Quantitative Aptitude PYQs

Shortcut Trick

Income = ₹10,200 | Savings = 30% = ₹3,060 | Expenditure = 70% = ₹7,140

Income Increase (+14%) = 10,200 × 0.14 = +₹1,428

Expenditure Increase (+40%) = 7,140 × 0.40 = +₹2,856

Change in Savings = Increase in Income − Increase in Expenditure = 1,428 − 2,856 = −₹1,428

∴ The correct answer is decrease by ₹1,428.

Alternate Method

Given: Initial Income = ₹10,200, Savings % = 30%

Formula Used: Income = Expenditure + Savings

⇒ Initial Savings = 30% of 10,200 = 0.30 × 10,200 = ₹3,060

⇒ Initial Expenditure = 10,200 − 3,060 = ₹7,140

⇒ New Income = 10,200 + 14% of 10,200 = 10,200 + 1,428 = ₹11,628

⇒ New Expenditure = 7,140 + 40% of 7,140 = 7,140 + 2,856 = ₹9,996

⇒ New Savings = 11,628 − 9,996 = ₹1,632

⇒ Change in Savings = 1,632 − 3,060 = −₹1,428

∴ The correct answer is decrease by ₹1,428.

Additional Information

Income-Expenditure Relationship

Always remember the fundamental equation: Income = Expenditure + Savings. Any percentage change in one side must balance the other.

Alligation Method for Savings

This problem can also be solved using alligation where the ratio of Expenditure to Savings is the inverse of the ratio of their respective percentage changes relative to the income change.

Successive Percentage Change

When a value increases by x% and then y%, the net change is given by (x + y + xy/100)%, though for split components like Income, direct calculation is safer.

Shortcut Trick

Base Area (πr²) = Volume ÷ Height = 1848 ÷ 12 = 154 cm².

When base area of a circle is 154, the radius r is always 7 cm.

Curved Surface Area (CSA) = 2 × (πr²) × (h ÷ r) = 2 × 154 × (12 ÷ 7) = 528 cm².

∴ The correct answer is 528 cm².

Alternate Method

Given: Volume = 1848 cm³, Height (h) = 12 cm.

Formula Used: Volume of Cylinder (V) = πr²h | Curved Surface Area (CSA) = 2πrh

⇒ 1848 = (22 ÷ 7) × r² × 12

⇒ r² = (1848 × 7) ÷ (22 × 12)

⇒ r² = 154 × 7 ÷ 22

⇒ r² = 7 × 7 = 49

⇒ r = √49 = 7 cm

⇒ CSA = 2 × (22 ÷ 7) × 7 × 12

⇒ CSA = 2 × 22 × 12

⇒ CSA = 44 × 12 = 528 cm²

∴ The correct answer is 528 cm².

Additional Information

Total Surface Area (TSA) of Cylinder

TSA includes the curved surface area plus the area of the two circular bases: TSA = 2πrh + 2πr² = 2πr(h + r).

Standard Circle Values

For a radius (r) of 7 cm, the circumference is 44 cm and the base area is 154 cm².

Volume of a Hollow Cylinder

If a cylinder has an outer radius R and an inner radius r, its volume is given by V = πh(R² − r²).

Shortcut Trick

Time taken is doubled from 40 minutes to 80 minutes.

Since the typing rate is constant, the number of pages will also double.

Number of pages = 2 × 2 = 4 pages.

∴ The correct answer is 4 pages.

Alternate Method

Given:

Time₁ (T₁) = 40 minutes, Work₁ (W₁) = 2 pages

Time₂ (T₂) = 80 minutes, Work₂ (W₂) = ?

Formula Used:

Work is directly proportional to Time: (W₁ / T₁) = (W₂ / T₂)

Calculations:

⇒ 2 / 40 = W₂ / 80

⇒ W₂ = (2 × 80) ÷ 40

⇒ W₂ = 160 ÷ 40

⇒ W₂ = 4 pages

∴ The correct answer is 4 pages.

Additional Information

Direct Proportion

When efficiency is constant, the work done is directly proportional to the time spent (W ∝ T).

