SSC CHSL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

Let the original number be 100.

The required correct value is obtained by successively increasing the number by 20% and 30%. Using the successive percentage change formula: Net percentage increase = a + b + (a × b)/100

Substituting a = 20 and b = 30: Net increase = 20 + 30 + (20 × 30)/100 = 56% Required correct value = 100 + 56 = 156

The actual resultant value is obtained by decreasing the original number by 22%: Resultant value = 100 - 22 = 78

The percentage of the required correct value that the resultant value represents is: (Resultant value ÷ Required correct value) × 100 Substituting the calculated values: (78 ÷ 156) × 100 = ½ × 100 = 50%

Explanation

Let the marked price be MP and the cost price be CP.

The selling price (SP) is given as 8/15 of the marked price: SP = (8/15) × MP

Since there is a loss of 20%, the selling price is also equal to 80% of the cost price: SP = (80/100) × CP = (4/5) × CP

Equating the two expressions for SP: (8/15) × MP = (4/5) × CP

To find the ratio of marked price to cost price, we solve for MP / CP: MP / CP = (4/5) × (15/8) = 3/2

Explanation

In triangle RST, Z is a point on side ST such that SZ = ZT, which makes RZ the median to side ST. Since angle RZS = 90°, RZ is also the altitude to side ST.

In right triangles RZS and RZT, RZ is a common side, angle RZS = angle RZT = 90°, and SZ = ZT. By Side-Angle-Side congruence, triangle RZS is congruent to triangle RZT.

Since corresponding parts of congruent triangles are equal, the side RS must equal RT. Given that RS = 10 cm, the length of RT is also 10 cm.

Explanation

Using the algebraic identity: (x - y)² = x² + y² - 2xy

Substitute the given values x² + y² = 29 and x - y = 3 into the identity: 3² = 29 - 2xy 9 = 29 - 2xy

Rearranging the terms to find 2xy: 2xy = 29 - 9 2xy = 20

Multiplying both sides by 2 gives the value of 4xy: 4xy = 2 × 20 = 40

Explanation

When two objects move in opposite directions, their relative speed is the sum of their individual speeds.

Relative speed = 81 + 63 = 144 km/hr

To convert this speed into meters per second, multiply by 5/18: Relative speed = 144 × 5/18 = 40 m/s

The total distance that needs to be covered for the two trains to completely cross each other is the sum of their individual lengths.

Total distance = 125 + 375 = 500 meters

Using the formula for time, Time = Distance / Speed: Time = 500 / 40 = 12.5 seconds

Explanation

To find the value of (a⁶ + 1)/a³, we can rewrite the expression as a³ + 1/a³.

We are given: a⁴ + 1/a⁴ = 194

Add 2 to both sides of the equation to form a perfect square: a⁴ + 1/a⁴ + 2 = 196 (a² + 1/a²)² = 196 a² + 1/a² = √196 = 14

Again, add 2 to both sides of this equation: a² + 1/a² + 2 = 16 (a + 1/a)² = 16 a + 1/a = √16 = 4

Now, using the algebraic identity for the sum of cubes: a³ + 1/a³ = (a + 1/a)³ - 3(a + 1/a)

Substitute the value of (a + 1/a) = 4 into the identity: a³ + 1/a³ = 4³ - 3(4) a³ + 1/a³ = 64 - 12 = 52

Explanation

Efficiency is inversely proportional to the time taken to complete a work. Since A takes 20 days and B takes 30 days, the ratio of their efficiencies is 30:20, which simplifies to 3:2. This confirms statement I is correct.

The time taken by both working together is calculated using the formula T = (x × y) / (x + y), where x and y are the individual days taken. Substituting the values gives T = (20 × 30) / (20 + 30) = 600 / 50 = 12 days. This confirms statement II is correct.

Explanation

The perimeter P of a rectangle is given by the formula P = 2(l + b), where l is the length and b is the breadth.

According to the given conditions: l = b + 24, which can be rewritten as b = l - 24

The perimeter is 128 cm: 2(l + b) = 128 l + b = 64

Substituting b = l - 24 into the equation: l + (l - 24) = 64 2l - 24 = 64 2l = 88 l = 44 cm

Explanation

Simple interest is calculated using the formula: Simple Interest = (Principal × Rate × Time) / 100 Substituting the given values: Simple Interest = (6000 × 10 × 2) / 100 = 1200 Thus, Statement I is correct.

