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Shortcut Trick

Ratio of Profit shares A : B : C = 1/8 : 1/3 : 13/24 = 3 : 8 : 13 (taking LCM 24).

Investment × Time = Profit ⇒ (A_Inv × 4) : (C_Inv × 8) = 3 : 13.

⇒ (A_Inv × 4) ÷ (1560 × 8) = 3 ÷ 13.

⇒ 4 × A_Inv = (3 × 12480) ÷ 13 = 2880.

⇒ A_Inv = 2880 ÷ 4 = 720.

∴ The correct answer is ₹720.

Alternate Method

Given:

A's time (T_A) = 4 months, Profit share (P_A) = 1/8

B's time (T_B) = 6 months, Profit share (P_B) = 1/3

C's time (T_C) = 8 months, Investment (I_C) = ₹1560

Calculations:

Total profit share = 1

⇒ C's profit share = 1 − (1/8 + 1/3)

⇒ C's profit share = 1 − (3 + 8)/24 = 1 − 11/24 = 13/24

The ratio of profits of A and C:

⇒ P_A : P_C = 1/8 : 13/24 = 3/24 : 13/24 = 3 : 13

Since Investment × Time = Profit:

⇒ (I_A × T_A) ÷ (I_C × T_C) = P_A ÷ P_C

⇒ (I_A × 4) ÷ (1560 × 8) = 3 ÷ 13

⇒ 4 × I_A ÷ 12480 = 3 ÷ 13

⇒ 4 × I_A = (3 × 12480) ÷ 13

⇒ 4 × I_A = 3 × 960

⇒ 4 × I_A = 2880

⇒ I_A = 2880 ÷ 4 = 720

∴ The correct answer is ₹720.

Additional Information

Partnership Profit Rule

In a partnership, the profit is distributed among partners in the ratio of the product of their capital and time period (Profit = Investment × Time).

Compound Partnership

When partners invest different amounts for different periods of time, it is called a compound partnership. The ratio of their effective capitals determines the profit share.

Investment Calculation

Individual investment can be found using the formula: Investment = Profit ÷ Time. If profit shares and times are known, the investment ratio is (P₁/T₁) : (P₂/T₂).

Shortcut Trick

In a trapezium (where AB || DC), the length of the segment joining the midpoints of the diagonals is always equal to half the difference between the parallel sides.

Length of EF = (Parallel Side 1 − Parallel Side 2) ÷ 2

Length of EF = (DC − AB) ÷ 2

Length of EF = (35 − 23) ÷ 2 = 12 ÷ 2 = 6 cm

∴ The correct answer is 6 cm.

Alternate Method

Given:

Parallel sides: AB = 23 cm and DC = 35 cm

Non-parallel sides: BC = 91 cm and AD = 62 cm

Diagonals: AC and BD with midpoints E and F respectively.

Formula Used:

In a trapezium with parallel sides a and b, the length of the segment EF connecting the midpoints of the diagonals is given by:

EF = |a − b| ÷ 2

Calculations:

⇒ Parallel sides are AB = 23 cm and DC = 35 cm.

⇒ EF = (DC − AB) ÷ 2

⇒ EF = (35 − 23) ÷ 2

⇒ EF = 12 ÷ 2

⇒ EF = 6 cm

∴ The correct answer is 6 cm.

Additional Information

Median of a Trapezium

The segment joining the midpoints of the non-parallel sides (lateral sides) is called the median, and its length is half the sum of the parallel sides: (AB + DC) ÷ 2.

Area of a Trapezium

The area is calculated as Area = 1/2 × (Sum of parallel sides) × Height.

Diagonal Property

In any trapezium, the diagonals divide each other in the same ratio, which is equal to the ratio of the parallel sides.

Shortcut Trick

Ratio of Orange Juice to Water = 2 : 5

If 2 units = 5 litres of Juice, then the scaling factor is 5 ÷ 2 = 2.5

Required Water = 5 units × 2.5 = 12.5 litres

∴ The correct answer is 12.5 litres.

