area of a circle = πr²

Percentage Increase = [(New Area − Original Area) ÷ Original Area] × 100

r = 5 m

R = 6 m

1 m

Calculations:

⇒ Radius of the original garden (r) = 10 ÷ 2 = 5 m

⇒ Original Area of the garden = π × (5)² = 25π m²

⇒ New radius including the path (R) = 5 + 1 = 6 m

⇒ New Area including the path = π × (6)² = 36π m²

⇒ Increase in Area = 36π − 25π = 11π m²

⇒ Percentage Increase = (11π ÷ 25π) × 100

⇒ Percentage Increase = (11 ÷ 25) × 100 = 11 × 4 = 44%

∴ The correct answer is 44%.

Additional Information

Area of a Circular Path

The area of a uniform path of width w around a circle of inner radius r is formulated as π[(r + w)² − r²] = πw(2r + w).

Successive Percentage Change

When a parameter (like radius) increases by x%, quantities proportional to its square (like area) undergo successive increases. The net change is calculated using: (2x + x² ÷ 100)%.

Percentage Change Rule

Always calculate the percentage increase or decrease relative to the original base value using: % Change = (Absolute Difference ÷ Original Value) × 100.

Shortcut Trick

Total area of the circle = πr² = (22/7) × 14 × 14 = 616 cm².

Sector area as a fraction of the total area = 154 / 616 = 1/4.

Angle of the sector = Fraction of area × 360°.

Angle = (1/4) × 360° = 90°.

∴ The correct answer is 90°.

Alternate Method

Given:

Radius (r) = 14 cm

Area of the sector = 154 cm²

Formula Used:

Area of a sector = (θ / 360°) × πr²

O

r = 14 cm r = 14 cm

θ = ? Area = 154

Calculations:

⇒ 154 = (θ / 360°) × (22 / 7) × (14)²

⇒ 154 = (θ / 360°) × 22 × 2 × 14

⇒ 154 = (θ / 360°) × 616

⇒ θ / 360° = 154 / 616

⇒ θ / 360° = 1/4

⇒ θ = 360° / 4

⇒ θ = 90°

∴ The correct answer is 90°.

Additional Information

Length of an Arc

The length of an arc (l) of a circle can be calculated as: l = (θ / 360°) × 2πr.

Area of Sector using Arc Length

If the arc length is already known, the area of the sector can be found directly using the formula: Area = (1 / 2) × l × r.

Area of a Segment

The area of a segment of a circle is defined as: Area of the corresponding sector − Area of the corresponding triangle formed by the chord and the bounding radii.

Shortcut Trick

Total Cost = (Percentage Painted) × (Area of Circle) × Rate

⇒ Total Cost = 90% × (π × r²) × 60

⇒ Total Cost = 0.9 × 3.14 × 2² × 60

⇒ Total Cost = 0.9 × 12.56 × 60 = 678.24

⇒ Total Cost ≈ ₹678.2

∴ The correct answer is ₹678.2.

Alternate Method

90% Painted 10% r = 2 m Total Cost = Painted Area × Cost per m²

Given:

Radius of the circular signboard (r) = 2 m

Cost of painting = ₹60 per m²

Unpainted portion = 10%

Formula Used:

Area of a circle = πr²

Calculation:

⇒ Total area of the board = 3.14 × 2² = 12.56 m²

⇒ Painted portion = 100% − 10% = 90%

⇒ Area to be painted = 90% of 12.56 = 0.9 × 12.56 = 11.304 m²

⇒ Total cost of painting = Painted area × Cost per m²

⇒ Total cost = 11.304 × 60 = ₹678.24

⇒ Total cost ≈ ₹678.2

∴ The correct answer is ₹678.2.

Additional Information

Area of a Sector

A sector represents a fraction of the circle. The formula is Area = (θ ÷ 360°) × πr², where θ is the central angle.

Circumference of a Circle

The total length of the boundary of a circle is given by C = 2πr.

Percentage Application in Geometry

Instead of finding the excluded area and subtracting, directly calculate the effective percentage (e.g., 100% − 10% = 90%) and multiply it with the total area.

Shortcut Trick

For any linear equation in the standard form Ax + By = C, the y-intercept can be calculated directly by dividing the constant term by the y-coefficient.

