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Shortcut Trick

Present Ratio (Father : Son) = 6 : 1 (Difference = 5 units)

Ratio after 5 years = 7 : 2 (Difference = 5 units)

Increase in units for both = 7 − 6 = 1 unit (or 2 − 1 = 1 unit).

This 1 unit increase corresponds to the 5-year time gap.

⇒ 1 unit = 5 years.

Son's present age = 1 unit = 5 years.

∴ The correct answer is 5 years.

Alternate Method

Given:

Present ratio of Father and Son = 6 : 1

Ratio after 5 years = 7 : 2

Calculations:

Let the present age of the Father be 6x and the Son be x.

⇒ Age of Father after 5 years = 6x + 5

⇒ Age of Son after 5 years = x + 5

According to the given ratio after 5 years:

⇒ (6x + 5) ÷ (x + 5) = 7 ÷ 2

⇒ 2 × (6x + 5) = 7 × (x + 5)

⇒ 12x + 10 = 7x + 35

⇒ 12x − 7x = 35 − 10

⇒ 5x = 25

⇒ x = 5

Present age of the Son = x = 5 years.

∴ The correct answer is 5 years.

Additional Information

Constant Age Difference

The difference between the ages of two individuals remains the same throughout their lives. In ratio problems, ensure the unit difference between the two terms is balanced across time periods.

Ratio Balancing Method

If the differences between ratio terms (e.g., 6−1=5 and 7−2=5) are already equal, the change in units directly represents the time interval. If not equal, multiply ratios by the other ratio's difference to balance them.

Linear Equation in Ages

For 'n' years ago, subtract n from the present age; for 'n' years hence, add n to the present age to form the required equation.

Given:

Expression: {1 / 0.01} × {[(5 / 8) + 0.25] × (4 / 5)}

Formula used:

BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Calculations:

{1 / 0.01} × {[(5 / 8) + 0.25] × (4 / 5)}

⇒ 100 × {[(5 / 8) + (2 / 8)] × (4 / 5)}

⇒ 100 × {7 / 8 × 4 / 5}

⇒ 100 × 28 / 40

⇒ 100 × 7 / 10

⇒ 10 × 7

⇒ 70

∴ The correct answer is option (2).

Shortcut Trick

The sum of cubes property states that if a + b + c = 0, then a³ + b³ + c³ = 3abc.

Here, the sum of the base values is 25 + (−20) + (−5) = 0.

Applying the trick: 3 × 25 × (−20) × (−5) = 75 × 100 = 7500.

∴ The correct answer is 7500.

Alternate Method

Given:

Expression: 25³ + (−20)³ + (−5)³

Let a = 25, b = −20, and c = −5.

Formula Used:

The Algebraic Identity used is: a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca).

Calculations:

⇒ First, calculate the sum of the variables: a + b + c = 25 + (−20) + (−5)

⇒ a + b + c = 25 − 20 − 5 = 0

⇒ Substitute (a + b + c) = 0 into the identity: a³ + b³ + c³ − 3abc = 0 × (a² + b² + c² − ab − bc − ca)

⇒ a³ + b³ + c³ = 3abc

⇒ Value = 3 × 25 × (−20) × (−5)

⇒ Value = 75 × 100

⇒ Value = 7500

∴ The correct answer is 7500.

Additional Information

Conditional Identity

When the sum of three variables is zero, their sum of cubes is always equal to three times their product (3abc).

Sum of Cubes Formula

The factorized form is a³ + b³ = (a + b)(a² − ab + b²), useful for binomial cubic expressions.

Trinomial Square Identity

The square of a sum is given by (a + b + c)² = a² + b² + c² + 2(ab + bc + ca).

Shortcut Trick

Scheme A: Net Discount = 45 + 44 − (45 × 44)/100 = 69.2%

Scheme B: Net Discount = 100 − (64% × 59% × 55%) ≈ 79.2%

Scheme C: Total Discount = [(4 × 49) + (6 × 19)] / 10 = 31%

Scheme D: Discount = 7 ÷ (10 + 7) × 100 ≈ 41.2%

Comparing all, Scheme C offers the minimum discount (31%).

∴ The correct answer is Scheme C.

Alternate Method

Assume the Marked Price (MP) of each toy is ₹100.

For a customer buying 10 toys, the total MP = 10 × 100 = ₹1000.

