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Shortcut Trick

Current sum of ages = Sum 5 years ago + (2 × 5) = 92 + 10 = 102 years.

Age difference = 84 years (given).

Smita (younger person) age = (Sum − Difference) ÷ 2

Smita age = (102 − 84) ÷ 2 = 18 ÷ 2 = 9 years.

∴ The correct answer is 9 years.

Alternate Method

Given:

Difference between Lata and Smita = 84 years.

Sum of their ages 5 years ago = 92 years.

Calculations:

⇒ Let Lata present age be L and Smita present age be S.

⇒ L − S = 84 ... (1)

⇒ (L − 5) + (S − 5) = 92

⇒ L + S − 10 = 92

⇒ L + S = 102 ... (2)

⇒ Subtracting equation (1) from equation (2):

⇒ (L + S) − (L − S) = 102 − 84

⇒ 2S = 18

⇒ S = 18 ÷ 2 = 9 years.

∴ The correct answer is 9 years.

Additional Information

Age Difference Consistency

The age difference between two individuals always remains the same, regardless of how many years pass or how many years ago it was.

Sum of Ages Concept

To find the present sum from the sum 'n' years ago for 'k' people, use: Present Sum = Past Sum + (n × k).

Linear Equations in Ages

Given Sum (A+B) and Difference (A−B), the larger value A = (Sum + Diff) ÷ 2 and the smaller value B = (Sum − Diff) ÷ 2.

Shortcut Trick

To simplify mixed fractions quickly, separate the integers (whole numbers) from the proper fractions.

Integer part: 2 + 1 − 1 = 2

Fractional part: 1/2 + 3/4 − 1/4 = 1/2 + (3/4 − 1/4) = 1/2 + 2/4 = 1/2 + 1/2 = 1

Final result: 2 + 1 = 3

∴ The correct answer is 3.

Alternate Method

Given: 2(1)/(2) + 1(3)/(4) − 1(1)/(4)

Convert the mixed fractions into improper fractions to perform the operation.

⇒ 2(1)/(2) = (2 × 2 + 1) / 2 = 5/2

⇒ 1(3)/(4) = (1 × 4 + 3) / 4 = 7/4

⇒ 1(1)/(4) = (1 × 4 + 1) / 4 = 5/4

⇒ 5/2 + 7/4 − 5/4

Taking the Least Common Multiple (LCM) of the denominators (2, 4, 4), which is 4:

⇒ (5 × 2) / (2 × 2) + 7/4 − 5/4

⇒ 10/4 + 7/4 − 5/4

⇒ (10 + 7 − 5) / 4

⇒ 12 / 4

⇒ 3

∴ The correct answer is 3.

Additional Information

BODMAS Rule

The BODMAS rule dictates the order of operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

Mixed Fraction Conversion

A mixed fraction (A b/c) is converted to an improper fraction using the formula: ((A × c) + b) ÷ c.

LCM in Fractions

To add or subtract fractions with unlike denominators, you must find a common denominator, typically the Lowest Common Multiple (LCM) of the given denominators.

Shortcut Trick

According to the Angle Bisector Theorem, the ratio of the sides containing the bisected angle is proportional to the segments of the opposite side.

PQ ÷ PR = QA ÷ AR

12 ÷ PR = 4.2 ÷ 3.8

PR = (12 × 3.8) ÷ 4.2 = 45.6 ÷ 4.2 ≈ 10.857 cm

Rounding to one decimal place, we get 10.9 cm.

∴ The correct answer is 10.9 cm.

Alternate Method

Given:

Length of PQ = 12 cm

Length of QA = 4.2 cm

Length of AR = 3.8 cm

Formula Used:

Internal Angle Bisector Theorem: In triangle PQR, if PA is the bisector of ∠P, then PQ/PR = QA/AR.

Calculations:

⇒ PQ/PR = QA/AR

⇒ 12/PR = 4.2/3.8

⇒ 12/PR = 42/38

⇒ 12/PR = 21/19

⇒ PR = (12 × 19) ÷ 21

⇒ PR = (4 × 19) ÷ 7

⇒ PR = 76 ÷ 7

⇒ PR = 10.85714...

⇒ PR ≈ 10.9 cm (rounded to one decimal place)

∴ The correct answer is 10.9 cm.

Additional Information

Angle Bisector Length Formula

The length of the internal angle bisector PA can be calculated using the formula: PA² = PQ × PR − QA × AR.

External Angle Bisector Theorem

If the external bisector of ∠P meets the extended side QR at point A', then PQ/PR = QA'/RA'.

Incenter and Ratio

The incenter I of a triangle divides the angle bisector PA in the ratio PI : IA = (PQ + PR) : QR.

Shortcut Trick

Equating the exponents of the first two terms: 2a − b = a + b ⇒ a = 2b.

Convert the square root to a power: √27 = (3³)^1/2 = 3^3/2.

Equate the power of the second term to the constant: a + b = 3/2.

Substitute a = 2b into the equation: 2b + b = 3/2 ⇒ 3b = 3/2.

