Quantitative Aptitude PYQs

Shortcut Trick

Angle between bisectors of ∠C and ∠D in a quadrilateral = (∠A + ∠B) ÷ 2

Given: ∠A = 36° and ∠B = 93°

∠DOC = (36° + 93°) ÷ 2

∠DOC = 129° ÷ 2 = 64.5°

∴ The correct answer is 64.5°.

Alternate Method

Given: ∠A = 36°, ∠B = 93°, CO and DO are bisectors of ∠C and ∠D.

Formula: Sum of angles in a quadrilateral = 360° | Sum of angles in a triangle = 180°

⇒ ∠A + ∠B + ∠C + ∠D = 360°

⇒ 36° + 93° + ∠C + ∠D = 360°

⇒ 129° + ∠C + ∠D = 360°

⇒ ∠C + ∠D = 360° − 129° = 231°

⇒ Divide by 2: (∠C + ∠D) / 2 = 231° / 2 = 115.5°

⇒ In ΔDOC, ∠DOC + ∠OCD + ∠ODC = 180°

⇒ ∠DOC + (∠C / 2) + (∠D / 2) = 180°

⇒ ∠DOC + 115.5° = 180°

⇒ ∠DOC = 180° − 115.5° = 64.5°

∴ The correct answer is 64.5°.

Additional Information

Sum of Interior Angles

For any n-sided polygon, the sum of interior angles is (n − 2) × 180°.

Angle Bisector Relation

In a quadrilateral, the angle between the internal bisectors of two opposite angles is half the difference of the other two angles.

Cyclic Quadrilateral

In a cyclic quadrilateral, opposite angles are supplementary, meaning their sum is always 180°.

Shortcut Trick

Smallest two-digit multiple of 6 = 12

Largest two-digit multiple of 6 = 96

Total numbers (n) = [(Last Term − First Term) ÷ Common Difference] + 1

Total numbers (n) = [(96 − 12) ÷ 6] + 1

n = [84 ÷ 6] + 1 = 14 + 1 = 15

∴ The correct answer is 15.

Alternate Method

Given: Two-digit numbers range from 10 to 99. We need to find multiples of 6 in this range.

Formula used: n^th term of an Arithmetic Progression (A.P.), a_n = a + (n − 1)d

⇒ First term (a) = 12

⇒ Last term (a_n) = 96

⇒ Common difference (d) = 6

⇒ 96 = 12 + (n − 1) × 6

⇒ 96 − 12 = (n − 1) × 6

⇒ 84 = (n − 1) × 6

⇒ n − 1 = 84 ÷ 6

⇒ n − 1 = 14

⇒ n = 14 + 1 = 15

∴ The correct answer is 15.

Additional Information

Arithmetic Progression (AP)

A sequence of numbers where the difference between consecutive terms is constant, denoted as 'd'.

Sum of n terms of AP

The sum S_n can be calculated using S_n = (n ÷ 2)[2a + (n − 1)d] or S_n = (n ÷ 2)[a + l].

Divisibility Rule of 6

A number is divisible by 6 if it is divisible by both 2 (ends in even digit) and 3 (sum of digits is divisible by 3).

Shortcut Trick

Efficiency ratio of Rishi : Shan = 100 : 110 = 10 : 11.

Since Efficiency is inversely proportional to Time, Time ratio = 11 : 10.

Rishi takes 11 units of time, which is given as 11 days.

∴ 1 unit = 1 day.

Shan takes 10 units of time = 10 × 1 = 10 days.

∴ The correct answer is 10.

Alternate Method

Given: Rishi completes work in 11 days. Shan is 10% more efficient than Rishi.

Let efficiency of Rishi = 10 units/day.

⇒ Efficiency of Shan = 10 + (10% of 10) = 11 units/day.

⇒ Total Work = Efficiency of Rishi × Time taken by Rishi

⇒ Total Work = 10 × 11 = 110 units.

⇒ Time taken by Shan = Total Work ÷ Efficiency of Shan

⇒ Time = 110 ÷ 11 = 10 days.