Formula Used: Effective Discount % = (Total Discount ÷ Total MP) × 100

⇒ Scheme A: Net discount = 45 + 44 − (45 × 44)/100 = 89 − 19.8 = 69.2%

⇒ Scheme B: Final SP factor = 0.64 × 0.59 × 0.55 = 0.20768 → Discount = (1 − 0.20768) × 100 = 79.232%

⇒ Scheme C: Discount on first 4 = 4 × 49 = ₹196; Discount on next 6 = 6 × 19 = ₹114. Total = ₹310. Discount % = (310/1000) × 100 = 31%

⇒ Scheme D: Discount % = [Free items ÷ Total items] × 100 = [7 ÷ 17] × 100 ≈ 41.18%

⇒ The least beneficial scheme for the customer is the one with the lowest discount percentage.

⇒ 31% < 41.18% < 69.2% < 79.23%

∴ The correct answer is Scheme C.

Additional Information

Successive Discounts Formula

For two successive discounts of x% and y%, the single equivalent discount is [x + y − (xy/100)]%.

Buy X Get Y Free

In such schemes, the effective discount percentage is given by [Y ÷ (X + Y)] × 100.

Customer Benefit Rule

A scheme is most beneficial to a customer when the total discount is maximum and least beneficial when the total discount is minimum.

Shortcut Trick

When multiple identical cuboids are stacked, the total Volume is simply the sum of their individual volumes.

Volume of one cuboid = 10.5 × 6.4 × 2 = 134.4 cm³

Total Volume = 3 × 134.4 = 403.2 cm³

∴ The correct answer is 403.2 cm³.

Alternate Method

Given:

Length of each cuboid (l) = 10.5 cm

Breadth of each cuboid (b) = 6.4 cm

Height of each cuboid (h) = 2 cm

Number of cuboids stacked = 3

Formula Used:

Volume of a cuboid = Length × Breadth × Height

When three cuboids are stacked one on top of the other:

⇒ New Length (L) = 10.5 cm

⇒ New Breadth (B) = 6.4 cm

⇒ New Height (H) = 2 + 2 + 2 = 6 cm

⇒ Volume of new cuboid = L × B × H

⇒ Volume = 10.5 × 6.4 × 6

⇒ Volume = 67.2 × 6

⇒ Volume = 403.2 cm³

∴ The correct answer is 403.2 cm³.

Additional Information

Total Surface Area of Cuboid

The total area of all six faces is given by 2 × (lb + bh + hl).

Diagonal of a Cuboid

The longest rod that can be placed inside a cuboid is the diagonal, calculated as √(l² + b² + h²).

Lateral Surface Area

The area of the four side walls is given by 2 × h × (l + b).

Shortcut Trick

In an average calculation, the sum of deviations from the average must be zero.

Deviations from 18 are: (6.5 − 18) = −11.5, (10.8 − 18) = −7.2, and (19.9 − 18) = +1.9.

Sum of known deviations = −11.5 − 7.2 + 1.9 = −16.8.

To make the net deviation zero, the deviation of a must be +16.8.

Value of a = 18 + 16.8 = 34.8.

∴ The correct answer is 34.8.

Alternate Method

Given:

Numbers: 6.5, 10.8, 19.9, and a

Total count (n) = 4

Average = 18

Formula Used:

Average = (Sum of all observations) ÷ (Total number of observations)

Calculations:

⇒ (6.5 + 10.8 + 19.9 + a) ÷ 4 = 18

⇒ 6.5 + 10.8 + 19.9 + a = 18 × 4

⇒ 37.2 + a = 72

⇒ a = 72 − 37.2

⇒ a = 34.8

∴ The correct answer is 34.8.

Additional Information

Arithmetic Mean

The Arithmetic Mean is the central value of a set of numbers, calculated by adding them together and dividing by the count.

Sum Property

The total sum of any dataset can be found using the formula: Sum = Average × Number of terms.

Deviation Rule

The sum of deviations of all observations from their arithmetic mean is always zero (Σ(x − Mean) = 0).

Shortcut Trick

Effective rate of Compound Interest for 2 years at 10% = x + y + (xy/100)

⇒ Effective Rate = 10 + 10 + (10 × 10)/100 = 21%

⇒ Compound Interest = 21% of ₹10,000

⇒ Interest = 0.21 × 10,000 = ₹2,100

∴ The correct answer is ₹2,100.

Alternate Method

Given:

Principal (P) = ₹10,000

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Compound Interest (CI) = A − P

Calculations:

⇒ A = 10,000 × (1 + 10/100)²

⇒ A = 10,000 × (11/10) × (11/10)

⇒ A = 10,000 × (121 / 100)

⇒ A = 100 × 121 = ₹12,100

⇒ Compound Interest (CI) = 12,100 − 10,000

⇒ CI = ₹2,100

∴ The correct answer is ₹2,100.

Additional Information

Simple Interest vs Compound Interest

Simple interest is calculated only on the Principal, while compound interest is calculated on the Principal plus the accumulated interest of previous periods.

