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Shortcut Trick

Profit 25% = 1/4 in fraction form.

For a dishonest dealer selling at cost price, the Weight Ratio is the inverse of the Price Ratio.

Price Ratio (CP : SP) = 4 : 5 ⇒ Weight Ratio (False : True) = 4 : 5.

Given True Weight (5 units) = 1000 gm.

1 unit = 200 gm ⇒ Actual Weight (4 units) = 4 × 200 = 800 gm.

∴ The correct answer is 800 gm.

Alternate Method

Given:

Claimed sale at cost price (CP).

Profit percentage = 25%.

Standard True Weight = 1 kg = 1000 gm.

Formula Used:

Profit % = [(True Weight − False Weight) ÷ False Weight] × 100

Calculations:

Let the false weight used be x gm.

⇒ 25 = [(1000 − x) ÷ x] × 100

⇒ 25 ÷ 100 = (1000 − x) ÷ x

⇒ 1 ÷ 4 = (1000 − x) ÷ x

⇒ x = 4 × (1000 − x)

⇒ x = 4000 − 4x

⇒ 5x = 4000

⇒ x = 800 gm

∴ The correct answer is 800 gm.

Additional Information

Dishonest Dealer Concept

When a shopkeeper sells at CP but uses less weight, the Profit % = (Error ÷ Actual Value Given) × 100.

Dual Profit Scenario

If the dealer marks up the price by P% and also uses a false weight, Net Profit % = [((100+P) × True Weight) ÷ False Weight] − 100.

Inverse Proportion Rule

Profit % and weight given are inversely related; to gain 25% (1/4), the weight must be reduced to 4/5 of the original.

Shortcut Trick

Total interest percentage = Rate × Time = 16% × 3 = 48%.

Since 48% of the Principal = ₹1,320.

Principal (100%) = (1320 ÷ 48) × 100 = 27.5 × 100 = ₹2,750.

∴ The correct answer is ₹2,750.

Alternate Method

Given:

Simple Interest (SI) = ₹1,320

Time (T) = 3 years

Rate (R) = 16% per annum

Formula used:

Principal (P) = (SI × 100) ÷ (R × T)

⇒ P = (1320 × 100) ÷ (16 × 3)

⇒ P = 132000 ÷ 48

⇒ P = 2750

∴ The correct answer is ₹2,750.

Additional Information

Total Amount in Simple Interest

The total amount (A) is the sum of the principal and the interest earned: A = P + SI = P × (1 + RT/100).

SI vs CI for First Year

For the first year, if the interest is compounded annually, the Simple Interest and Compound Interest are always equal.

Difference between CI and SI for 2 Years

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

Shortcut Trick

In problems involving Sum (S) and Difference (D) of two values, the larger value (Mother's age) is calculated as:

Mother's age = (Sum + Difference) ÷ 2

⇒ Mother's age = (52 + 24) ÷ 2

⇒ Mother's age = 76 ÷ 2 = 38 years.

∴ The correct answer is 38 years.

Alternate Method

Given:

Sum of Mother's age (M) and Daughter's age (D) = 52 years

Difference between their ages = 24 years

Let the Mother's age be M and the Daughter's age be D.

According to the question:

1. M + D = 52

2. M − D = 24

By adding equation (1) and equation (2):

⇒ (M + D) + (M − D) = 52 + 24

⇒ 2M = 76

⇒ M = 76 ÷ 2

⇒ M = 38 years.

∴ The correct answer is 38 years.

Additional Information

Linear Equations in Two Variables

Equations of the form x + y = A and x − y = B can be solved quickly by finding x = (A + B) ÷ 2 and y = (A − B) ÷ 2.

Age Difference Property

The difference between the ages of two people remains constant regardless of how many years pass or how many years ago it was.

Finding the Smaller Value

To find the younger person's age (Daughter), subtract the difference from the sum and divide by 2: (52 − 24) ÷ 2 = 14 years.

Shortcut Trick

Convert all numbers into like decimals: 7.0, 12.4, and 1.0.

Remove the decimal point to treat them as whole numbers: 70, 124, and 10.

Calculate the LCM of 70, 124, and 10 using prime factorization.

LCM(70, 124, 10) = 4340.

Divide the result by 10 to place the decimal point back: 4340 ÷ 10 = 434.

∴ The correct answer is 434.

Alternate Method

Given: The numbers are 7, 12.4, and 1.

Formula Used: LCM of Fractions = LCM(Numerators) ÷ HCF(Denominators).

⇒ Represent numbers as fractions: 7 = 7/1, 12.4 = 124/10 = 62/5, 1 = 1/1.