∴ The correct answer is 10.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete a work. If the ratio of efficiency is x : y, the ratio of time taken is y :

x. Work Formula

Total Work = Efficiency × Time. This remains constant for a specific piece of work regardless of who performs it.

Percentage Efficiency Increase

If A is k% more efficient than B, then Efficiency_A = Efficiency_B × (1 + k/100).

Shortcut Trick

Total remaining fraction = (1 − 1/5) × (1 − 1/4) = 4/5 × 3/4 = 3/5

Here, 3 units represent the remaining amount ₹525.

1 unit = 525 ÷ 3 = 175

Original amount (5 units) = 175 × 5 = ₹875

∴ The correct answer is ₹875.

Alternate Method

Given:

Loss percentage = 20%

Expenditure from remainder = 25%

Final amount left = ₹525

Formula Used:

Final Value = Initial Value × (100 − R₁)/100 × (100 − R₂)/100

Calculations:

Let the original amount be ₹ x.

⇒ Amount left after 20% loss = x − (20/100) × x = 0.8x

⇒ Amount spent from remainder = 25% of 0.8x = (25/100) × 0.8x = 0.2x

⇒ Final amount left = 0.8x − 0.2x = 0.6x

⇒ 0.6x = 525

⇒ x = 525 ÷ 0.6

⇒ x = 5250 ÷ 6

⇒ x = 875

∴ The correct answer is ₹875.

Additional Information

Successive Percentage Change

If a value changes by a% and then b%, the net effective change is given by (a + b + ab/100)%.

Successive Discounts

Two successive discounts of x% and y% are equivalent to a single discount of (x + y − xy/100)%.

Percentage to Fraction Conversion

To simplify calculations, use 20% = 1/5 and 25% = 1/4 to find the multipliers directly.

Shortcut Trick

When distance is constant, the ratio of time is the inverse of the ratio of speed.

Ratio of speeds = 2 : 3 : 5

Ratio of times = (3 × 5) : (2 × 5) : (2 × 3)

Ratio of times = 15 : 10 : 6

∴ The correct answer is 15 : 10 : 6.

Alternate Method

Given:

Ratio of speeds of three cars (S₁ : S₂ : S₃) = 2 : 3 : 5

Distance (D) = Constant for all three cars

Formula Used:

Time (T) = Distance (D) ÷ Speed (S)

Calculations:

⇒ T₁ : T₂ : T₃ = (D/S₁) : (D/S₂) : (D/S₃)

⇒ T₁ : T₂ : T₃ = (1/2) : (1/3) : (1/5)

⇒ To simplify, multiply each term by the LCM of 2, 3, and 5.

⇒ LCM(2, 3, 5) = 30

⇒ T₁ : T₂ : T₃ = (1/2 × 30) : (1/3 × 30) : (1/5 × 30)

⇒ T₁ : T₂ : T₃ = 15 : 10 : 6

∴ The correct answer is 15 : 10 : 6.

Additional Information

Inverse Proportionality

When the product of two variables is constant (Distance = Speed × Time), the variables are inversely proportional; as speed increases, time decreases.

Average Speed

For two equal distances covered at speeds x and y, Average Speed = 2xy ÷ (x + y).

Unit Conversion

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

Shortcut Trick

A number is divisible by 18 if it is divisible by both 2 and 9.

Divisibility by 2 requires the last digit X to be even (0, 2, 4, 6, 8).

Among the given options (5, 3, 2, 1), only 2 is an even number.

Thus, X must be 2 to satisfy the first condition of divisibility by 18.

∴ The correct answer is 2.

Alternate Method

Given number: 8732326687X

Condition: Number is divisible by 18.

Method: Check divisibility rules for 2 and 9 (since 18 = 2 × 9).

⇒ Divisibility by 9: Sum of digits must be divisible by 9.

⇒ Sum = 8 + 7 + 3 + 2 + 3 + 2 + 6 + 6 + 8 + 7 + X

⇒ Sum = 52 + X

⇒ The nearest multiple of 9 greater than 52 is 54 (9 × 6 = 54).