Alternate Method

Given:

Initial Orange juice (J₁) = 2 litres

Initial Water (W₁) = 5 litres

New Orange juice (J₂) = 5 litres

Formula Used:

Since the mixture composition remains in the same proportion: J₁/W₁ = J₂/W₂

Calculations:

⇒ 2 / 5 = 5 / W₂

⇒ 2 × W₂ = 5 × 5

⇒ 2 × W₂ = 25

⇒ W₂ = 25 ÷ 2

⇒ W₂ = 12.5 litres

∴ The correct answer is 12.5 litres.

Additional Information

Direct Proportion

Two quantities are in direct proportion if an increase in one leads to a proportional increase in the other, represented as y = kx.

Inverse Proportion

Two quantities are in inverse proportion if an increase in one leads to a proportional decrease in the other, represented as xy = k.

Unitary Method

A technique where you first find the value of a single unit and then multiply by the required number of units to find the total value.

Given:

{(13/33) ÷ (13/7)} ÷ (2/8 × 16/6 + 5/11) + 7/3 ÷ (37/20 of 20/3)

Formula used:

The BODMAS rule determines the order of operations: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Calculation:

⇒ {(13/33) × (7/13)} ÷ ((1/4 × 8/3) + 5/11) + 7/3 ÷ (37/20 × 20/3)

⇒ (7/33) ÷ (2/3 + 5/11) + 7/3 ÷ (37/3)

⇒ (7/33) ÷ ((22 + 15)/33) + (7/3 × 3/37)

⇒ (7/33 × 33/37) + 7/37

⇒ 7/37 + 7/37

⇒ 14/37

∴ The correct answer is option (3).

Shortcut Trick

In any Arithmetic Progression (like consecutive even numbers), the Average is always the mean of the Smallest and Largest terms.

Formula: Average = (Smallest + Largest) ÷ 2

⇒ Sum of Smallest and Largest = 2 × Average

⇒ Sum = 2 × 32 = 64

∴ The correct answer is 64.

Alternate Method

Given:

⇒ Total terms = 7

⇒ Average of the consecutive even numbers = 32

Formula Used:

⇒ Average = (Sum of all terms) ÷ (Total number of terms)

Calculations:

⇒ For an odd number of terms (7), the Average is equal to the middle term (4^th term).

⇒ 4^th term = 32

⇒ 1^st term (Smallest) = 32 − (3 × 2) = 26

⇒ 7^th term (Largest) = 32 + (3 × 2) = 38

⇒ Sum of Smallest and Largest = 26 + 38 = 64

∴ The correct answer is 64.

Additional Information

Arithmetic Progression (AP) Property

In any AP, the average of 'n' terms is always the average of the first term (a) and the last term (l). Formula: Average = (a + l) ÷ 2.

Middle Term Concept

For a series of 'n' consecutive terms where 'n' is an odd number, the average is the exactly the {(n+1)/2}^th term.

Consecutive Even/Odd Numbers

The common difference (d) between consecutive even or consecutive odd numbers is always 2.

Given:

y = 31 × 13 + 18 × 86 − 86

Formula used:

BODMAS rule and Distributive property: (a × c) − (b × c) = (a − b) × c

Calculations:

y = 31 × 13 + 18 × 86 − 86

⇒ y = (31 × 13) + (18 − 1) × 86

⇒ y = 403 + 17 × 86

⇒ y = 403 + 1462

⇒ y = 1865

∴ The correct answer is option (3).

Shortcut Trick

Total Surface Area (TSA) of a cube = 6 × side²

Given side = 64 cm, TSA = 6 × (64)²

Last Digit Check: 6 × (unit digit 4)² → 6 × 16 → ends in 6.

Only one option (24,576) ends in 6.

∴ The correct answer is 24,576 cm².

Alternate Method

Given: Side of the cube (a) = 64 cm

Formula: Total Surface Area = 6 × a²

⇒ Side of the cube (a) = 64 cm

⇒ Total Surface Area = 6 × (64)²

⇒ TSA = 6 × (64 × 64)

⇒ TSA = 6 × 4096

⇒ TSA = 24,576 cm²

∴ The correct answer is 24,576 cm².