Formula: y-intercept = C ÷ B

Here, the given equation is 4x + 5y = 20, where C = 20 and B = 5.

y-intercept = 20 ÷ 5 = 4

∴ The correct answer is 4.

Alternate Method

Given:

The linear equation of the line is 4x + 5y = 20.

Formula Used:

The y-intercept is the exact point where the graph of the line crosses the y-axis. At this specific point, the x-coordinate is always 0.

X Y O

(0, 4) y-intercept (5, 0) x-intercept

4x + 5y = 20

Calculations:

⇒ To find the y-intercept, substitute x = 0 in the given equation.

⇒ 4(0) + 5y = 20

⇒ 0 + 5y = 20

⇒ 5y = 20

⇒ y = 20 ÷ 5

⇒ y = 4

∴ The correct answer is 4.

Additional Information

Intercept Form of a Line

The equation of a straight line can be expressed as x/a + y/b = 1, where a represents the x-intercept and b represents the y-intercept.

Slope-Intercept Form

A line can be represented as y = mx + c, where m signifies the slope of the line and c represents the y-intercept.

Finding the x-intercept

The x-intercept is the exact point where the graph of the line intersects the x-axis. It is determined by setting y = 0 in the equation.

Shortcut Trick

For any given algebraic expression where x + 1/x = k, the value of x³ + 1/x³ is directly given by k³ − 3k.

Here, the given value is k = 4.

Substituting the value in the shortcut formula, we get: 4³ − 3 × 4

Value = 64 − 12 = 52

∴ The correct answer is 52.

Alternate Method

Given:

x + 1/x = 4

Formula Used:

(a + b)³ = a³ + b³ + 3ab(a + b)

Calculation:

⇒ Cubing both sides of the given equation, we get:

⇒ (x + 1/x)³ = 4³

⇒ x³ + 1/x³ + 3 × x × (1/x) × (x + 1/x) = 64

⇒ x³ + 1/x³ + 3 × 1 × 4 = 64

⇒ x³ + 1/x³ + 12 = 64

⇒ x³ + 1/x³ = 64 − 12

⇒ x³ + 1/x³ = 52

∴ The correct answer is 52.

Additional Information

Algebraic Identity 1

If x − 1/x = k, then the direct result is x³ − 1/x³ = k³ + 3k.

Algebraic Identity 2

If x + 1/x = k, then the square values relate as x² + 1/x² = k² − 2.

Algebraic Identity 3

If x − 1/x = k, then the square values relate as x² + 1/x² = k² + 2.

Shortcut Trick

When adding decimals with a single 1 in different place values, just position 1 in each corresponding decimal place.

0.1 has 1 in the tenths place.

0.01 has 1 in the hundredths place.

0.001 has 1 in the thousandths place.

⇒ Combining them directly gives 0.111

∴ The correct answer is 0.111.

Alternate Method

Given:

Expression: 0.1 + 0.01 + 0.001

Method Used:

Convert the given decimals into fractions with a common denominator of 1000, and then add them together.

Calculation:

⇒ 0.1 = 1 ÷ 10 = 100 ÷ 1000

⇒ 0.01 = 1 ÷ 100 = 10 ÷ 1000

⇒ 0.001 = 1 ÷ 1000

⇒ Sum = (100 ÷ 1000) + (10 ÷ 1000) + (1 ÷ 1000)

⇒ Sum = (100 + 10 + 1) ÷ 1000

⇒ Sum = 111 ÷ 1000

⇒ Sum = 0.111

∴ The correct answer is 0.111.

Additional Information

BODMAS Rule

It determines the correct sequence of operations in any mathematical expression: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Decimal Fractions

Fractions in which denominators are powers of 10 (like 10, 100, 1000) are known as decimal fractions. Example: 1 ÷ 10 = 0.1

Like Decimals

Decimals having the exact same number of decimal places are called like decimals. Adding trailing zeros converts unlike decimals to like decimals, making addition simpler (e.g., 0.100 + 0.010 + 0.001).

Shortcut Trick

To find the compound ratio N : P directly, we can multiply the given intermediate ratios N : O and O : P.

N / P = (N / O) × (O / P)

N / P = (6 / 7) × (9 / 11)

N / P = 54 / 77

∴ The correct answer is 54 : 77.