⇒ 52 + X = 54

⇒ X = 54 − 52 = 2

⇒ Divisibility by 2: The last digit X must be even. Since X = 2 is even, this condition is satisfied.

∴ The correct answer is 2.

Additional Information

Divisibility by 9

A number is divisible by 9 if the sum of its digits is a multiple of 9.

Divisibility by 2

A number is divisible by 2 if its last digit is even, i.e., 0, 2, 4, 6, or 8.

Composite Divisibility Rule

If a number is divisible by two co-prime numbers a and b, then it is also divisible by their product (a × b).

Shortcut Trick

Ratio of speeds A : B = 5 : 2

Ratio of distances A : B = 5 : 2 (since time is constant)

Difference in distance units = 5 − 2 = 3 units

According to question, 3 units = 80 m (head start)

Winning post distance (A's distance) = 5 units = (80 ÷ 3) × 5 = 133¹/₃ m

∴ The correct answer is 133¹/₃ m.

Alternate Method

Given:

Speed of A = 5/2 × Speed of B

Head start given to B = 80 m

Formula Used:

Distance = Speed × Time

Calculations:

⇒ Speed of A / Speed of B = 5 / 2

⇒ Let the distance of the winning post from A's start be D meters.

⇒ Time taken by A = D ÷ Speed_A

⇒ Time taken by B = (D − 80) ÷ Speed_B

⇒ Since they reach at the same time: D / Speed_A = (D − 80) / Speed_B

⇒ D / (D − 80) = Speed_A / Speed_B = 5 / 2

⇒ 2 × D = 5 × (D − 80)

⇒ 2D = 5D − 400

⇒ 3D = 400

⇒ D = 400 / 3 = 133¹/₃ m

∴ The correct answer is 133¹/₃ m.

Additional Information

Constant Time Property

If two bodies take the same time to cover different distances, the ratio of their distances is equal to the ratio of their speeds: D₁/D₂ = S₁/S₂.

Concept of Head Start

In a race of length L, if A gives B a start of x meters, A has to run L meters while B has to run only (L − x) meters.

Dead Heat

A race is said to end in a dead heat if all participants reach the winning post at exactly the same time.

Shortcut Trick

In a 30°−60°−90° triangle, the ratio of sides Height : Base : Hypotenuse is 1 : √3 : 2.

Side opposite to 60° (Base) = 48 m.

√3 units = 48 m

1 unit (Height) = 48/√3 = (16 × 3)/√3 = 16√3 m.

∴ The correct answer is 16√3.

Alternate Method

Given:

Distance of the point from the foot of the tower (Base) = 48 m

Angle of elevation (θ) = 30°

Formula used: tan θ = Perpendicular (Height) / Base

⇒ Let the height of the tower be h.

⇒ tan 30° = h / 48

⇒ 1/√3 = h / 48

⇒ h = 48 / √3

⇒ h = (16 × 3) / √3

⇒ h = 16√3 m

∴ The correct answer is 16√3.

Additional Information

Trigonometric Ratios for Standard Angles

Standard values: tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3.

Angle of Elevation

It is the angle formed by the line of sight with the horizontal when the object being viewed is above the horizontal level.

30°−60°−90° Triangle Property

The side opposite to 30° is x, the side opposite to 60° is x√3, and the hypotenuse is 2x.

Shortcut Trick

Factorize the numbers using the largest common factor:

45 = 9 × 5

63 = 9 × 7

81 = 9 × 9

LCM = 9 × (Remaining factors: 5 × 7 × 9)

LCM = 9 × 315 = 2835

∴ The correct answer is 2835.

Alternate Method

Given: Numbers 45, 63, and 81.

Formula Used: LCM by Prime Factorization Method (Product of highest powers of all prime factors).

