Quantitative Aptitude PYQs

Shortcut Trick

Apply the BODMAS rule to evaluate multiplication and division first.

First part: 0.7 × 72 ÷ 14 = 50.4 ÷ 14 = 3.6

Second part: 8.34 × 0.15 = 1.251

Substitute the parts back into the equation: 3.6 + 6.25 − 1.251 + 5.95

Add the positive terms together and subtract: 15.80 − 1.251 = 14.549

∴ The correct answer is 14.549.

Alternate Method

Given:

Expression: 0.7 × 72 ÷ 14 + 6.25 − 8.34 × 0.15 + 5.95

Concept Used:

BODMAS Rule: Brackets, Order, Division, Multiplication, Addition, and Subtraction.

Calculations:

⇒ 0.7 × 72 ÷ 14 + 6.25 − 8.34 × 0.15 + 5.95

⇒ (0.7 × 72) ÷ 14 + 6.25 − (8.34 × 0.15) + 5.95

⇒ 50.4 ÷ 14 + 6.25 − 1.251 + 5.95

⇒ 3.6 + 6.25 − 1.251 + 5.95

Group all the positive values together first to make the addition easier:

⇒ (3.6 + 6.25 + 5.95) − 1.251

⇒ 15.80 − 1.251

⇒ 14.549

∴ The correct answer is 14.549.

Additional Information

BODMAS Rule

It specifies the sequence of mathematical operations: Brackets first, then Orders (powers/roots), followed by Division and Multiplication (left to right), and finally Addition and Subtraction (left to right).

Decimal Addition and Subtraction

When adding or subtracting decimal numbers, always align the decimal points precisely before proceeding with the calculation to prevent place-value errors.

Decimal Multiplication

Multiply the numbers as whole integers, then place the decimal point in the resulting product so that it has the exact total number of decimal places present in both of the original factors.

Shortcut Trick

Income ratio (David : Ashish) = 5 : 8

Expenditure ratio = 2 : 4 = 1 : 2

Since savings are equal, make the difference between income and expenditure parts equal.

Multiply expenditure ratio (1:2) by 3 → 3 : 6. Now, Savings = Income − Expenditure = 5−3 = 2 units and 8−6 = 2 units.

2 units = ₹81,000 ⇒ 1 unit = ₹40,500.

David income = 5 units = 5 × 40,500 = ₹2,02,500.

∴ The correct answer is ₹2,02,500.

Alternate Method

Given: Ratio of Income = 5 : 8, Ratio of Expenditure = 2 : 4, Savings of each = ₹81,000.

Formula: Income − Savings = Expenditure

⇒ Let David income = 5x and Ashish income = 8x.

⇒ According to question: (5x − 81,000) / (8x − 81,000) = 2 / 4

⇒ (5x − 81,000) / (8x − 81,000) = 1 / 2

⇒ 2(5x − 81,000) = 1(8x − 81,000)

⇒ 10x − 1,62,000 = 8x − 81,000

⇒ 2x = 1,62,000 − 81,000

⇒ 2x = 81,000

⇒ x = 40,500

⇒ David income = 5x = 5 × 40,500 = 2,02,500

∴ The correct answer is ₹2,02,500.

Additional Information

Concept of Savings

Savings is the surplus of income over expenditure: Savings = Income − Expenditure.

Ratio and Proportion Application

If two ratios are given and a constant amount is added/subtracted, we use the cross-multiplication method or x-variable method to solve for the original values.

Income-Expenditure Relationship

Income is always equal to the sum of Expenditure and Savings (I = E + S).

Shortcut Trick

Total Surface Area of n equal parts formed by m diametrical planes passing through a common axis = 4πr² + 2mπr².

Each diametrical cut adds 2 circular faces (each πr²) to the total area.

Given m = 3 planes and r = 14 cm.

Total Surface Area = 4πr² + 2(3)πr² = 10πr².

Calculation ⇒ 10 × (22/7) × 14 × 14 = 6160 cm².

∴ The correct answer is 6160 cm².

