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Shortcut Trick

Using the direct identity: (Difference)² = (Sum)² − 4 × Product

Given: Sum = 51 and Product = 230

(Difference)² = 51² − 4 × 230

(Difference)² = 2601 − 920 = 1681

Difference = √1681 = 41

∴ The correct answer is 41.

Alternate Method

Given: Sum of two numbers = 51, Product = 230.

Formula Used: Quadratic equation x² − (Sum)x + Product = 0.

⇒ Let the numbers be x and y.

⇒ x² − 51x + 230 = 0

⇒ x² − 46x − 5x + 230 = 0

⇒ x(x − 46) − 5(x − 46) = 0

⇒ (x − 46)(x − 5) = 0

⇒ The numbers are 46 and 5.

⇒ Positive difference = 46 − 5 = 41.

∴ The correct answer is 41.

Additional Information

Algebraic Identity for Difference

The square of the difference is related to the sum and product by (a − b)² = (a + b)² − 4ab.

Sum and Product of Roots

For a quadratic equation ax² + bx + c = 0, the sum of roots is −b/a and the product of roots is c/a.

Symmetry in Numbers

If two numbers have a fixed sum, their product is maximum when the numbers are equal (difference is zero).

Shortcut Trick

Age difference remains constant: 28 years.

Ratio 7 years ago = 5 : 1. Difference = 4 units.

4 units = 28 years ⇒ 1 unit = 7 years.

Total units 7 years ago = 5 + 1 = 6 units.

Sum 7 years ago = 6 × 7 = 42 years.

Present sum = 42 + (2 × 7) = 56 years.

∴ The correct answer is 56 years.

Alternate Method

Given: Present age difference = 28 years. Ratio 7 years ago = 5 : 1.

Let the present age of the son be S years.

Then, the present age of Ramu = S + 28 years.

According to the condition (7 years ago):

⇒ (S + 28 − 7) = 5 × (S − 7)

⇒ S + 21 = 5S − 35

⇒ 5S − S = 21 + 35

⇒ 4S = 56

⇒ S = 14 years

Ramu's age = 14 + 28 = 42 years.

⇒ Sum of their ages = 42 + 14 = 56 years.

∴ The correct answer is 56 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two people always remains the same, regardless of how many years pass.

Ratio of Ages

If the ratio of ages is x : y and the difference is D, then (x − y) units = D. This helps in finding actual values quickly.

Sum of Ages Change

If there are 'n' people, the sum of their ages increases by (n × t) after 't' years.

Shortcut Trick

Ratio of first to second number (a:b) = 2 : 1.

For fourth proportional, a/b = c/d ⇒ 2/1 = c/120 ⇒ third number (c) = 240.

Since c = 5a, then first number (a) = 240 ÷ 5 = 48.

Since a = 2b, then second number (b) = 48 ÷ 2 = 24.

Total Sum = 48 + 24 + 240 = 312.

∴ The correct answer is 312.

Alternate Method

Given:

Let the three numbers be a, b, and c.

First number (a) = 2 × Second number (b)

Third number (c) = 5 × First number (a)

Fourth proportional (d) = 120

Formula Used:

If four numbers a, b, c, and d are in proportion, then a/b = c/d.

Calculations:

⇒ Let the second number b = x.

⇒ First number a = 2x.

⇒ Third number c = 5 × (2x) = 10x.

⇒ According to the proportion rule: a/b = c/d

⇒ (2x) / x = (10x) / 120

⇒ 2 = 10x / 120

⇒ 2 × 120 = 10x

⇒ 240 = 10x ⇒ x = 24

⇒ a = 2 × 24 = 48

⇒ b = 24

⇒ c = 10 × 24 = 240

⇒ Sum of all three numbers = a + b + c = 48 + 24 + 240 = 312.

∴ The correct answer is 312.

Additional Information

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d is calculated as d = (b × c) ÷ a.

Mean Proportional

The mean proportional between two numbers a and b is the square root of their product, given by √(a × b).

Third Proportional

For two numbers a and b, the third proportional c satisfies the relation a/b = b/c, which implies c = b² ÷ a.

Continued Proportion

Numbers a, b, c, d... are in continued proportion if a/b = b/c = c/d and so on.

Shortcut Trick

Speed of 54 km/h means covering 54 km in 60 minutes.

Given Time = 20 minutes, which is exactly 1/3 of an hour (60 ÷ 3 = 20).

Therefore, Distance covered = 1/3 of the hourly speed.

Distance = 54 ÷ 3 = 18 km.

∴ The correct answer is 18 km.

Alternate Method

Given:

Speed (S) = 54 km/h

Time (T) = 20 minutes

Formula Used:

Distance (D) = Speed × Time

⇒ Convert Time from minutes to hours:

⇒ Time = 20 ÷ 60 hr = 1/3 hr

⇒ Now, substitute values in the formula:

⇒ Distance = 54 × (1/3)

⇒ Distance = 18 km

∴ The correct answer is 18 km.