⇒ Prime factors of 45 = 3 × 3 × 5 = 3² × 5¹

⇒ Prime factors of 63 = 3 × 3 × 7 = 3² × 7¹

⇒ Prime factors of 81 = 3 × 3 × 3 × 3 = 3⁴

⇒ LCM = Highest powers of involved primes (3, 5, 7)

⇒ LCM = 3⁴ × 5¹ × 7¹

⇒ LCM = 81 × 5 × 7

⇒ LCM = 81 × 35 = 2835

∴ The correct answer is 2835.

Additional Information

HCF (Highest Common Factor)

The HCF is the largest positive integer that divides each of the numbers without a remainder. For 45, 63, and 81, the HCF is 9.

Relationship for Two Numbers

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. This rule does not directly apply to more than two numbers.

LCM of Fractions

LCM of Fractions = (LCM of Numerators) ÷ (HCF of Denominators). This is essential for simplifying fraction-based arithmetic.

Shortcut Trick

Use multiplying factors for discount and VAT: MP × (100 − D)% × (100 + VAT)% = Final SP

x × 0.46 × 1.52 = 437

x = 437 / (0.46 × 1.52)

x = 437 / 0.6992 = 625

∴ The correct answer is 625.

Alternate Method

Given:

Marked Price (MP) = ₹x

Discount = 54% | VAT = 52% | Final SP = ₹437

Formula Used:

Price after tax = Price before tax × (1 + Tax/100)

Selling Price before tax = MP × (1 − Discount/100)

⇒ Price after discount = Final SP ÷ (1 + VAT/100)

⇒ Price after discount = 437 ÷ 1.52 = 287.5

⇒ Marked Price (x) = Price after discount ÷ (1 − Discount/100)

⇒ x = 287.5 ÷ (1 − 0.54)

⇒ x = 287.5 ÷ 0.46

⇒ x = 625

∴ The correct answer is 625.

Additional Information

Successive Percentage Change

When a value is changed by a% and then by b%, the effective change is [a + b + (ab/100)]%.

Marked Price vs Selling Price

Discount is always calculated on the Marked Price (MP), while Profit or Loss is calculated on the Cost Price (CP).

VAT and Taxes

Value Added Tax (VAT) or GST is usually applied to the net selling price after all discounts have been deducted.

Shortcut Trick

Convert 1/5 to percentage: 1/5 × 100 = 20%.

Both percentages are calculated on the same base value (45).

Difference in percentage = 60% − 20% = 40%.

Required difference = 40% of 45.

Required difference = 0.4 × 45 = 18.

∴ The correct answer is 18.

Alternate Method

Given: Main number = 45

Formula Used: Value = (Percentage/100) × Number | Value = Fraction × Number

⇒ First value = 60% of 45

⇒ First value = (60/100) × 45 = 0.6 × 45 = 27

⇒ Second value = 1/5 of 45

⇒ Second value = (1/5) × 45 = 9

⇒ Difference = 27 − 9

⇒ Difference = 18

∴ The correct answer is 18.

Additional Information

Percentage to Fraction Conversion

To convert a percentage to a fraction, divide by 100. Example: 60% = 60/100 = 3/5.

Fraction to Percentage Conversion

To convert a fraction to a percentage, multiply by 100. Example: 1/5 × 100 = 20%.

Common Base Property

If multiple percentages or fractions are applied to the same number, we can subtract or add the proportions directly before multiplying with the base value.

Shortcut Trick

Ratio of prices = 3 : 5 : 7 ⇒ Total ratio units = 3 + 5 + 7 = 15 units.

Total price = Average × Number of items = 16,425 × 3.

15 units = 16,425 × 3 ⇒ 5 units = 16,425.

1 unit = 3,285.

Costliest item (7 units) = 7 × 3,285 = 22,995.

∴ The correct answer is 22,995.

Alternate Method

Given:

Average price of 3 items = ₹16,425

Ratio of prices = 3 : 5 : 7

Formula Used:

Total Value = Average × Number of items

Value of a part = (Ratio of that part / Sum of Ratios) × Total Value

Calculations:

Let the prices of the three items be 3x, 5x, and 7x.