Alternate Method

Given:

Radius of sphere (r) = 14 cm

Number of diametrical planes = 3

Number of equal parts = 6

Formula Used:

Surface area of sphere = 4πr²

Area of one circular cross-section = πr²

Calculations:

⇒ Original surface area of sphere = 4 × (22/7) × 14 × 14

⇒ Original area = 4 × 22 × 2 × 14 = 2464 cm²

⇒ Each diametrical cut creates 2 new circular faces.

⇒ Total new faces created by 3 planes = 3 × 2 = 6 faces.

⇒ Total area of new faces = 6 × πr²

⇒ New faces area = 6 × (22/7) × 14 × 14 = 6 × 22 × 2 × 14

⇒ New faces area = 3696 cm²

⇒ Total surface area of all pieces = Original area + New faces area

⇒ Total surface area = 2464 + 3696 = 6160 cm²

∴ The correct answer is 6160 cm².

Additional Information

Surface Area of Hemisphere

Curved surface area (CSA) = 2πr² and Total surface area (TSA) = 3πr².

Volume of Sphere

The volume of a sphere is given by V = (4/3)πr³.

Cutting Rule for Solids

Every time a solid is cut, the total volume remains constant, but the total surface area increases by the area of the newly exposed surfaces.

Shortcut Trick

Total Deduction = 65 + 39 + 77 = ₹181.

Remaining Sum = 4,445 − 181 = ₹4,264.

Total Ratio = 18 + 3 + 20 = 41 units.

Value of 1 unit = 4,264 ÷ 41 = ₹104.

Difference (B & C) = (20 − 3) units + (77 − 39) = 17 × 104 + 38 = 1,768 + 38 = ₹1,806.

∴ The correct answer is ₹1,806.

Alternate Method

Given: Total sum = ₹4,445. Deductions from A, B, C are ₹65, ₹39, ₹77 respectively. Ratio after deduction = 18 : 3 : 20.

⇒ Sum of remaining shares = 4,445 − (65 + 39 + 77)

⇒ Sum of remaining shares = 4,445 − 181 = ₹4,264

⇒ Total ratio parts = 18 + 3 + 20 = 41 units

⇒ Value of 1 unit = 4,264 ÷ 41 = ₹104

⇒ B’s share after deduction = 3 × 104 = ₹312

⇒ C’s share after deduction = 20 × 104 = ₹2,080

⇒ B’s original share = 312 + 39 = ₹351

⇒ C’s original share = 2,080 + 77 = ₹2,157

⇒ Difference = 2,157 − 351 = ₹1,806

∴ The correct answer is ₹1,806.

Additional Information

Ratio and Proportion Basics

If a total sum S is divided in the ratio a:b:c, then the share of B is [b / (a+b+c)] × S.

Handling Deductions in Ratios

When fixed amounts are deducted from shares to obtain a ratio, always subtract the sum of deductions from the total capital before applying the ratio units.

Difference between Shares

Difference between C and B can be calculated as: (Difference in ratio units × value of 1 unit) + (Difference in deducted amounts).

Shortcut Trick

Ratio of Girls to Boys = 2 : 5

Given, 2 units = 200 (Number of girls)

⇒ 1 unit = 100

∴ 5 units = 5 × 100 = 500 (Number of boys)

∴ The correct answer is 500 boys.

Alternate Method

Given:

Ratio of Girls : Boys = 2 : 5

Total number of girls = 200

Formula Used:

Value of a part = (Total value ÷ Sum of ratios) × Specific ratio part

Calculations:

⇒ Let the number of girls be 2x and boys be 5x.

⇒ 2x = 200

⇒ x = 200 ÷ 2

⇒ x = 100

⇒ Number of boys = 5x = 5 × 100

⇒ Number of boys = 500

∴ The correct answer is 500 boys.

Additional Information

Ratio Definition

A ratio compares two quantities by division. If a : b = c : d, then a × d = b × c.

Proportion Rule

To find an unknown in a ratio, equate the ratio of units to the ratio of actual values: (Part A / Part B) = (Value A / Value B).

Compound Ratio

The compound ratio of two ratios a : b and c : d is (a × c) : (b × d).