Alternate Method

Given:

M : N = 4 : 5

N : O = 6 : 7

O : P = 9 : 11

N O P

6 7 → 7

9 ← 9 11

54 63 77

Calculations:

⇒ To find the combined ratio, make the common term (O) equal in both ratios by taking the LCM of 7 and 9, which is 63.

⇒ Multiply the N : O ratio by 9 ⇒ N : O = (6 × 9) : (7 × 9) = 54 : 63

⇒ Multiply the O : P ratio by 7 ⇒ O : P = (9 × 7) : (11 × 7) = 63 : 77

⇒ Now, the combined ratio N : O : P = 54 : 63 : 77

⇒ Extracting the required values N and P from the combined ratio, we get N : P = 54 : 77

(Note: M : N is extraneous information and is not required for solving the problem.)

∴ The correct answer is 54 : 77.

Additional Information

Compound Ratio

When two or more ratios are multiplied together, the resulting ratio is called a compound ratio. For a : b and c : d, the compound ratio is ac : bd.

Combining Ratios

If A : B = x : y and B : C = p : q, you can combine them by equating the common term 'B'. Multiply the first ratio by p and the second by y to get A : B : C = xp : yp : yq.

Extraneous Data in Exams

Questions often contain additional data (like M : N in this problem) that is not required to find the final answer. Reading the question carefully to identify the core requirement saves valuable exam time.

Shortcut Trick

Convert the mixed fraction directly to a decimal: 2¹/₄ = 2.25

Dividing any number by 0.5 is mathematically equivalent to multiplying that number by 2.

⇒ 2.25 × 2 = 4.5

∴ The correct answer is 4.5.

Alternate Method

Given:

Expression: 2¹/₄ ÷ 0.5

Calculations:

⇒ Convert the mixed fraction into an improper fraction:

⇒ 2¹/₄ = (2 × 4 + 1) / 4 = 9/4

⇒ Convert the decimal 0.5 into a fraction:

⇒ 0.5 = 5/10 = 1/2

⇒ Substitute the fractions into the given expression:

⇒ 9/4 ÷ 1/2

⇒ To divide by a fraction, multiply by its reciprocal:

⇒ 9/4 × 2/1

⇒ Multiply the numerators and denominators:

⇒ 18/4

⇒ Simplify the resulting fraction:

⇒ 9/2 = 4.5

∴ The correct answer is 4.5.

Additional Information

Fraction Division Rule

To divide by a fraction, multiply the first number by the reciprocal of the second fraction: (a/b) ÷ (c/d) = (a/b) × (d/c).

Mixed Fraction to Improper Fraction

Multiply the whole number part by the denominator, add the numerator, and place the result over the original denominator.

Decimal to Fraction Conversion

Decimals can be easily converted to fractions by placing the number over its place value (e.g., 0.5 = 5/10 = 1/2, or 0.25 = 25/100 = 1/4).

Shortcut Trick

Let the given numbers be a = 10, b = 12, c = 13, and d = 16.

Direct formula for the number to be subtracted: x = (a × d − b × c) ÷ ((a + d) − (b + c))

⇒ x = (10 × 16 − 12 × 13) ÷ ((10 + 16) − (12 + 13))

⇒ x = (160 − 156) ÷ (26 − 25)

⇒ x = 4 ÷ 1 = 4

∴ The correct answer is 4.

Alternate Method

Given:

The numbers are 10, 12, 13, and 16.

Let the number to be subtracted from each term be x.

Formula Used:

If four numbers A, B, C, and D are in proportion, then A : B :: C : D or A ÷ B = C ÷ D.

10 − x

12 − x

=

13 − x

16 − x

Proportion: (10 − x) : (12 − x) :: (13 − x) : (16 − x)

Calculations:

⇒ (10 − x) ÷ (12 − x) = (13 − x) ÷ (16 − x)

⇒ Cross-multiplying both the sides:

⇒ (10 − x) × (16 − x) = (13 − x) × (12 − x)

⇒ 160 − 10x − 16x + x² = 156 − 13x − 12x + x²

⇒ 160 − 26x = 156 − 25x

⇒ 26x − 25x = 160 − 156

⇒ x = 4

∴ The correct answer is 4.