⇒ Total Price = 3x + 5x + 7x = 15x

⇒ Average = Total Price ÷ 3

⇒ 16,425 = 15x ÷ 3

⇒ 16,425 = 5x

⇒ x = 16,425 ÷ 5

⇒ x = 3,285

⇒ Price of the costliest item = 7x

⇒ Price = 7 × 3,285

⇒ Price = 22,995

∴ The correct answer is 22,995.

Additional Information

Average Concept

The average of a set of observations is the sum of observations divided by the total number of observations. Sum = Average × n.

Ratio and Proportion

If a total amount A is divided into parts according to ratio a : b : c, then the value of the largest part is [max(a,b,c) / (a+b+c)] × A.

Weighted Average

When items have different weights w₁, w₂, the combined average is (w₁x₁ + w₂x₂) ÷ (w₁ + w₂).

Shortcut Trick

Total Surface Area of a cylinder is 2πr(h + r), which involves π (22/7).

The answer must be a multiple of 11.

Check options for 11 divisibility: 1144 → (1+4) − (1+4) = 0 (Divisible by 11).

Other options: 1121 (1), 1175 (2), 1106 (6) are not divisible by 11.

∴ The correct answer is 1144 cm².

Alternate Method

Given:

Volume of cylinder (V) = 2926 cm³

Height (h) = 19 cm

Formula Used:

Volume (V) = πr²h

Total Surface Area (TSA) = 2πr(h + r)

Calculations:

⇒ Volume = πr²h = 2926

⇒ (22/7) × r² × 19 = 2926

⇒ r² = (2926 × 7) / (22 × 19)

⇒ r² = (154 × 7) / 22

⇒ r² = 7 × 7

⇒ r = 7 cm

⇒ TSA = 2πr(h + r)

⇒ TSA = 2 × (22/7) × 7 × (19 + 7)

⇒ TSA = 44 × 26

⇒ TSA = 1144 cm²

∴ The correct answer is 1144 cm².

Additional Information

Curved Surface Area (CSA)

The lateral surface area of a cylinder excluding its top and bottom bases is given by 2πrh.

Total Surface Area (TSA)

The sum of the area of the curved surface and the areas of the two circular bases: TSA = 2πrh + 2πr² = 2πr(h + r).

Volume of Cylinder

The capacity or space occupied by a solid cylinder is V = πr²h.

Shortcut Trick

Successive Discount Formula = x + y − (x × y)/100

Where x = 5% and y = 8%

Net Discount = 5 + 8 − (5 × 8)/100

Net Discount = 13 − 40/100 = 13 − 0.4 = 12.6%

∴ The correct answer is 12.6%.

Alternate Method

Given: First discount (d₁) = 5%, Second discount (d₂) = 8%

Formula: Final Selling Price = MP × (100 − d₁)/100 × (100 − d₂)/100

Let the Marked Price (MP) be 100 units.

⇒ Price after 1^st discount = 100 − (5% of 100) = 100 − 5 = 95 units.

⇒ Final Selling Price (SP) after 2^nd discount = 95 − (8% of 95)

⇒ SP = 95 − (8 × 95)/100 = 95 − 7.6 = 87.4 units.

⇒ Total Discount = MP − SP = 100 − 87.4 = 12.6 units.

⇒ Single Equivalent Discount % = (12.6/100) × 100 = 12.6%

∴ The correct answer is 12.6%.

Additional Information

Successive Discount for Three Rates

For three discounts a%, b%, and c%, the net discount is calculated as 100 − [100 × (1 − a/100) × (1 − b/100) × (1 − c/100)].

Profit/Loss and Discounts

Marked Price (MP) = CP × (100 + Profit%)/(100 − Discount%). This relates the cost price to the marked price via the profit and discount percentages.

Successive Percentage Change

The net change for successive increases/decreases x and y is given by the formula x + y + (xy/100), where negative signs are used for decreases.

Shortcut Trick

Assume the average for all 4 numbers is 17.

Net deviation from the average must be 0.

Deviations: (6.5 − 17) = −10.5, (10.7 − 17) = −6.3, (20.2 − 17) = +3.2.