Additional Information

Unit Conversion Rules

To convert km/h to m/s, multiply the speed by 5/18. To convert m/s to km/h, multiply the speed by 18/5.

Average Speed Formula

If a journey is covered at different speeds, Average Speed = Total Distance ÷ Total Time. For two equal distances with speeds x and y, Average Speed = 2xy ÷ (x + y).

Relative Speed

When two objects move in the same direction, Relative Speed = S₁ − S₂. In opposite directions, Relative Speed = S₁ + S₂.

Shortcut Trick

The equation p² − q² = 45 can be factored as (p + q)(p − q) = 45.

Since p and q are prime numbers, we test factor pairs of 45: (9, 5), (15, 3), and (45, 1).

For the pair (9, 5): p + q = 9 and p − q = 5.

Solving these: 2p = 14 ⇒ p = 7; 2q = 4 ⇒ q = 2.

Since both 7 and 2 are distinct positive prime numbers, p + q = 9.

∴ The correct answer is 9.

Alternate Method

Given: p and q are distinct positive prime numbers such that p > q and p² − q² = 45.

Formula Used: Algebraic Identity a² − b² = (a + b)(a − b).

⇒ p² − q² = 45

⇒ (p + q)(p − q) = 45

⇒ Possible factor pairs (p + q, p − q) where (p + q) > (p − q):

⇒ 1) p + q = 45, p − q = 1 ⇒ 2p = 46 ⇒ p = 23, q = 22 (22 is not prime)

⇒ 2) p + q = 15, p − q = 3 ⇒ 2p = 18 ⇒ p = 9 (9 is not prime)

⇒ 3) p + q = 9, p − q = 5 ⇒ 2p = 14 ⇒ p = 7, q = 2 (Both are prime)

⇒ The values p = 7 and q = 2 satisfy the condition of being distinct primes.

⇒ Therefore, p + q = 7 + 2 = 9.

∴ The correct answer is 9.

Additional Information

Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The only even prime number is 2.

Difference of Squares

The algebraic identity a² − b² = (a − b)(a + b) is used to factorize the difference of two squared terms into the product of their sum and difference.

Factors of 45

The integer factors of 45 are 1, 3, 5, 9, 15, and 45. In problems involving prime variables, testing these factor pairs is the most efficient method.

Shortcut Trick

Cash discount = 15%.

Quantity discount for getting 100 gm free on 1.9 kg = [100 / (1900 + 100)] × 100 = 5%.

Using successive discount formula: Net Discount = d₁ + d₂ − (d₁ × d₂ / 100).

Net Discount = 15 + 5 − (15 × 5 / 100) = 20 − 0.75 = 19.25%.

∴ The correct answer is 19.25%.

Alternate Method

Given: Discount on rice = 15%, Purchase quantity = 1.9 kg, Free quantity = 100 gm.

Formula Used: Overall Discount % = (Total Discount Value / Total Marked Price) × 100.

Let the Marked Price (MP) of 1 kg rice be ₹1000.

⇒ Total rice received by customer = 1.9 kg + 0.1 kg = 2.0 kg.

⇒ Total Marked Price of 2.0 kg rice = 2 × 1000 = ₹2000.

⇒ Price of 1.9 kg rice = 1.9 × 1000 = ₹1900.

⇒ Selling Price (SP) after 15% discount = 1900 × (100 − 15) / 100 = 1900 × 0.85 = ₹1615.

⇒ Total Discount offered = Total MP − SP = 2000 − 1615 = ₹385.

⇒ Overall Discount Percentage = (385 / 2000) × 100 = 19.25%.

∴ The correct answer is 19.25%.

Additional Information

Successive Discounts

When two discounts a% and b% are applied one after another, the single equivalent discount is calculated as [a + b − (ab / 100)]%.

Buy X Get Y Free

In a 'Buy X Get Y Free' scheme, the discount percentage is calculated using the formula: [Y / (X + Y)] × 100.

Marked Price and Selling Price Relationship

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

Shortcut Trick

Calculate the effective interest rate for the given 5-month period.

Monthly Interest Rate = 12% ÷ 12 = 1% per month.

Total Effective Rate for 5 months = 1% × 5 = 5%.

Since 5% of the Principal corresponds to ₹15,000.

Invested Amount (100%) = (15,000 ÷ 5) × 100 = ₹3,00,000.

∴ The correct answer is ₹3,00,000.

Alternate Method

Given: Simple Interest (SI) = ₹15,000, Rate of Interest (R) = 12% per annum, Time (T) = 5 months.

Formula Used: SI = (P × R × T) ÷ 100

⇒ Time in years (T) = 5 ÷ 12 years

⇒ 15,000 = (P × 12 × 5/12) ÷ 100

⇒ 15,000 = (P × 5) ÷ 100

⇒ 15,000 = P ÷ 20

⇒ P = 15,000 × 20

⇒ P = ₹3,00,000

∴ The correct answer is ₹3,00,000.