Compound interest is calculated using the formula: Amount = Principal × (1 + Rate / 100)ᵀⁱᵐᵉ Substituting the given values: Amount = 6000 × (1 + 10 / 100)² = 6000 × (1.1)² = 6000 × 1.21 = 7260 Compound Interest = Amount - Principal = 7260 - 6000 = 1260 Thus, Statement II is also correct.

Explanation

Given the trigonometric relation tan θ + cot θ = 2. Since cot θ is the reciprocal of tan θ, this relation can be written as:

tan θ + 1/tan θ = 2

Multiplying the entire equation by tan θ yields the quadratic equation:

tan² θ - 2 tan θ + 1 = 0

This is a perfect square trinomial:

(tan θ - 1)² = 0

Solving for tan θ gives:

tan θ = 1

Since tan θ = 1, then cot θ = 1/tan θ = 1.

Substituting these values into the required expression:

tan θ - cot θ = 1 - 1 = 0

Explanation

The algebraic identity for the difference of two cubes is: a³ - b³ = (a - b)³ + 3ab(a - b)

Substitute the given values a - b = 5 and ab = 150 into this formula: a³ - b³ = 5³ + 3 × 150 × 5 a³ - b³ = 125 + 2250 a³ - b³ = 2375

Explanation

The total distance covered by a wheel is the product of the number of revolutions and the circumference of the wheel.

Distance = Number of revolutions × 2πr

Here, the total distance is 44 km, which is equal to 44000 m. The number of revolutions is 500.

44000 = 500 × 2 × (22/7) × r 44000 = 1000 × (22/7) × r 44 = (22/7) × r r = 44 × 7 / 22 r = 14 m

Explanation

A perpendicular drawn from the center of a circle to a chord bisects the chord.

If we draw a perpendicular from the center O to the chord MN at point D, then D bisects MN. Therefore: MD = MN / 2 = 40 / 2 = 20 cm

The perpendicular distance from the center to the chord is OD = 15 cm.

In the right-angled triangle ODM, the radius r (which is the hypotenuse OM) is calculated using the Pythagorean theorem: r² = OD² + MD² r² = 15² + 20² r² = 225 + 400 r² = 625 r = √625 = 25 cm

Alternatively, using the standard Pythagorean triplet (3, 4, 5) scaled by a factor of 5 directly gives the sides as (15, 20, 25), where 25 cm is the hypotenuse and represents the radius.

Explanation

Using the standard trigonometric values: cosec 90° = 1 cos 60° = ½

Substituting these values into the given expression: cosec 90° / (1 - cos² 60°) = 1 / (1 - (½)²)

Simplifying the denominator: 1 - (½)² = 1 - ¼ = ¾

Now, evaluating the fraction: 1 / (¾) = 4/3

Explanation

To find the value of the given expression, we use the BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction), where the division symbol ÷ is used in place of the misprinted + signs in the first two terms:

1200 ÷ 15 × [45 ÷ (17 - 2)] of 2 + [-2(1 + 2)]

First, solve the innermost round brackets: 17 - 2 = 15 1 + 2 = 3

Substitute these back into the expression: 1200 ÷ 15 × [45 ÷ 15] of 2 + [-2(3)]

Next, solve the square brackets: 45 ÷ 15 = 3 -2 × 3 = -6

The expression becomes: 1200 ÷ 15 × 3 of 2 + [-6]

Now, evaluate the "of" operator, which has higher precedence than division and multiplication: 3 of 2 = 3 × 2 = 6

This simplifies the expression to: 1200 ÷ 15 × 6 - 6

Perform the division and multiplication operations from left to right: 1200 ÷ 15 = 80 80 × 6 = 480

Finally, perform the subtraction: 480 - 6 = 474

Explanation

To evaluate Statement I, calculate the percentage of male teachers in school L with respect to female teachers in school P: Percentage = (160 ÷ 110) × 100% = 145.45% This makes Statement I incorrect.

To evaluate Statement II, find the sum of female teachers in schools M and N: 170 + 120 = 290 Next, find the sum of male teachers in schools N and P: 170 + 150 = 320 The ratio of these two sums is: 290 : 320 = 29 : 32 This makes Statement II correct.

Explanation

Let the total number of votes polled in the election be 100%.