Difference for 2 Years

The difference between CI and SI for 2 years is given by the formula: Difference = P × (R/100)².

Rule of 72

To find the approximate time required to double your money at a given interest rate, divide 72 by the annual rate of return.

Shortcut Trick

The Surface Area of a sphere is given by 4πr².

Since π contains the factor 22 (2 × 11), the result must be a multiple of 11.

For r = 7 cm, Surface Area = 4 × (22/7) × 7 × 7 = 88 × 7.

88 × 7 = 616 sq cm.

∴ The correct answer is 616 sq cm.

Alternate Method

Given:

Radius of the sphere (r) = 7 cm

Value of π = 22/7

Formula Used:

Surface Area of a Sphere = 4πr²

Calculations:

⇒ Surface Area = 4 × (22/7) × (7)²

⇒ Surface Area = 4 × (22/7) × 7 × 7

⇒ Surface Area = 4 × 22 × 7

⇒ Surface Area = 88 × 7

⇒ Surface Area = 616 sq cm

∴ The correct answer is 616 sq cm.

Additional Information

Volume of a Sphere

The total space occupied by a sphere is given by the formula V = (4/3)πr³.

Curved Surface Area of a Hemisphere

A hemisphere is half of a sphere; its curved surface area is half that of a sphere: CSA = 2πr².

Total Surface Area of a Hemisphere

For a solid hemisphere, the total surface area includes the curved part and the flat circular base: TSA = 3πr².

Shortcut Trick

Net difference in percentage = Gain% − (Loss%) = 5% − (−10%) = 15%

According to the question, 15% of Cost Price = ₹60

1% of Cost Price = 60 ÷ 15 = ₹4

100% (Cost Price) = 4 × 100 = ₹400

∴ The correct answer is ₹400.

Alternate Method

Given:

Initial Loss = 10%

New Gain = 5%

Difference in Selling Price = ₹60

Formula Used:

Selling Price (SP) = Cost Price (CP) × (100 ± Profit/Loss%)/100

Let the Cost Price (CP) of the article be ₹x.

⇒ Initial Selling Price (SP₁) = x − 10% of x = 0.90x

⇒ New Selling Price (SP₂) = x + 5% of x = 1.05x

⇒ Difference: SP₂ − SP₁ = 60

⇒ 1.05x − 0.90x = 60

⇒ 0.15x = 60

⇒ x = 60 ÷ 0.15

⇒ x = 400

∴ The correct answer is ₹400.

Additional Information

Direct CP Formula

When there is a shift from loss to gain, Cost Price = [Difference in SP ÷ (Loss% + Gain%)] × 100.

Profit and Loss Percentage

Always calculate Profit% and Loss% on the Cost Price unless stated otherwise. Formula: (Profit/Loss ÷ CP) × 100.

Marked Price (MP)

The price at which a Discount is offered is the Marked Price. SP = MP × (100 − Discount%)/100.

Shortcut Trick

Simplified ratio X : Y : Z = 1/2 : 1/4 : 5/16 → Multiply by LCM (16) = 8 : 4 : 5.

Total parts = 8 + 4 + 5 = 17 units.

17 units = ₹9,520 ⇒ 1 unit = 9520 ÷ 17 = ₹560.

Share of Y = 4 units = 4 × 560 = ₹2,240.

This corresponds to Option 3.

∴ The correct answer is ₹2,240.

Alternate Method

Given:

Total sum = ₹9,520

Ratio of X : Y : Z = 1/2 : 1/4 : 5/16

Formula Used:

Share of an individual = (Individual ratio part ÷ Sum of ratio parts) × Total Amount

Calculations:

⇒ To simplify the Ratio, find the LCM of denominators 2, 4, and 16, which is 16.

⇒ Ratio X : Y : Z = (1/2 × 16) : (1/4 × 16) : (5/16 × 16)

⇒ Ratio X : Y : Z = 8 : 4 : 5

⇒ Sum of ratios = 8 + 4 + 5 = 17 units

⇒ 17 units = ₹9,520

⇒ 1 unit = 9520 ÷ 17 = ₹560

⇒ Share of Y = 4 units = 4 × 560 = ₹2,240

⇒ Therefore, Option 3 is the correct result.

∴ The correct answer is ₹2,240.

Additional Information

Ratio Simplification

To convert a fractional ratio into a whole number ratio, multiply each term by the Least Common Multiple (LCM) of all the denominators.

Unitary Method in Ratios

Divide the total quantity by the sum of the ratio parts to find the value of a single unit (1 part = Total ÷ Sum of Ratios).

Direct Proportion

The share of any individual is directly proportional to their respective ratio part when a fixed sum is divided among multiple entities.