Shortcut Trick

Evaluate the terms mentally by applying the BODMAS rule:

⇒ 2² = 4, √(8²) = 8, 2√4 = 4

Substitute directly into the given expression:

⇒ 4 + 8 − 4 − 11 = −3

∴ The correct answer is −3.

Alternate Method

Given:

2² + √(8²) − 2√4 − 11

Concept Used:

BODMAS Rule: First evaluate Orders (powers and roots), then Division, Multiplication, Addition, and Subtraction in sequence.

Calculations:

⇒ 2² = 4

⇒ √(8²) = √64 = 8

⇒ √4 = 2

Substitute the evaluated terms back into the given expression:

⇒ 4 + 8 − 2 × 2 − 11

⇒ 4 + 8 − 4 − 11

⇒ 12 − 15

⇒ −3

∴ The correct answer is −3.

Additional Information

BODMAS Rule

It dictates the correct order of mathematical operations to follow: Brackets, Orders (powers and roots), Division, Multiplication, Addition, and Subtraction.

Properties of Square Roots

The principal square root of a perfect square directly gives the absolute base value: √(x²) = |x|.

Simplification of Surds

Expressions containing surds like √4 should be simplified into rational numbers (such as 2) before proceeding with addition or subtraction steps.

Shortcut Trick

The product of two consecutive positive integers is always close to the perfect square of the smaller number.

The nearest perfect square less than 272 is 256, which is 16².

The two consecutive integers are 16 and 17, as 16 × 17 = 272.

The greater of the two numbers is 17.

∴ The correct answer is 17.

Alternate Method

Given:

The product of two consecutive positive natural numbers = 272

Formula Used:

Quadratic equation standard form: ax² + bx + c = 0

Calculations:

⇒ Let the smaller positive natural number be x.

⇒ The next consecutive natural number will be (x + 1).

⇒ According to the problem, x × (x + 1) = 272

⇒ x² + x = 272

⇒ x² + x − 272 = 0

⇒ We need to split the middle term such that the product is −272 and the sum is 1.

⇒ x² + 17x − 16x − 272 = 0

⇒ x(x + 17) − 16(x + 17) = 0

⇒ (x − 16)(x + 17) = 0

⇒ So, x = 16 or x = −17

⇒ Since the number must be a positive natural number, we reject −17 and take x = 16.

⇒ The greater number is x + 1 = 16 + 1 = 17.

∴ The correct answer is 17.

Additional Information

Consecutive Numbers

Consecutive numbers are numbers that follow each other in order, from smallest to largest, without gaps. For natural numbers, the difference between consecutive terms is exactly 1 (e.g., n, n + 1, n + 2).

Quadratic Equation Roots

For a standard quadratic equation ax² + bx + c = 0, the nature of roots is determined by the discriminant D = b² − 4ac. The roots can be calculated using factorization or the formula: x = (−b ± √D) ÷ 2a.

Shortcut Trick

Apply the BODMAS rule directly, solving from the innermost bracket outwards:

⇒ Round bracket and multiply: 5 × (2 − 7) = 5 × (−5) = −25

⇒ Curly bracket: {27 − 25} = 2

⇒ Final division: 290 ÷ 2 = 145

∴ The correct answer is 145.

Alternate Method

Given:

Expression: [290 ÷ {27 + 5 × (2 − 7)}]

Concept Used:

The BODMAS Rule is applied to correctly evaluate mathematical expressions in the exact order of Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Calculations:

⇒ [290 ÷ {27 + 5 × (2 − 7)}]

⇒ Solving the round bracket first:

⇒ [290 ÷ {27 + 5 × (−5)}]

⇒ Performing multiplication inside the curly bracket:

⇒ [290 ÷ {27 − 25}]

⇒ Solving the curly bracket:

⇒ [290 ÷ 2]

⇒ Performing final division:

⇒ 145

∴ The correct answer is 145.

Additional Information

BODMAS Rule Priority

Always evaluate expressions in this exact order: Brackets → Of → Division → Multiplication → Addition → Subtraction.

Order of Brackets

When multiple brackets are present, solve them sequentially from innermost to outermost: Vinculum (Bar) → Round Brackets () → Curly Brackets {} → Square Brackets [].