Sum of deviations = −10.5 − 6.3 + 3.2 = −13.6.

To balance, value of a = 17 + 13.6 = 30.6.

∴ The correct answer is 30.6.

Alternate Method

Given:

Numbers = 6.5, 10.7, 20.2, and a

Average = 17

Total count (n) = 4

Formula:

Average = (Sum of observations) ÷ (Number of observations)

⇒ Average = (6.5 + 10.7 + 20.2 + a) ÷ 4

⇒ 17 = (37.4 + a) ÷ 4

⇒ 17 × 4 = 37.4 + a

⇒ 68 = 37.4 + a

⇒ a = 68 − 37.4

⇒ a = 30.6

∴ The correct answer is 30.6.

Additional Information

Arithmetic Mean

The most common type of average, calculated by dividing the sum of all values by the total number of values.

Sum of Deviations

The sum of deviations of a set of observations from their arithmetic mean is always zero: ∑(x_i − Average) = 0.

Weighted Average

Used when different values have different levels of importance or frequency: Average = ∑(w_i × x_i) ÷ ∑w_i.

Shortcut Trick

Net Amount = 8442 − (7 + 73 + 68) = 8294

Total Ratio Units = 6 + 7 + 16 = 29 units

Value of 1 unit = 8294 ÷ 29 = 286

Ratio difference (C − B) = 16 − 7 = 9 units

Difference in shares = (9 × 286) + (68 − 73) = 2574 − 5 = 2569

∴ The correct answer is ₹2,569.

Alternate Method

Given:

Total Amount = ₹8,442

Deductions: S = ₹7, B = ₹73, C = ₹68

New Ratio S : B : C = 6 : 7 : 16

Calculations:

⇒ Total Deductions = 7 + 73 + 68 = ₹148

⇒ Remaining amount = 8442 − 148 = ₹8,294

⇒ 29 units = 8294

⇒ 1 unit = 8294 ÷ 29 = 286

⇒ B's original share = (7 × 286) + 73 = 2002 + 73 = ₹2,075

⇒ C's original share = (16 × 286) + 68 = 4576 + 68 = ₹4,644

⇒ Difference = 4644 − 2075 = ₹2,569

∴ The correct answer is ₹2,569.

Additional Information

Ratio Division with Constants

If (A−x):(B−y):(C−z) = a:b:c, then the sum (A+B+C) − (x+y+z) is distributed in the ratio a:b:c.

Difference between Shares

Difference between A and B = |(a × k + x) − (b × k + y)|, where k is the value of one ratio unit.

Compounded Ratio

The compounded ratio of a:b and c:d is (a×c):(b×d).

Shortcut Trick

From 2K + 2W = 48, we get 1 Keychain (K) + 1 Wallet (W) = 24.

Multiply by 3: 3K + 3W = 72.

Subtracting this from (4K + 3W = 75): (4K − 3K) = 75 − 72 ⇒ K = 3.

Since K + W = 24, then W = 24 − 3 = 21.

Cost of 3K + 1W = 3(3) + 21 = 9 + 21 = 30.

∴ The correct answer is ₹ 30.

Alternate Method

Given:

4 × Cost of Keychain (K) + 3 × Cost of Wallet (W) = ₹ 75 --- (i)

2 × Cost of Keychain (K) + 2 × Cost of Wallet (W) = ₹ 48 --- (ii)

Formula Used:

Linear equations in two variables: a₁x + b₁y = c₁ and a₂x + b₂y = c₂

Calculations:

⇒ From (ii), divide by 2: K + W = 24

⇒ W = 24 − K --- (iii)

⇒ Substitute (iii) in (i): 4K + 3(24 − K) = 75

⇒ 4K + 72 − 3K = 75

⇒ K + 72 = 75

⇒ K = 75 − 72 = 3

⇒ Put K = 3 in (iii): W = 24 − 3 = 21

⇒ Required Cost = 3K + 1W

⇒ Required Cost = 3(3) + 21

⇒ Required Cost = 9 + 21 = 30

∴ The correct answer is ₹ 30.