The percentage of votes received by Party A is 30% and by Party B is 25%.

The remaining votes received by Party C is equal to 100% - (30% + 25%) = 45%.

The difference in percentage of votes between Party C and Party A is 45% - 30% = 15%.

According to the question, Party C got 9000 more votes than Party A. Therefore, 15% of the total votes corresponds to 9000.

Using the unitary method, 1% of the total votes is 9000 ÷ 15 = 600.

The number of votes received by Party B, which is 25% of the total votes, is 25 × 600 = 15000.

Explanation

The given expression can be rearranged as follows: (sec²x + cosec²x) - (tan²x + cot²x) = (sec²x - tan²x) + (cosec²x - cot²x)

Using the standard trigonometric identities: sec²x - tan²x = 1 cosec²x - cot²x = 1

Substituting these identities into the equation: 1 + 1 = 2

Explanation

To find the number of prime numbers between 20 and 50, we list the numbers in this range that have exactly two distinct factors, namely 1 and the number itself.

The prime numbers in this range are 23, 29, 31, 37, 41, 43, and 47.

Counting these values, we find there are exactly 7 prime numbers between 20 and 50. All other numbers in this range are composite because they can be factored further by prime numbers such as 2, 3, 5, or 7.

Explanation

The algebraic identity for the square of a binomial is: (a + b)² = a² + b² + 2ab

By setting a = 2p and b = 5q, the expansion becomes: (2p + 5q)² = (2p)² + (5q)² + 2(2p)(5q) (2p + 5q)² = 4p² + 25q² + 20pq

Substitute the given values, 2p + 5q = 12 and pq = 3, into this equation: 12² = 4p² + 25q² + 20(3) 144 = 4p² + 25q² + 60

Rearranging the terms to solve for the required expression: 4p² + 25q² = 144 - 60 4p² + 25q² = 84

Explanation

Correct Answer: D

According to the triangle inequality theorem, the length of any side of a triangle must be strictly greater than the absolute difference of the other two sides and strictly less than their sum.

Let the length of AC be x.

|BC - AB| < x < BC + AB

Substituting the given values AB = 6 cm and BC = 11 cm into the inequality:

|11 - 6| < x < 11 + 6 5 < x < 17

The possible integral values for x are the integers strictly between 5 and 17, which are 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16.

The number of such integral values is calculated as: (17 - 5) - 1 = 11

Alternatively, using the direct formula for the number of possible integer values of the third side when two sides are known: Number of values = 2 × min(side1, side2) - 1 Number of values = 2 × 6 - 1 = 11

Explanation

To determine which statements are incorrect, we evaluate each statement individually.

For Statement I, we calculate the total profit of D, E, and F, which is 200 + 150 + 300 = 650. The total cost price of G, H, and I is 500 + 300 + 800 = 1600. The required percentage is (650 / 1600) × 100 = 40.625%. Since the statement claims this value is 30.63%, Statement I is incorrect.

For Statement II, the total cost price of all articles is 700 + 600 + 400 + 500 + 300 + 800 = 3300. The total profit of all articles is 200 + 150 + 300 + 400 + 250 + 350 = 1650. The ratio of total cost price to total profit is 3300 : 1650, which simplifies to 2 : 1. Since the statement claims the ratio is 1 : 2, Statement II is incorrect.

Explanation

Let the present ages of the father, mother, and son be represented by 8x, 5x, and 2x respectively.

Since the present age of the father is 48 years: 8x = 48 x = 6

Using this value of x, the present ages of the mother and the son are: Mother = 5 × 6 = 30 years Son = 2 × 6 = 12 years

The sum of their present ages is 48 + 30 + 12 = 90 years.

The present average age of the family of 3 members is 90 / 3 = 30 years.

Since the age of each of the 3 members increases by 3 years after 3 years, the average age of the family also increases by exactly 3 years.

Average age after 3 years = 30 + 3 = 33 years.