Sign Rules in Multiplication

Multiplying a positive number by a negative number always yields a negative product (+a × −b = −ab).

Shortcut Trick
Divide the given number 81661 by the divisor 73 to find the remainder.
81661 ÷ 73 gives a quotient of 1118 and a remainder of 47.
To make it perfectly divisible, the smallest number to be added is the difference between the divisor and the remainder.
⇒ Required number = Divisor − Remainder = 73 − 47 = 26
∴ The correct answer is 26.

Alternate Method
Given:
Number = 81661
Divisor = 73
Formula Used:
Dividend = (Divisor × Quotient) + Remainder
Smallest number to be added = Divisor − Remainder
Calculation:
⇒ First, we divide the given number 81661 by 73.
⇒ 81661 = 73 × 1118 + 47
⇒ Here, we observe that the remainder is 47.
⇒ If we subtract 47 from 81661, the resulting number (81614) will be perfectly divisible by 73.
⇒ However, the question asks for the smallest number to be added.
⇒ To reach the next multiple of 73, we must add the remaining difference to the remainder.
⇒ Required number to add = Divisor − Remainder
⇒ Required number = 73 − 47 = 26
⇒ Verification: 81661 + 26 = 81687, and 81687 ÷ 73 = 1119 (perfectly divisible).
∴ The correct answer is 26.

Additional Information
Number to be Subtracted
To find the smallest number to be subtracted so that the number is completely divisible, the answer is simply the remainder itself.
Euclid's Division Lemma
For any two positive integers a and b, there exist unique integers q and r such that a = b × q + r, where 0 ≤ r < b.
Divisibility and Multiples
The distance to the next highest multiple of a divisor b from a given dividend a is always equal to b − r, where r is the remainder obtained when a is divided by b.

Shortcut Trick

In ΔABC, sum of angles is 180° ⇒ ∠BCA = 180° − (113° + 26°) = 41°.

Since BC ∥ AD, alternate interior angles are equal ⇒ ∠CAD = ∠BCA = 41°.

In ΔACD, given AC = CD, angles opposite to equal sides are equal ⇒ ∠ADC = ∠CAD = 41°.

In ΔACD, sum of angles is 180° ⇒ ∠ACD = 180° − (41° + 41°) = 98°.

∴ The correct answer is 98°.

Alternate Method

A
B
C
D

113°
26°
41°
41°
41°
98°

Given:

BC ∥ AD, AC = CD

∠ABC = 113° and ∠BAC = 26°

Calculations:

⇒ In ΔABC, the sum of all interior angles is 180°

⇒ ∠BCA = 180° − (∠ABC + ∠BAC)

⇒ ∠BCA = 180° − (113° + 26°) = 180° − 139° = 41°

⇒ Since BC ∥ AD, the Alternate Interior Angles are equal.

⇒ ∠CAD = ∠BCA = 41°

⇒ In ΔACD, it is given that AC = CD. Thus, angles opposite to the equal sides are also equal.

⇒ ∠ADC = ∠CAD = 41°

⇒ In ΔACD, the sum of interior angles is 180°

⇒ ∠ACD + ∠CAD + ∠ADC = 180°

⇒ ∠ACD + 41° + 41° = 180°

⇒ ∠ACD + 82° = 180°

⇒ ∠ACD = 180° − 82° = 98°

∴ The correct answer is 98°.

Additional Information

Trapezium Properties

A trapezium is a quadrilateral with at least one pair of parallel sides. The consecutive interior angles located between the parallel sides are always supplementary.

Alternate Interior Angles Theorem

When two parallel lines are cut by a transversal line, the alternate interior angles formed on opposite sides of the transversal are strictly equal to each other.

Isosceles Triangle Property

If any two sides of a triangle are equal in length, it forms an isosceles triangle. Consequently, the angles opposite to these equal sides will always be equal in measure.

Angle Sum Property

The sum of all three interior angles enclosed within any triangle universally adds up to exactly 180°.

Shortcut Trick

The discount amount can be directly calculated as a straight percentage of the marked price.