Unitary Method

Find the value of a single unit (pages per minute) first: 2 ÷ 40 = 0.05 pages/min, then multiply by the required time: 0.05 × 80 = 4 pages.

Work and Efficiency

The general formula is Work = Efficiency × Time. If Efficiency is same, W₁/T₁ = W₂/T₂.

Shortcut Trick

Total Distance = 80 + 36 = 116 km.

Total Time = (80 ÷ 40) + (36 ÷ 18) = 2 + 2 = 4 hours.

Average Speed = Total Distance ÷ Total Time = 116 ÷ 4 = 29 km/hr.

∴ The correct answer is 29 km/hr.

Alternate Method

Given:

Distance 1 (D₁) = 80 km, Speed 1 (S₁) = 40 km/hr

Distance 2 (D₂) = 36 km, Speed 2 (S₂) = 18 km/hr

Formula Used:

Average Speed = (Total Distance) ÷ (Total Time)

Time = Distance ÷ Speed

Calculations:

⇒ Time taken for first part (T₁) = 80 ÷ 40 = 2 hours

⇒ Time taken for second part (T₂) = 36 ÷ 18 = 2 hours

⇒ Total Distance = 80 + 36 = 116 km

⇒ Total Time = 2 + 2 = 4 hours

⇒ Average Speed = 116 ÷ 4 = 29 km/hr

∴ The correct answer is 29 km/hr.

Additional Information

Average Speed (Equal Distances)

If a body covers two equal distances at speeds x and y, the average speed is given by (2xy) ÷ (x + y).

Relative Speed

When two objects move in the same direction, relative speed is the difference of their speeds. In opposite directions, it is the sum of their speeds.

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

Net Change Formula: a + b + (a × b) / 100

Here, Price (a) = −16% and Sale (b) = +80%

Net Change = −16 + 80 + (−16 × 80) / 100

Net Change = 64 − 12.8 = 51.2%

∴ The correct answer is 51.2%.

Alternate Method

Given: Reduction in price = 16%, Increase in sale = 80%

Formula: Revenue = Price × Sale

⇒ Let initial price = 100 and initial sale = 100

⇒ Initial Revenue = 100 × 100 = 10,000

⇒ New price = 100 − 16 = 84

⇒ New sale = 100 + 80 = 180

⇒ New Revenue = 84 × 180 = 15,120

⇒ Increment = 15,120 − 10,000 = 5,120

⇒ Net per cent increment = (5,120 ÷ 10,000) × 100 = 51.2%

∴ The correct answer is 51.2%.

Additional Information

Successive Percentage Change

When a value changes by x% and then by y%, the net effective change is given by [x + y + (xy/100)]%.

Product Stability Rule

If the product of two variables remains constant, an increase in one variable results in a reciprocal decrease in the other.

Revenue Calculation

Total Revenue is always the product of Price per unit and the Quantity sold (Revenue = P × Q).

Shortcut Trick

For an isosceles right-angled triangle with area 50 square units:

Area = (1/2) × base × height = (1/2) × side² = 50

⇒ side² = 100 ⇒ side = 10 units.

Hypotenuse = √(10² + 10²) = 10√2 ≈ 14.14 units.

The sides are 10, 10, and 14.14 units.

None of the sides is 8 units.

∴ The correct answer is An isosceles right-angled triangle with area 50 square units has one side with length 8 units.

Alternate Method

Given: Four statements about triangles to verify.

Verification of Options:

Option 1: Let the equal sides of the isosceles right triangle be x.

⇒ Area = (1/2) × x² = 50

⇒ x² = 100 ⇒ x = 10 units. Hypotenuse = 10√2 units. Statement is FALSE.

Option 2: By the Isoperimetric Theorem, for a fixed area, the equilateral triangle has the minimum perimeter. Statement is TRUE.

Option 3: Let the largest side be 4. Max area occurs for an equilateral triangle with side 4.

⇒ Max Area = (√3 / 4) × 4² = 4√3 ≈ 6.928 square units.

⇒ 6.928 < 8. Statement is TRUE.