Additional Information

Simple Interest Formula

SI = (P × R × T) ÷ 100, where P is Principal, R is Rate per annum, and T is Time in years.

Total Amount (A)

The total sum returned after the time period is A = P + SI = P(1 + RT/100).

Time Conversion

Always convert time into years if the rate is per annum (Months ÷ 12, Days ÷ 365).

Shortcut Trick

The sum of the ratio parts is 2 + 3 = 5.

Since consecutive angles of a parallelogram are supplementary, their sum is 180°.

⇒ 5 units = 180°

⇒ 1 unit = 36°

⇒ Smaller angle = 2 units = 2 × 36° = 72°.

∴ The correct answer is 72°.

Alternate Method

Given:

Ratio of consecutive angles = 2 : 3

Formula Used:

Sum of consecutive interior angles of a parallelogram = 180°

Calculations:

Let the two consecutive angles be 2x and 3x.

⇒ 2x + 3x = 180°

⇒ 5x = 180°

⇒ x = 180° ÷ 5

⇒ x = 36°

⇒ Smaller angle = 2x = 2 × 36°

⇒ Smaller angle = 72°

⇒ Larger angle = 3x = 3 × 36° = 108°

∴ The correct answer is 72°.

Additional Information

Opposite Angles Property

In any parallelogram, opposite angles are always equal. If one angle is 72°, its opposite angle is also 72°.

Sum of Interior Angles

The sum of all four interior angles of a parallelogram (or any quadrilateral) is always 360°.

Parallel Lines and Transversals

Consecutive angles in a parallelogram are co-interior angles formed by a transversal intersecting two parallel lines, which are always supplementary (sum to 180°).

Shortcut Trick

Total Distance to cover = Sum of lengths = 250 + 300 = 550 m

Relative Speed (Opposite) = 72 + 108 = 180 km/hr

Convert Speed to m/s: 180 × 5/18 = 50 m/s

Time = Total Distance ÷ Relative Speed = 550 ÷ 50 = 11 sec

∴ The correct answer is 11 sec.

Alternate Method

Given:

Length of first train (L₁) = 250 m

Length of second train (L₂) = 300 m

Speed of first train (S₁) = 72 km/hr

Speed of second train (S₂) = 108 km/hr

Formula Used:

Time (t) = (L₁ + L₂) ÷ (Relative Speed)

Calculations:

⇒ Total Distance = L₁ + L₂ = 250 + 300 = 550 m

⇒ Relative Speed = S₁ + S₂ = 72 + 108 = 180 km/hr

⇒ Conversion to m/s = 180 × (5/18) m/s

⇒ Relative Speed = 10 × 5 = 50 m/s

⇒ Time to cross = 550 ÷ 50

⇒ Time = 11 seconds

∴ The correct answer is 11 sec.

Additional Information

Relative Speed Rules

When two objects move in the opposite direction, Relative Speed = S₁ + S₂. When they move in the same direction, Relative Speed = |S₁ − S₂|.

Unit Conversion

To convert speed from km/hr to m/s, multiply by 5/18. To convert from m/s to km/hr, multiply by 18/5.

Crossing Distance

When a train crosses another train (stationary or moving), the total distance to be covered is always the sum of the lengths of both trains.

Shortcut Trick

If the ratio of two numbers is a : b and their HCF is H, then their LCM is given by the formula: LCM = H × a × b (provided a and b are co-prime).

Given Ratio = 8 : 3 and HCF = 8.

Since 8 and 3 are co-prime (their common factor is 1):

⇒ LCM = 8 × 8 × 3 = 192.

∴ The correct answer is 192.

Alternate Method

Given:

Ratio of two numbers = 8 : 3

Highest Common Factor (HCF) = 8

Formula Used:

Number = Ratio part × HCF

LCM × HCF = Product of the two numbers

⇒ Let the two numbers be 8x and 3x.

⇒ The HCF of 8x and 3x is x (since 8 and 3 have no common factors).

⇒ According to the question, x = 8.

⇒ First number = 8 × 8 = 64

⇒ Second number = 3 × 8 = 24

⇒ Using the product rule: LCM × HCF = 64 × 24

⇒ LCM × 8 = 1536

⇒ LCM = 1536 ÷ 8 = 192

∴ The correct answer is 192.

Additional Information

Product Rule of HCF and LCM

For any two numbers a and b, the product of the numbers is equal to the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).

Co-prime Numbers

Two numbers are said to be co-prime if their HCF is 1. For co-prime numbers, their LCM is simply the product of the two numbers.

HCF and LCM of Fractions

HCF = (HCF of Numerators) ÷ (LCM of Denominators) | LCM = (LCM of Numerators) ÷ (HCF of Denominators).