Discount = 9% of 32000

Discount = (9 ÷ 100) × 32000

Discount = 9 × 320 = 2880

∴ The correct answer is ₹2,880.

Alternate Method

Given:

Marked Price (Price of printer) = ₹32,000

Trade Discount percentage = 9%

Formula Used:

Discount Amount = Marked Price × (Discount Percentage ÷ 100)

Marked Price (MP)
₹32,000

9% Discount

Discount Amount
₹2,880

Discount = 9% of ₹32,000 = ₹2,880

Calculations:

⇒ Discount Amount = 32000 × (9 ÷ 100)

⇒ Discount Amount = 320 × 9

⇒ Discount Amount = 2880

∴ The correct answer is ₹2,880.

Additional Information

Selling Price (SP)

Selling Price is the final amount at which the item is sold after deducting the discount. Formula: SP = Marked Price − Discount.

Successive Discounts

When two discounts of a% and b% are given successively, the equivalent single discount can be calculated using the formula: [a + b − (a × b ÷ 100)]%.

Trade Discount

A trade discount is a direct percentage reduction in the listed price (marked price) of an item, commonly provided by a manufacturer or wholesaler to a retailer.

Shortcut Trick

Total Work = LCM(14, 24, 17) = 2856 units.

Combined Efficiency = (2856 ÷ 14) + (2856 ÷ 24) + (2856 ÷ 17) = 204 + 119 + 168 = 491 units/hour.

Time taken together = Total Work ÷ Combined Efficiency = 2856 ÷ 491.

Time = 5 ⁴⁰¹/₄₉₁ hours.

∴ The correct answer is 5 ⁴⁰¹/₄₉₁ hours.

Alternate Method

Given:

Time taken by Pipe A = 14 hours

Time taken by Pipe B = 24 hours

Time taken by Pipe C = 17 hours

Formula Used:

Total Time = Total Work ÷ Total Efficiency

Calculations:

⇒ Total Efficiency (A + B + C) = 204 + 119 + 168 = 491 units/hour

⇒ Time taken together = 2856 ÷ 491

⇒ Time = 5 ⁴⁰¹/₄₉₁ hours

∴ The correct answer is 5 ⁴⁰¹/₄₉₁ hours.

Additional Information

Pipes & Cisterns Efficiency Concept

Total Work is generally assumed as the LCM of individual completion times to avoid complex fractional calculations. Efficiency is defined as the amount of work done per unit time.

Inlet and Outlet Rules

An inlet pipe contributes positive work (+), while an outlet pipe or leak contributes negative work (−). The net combined efficiency is always the algebraic sum of all active pipes operating simultaneously.

Shortcut Trick

When there is neither profit nor loss, absolute loss equals absolute profit.

CP₁ × 10% = CP₂ × 15% ⇒ CP₁ : CP₂ = 15 : 10 = 3 : 2

Total ratio parts = 3 + 2 = 5 parts

5 parts = ₹1200 ⇒ 1 part = ₹240

CP of the first frock (3 parts) = 3 × 240 = ₹720

∴ The correct answer is ₹720.

Alternate Method

First Frock
Second Frock

−10%

+15%

0%

15

10

Ratio ⇒ 3 : 2
Total Parts = 5 ⇒ ₹1200

Given:

Total CP of two frocks = ₹1200

Loss on the first frock = 10%

Gain on the second frock = 15%

Overall transaction = No profit and no loss

Formula Used:

When overall outcome is 0% profit/loss, Loss Amount = Profit Amount.

Calculations:

Let the CP of the first frock be x.

Then, the CP of the second frock is (1200 − x).

⇒ Loss on 1^st = Gain on 2^nd

⇒ 10% of x = 15% of (1200 − x)

⇒ (10 ÷ 100) × x = (15 ÷ 100) × (1200 − x)

⇒ 10x = 15 × (1200 − x)

⇒ 10x = 18000 − 15x

⇒ 25x = 18000

⇒ x = 18000 ÷ 25 = 720

∴ The correct answer is ₹720.