Option 4: Let sides be n, n+1, n+2. For a right triangle:

⇒ n² + (n+1)² = (n+2)²

⇒ n² + n² + 2n + 1 = n² + 4n + 4 ⇒ n² − 2n − 3 = 0

⇒ (n − 3)(n + 1) = 0 ⇒ n = 3. Sides are 3, 4, 5. Statement is TRUE.

∴ The correct answer is An isosceles right-angled triangle with area 50 square units has one side with length 8 units.

Additional Information

Isosceles Right Triangle Properties

The sides are in the ratio 1 : 1 : √2. If the legs are a, the hypotenuse is a√2 and the area is a²/2.

Pythagorean Triples

Sets of three positive integers (a, b, c) such that a² + b² = c². The only triple with consecutive integers is (3, 4, 5).

Triangle Inequality and Area

For any triangle with sides a, b, c, the area is always less than (1/2) × (largest side)².

Shortcut Trick

The relationship between Cost Price (CP) and Marked Price (MP) is given by the ratio CP : MP = (100 − Discount%) : (100 + Profit%).

Given: Discount = 50% and Profit = 71%.

CP : MP = (100 − 50) : (100 + 71).

CP : MP = 50 : 171.

∴ The correct answer is 50 : 171.

Alternate Method

Given: Discount = 50%, Profit = 71%.

Formulas: SP = MP × (100 − D%) ÷ 100 | SP = CP × (100 + P%) ÷ 100.

⇒ Let the Marked Price (Listed Price) be 100.

⇒ Selling Price (SP) = 100 × (100 − 50) ÷ 100 = 50.

⇒ Cost Price (CP) × (100 + 71) ÷ 100 = 50.

⇒ CP × 171 ÷ 100 = 50 ⇒ CP = 5000 ÷ 171.

⇒ Ratio CP : MP = (5000 ÷ 171) : 100 = 5000 : 17100 = 50 : 171.

∴ The correct answer is 50 : 171.

Additional Information

Relationship between CP and MP

The formula CP/MP = (100 − D%)/(100 + P%) is derived by equating the Selling Price from both the CP and MP perspectives.

Marked Price (List Price)

The price printed on the label of an article. Discounts are always calculated on this price.

Shortcut Trick

Ratio of Profit = Ratio of Investment (when time is constant).

Total Profit = ₹9,000 | Roshan's Profit = ₹1,300.

Investment Ratio (Roshan : Total) = 1,300 : 9,000 = 13 : 90.

Total Investment = ₹13,500.

Roshan's Investment = (13 ÷ 90) × 13,500 = 13 × 150 = ₹1,950.

∴ The correct answer is ₹1,950.

Alternate Method

Given: Total Investment = ₹13,500, Total Profit = ₹9,000, Roshan's Profit = ₹1,300.

Formula: Investment_Roshan = (Profit_Roshan ÷ Total Profit) × Total Investment

⇒ Total Profit share units = 9,000

⇒ Roshan's profit share = 1,300

⇒ Let Roshan's investment be X.

⇒ X ÷ 13,500 = 1,300 ÷ 9,000

⇒ X = (1,300 × 13,500) ÷ 9,000

⇒ X = 1,300 × 1.5 = ₹1,950

∴ The correct answer is ₹1,950.

Additional Information

Partnership Principle

When multiple partners invest for the same time period, the profit is distributed in the direct ratio of their capital investments.

Variable Time Period

If time periods differ, Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂).

Shortcut Trick

Check the options for divisibility after subtracting the remainder 2.

Number − 2 must be divisible by 9 (Sum of digits divisible by 9).

(1) 9984 − 2 = 9982 (Sum = 28, No)

(2) 9948 − 2 = 9946 (Sum = 28, No)

(3) 9998 − 2 = 9996 (Sum = 33, No)

(4) 9956 − 2 = 9954 (Sum = 27, Yes)

∴ The correct answer is 9956.

Alternate Method

Given: Divisors are 14, 9, and 6. Remainder in each case is 2.