⇒ To find the LCM of 7/1, 62/5, and 1/1, find the LCM of numerators (7, 62, 1) and HCF of denominators (1, 5, 1).

⇒ LCM(7, 62, 1) = 7 × 62 = 434 (since 7 and 62 have no common factors).

⇒ HCF(1, 5, 1) = 1.

⇒ Required LCM = 434 ÷ 1 = 434.

∴ The correct answer is 434.

Additional Information

LCM of Fractions

The Least Common Multiple of a set of fractions is given by the formula: LCM = LCM of Numerators ÷ HCF of Denominators.

HCF of Fractions

The Highest Common Factor of a set of fractions is given by the formula: HCF = HCF of Numerators ÷ LCM of Denominators.

LCM of Decimals

To find the LCM of decimals, first convert all numbers into like decimals (equal decimal places) and then find the LCM as if they were integers, finally adjusting the decimal point.

Shortcut Trick

Net multiplier = (1 + 0.20) × (1 − 0.10) × (1 + 0.15)

Net multiplier = 1.20 × 0.90 × 1.15 = 1.08 × 1.15 = 1.242

Overall percentage change = (1.242 − 1) × 100 = 24.2%

∴ The correct answer is 24.2% increase.

Alternate Method

Given: Year 1 change = 20% increase; Year 2 change = 10% decrease; Year 3 change = 15% increase.

Formula: Final Revenue = Initial Revenue × (100 + r₁)/100 × (100 + r₂)/100 × (100 + r₃)/100

⇒ Let the initial Revenue be 100 units.

⇒ Revenue after 1^st year = 100 + (20% of 100) = 120 units.

⇒ Revenue after 2^nd year = 120 − (10% of 120) = 120 − 12 = 108 units.

⇒ Revenue after 3^rd year = 108 + (15% of 108) = 108 + 16.2 = 124.2 units.

⇒ Overall percentage change = 124.2 − 100 = 24.2%.

∴ The correct answer is 24.2% increase.

Additional Information

Successive Percentage Formula

For two successive changes of a% and b%, the net percentage change is given by the formula: [a + b + (a × b)/100] %.

Multiplying Factor Method

An x% increase corresponds to a multiplier of (1 + x/100), while an x% decrease corresponds to a multiplier of (1 − x/100). The product of these factors gives the final result.

Base Value Adjustment

In successive changes, each percentage change is applied to the updated value from the previous step, not the original starting value.

Shortcut Trick

Metal Volume = π(R² − r²)h = 396

Given thickness (R − r) = 1 ⇒ (R − r)(R + r) = 9 ⇒ R + r = 9

Solving R − r = 1 and R + r = 9 gives inner radius (r) = 4 units.

Inner volume = πr²h = (22/7) × 16 × 14 = 704 cubic units.

∴ The correct answer is 704 cubic units.

Alternate Method

Given: Length (h) = 14 units, Thickness (t) = 1 unit, Metal Volume = 396 cubic units.

Formula Used: Volume of metal = πh(R² − r²); Inner volume = πr²h

Let the inner radius be r units and outer radius be R units.

⇒ Thickness = R − r = 1 unit ⇒ R = r + 1

⇒ Volume of metal = π × (R² − r²) × h

⇒ 396 = (22/7) × ((r + 1)² − r²) × 14

⇒ 396 = 22 × (r² + 2r + 1 − r²) × 2

⇒ 396 = 44 × (2r + 1)

⇒ 2r + 1 = 396 ÷ 44

⇒ 2r + 1 = 9

⇒ 2r = 8 ⇒ r = 4 units

⇒ Inner volume = πr²h

⇒ Inner volume = (22/7) × 4² × 14

⇒ Inner volume = 22 × 16 × 2 = 704 cubic units

∴ The correct answer is 704 cubic units.

Additional Information

Volume of Hollow Cylinder

The volume of the material in a hollow cylinder is given by V = πh(R² − r²), where R is the outer radius and r is the inner radius.

Total Surface Area (TSA)

The Total Surface Area of a hollow cylinder is the sum of the inner CSA, outer CSA, and the area of the two circular rings: TSA = 2πh(R + r) + 2π(R² − r²).

Thickness Property

The thickness (t) is the difference between the outer radius and the inner radius: t = R − r.

Shortcut Trick

Rise in water level (h) = (Total volume of 30 spheres) ÷ (Area of cylinder base)

h = [30 × (4/3) × π × 7³] ÷ [π × 20²]

h = [40 × 343] ÷ 400

h = 13720 ÷ 400 = 34.3 cm

∴ The correct answer is 34.3 cm.