Additional Information

No Profit No Loss Case

When a transaction results in neither a net gain nor a net loss, the absolute financial loss incurred on one item exactly counterbalances the absolute financial gain on the other. Therefore, CP₁ × Loss% = CP₂ × Profit%.

Alligation Method for Profit and Loss

Applying the rule of alligation on profit and loss percentages directly yields the ratio of the Cost Prices (CP) of the respective items. Loss values are taken as negative, while profit values are positive.

Successive Percentage Change

If the price of a single item is changed successively by P% profit and L% loss, the net effect can be evaluated using the equivalent percentage formula: (P − L − (P×L)÷100)%.

Shortcut Trick

Let the total distance be the LCM of 7 hours and 5 hours = 35 units.

Speed of Train A = 35 ÷ 7 = 5 units/hr and Speed of Train B = 35 ÷ 5 = 7 units/hr.

Distance covered by A alone in 2 hours (8:05 a.m. to 10:05 a.m.) = 5 × 2 = 10 units.

Time to cover the remaining 25 units together = 25 ÷ (5 + 7) = 25/12 hours = 2 hours 5 mins.

Meeting time = 10:05 a.m. + 2 hours 5 mins = 12:10 p.m.

∴ The correct answer is 12:10 p.m.

Alternate Method

Station M
Station N

Meeting Point
12:10 p.m.

Total Distance = 35 units

Train A
8:05 a.m.

Train B
10:05 a.m.

Given:

Train A starts at 8:05 a.m. and reaches at 3:05 p.m.

Train B starts at 10:05 a.m. and reaches at 3:05 p.m.

Formula Used:

Speed = Distance ÷ Time

Time to meet = Remaining Distance ÷ Relative Speed

Calculations:

⇒ Total time taken by Train A = 7 hours

⇒ Total time taken by Train B = 5 hours

⇒ Let the total distance between station M and station N be 35 units (LCM of 7 and 5).

⇒ Speed of Train A = 35 ÷ 7 = 5 units/hr.

⇒ Speed of Train B = 35 ÷ 5 = 7 units/hr.

⇒ Distance traveled by Train A in 2 hours (8:05 a.m. to 10:05 a.m.) = 5 × 2 = 10 units.

⇒ Remaining distance when Train B starts = 35 − 10 = 25 units.

⇒ Relative speed = Speed of A + Speed of B = 5 + 7 = 12 units/hr.

⇒ Time taken to meet = 25 ÷ 12 hours = 2 hours and (1/12 × 60) minutes = 2 hours 5 minutes.

⇒ Meeting time = 10:05 a.m. + 2 hours 5 minutes = 12:10 p.m.

∴ The correct answer is 12:10 p.m.

Additional Information

Relative Speed in Opposite Directions

When two bodies move towards each other, their relative speed is the sum of their individual speeds (S₁ + S₂).

Relative Speed in Same Direction

When two bodies move in the same direction, their relative speed is the difference of their individual speeds (|S₁ − S₂|).

Time and Speed Relationship

For a constant distance, speed is inversely proportional to the time taken (S ∝ 1/T).

Shortcut Trick

The total surface area of a cuboid is given by 2 × (lb + bh + hl).

Substitute the given dimensions directly into the formula.

Total Surface Area = 2 × (8 × 5 + 5 × 3 + 3 × 8)

Total Surface Area = 2 × (40 + 15 + 24) = 2 × 79

∴ The correct answer is 158 cm².

Alternate Method

Given:

Length (l) = 8 cm

Breadth (b) = 5 cm

Height (h) = 3 cm

8 cm
5 cm
3 cm
Total Surface Area = 158 cm2

Formula Used:

Total Surface Area of a Cuboid = 2 × (l × b + b × h + h × l)

Calculations:

⇒ Surface Area = 2 × (8 × 5 + 5 × 3 + 3 × 8)

⇒ Surface Area = 2 × (40 + 15 + 24)

⇒ Surface Area = 2 × (79)

⇒ Surface Area = 158 cm²

∴ The correct answer is 158 cm².

Additional Information

Lateral Surface Area of Cuboid

LSA = 2 × h × (l + b), which calculates the area of the 4 vertical walls excluding the top and bottom faces.