Formula: Required Number = (k × LCM of divisors) + remainder

⇒ LCM(14, 9, 6):

⇒ 14 = 2 × 7, 9 = 3², 6 = 2 × 3

⇒ LCM = 2 × 3² × 7 = 126

⇒ Largest 4-digit number = 9999

⇒ 9999 ÷ 126 ⇒ Quotient = 79, Remainder = 45

⇒ Largest 4-digit multiple of 126 = 9999 − 45 = 9954

⇒ Required Number = 9954 + 2 = 9956

∴ The correct answer is 9956.

Additional Information

LCM Method for Remainders

When a number leaves the same remainder 'r' when divided by x, y, and z, the general form is (LCM(x, y, z) × n) + r.

Divisibility Rule of 9

A number is divisible by 9 if the sum of its digits is divisible by 9. This is the fastest way to eliminate options in remainder problems involving 9.

Finding Multiples near N

To find the largest number near N divisible by L, subtract (N mod L) from N.

Shortcut Trick

For any Arithmetic Progression (AP), the average is the middle term.

The first 9 multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27.

Since there are 9 terms, the 5^th term is the middle term.

Average = 5^th multiple = 3 × 5 = 15.

∴ The correct answer is 15.

Alternate Method

Given: First 9 multiples of 3.

Formula Used: Average of an Arithmetic Progression = (First Term + Last Term) ÷ 2.

⇒ First term (a) = 3 × 1 = 3

⇒ Last term (l) = 3 × 9 = 27

⇒ Total number of terms (n) = 9

⇒ Average = (3 + 27) ÷ 2

⇒ Average = 30 ÷ 2

⇒ Average = 15

∴ The correct answer is 15.

Additional Information

Average of First n Multiples

The average of the first n multiples of any number x is given by the formula: x × (n + 1) ÷ 2.

Average of Continuous Numbers

For a set of numbers in Arithmetic Progression, the average is always the arithmetic mean of the first and last terms.

Sum of First n Multiples

The sum of the first n multiples of x is calculated as: [x × n(n + 1)] ÷ 2.

Shortcut Trick

Net Rate Difference per year = 7.5% − 3.5% = 4%.

Total Interest Difference for 2 years = 4% × 2 = 8%.

Positive difference = 8% of ₹2,60,000 = 2,600 × 8 = ₹20,800.

∴ The correct answer is ₹20,800.

Alternate Method

Given:

Principal (P) = ₹2,60,000

Rate 1 (R₁) = 3.5% | Rate 2 (R₂) = 7.5%

Time (T) = 2 years

Formula Used:

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

Calculations:

⇒ SI paid to Bank A = (2,60,000 × 3.5 × 2) ÷ 100

⇒ SI_A = 2,600 × 7 = ₹18,200

⇒ SI paid to Bank B = (2,60,000 × 7.5 × 2) ÷ 100

⇒ SI_B = 2,600 × 15 = ₹39,000

⇒ Positive Difference = SI_B − SI_A

⇒ Difference = 39,000 − 18,200 = ₹20,800

∴ The correct answer is ₹20,800.

Additional Information

Simple Interest Difference Rule

When the principal and time are the same, the difference in SI is given by ΔSI = (P × ΔR × T) ÷ 100, where ΔR is the difference in rates.

Total Amount Formula

The total amount (A) to be repaid is A = P + SI = P × [1 + (R × T) ÷ 100].

Effective Rate of Interest

The effective interest for the total duration T is simply R × T in simple interest, unlike compound interest where it follows successive percentage change.

Shortcut Trick

Net change formula for Successive percentage: Profit% = m − d − (m × d) ÷ 100

Where m = markup percentage (40%) and d = discount percentage (20%).

Profit% = 40 − 20 − (40 × 20) ÷ 100

Profit% = 20 − 8 = 12%.

∴ The correct answer is 12%.

Alternate Method

Given: Cost Price (CP) = ₹4,000, Markup% = 40%, Discount% = 20%.

Formulas: MP = CP × (100 + Markup%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | Profit% = (SP − CP) ÷ CP × 100.

⇒ Marked Price (MP) = 4000 × (140 ÷ 100) = ₹5,600.