Solve for b: b = 1/2.

∴ The correct answer is 1/2.

Alternate Method

Given:

3^2a−b = 3^a+b = √27

Formula Used:

1. √x = x^1/2
2. If a^m = aⁿ, then m = n.

Calculations:

⇒ √27 = (3³)^1/2 = 3^3/2

⇒ From the given equality, we have two equations by equating the indices:

⇒ 2a − b = 3/2 ... (i)

⇒ a + b = 3/2 ... (ii)

⇒ Adding equation (i) and (ii):

⇒ (2a − b) + (a + b) = 3/2 + 3/2

⇒ 3a = 3

⇒ a = 1

⇒ Substitute a = 1 in equation (ii):

⇒ 1 + b = 3/2

⇒ b = 3/2 − 1

⇒ b = 1/2

∴ The correct answer is 1/2.

Additional Information

Laws of Indices

When multiplying same bases, add exponents: a^m × aⁿ = a^m+n. When dividing, subtract exponents: a^m ÷ aⁿ = a^m−n.

Power Equality Rule

If a^x = b^x and x ≠ 0, then a = b. Conversely, if the bases are the same, the powers must be equal.

Radical to Exponent

Any root can be expressed as a fractional power: ⁿ√x = x^1/n. For square roots, n = 2.

Shortcut Trick

Use the multiplication factor for successive reductions:

Effective Price = 50,000 × 0.90 (10% off) × 0.88 (12% off) × 0.95 (5% off)

⇒ 50,000 × (9/10) × (22/25) × (19/20)

⇒ 10 × 9 × 22 × 19 = 37,620

∴ The correct answer is ₹37,620.

Alternate Method

Given:

Marked Price (MP) of the laptop = ₹50,000

First Discount = 10%

Second Discount = 12%

Credit Card Cashback = 5%

Step-by-step Calculation:

⇒ Price after 10% discount = 50,000 − (10% of 50,000)

⇒ Price after 1^st discount = 50,000 − 5,000 = ₹45,000

⇒ Price after 12% festival discount = 45,000 − (12% of 45,000)

⇒ 12% of 45,000 = 12 × 450 = 5,400

⇒ Price after 2^nd discount = 45,000 − 5,400 = ₹39,600

⇒ Effective price after 5% cashback = 39,600 − (5% of 39,600)

⇒ 5% of 39,600 = 39,600 / 20 = 1,980

⇒ Effective Payment = 39,600 − 1,980 = ₹37,620

∴ The correct answer is ₹37,620.

Additional Information

Successive Discount Formula

For two successive discounts of a% and b%, the single equivalent discount is (a + b − ab/100)%.

Effective Multiplier Method

The final price is calculated by multiplying the original price by (1 − d₁/100) × (1 − d₂/100) × ... × (1 − d_n/100).

Cashback vs Discount

A discount reduces the price at the time of purchase, whereas a cashback is effectively a post-purchase reduction that lowers the net cost.

Shortcut Trick

Time Ratio (Kirti : Vaishnavi) = 12 months : 9 months = 4 : 3

Profit Ratio (Kirti : Vaishnavi) = 7 : 6

Capital Ratio = Profit ÷ Time = (7/4) : (6/3) = 7 : 8

Since 7 units = ₹35,000, then 1 unit = ₹5,000.

Vaishnavi's Capital = 8 units = 8 × 5,000 = ₹40,000.

∴ The correct answer is ₹40,000.

Alternate Method

Given:

Kirti's investment (C₁) = ₹35,000

Kirti's time period (T₁) = 12 months

Vaishnavi's time period (T₂) = 12 − 3 = 9 months

Profit ratio = 7 : 6

Formula Used:

Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)

Calculations:

⇒ 7 / 6 = (35000 × 12) / (X × 9)

⇒ 7 / 6 = (35000 × 4) / (X × 3)

⇒ 7 / 6 = 140000 / 3X

⇒ 7 × 3X = 140000 × 6

⇒ 21X = 840000

⇒ X = 840000 ÷ 21

⇒ X = ₹40,000

∴ The correct answer is ₹40,000.

Additional Information

Compound Partnership

When partners invest different amounts for different time periods, it is called a compound partnership. The profit is always divided in the ratio of the product of Capital and Time.

Capital Ratio Formula

If Profit ratio (P₁:P₂) and Time ratio (T₁:T₂) are known, the Capital Ratio is calculated as (P₁/T₁) : (P₂/T₂).

Working Partner

A partner who manages the business is a Working Partner and may receive a fixed percentage of the profit before the remaining profit is distributed in the investment ratio.

Shortcut Trick

Total Interest percentage for 2 years = 10% × 2 = 20%

According to the question, 20% of the sum = ₹720

Then, 100% of the sum = (720 ÷ 20) × 100

Principal = 36 × 100 = ₹3600

∴ The correct answer is ₹3600.