Volume of Cuboid

Volume = l × b × h, representing the total 3-dimensional space capacity enclosed by the cuboid.

Diagonal of Cuboid

Length of the longest rod that can completely fit inside the cuboid = √(l² + b² + h²).

Shortcut Trick

Total work = LCM(22, 11, 8) = 88 units.

Efficiency of Priya = 88 ÷ 22 = 4 units/day.

Efficiency of Nisha = (Priya + Varsha + Nisha) − (Priya + Varsha) = (88 ÷ 8) − (88 ÷ 11) = 11 − 8 = 3 units/day.

Combined efficiency of Priya and Nisha = 4 + 3 = 7 units/day.

Time required by Priya and Nisha = 88 ÷ 7 days.

∴ The correct answer is 88/7 days.

Alternate Method

Given:

Time taken by Priya = 22 days

Time taken by Priya and Varsha together = 11 days

Time taken by Priya, Varsha, and Nisha together = 8 days

Method Used:

Total Work = LCM of given days

Efficiency = Total Work ÷ Number of Days

Calculations:

⇒ Efficiency of Nisha = (Efficiency of Priya, Varsha and Nisha) − (Efficiency of Priya and Varsha)

⇒ Efficiency of Nisha = 11 − 8 = 3 units/day

⇒ Combined efficiency of Priya and Nisha = Efficiency of Priya + Efficiency of Nisha

⇒ Combined efficiency = 4 + 3 = 7 units/day

⇒ Time taken by Priya and Nisha together = Total Work ÷ Combined Efficiency

⇒ Time taken = 88 ÷ 7 days

∴ The correct answer is 88/7 days.

Additional Information

Pipes & Cisterns

Works on the same concept as Time and Work. Inlet pipes have positive efficiency, while outlet pipes (leaks) have negative efficiency.

Alternate Day Work

When individuals work on alternate days, find the combined work done in one complete cycle (e.g., 2 days) and multiply by the maximum possible integer before calculating the remaining work manually.

Work & Wages

Wages are always distributed in the direct ratio of the total work done by each person. If they work for the same number of days, wages are directly proportional to their efficiencies.

Shortcut Trick

Divide the entire equation by 5 directly to simplify the terms.

⇒ (5x² − 7) − (5x² + 2x − 4) = 3

⇒ 5x² − 7 − 5x² − 2x + 4 = 3

⇒ −2x − 3 = 3

⇒ −2x = 6 ⇒ x = −3

∴ The correct answer is −3.

Alternate Method

Given:

5(5x² − 7) − 5(5x² + 2x − 4) = 15

Calculation:

⇒ Expand the brackets on the left side.

⇒ 25x² − 35 − 25x² − 10x + 20 = 15

⇒ The 25x² terms cancel out completely.

⇒ −10x − 15 = 15

⇒ Add 15 to both sides.

⇒ −10x = 15 + 15

⇒ −10x = 30

⇒ x = 30 ÷ (−10)

⇒ x = −3

∴ The correct answer is −3.

Additional Information

Linear Equation in One Variable

An equation of the form ax + b = 0 where a ≠ 0. It always has exactly one unique solution.

BODMAS Rule

Always simplify brackets first, then calculate orders (powers/roots), followed by division, multiplication, addition, and finally subtraction.

Quadratic vs Linear

Although the given equation initially contains x² terms, they cancel out entirely during simplification, reducing the expression to a basic linear equation.

Shortcut Trick

Total Simple Interest percentage = Rate × Time = 4% × 5 = 20%

Total Amount = Principal (100%) + Simple Interest (20%) = 120% of Principal

⇒ 120% of P = 2280

⇒ P = (2280 × 100) ÷ 120

⇒ P = 1900

∴ The correct answer is ₹1,900.