⇒ Selling Price (SP) = 5600 × (80 ÷ 100) = ₹4,480.

⇒ Profit = 4480 − 4000 = ₹480.

⇒ Profit% = (480 ÷ 4000) × 100 = 12%.

∴ The correct answer is 12%.

Additional Information

Relationship between CP and MP

The ratio of Marked Price to Cost Price can be expressed as MP/CP = (100 + Profit%)/(100 − Discount%).

Effective Profit

Profit is always calculated on the Cost Price unless stated otherwise, while Discount is always calculated on the Marked Price.

Shortcut Trick

Check the ratio difference: P : Q present ratio is 9 : 8 (diff = 1 unit).

After 2 years, the ratio is 10 : 9 (diff = 1 unit).

Since the difference is the same, 1 unit increase in ratio corresponds to 2 years.

Present age of P = 9 units = 9 × 2 = 18 years.

∴ The correct answer is 18 years.

Alternate Method

Given: Present age ratio P : Q = 9 : 8. Ratio after 2 years = 10 : 9.

Formula Used: Ratio & Proportion method.

⇒ Let the present ages of P and Q be 9x and 8x.

⇒ After 2 years: (9x + 2) ÷ (8x + 2) = 10 ÷ 9

⇒ 9 × (9x + 2) = 10 × (8x + 2)

⇒ 81x + 18 = 80x + 20

⇒ 81x − 80x = 20 − 18

⇒ x = 2

⇒ Present age of P = 9x = 9 × 2 = 18 years.

∴ The correct answer is 18 years.

Additional Information

Age Ratio Rule

The difference between the ages of two people remains constant regardless of the number of years passed.

Ratio Balancing

If the difference in units between ratios is not equal, multiply the ratios by the difference of the other ratio to balance them.

Shortcut Trick

Equate the common entity Red tokens by multiplying the first ratio by 2 and the second ratio by 15.

Red : Green = 15 × 2 : 19 × 2 = 30 : 38

Pink : Red = 17 × 15 : 2 × 15 = 255 : 30

Combined Ratio Green : Pink = 38 : 255

∴ The correct answer is 38:255.

Alternate Method

Given:

Ratio of Red (R) to Green (G) = 15 : 19

Ratio of Pink (P) to Red (R) = 17 : 2

Formula Used:

If A : B and B : C are given, then A : B : C can be found by making the common term B equal in both ratios.

Calculations:

⇒ Red / Green = 15 / 19 ⇒ Green / Red = 19 / 15

⇒ Pink / Red = 17 / 2

⇒ To find Green : Pink, we divide (Green / Red) by (Pink / Red):

⇒ Green / Pink = (Green / Red) ÷ (Pink / Red) = (19 / 15) ÷ (17 / 2)

⇒ Green / Pink = (19 / 15) × (2 / 17)

⇒ Green / Pink = (19 × 2) / (15 × 17)

⇒ Green / Pink = 38 / 255

∴ The correct answer is 38:255.

Additional Information

Compound Ratio

The compound ratio of (a : b) and (c : d) is (ac : bd), obtained by multiplying corresponding terms.

Inverse Ratio

The inverse ratio of a : b is b : a. This is useful when the required comparison is the reciprocal of the given data.

Ratio Balancing

To combine A:B and B:C, multiply the ratios by factors that make the B-value equal to the LCM of the two B-values.

Shortcut Trick

Apply BODMAS to solve the brackets and operations directly:

⇒ 12 + {6 × 9 + 2} − (2) × 5

⇒ 12 + {54 + 2} − 10

⇒ 12 + 56 − 10

⇒ 68 − 10 = 58

∴ The correct answer is 58.

Alternate Method

Given Expression:

12 + {6 × 27 ÷ 3 + 2} − (16 ÷ 8) × 5

Method Used:

The expression must be solved using the BODMAS rule, which sets the order of operations as Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Calculation:

12 + {6 × 27 ÷ 3 + 2} − (16 ÷ 8) × 5

⇒ 12 + {6 × 27 ÷ 3 + 2} − 2 × 5

⇒ 12 + {6 × 9 + 2} − (2) × 5

⇒ 12 + {54 + 2} − 10

⇒ 12 + 56 − 10

⇒ 68 − 10 = 58

∴ The correct answer is 58.