Alternate Method

Given:

Simple Interest (SI) = ₹720

Time (T) = 2 years

Rate (R) = 10% per annum

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Calculations:

⇒ 720 = (P × 10 × 2) ÷ 100

⇒ 720 = 20P ÷ 100

⇒ 720 = P ÷ 5

⇒ P = 720 × 5

⇒ P = 3600

∴ The correct answer is ₹3600.

Additional Information

Amount in Simple Interest

The total amount (A) is the sum of the Principal (P) and the Simple Interest (SI): A = P + SI.

Difference between SI and CI

For a period of 2 years, the difference (D) between Compound Interest and Simple Interest is given by: D = P(R ÷ 100)².

Rate of Interest calculation

If the interest, principal, and time are known, the rate can be found using: R = (SI × 100) ÷ (P × T).

Shortcut Trick

Difference in percentage = 91% − 36% = 55%

Total difference in marks = 187 (Passed) + 60.5 (Failed) = 247.5

Since 55% = 247.5, then 1% = 247.5 ÷ 55 = 4.5

Maximum marks (100%) = 4.5 × 100 = 450

∴ The correct answer is 450 marks.

Alternate Method

Given:

Student scores 91% and passes by 187 marks.

Student scores 36% and fails by 60.5 marks.

Calculations:

Let the Maximum Marks be M.

⇒ Passing Marks = 91% of M − 187 ... (1)

⇒ Passing Marks = 36% of M + 60.5 ... (2)

Equating (1) and (2):

⇒ 0.91M − 187 = 0.36M + 60.5

⇒ 0.91M − 0.36M = 187 + 60.5

⇒ 0.55M = 247.5

⇒ M = 247.5 ÷ 0.55

⇒ M = 2475 ÷ 5.5 = 450

∴ The correct answer is 450 marks.

Additional Information

Percentage Score Calculation

Marks Scored = (Percentage ÷ 100) × Maximum Marks. This is the fundamental base for all exam-based percentage problems.

Passing Marks Relation

Passing Marks = Scored Marks + Marks failed by = Scored Marks − Marks passed by. Use this to equate two different scoring scenarios.

Net Percentage Gap

When comparing two scores, the difference in their percentages always corresponds directly to the difference in the absolute marks obtained.

Shortcut Trick

For the expansion of (x + a)(x + b)(x + c), the coefficient of x is given by the sum of products of constants taken two at a time: (ab + bc + ca).

Here, a = 9, b = −6, and c = 5.

Coefficient of x = (9 × −6) + (−6 × 5) + (5 × 9)

Coefficient of x = −54 − 30 + 45 = −39.

∴ The correct answer is −39.

Alternate Method

Given: Algebraic expression (x + 9)(x − 6)(x + 5)

Formula Used: Product of binomials and trinomials expansion.

⇒ First, multiply (x + 9) and (x − 6):

⇒ (x + 9)(x − 6) = x(x − 6) + 9(x − 6)

⇒ x² − 6x + 9x − 54 = x² + 3x − 54

⇒ Now, multiply this result by (x + 5):

⇒ (x² + 3x − 54)(x + 5)

⇒ x(x² + 3x − 54) + 5(x² + 3x − 54)

⇒ x³ + 3x² − 54x + 5x² + 15x − 270

⇒ x³ + 8x² − 39x − 270

⇒ Identifying the term with x, we get −39x.

∴ The correct answer is −39.

Additional Information

General Expansion of (x + a)(x + b)(x + c)

The full expansion is x³ + (a + b + c)x² + (ab + bc + ca)x + abc.

Coefficient of x²

The coefficient of x² is the sum of the constants: (a + b + c).

Constant Term

The constant term (term without x) in this expansion is the product of all constants: (a × b × c).

Shortcut Trick

Present sum of ages = Sum 5 years ago + (2 persons × 5 years) = 60 + 10 = 70 years.

Since Ayushi is older, Nidhi's age = (Sum − Difference) ÷ 2.

Nidhi's present age = (70 − 14) ÷ 2 = 56 ÷ 2 = 28 years.

∴ The correct answer is 28 years.

Alternate Method

Given: Difference in present ages = 14 years. Sum of ages 5 years ago = 60 years. Ayushi is older than Nidhi.

Formula Used: Present Sum = Past Sum + (Total Persons × Years Elapsed)

Let the present age of Ayushi be A and Nidhi be N.

⇒ A − N = 14 --- (Eq. 1)

⇒ (A − 5) + (N − 5) = 60

⇒ A + N − 10 = 60

⇒ A + N = 70 --- (Eq. 2)

⇒ Subtracting (Eq. 1) from (Eq. 2):

⇒ (A + N) − (A − N) = 70 − 14

⇒ 2N = 56

⇒ N = 28 years.

∴ The correct answer is 28 years.

Additional Information

Constant Age Difference

The difference between the ages of two individuals remains the same at any point in time (past, present, or future).

Sum-Difference Variables

If the sum (S) and difference (D) of two numbers are known, the larger number is (S + D) ÷ 2 and the smaller number is (S − D) ÷ 2.

Ages in 'n' Years

If the current sum of ages of 'k' people is S, then after 'n' years, the sum will be S + (k × n).