Additional Information

Elimination Method

A process of solving equations by adding or subtracting them to cancel out one variable, usually after multiplying one or both by constant factors.

Substitution Method

Expressing one variable in terms of the other from one equation and substituting this expression into the second equation.

Nature of Solutions

A system has a unique solution if a₁/a₂ ≠ b₁/b₂, no solution if a₁/a₂ = b₁/b₂ ≠ c₁/c₂, and infinite solutions if a₁/a₂ = b₁/b₂ = c₁/c₂.

Shortcut Trick

Present Ratio = 3 : 4 (Difference = 1); Past Ratio (4 yrs ago) = 7 : 10 (Difference = 3).

Equalize differences by multiplying present ratio by 3 → 9 : 12.

Change in units: 9 − 7 = 2 units (or 12 − 10 = 2 units).

2 units = 4 years ⇒ 1 unit = 2 years.

Present ages: A = 9 × 2 = 18 years and B = 12 × 2 = 24 years.

∴ The correct answer is 18 years and 24 years.

Alternate Method

Given: Present ratio of A and B = 3 : 4. Ratio 4 years ago = 7 : 10.

Formula: Age = Ratio Part × Common Multiplier (x).

Let the present ages of A and B be 3x and 4x respectively.

⇒ (3x − 4) / (4x − 4) = 7 / 10

⇒ 10 × (3x − 4) = 7 × (4x − 4)

⇒ 30x − 40 = 28x − 28

⇒ 30x − 28x = 40 − 28

⇒ 2x = 12

⇒ x = 6

⇒ Present age of A = 3 × 6 = 18 years

⇒ Present age of B = 4 × 6 = 24 years

∴ The correct answer is 18 years and 24 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two individuals remains constant regardless of the number of years passed or preceding.

Ratio Balancing Method

If the difference between ratio parts is not the same, multiply the ratios by the difference of the opposite ratio to balance the units.

Future and Past Ages

If present age is x, age after n years = x + n; age n years ago = x − n.

Shortcut Trick

Original Price = 100% = ₹120

New Price after 15% increase = 115% of Original Price

New Price = 1.15 × 120

New Price = ₹138

∴ The correct answer is ₹138.

Alternate Method

Given:

Original Price = ₹120

Percentage Increase = 15%

Formula Used:

New Price = Original Price + (Increase% × Original Price)

Calculations:

⇒ Amount of increase = 15% of 120

⇒ Amount of increase = (15 ÷ 100) × 120

⇒ Amount of increase = 0.15 × 120 = ₹18

⇒ New Price = 120 + 18

⇒ New Price = ₹138

∴ The correct answer is ₹138.

Additional Information

Percentage Increase Formula

New Value = Original Value × (1 + r/100), where r is the percentage increase rate.

Successive Percentage Change

If a value is increased by x% and then y%, the net effective percentage change is [x + y + (xy/100)]%.

Fractional Conversion

An increase of 15% is equivalent to a multiplier of 23/20 (since 15% = 3/20 and 1 + 3/20 = 23/20).

Shortcut Trick

Volume of a cube = (Side)³

Given side = 19 m, which ends in the digit 9.

The unit digit of 9³ is 9 (9 × 9 × 9 = 729).

Therefore, the result must end with 9.

Only option (4) 6859 satisfies this condition.

∴ The correct answer is 6859.

Alternate Method

Given: Edge of the cube (a) = 19 m

Formula Used: Volume of cube = a³

⇒ Volume = 19³

⇒ Volume = 19 × 19 × 19

⇒ Volume = 361 × 19

⇒ Volume = 6859 m³

∴ The correct answer is 6859.

Additional Information

Total Surface Area of Cube

The total surface area of a cube with side 'a' is given by 6a².

Lateral Surface Area of Cube

The area of the four vertical faces is called the lateral surface area, calculated as 4a².

Diagonal of a Cube

The longest rod that can be placed inside a cube is its diagonal, which is equal to a√3.

Perimeter of Cube

The sum of all the edge lengths of a cube is 12a.