Additional Information

BODMAS Rule

It specifies the order of operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

Order of Brackets

Always simplify the innermost brackets first, following the order: parenthesis ( ), then curly braces { }, and finally square brackets [ ].

Priority of Operations

Division and Multiplication share the same priority level and are solved from left to right. Similarly, Addition and Subtraction share the same priority and are also solved from left to right.

Shortcut Trick

Use the direct formula: Q₁ × (100 − L₁) / Price₁ = Q₂ × (100 + P₂) / Price₂

Here, Q₁ = 6, L₁ = 40%, Price₁ = 1, P₂ = 20%, Price₂ = 1

⇒ 6 × (100 − 40) / 1 = Q₂ × (100 + 20) / 1

⇒ 6 × 60 = Q₂ × 120

⇒ 360 = 120 × Q₂ ⇒ Q₂ = 3

∴ The correct answer is 3 rubber-bands.

Alternate Method

Given:

Selling 6 rubber-bands for Re. 1 results in a loss of 40%.

Desired profit = 20%.

SP1 (40% Loss)
Value = 60% of CP
Qty = 6 per Re. 1

SP2 (20% Gain)
Value = 120% of CP
Qty = 3 per Re. 1

Value × 2
Qty ÷ 2

Calculations:

⇒ Let the Cost Price (CP) of 6 rubber-bands be x.

⇒ Selling Price (SP) of 6 rubber-bands = Re. 1

⇒ Loss = 40%

⇒ CP × (100 − 40) / 100 = SP

⇒ x × 60 / 100 = 1

⇒ x = 100 / 60 = 5/3 rupees

⇒ To gain 20%, the new SP for 6 rubber-bands must be:

⇒ New SP = CP × (100 + 20) / 100

⇒ New SP = (5/3) × (120/100) = (5/3) × (6/5) = 2 rupees

⇒ Thus, he must sell 6 rubber-bands for Rs. 2 to gain 20%.

⇒ Therefore, for Re. 1, he must sell = 6 ÷ 2 = 3 rubber-bands.

∴ The correct answer is 3 rubber-bands.

Additional Information

Price and Quantity Relation

When the total expenditure remains constant, Price and Quantity are inversely proportional to each other (Price ∝ 1 / Quantity).

Universal Formula

Quantity₁ × (100 ± P/L%)₁ ÷ Price₁ = Quantity₂ × (100 ± P/L%)₂ ÷ Price₂

Calculating CP from SP

CP = (SP × 100) / (100 − Loss%) or CP = (SP × 100) / (100 + Profit%).

Shortcut Trick

Increasing a number by 100% means the number is doubled, becoming 200% of itself.

200% of the number = 2780

100% of the number = 2780 ÷ 2 = 1390

∴ The correct answer is 1390.

Alternate Method

Original (x)
Increase (+x)

Total = 2x = 2780

Given:

A number, when increased by 100%, gives 2780.

Formula Used:

Final Value = Original Value × (1 + Percentage Increase ÷ 100)

Calculations:

⇒ Let the required number be x.

⇒ Final value = x + (100% of x)

⇒ Final value = x + x

⇒ 2x = 2780

⇒ x = 2780 ÷ 2

⇒ x = 1390

∴ The correct answer is 1390.

Additional Information

Percentage Increase

When a value is increased by R%, the new value becomes (100 + R)% of the original value.

Base Value Concept

In percentage problems, the original or initial value is always considered as the base value of 100%.

Fraction Equivalent

An increase of 100% is equivalent to adding the full value itself, which is a fractional increase of 1/1, making the final value twice the original.