Alternate Method

Given:

Amount (A) = ₹2,280

Rate of Interest (R) = 4% per annum

Time (T) = 5 years

Principal = P Amount = ₹2280 Time = 5 yrs, Rate = 4% p.a. + 20% Simple Interest Amount = P + (20% of P) = 120% of P

Formula Used:

A = P × [1 + (R × T) ÷ 100]

Calculations:

⇒ 2280 = P × [1 + (4 × 5) ÷ 100]

⇒ 2280 = P × [1 + 20 ÷ 100]

⇒ 2280 = P × [1 + 0.20]

⇒ 2280 = P × 1.20

⇒ P = 2280 ÷ 1.20

⇒ P = 1900

∴ The correct answer is ₹1,900.

Additional Information

Simple Interest (SI) Formula

Interest is calculated strictly on the initial principal amount: SI = (P × R × T) ÷ 100.

Compound Interest (CI) Formula

Interest is calculated on the principal along with accumulated interest: A = P × (1 + R ÷ 100)^T.

Difference Between SI and CI

For the 1^st year at a given rate, SI and CI are equal. For 2 years, the difference between CI and SI is P × (R ÷ 100)².

Shortcut Trick

The given number 726179 is an odd number, so it cannot be divided by even numbers like 28 and 20.

We only need to test divisibility for 29 and 23.

Using the divisibility rule for 23 (add 7 × last digit to the remaining number):

72617 + (9 × 7) = 72680 → 7268 + (0 × 7) = 7268 → 726 + (8 × 7) = 782

Further applying the rule: 78 + (2 × 7) = 92

Since 92 is completely divisible by 23 (23 × 4 = 92), the entire number is divisible by 23.

∴ The correct answer is 23.

Alternate Method

Given:

Number = 726179

Options provided: 29, 28, 23, 20

Calculation:

⇒ Since 726179 ends in 9, it is an odd number. Hence, we can eliminate the even options 28 and 20 directly.

⇒ Let us test divisibility by 23 through standard long division.

⇒ 726 ÷ 23 = 31 with a remainder of 13

⇒ Bring down 1 to make 131. 131 ÷ 23 = 5 with a remainder of 16

⇒ Bring down 7 to make 167. 167 ÷ 23 = 7 with a remainder of 6

⇒ Bring down 9 to make 69. 69 ÷ 23 = 3 with a remainder of 0

⇒ Since the final remainder is 0, 726179 is exactly divisible by 23.

⇒ The quotient is 31573.

∴ The correct answer is 23.

Additional Information

Divisibility Rule of 23

Add 7 times the unit digit to the remaining part of the number. If the result is divisible by 23, the original number is also divisible by 23.

Divisibility Rule of 29

Add 3 times the unit digit to the remaining part of the number. If the result is divisible by 29, the original number is also divisible by 29.

Divisibility Rule of 19

Add 2 times the unit digit to the remaining part of the number. If the result is divisible by 19, the original number is also divisible by 19.

Divisibility Rule of 17

Subtract 5 times the unit digit from the remaining part of the number. If the result is divisible by 17, the original number is also divisible by 17.

Shortcut Trick

The third proportional of two numbers a and b is given directly by the formula x = b² ÷ a.

Here, a = 54 and b = 18.

x = (18 × 18) ÷ 54

x = 324 ÷ 54 = 6

∴ The correct answer is 6.

Alternate Method

Given:

First number (a) = 54

Second number (b) = 18

Formula Used:

If x is the third proportional to a and b, they form a continuous proportion such that a : b :: b : x

⇒ a ÷ b = b ÷ x

÷ 3

÷ 3

54
:

18
::

18
:

x

First (a)
Second (b)
Second (b)
Third (x)

x = 18 ÷ 3 = 6

Calculations:

⇒ 54 : 18 = 18 : x

⇒ 54 ÷ 18 = 18 ÷ x

⇒ 3 = 18 ÷ x

⇒ x = 18 ÷ 3

⇒ x = 6

∴ The correct answer is 6.

Additional Information

Third Proportional

If a : b :: b : c, then c is called the third proportional to a and b. Formula: c = b² ÷ a.

Fourth Proportional

If a : b :: c : d, then d is the fourth proportional to a, b, and c. Formula: d = (b × c) ÷ a.

Mean Proportional

The mean proportional between two quantities a and b establishes an equal ratio between them. Formula: x = √(a × b).