Alternate Method

Given:

Radius of cylindrical tank (R) = 20 cm

Radius of each solid sphere (r) = 7 cm

Number of solid spheres (n) = 30

Formula Used:

Volume of cylinder = π × R² × h

Volume of sphere = (4/3) × π × r³

Calculations:

Let the rise in water level in the cylinder be h cm.

When the spheres are completely immersed, the volume of water displaced is equal to the total volume of the 30 spheres:

⇒ Volume of water risen in cylinder = 30 × Volume of one sphere

⇒ π × R² × h = 30 × (4/3) × π × r³

Canceling π from both sides:

⇒ R² × h = 10 × 4 × r³

⇒ 20² × h = 40 × 7³

⇒ 400 × h = 40 × 343

⇒ 400 × h = 13720

⇒ h = 13720 ÷ 400

⇒ h = 34.3 cm

∴ The correct answer is

Shortcut Trick

Let the total distance be the LCM of denominators (5 and 4) = 20 units.

Distance by train = 3/5 × 20 = 12 units.

Distance by car = 1/4 × 20 = 5 units.

Remaining distance on foot = 20 − (12 + 5) = 3 units.

Given that 3 units = 300 m ⇒ 1 unit = 100 m.

Total distance = 20 units = 2000 m = 2 km.

∴ The correct answer is 2 km.

Alternate Method

Let the total distance from home to office be d meters.

Distance traveled by train = 3/5 of d

Distance traveled by car = 1/4 of d

Distance traveled on foot = 300 m

⇒ Train distance + Car distance + Foot distance = Total distance

⇒ (3/5) × d + (1/4) × d + 300 = d

⇒ 300 = d − (3/5 + 1/4) × d

⇒ 300 = d − [(12 + 5) / 20] × d

⇒ 300 = d − (17/20) × d

⇒ 300 = (3/20) × d

⇒ d = (300 × 20) ÷ 3

⇒ d = 2000 m

Converting meters into kilometers:

⇒ d = 2000 ÷ 1000 = 2 km

∴ The correct answer is 2 km.

Additional Information

Fraction of a Whole

The sum of all fractional parts of any quantity is always equal to 1. The remaining fraction is calculated as 1 − (sum of known fractions).

Unit Conversion

To convert distance from meters to kilometers, divide the value by 1000 (since 1 km = 1000 m).

Distance-Speed-Time Relation

Distance is directly proportional to speed and time: Distance = Speed × Time.

Given:

2^x+1 × 4^y−2 = 32

x = 3y

Formula used:

a^m × aⁿ = a^m+n

(a^m)ⁿ = a^mn

Calculation:

Expressing the equation in base 2:

2^x+1 × (2²)^y−2 = 2⁵

⇒ 2^x+1 × 2^2(y−2) = 2⁵

⇒ 2^x+1 × 2^2y−4 = 2⁵

⇒ 2^x+1+2y−4 = 2⁵

⇒ 2^x+2y−3 = 2⁵

Comparing the exponents:

⇒ x + 2y − 3 = 5

⇒ x + 2y = 8

Substituting x = 3y from the given data:

⇒ 3y + 2y = 8

⇒ 5y = 8

⇒ y = 8/5

⇒ y = 1.6

∴ The correct answer is option (3).

Shortcut Trick

A's age after 8 years = 32 years.

A's present age = 32 − 8 = 24 years.

Ratio of present ages A : B = 4 : 5.

4 units → 24 years ⇒ 1 unit = 6 years.

B's present age = 5 units × 6 = 30 years.

∴ The correct answer is 30 years.

Alternate Method

Given:

Ratio of present ages of A and B = 4 : 5

Age of A after 8 years = 32 years

Calculations:

Let the present age of A be 4x and B be 5x.

According to the question, A's age after 8 years is 32.

⇒ 4x + 8 = 32

⇒ 4x = 32 − 8

⇒ 4x = 24

⇒ x = 24 ÷ 4

⇒ x = 6

Now, calculate B's present age:

⇒ B's present age = 5x

⇒ B's present age = 5 × 6 = 30 years.

Therefore, the value matches Option (2).

∴ The correct answer is 30 years.

Additional Information

Constant Age Difference

The difference between the ages of two persons always remains constant, regardless of how many years pass or have passed.

Ratio Scaling

If the ratio of ages is given as a:b, and the age of one person is known, the other can be found using the unitary method (Age ÷ Ratio Part × Required Part).

Future and Past Ages

To find age 'n' years after, add 'n' to the present age. To find age 'n' years ago, subtract 'n' from the present age.