Shortcut Trick
The product of two consecutive natural numbers n × (n + 1) is slightly greater than n2.
Find a perfect square closest to 210 to approximate the numbers.
We know that 142 = 196 and 152 = 225.
Since 196 < 210 < 225, the two consecutive numbers are exactly 14 and 15.
The greater of the two consecutive numbers is 15.
∴ The correct answer is 15.
Alternate Method
Given:
The product of two consecutive natural numbers is 210.
Calculation:
Let the two consecutive natural numbers be x and (x + 1).
⇒ x × (x + 1) = 210
⇒ x2 + x = 210
⇒ x2 + x − 210 = 0
⇒ x2 + 15x − 14x − 210 = 0
⇒ x(x + 15) − 14(x + 15) = 0
⇒ (x − 14)(x + 15) = 0
Since x is a natural number, it must be positive. Thus, x cannot be −15.
⇒ x = 14
The smaller number is 14, and the greater number is x + 1 = 15.
∴ The correct answer is 15.
Additional Information
Natural Numbers
Natural numbers are the set of positive integers starting from 1 (1, 2, 3, 4, ...). They do not include zero, fractions, or negative numbers.
Consecutive Numbers
Consecutive numbers are numbers that follow each other continuously from smallest to largest with a constant difference of 1 (e.g., n, n+1, n+2).
Quadratic Equation Roots
A quadratic equation ax2 + bx + c = 0 has two roots. If the constant term is negative while a is positive, the roots will have opposite signs.

Shortcut Trick
Let the variable × be denoted as x. Given, x = 1490.59, y = 1490.94 and xz = y6
Directly substitute x and y to get: (1490.59)z = (1490.94)6
Equating the exponents since the bases are equal: 0.59 × z = 0.94 × 6
z = 5.64 ÷ 0.59 = 564 ÷ 59
z ≈ 9.559...
∴ The correct answer is 9.56.

Alternate Method
Given:
x = 1490.59
y = 1490.94
xz = y6
Formula Used:
Law of Indices: (am)n = am × n
If am = an, then m = n (for a > 0, a ≠ 1)
Calculations:
⇒ Substitute the values of x and y in the equation xz = y6
⇒ (1490.59)z = (1490.94)6
⇒ 1490.59z = 1490.94 × 6
⇒ 1490.59z = 1495.64
Since the bases on both sides are equal, we can equate the exponents:
⇒ 0.59z = 5.64
⇒ z = 5.64 ÷ 0.59
⇒ z = 564 ÷ 59
⇒ z ≈ 9.5593
⇒ z ≈ 9.56
∴ The correct answer is 9.56.

Additional Information
Law of Exponents (Power of a Power)
When raising a power to another power, you multiply the exponents: (am)n = am × n
Law of Exponents (Equality Rule)
If two terms with the same positive base (other than 1) are equal, their exponents must be equal: If ax = ay, then x = y.
Law of Exponents (Product Rule)
When multiplying two terms with the same base, you add the exponents: am × an = am + n

Shortcut Trick

The average of the first n even numbers is directly calculated using the formula: (n + 1).

Here, the number of terms n = 151.

Average = 151 + 1 = 152.

∴ The correct answer is 152.

Alternate Method

2
1^st

4

6

302
151^st

Average = (2 + 302) ÷ 2 = 152

Given:

The sequence consists of the first 151 even numbers.

Formula Used:

For an Arithmetic Progression (A.P.), Average = (First term + Last term) ÷ 2

nth term of A.P., T_n = a + (n − 1)d

Calculation:

⇒ First even number (a) = 2

⇒ Common difference (d) = 2

⇒ Number of terms (n) = 151

⇒ Last term (l) = a + (n − 1)d

⇒ Last term = 2 + (151 − 1) × 2

⇒ Last term = 2 + 150 × 2 = 302

⇒ Average = (2 + 302) ÷ 2

⇒ Average = 304 ÷ 2 = 152

∴ The correct answer is 152.

Additional Information

Average of First n Natural Numbers

The average of the first n consecutive natural numbers is given by the formula: (n + 1) ÷ 2.

Average of First n Odd Numbers

The average of the first n consecutive odd numbers is exactly equal to n.

Sum of First n Even Numbers

The total sum of the first n even numbers is calculated as n(n + 1).

Average of Consecutive Numbers

For any sequence of continuous consecutive numbers, continuous even numbers, or continuous odd numbers, the average is always strictly the middle value.