Additional Information

Volume of a Cube

The volume represents the capacity of the cube and is calculated as V = a³.

Lateral Surface Area (LSA)

The area of the four vertical faces (excluding top and bottom) is LSA = 4a².

Diagonal of a Cube

The length of the longest rod that can be placed inside a cube is d = a√3.

Properties of a Cube

A cube has 12 edges, 8 vertices, and 6 identical square faces.

Shortcut Trick

Using the Net Multiplying Factor method for successive changes:

Net Factor = 1.20 (for +20%) × 0.75 (for −25%) × 1.10 (for +10%)

Net Factor = (1.2 × 0.75) × 1.1 = 0.9 × 1.1 = 0.99

Overall Change = (0.99 − 1) × 100 = −1%

A negative sign indicates a percentage decrease of 1%.

∴ The correct answer is 1% decrease.

Alternate Method

Given: Let the initial price of the machine be ₹100.

Step-by-step price changes:

⇒ Price after 20% increase = 100 + (20% of 100) = 120

⇒ Price after 25% decrease = 120 − (25% of 120) = 120 − 30 = 90

⇒ Price after 10% increase = 90 + (10% of 90) = 90 + 9 = 99

⇒ Overall percentage change = [(Final Price − Initial Price) ÷ Initial Price] × 100

⇒ Overall Change = [(99 − 100) ÷ 100] × 100 = −1%

∴ The correct answer is 1% decrease.

Additional Information

Successive Percentage Formula

For two successive changes a% and b%, the net percentage change is given by: a + b + (ab ÷ 100).

Multiplying Factor (MF)

For an x% increase, the MF is (1 + x/100). For an x% decrease, the MF is (1 − x/100). Net MF is the product of all individual factors.

Percentage to Fraction Conversion

20% = 1/5 | 25% = 1/4 | 10% = 1/10. These can be used to solve using ratio methods (Final/Initial = 6/5 × 3/4 × 11/10 = 99/100).

Given:

First addition = 3/5 unit

Second addition = 0.4 unit

Removal = 1/10 unit

Formula used:

Net change = (Addition₁ + Addition₂) − Removal

Calculations:

Converting fractions to decimals:

3/5 = 0.6

1/10 = 0.1

Net change = 0.6 + 0.4 − 0.1

⇒ Net change = 1.0 − 0.1

⇒ Net change = 0.9 unit

Since the result is positive, it represents an increase of 0.9 unit.

∴ The correct answer is option (1).

Shortcut Trick

Total effort for the first group = 9 × 8 = 72 units.

Total effort for the second group = 15 × 9 = 135 units.

Ratio of wages = Ratio of units = 72 : 135 = 8 : 15.

8 units → ₹2,880 ⇒ 1 unit = ₹360.

15 units → 15 × 360 = ₹5,400.

∴ The correct answer is ₹5,400.

Alternate Method

Given:

Case 1: Workers (M₁) = 9, Days (D₁) = 8, Wages (W₁) = ₹2,880

Case 2: Workers (M₂) = 15, Days (D₂) = 9, Wages (W₂) = ?

Formula Used:

(M₁ × D₁) ÷ W₁ = (M₂ × D₂) ÷ W₂

Calculations:

⇒ (9 × 8) ÷ 2,880 = (15 × 9) ÷ W₂

⇒ 72 ÷ 2,880 = 135 ÷ W₂

⇒ 1 ÷ 40 = 135 ÷ W₂

⇒ W₂ = 135 × 40

⇒ W₂ = 5,400

∴ The correct answer is ₹5,400.

Additional Information

Chain Rule Formula

For relationship between people (M), days (D), hours (H), and work or wages (W), use (M₁D₁H₁)/W₁ = (M₂D₂H₂)/W₂.

Direct Proportionality

Wages are directly proportional to the total number of man-days worked when the rate remains constant.

Efficiency and Wages

Wages ∝ Efficiency × Time. If efficiency changes, it must be multiplied in the numerator of the Chain Rule.