Additional Information

Fourth Proportional

If a, b, c, and d are in proportion (a : b :: c : d), then d is called the fourth proportional. It is calculated as d = (b × c) ÷ a.

Third Proportional

For two numbers a and b, the third proportional c is defined such that a : b :: b : c. It is calculated as c = b² ÷ a.

Mean Proportional

The mean proportional between two positive numbers a and b is given by √(a × b). It represents the middle term in a continuous proportion.

Componendo and Dividendo Rule

If a ÷ b = c ÷ d, then according to this mathematical rule: (a + b) ÷ (a − b) = (c + d) ÷ (c − d).

Shortcut Trick

Difference in percentage = 45% − 30% = 15%

Value of 15% of the number = 0.9

1% of the number = 0.9 ÷ 15 = 0.06

100% of the number = 0.06 × 100 = 6

∴ The correct answer is 6.

Alternate Method

Given:

45% of a number is 0.9 more than 30% of it.

30% 15%

0.9 Difference

55%

Total = 100% = x

Calculations:

Let the number be x.

⇒ 45% of x − 30% of x = 0.9

⇒ (45 − 30)% of x = 0.9

⇒ 15% of x = 0.9

⇒ (15 ÷ 100) × x = 0.9

⇒ x = 0.9 × (100 ÷ 15)

⇒ x = 90 ÷ 15

⇒ x = 6

∴ The correct answer is 6.

Additional Information

Percentage Difference

When comparing two percentages of the same base value, the difference between their absolute values is equal to the percentage of the difference applied to the base. Formula: a% of x − b% of x = (a − b)% of x.

Base of Percentage

A percentage is always calculated on a base value (considered as 100%). Finding the unknown base given a percentage value involves multiplying the value by 100 and dividing by the given percentage.

Percentage to Fraction Conversion

To convert a percentage into a fraction, divide the value by 100. For example, 15% is equivalent to 15 ÷ 100 or 3 ÷ 20 in simplified fractional form.

Shortcut Trick

In partnership, Profit Ratio = Ratio of (Investment × Time).

Working in terms of thousands (k) to simplify calculations:

Sohan's effective capital = (90 × 8) + (60 × 4) = 720 + 240 = 960

Rohan's effective capital = 150 × 7 = 1050

Profit Ratio = 960 : 1050 = 96 : 105 = 32 : 35

∴ The correct answer is 32:35.

Alternate Method

Given:

Sohan's initial investment = ₹90,000

Time for which Sohan invested his initial amount = 5 + 3 = 8 months

Amount withdrawn by Sohan after 8 months = ₹30,000

Remaining time for Sohan's adjusted investment = 12 − 8 = 4 months

Rohan's investment = ₹1,50,000

Rohan joined after 5 months, so his investment time = 12 − 5 = 7 months

Formula Used:

Profit Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

0m 5m 8m 12m

Sohan

₹90k × 8m

₹60k × 4m

Rohan

₹150k × 7m

Effective Capital = Investment × Time

Calculations:

⇒ Sohan's remaining investment after withdrawal = 90,000 − 30,000 = ₹60,000

⇒ Sohan's effective capital = (90,000 × 8) + (60,000 × 4)

⇒ Sohan's effective capital = 7,20,000 + 2,40,000 = 9,60,000

⇒ Rohan's effective capital = 1,50,000 × 7 = 10,50,000

⇒ Profit Ratio (Sohan : Rohan) = 9,60,000 : 10,50,000

⇒ Ratio = 96 : 105

Dividing both sides by 3:

⇒ Ratio = 32 : 35

∴ The correct answer is 32:35.

Additional Information

Partnership Profit Sharing

Profit is shared among partners strictly based on their effective capital. The effective capital is calculated as the product of the capital invested and the time duration of the investment.

Compound Partnership

When partners invest for varying time intervals, or they alter (add/withdraw) their invested capital mid-way through the business term, it represents a compound partnership. The total effective capital is the sum of (capital × time) for each sub-period.

Working vs Sleeping Partner

A working partner actively manages the business and frequently takes a specified salary or percentage of the profit before the remainder is divided. A sleeping partner simply invests funds without engaging in day-to-day operations and receives only their proportionate share.