Explanation

The equivalent single discount of two successive discounts of a% and b% is calculated using the formula:

Equivalent Discount = a + b - (a × b)/100

Substituting the given values, a = 12 and b = 10:

Equivalent Discount = 12 + 10 - (12 × 10)/100 Equivalent Discount = 22 - 120/100 Equivalent Discount = 22 - 1.2 Equivalent Discount = 20.8%

Explanation

Cost price (CP) of 1 dozen corns = Rs. 180

Since 1 dozen contains 12 items, the CP of 1 corn = 180 / 12 = Rs. 15

Selling price (SP) of 1 corn = Rs. 22

The formula for profit percentage is: Profit% = ((SP - CP) / CP) × 100

Substituting the values: Profit% = ((22 - 15) / 15) × 100 Profit% = (7 / 15) × 100 Profit% = 700 / 15 ≈ 46.6%

Explanation

The given algebraic expression is x² + 4y² + z² - 4xy + 4yz - 2xz.

This expression can be factored using the standard algebraic identity: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Comparing the terms of the given expression with the identity, we have: a² = (-x)² = x² b² = (2y)² = 4y² c² = z² = z²

The product terms are: 2ab = 2(-x)(2y) = -4xy 2bc = 2(2y)(z) = 4yz 2ca = 2(z)(-x) = -2xz

Substituting these values into the identity gives: (-x + 2y + z)² = x² + 4y² + z² - 4xy + 4yz - 2xz

Rearranging the terms inside the square, we get: (2y - x + z)²

Explanation

The largest circle that can be inscribed inside a square of side s has a diameter d equal to the side length of the square.

Given the side of the square s = 18 cm, the diameter of the largest circle is: d = 18 cm

The radius r of this circle is half of its diameter: r = d / 2 = 18 / 2 = 9 cm

The area A of a circle is calculated using the formula: A = πr²

Substituting the radius into the formula yields: A = π × 9² = 81π cm²

Explanation

Using the method of alligation, the ratio of senior employees to the remaining staff can be determined by comparing their average salaries to the overall average salary.

The difference between the average salary of senior employees and the overall average is 450 - 350 = 100. The difference between the overall average and the average salary of the remaining staff is 350 - 250 = 100.

Since the differences are equal, the ratio of senior employees to the remaining staff is 100:100, which simplifies to 1:1.

Given that there are 10 senior employees, there must also be 10 remaining staff.

Therefore, the total number of employees working in the sales department is 10 + 10 = 20.

Explanation

To find the required speed, we determine the remaining distance and the remaining time to reach the destination 30 minutes early.

Total distance is 250 km. Ram covers 3/5 of this distance: Distance covered = 3/5 × 250 = 150 km Remaining distance = 250 - 150 = 100 km

The total time allowed to reach the center is 6 hours. Since Ram wants to arrive 30 minutes (0.5 hours) early, his target total time is: Target total time = 6 - 0.5 = 5.5 hours

He has already traveled for 3.5 hours, so the remaining time available is: Remaining time = 5.5 - 3.5 = 2 hours

The required speed to cover the remaining distance of 100 km in 2 hours is: Required speed = Remaining distance / Remaining time = 100 / 2 = 50 km/h

Explanation

To find the ratio of investments, assume the total investment is the least common multiple of the denominators 5 and 4, which is 20 units.

P's investment = 1/5 of 20 = 4 units

Q's investment = 1/4 of 20 = 5 units

R's investment = 20 - (4 + 5) = 11 units

The ratio of the investments of P, Q, and R is 4:5:11.

Explanation

The given expression is (a + b + c)[(a - b)² + (b - c)² + (c - a)²].

Expanding the terms inside the square brackets: (a - b)² + (b - c)² + (c - a)² = a² - 2ab + b² + b² - 2bc + c² + c² - 2ca + a² = 2a² + 2b² + 2c² - 2ab - 2bc - 2ca = 2(a² + b² + c² - ab - bc - ca)

Multiplying this result by (a + b + c) gives: 2(a + b + c)(a² + b² + c² - ab - bc - ca)

Using the standard algebraic identity (a + b + c)(a² + b² + c² - ab - bc - ca) = a³ + b³ + c³ - 3abc, we can simplify the expression: 2(a³ + b³ + c³ - 3abc) = 2a³ + 2b³ + 2c³ - 6abc

Explanation

Using the formulas for boat and stream speed:

Upstream speed = Speed of boat - Speed of stream = 11 - 4 = 7 km/hr Downstream speed = Speed of boat + Speed of stream = 11 + 4 = 15 km/hr

Time taken to travel upstream = Upstream distance / Upstream speed = 21 / 7 = 3 hours Time taken to travel downstream = Downstream distance / Downstream speed = 45 / 15 = 3 hours

Total time taken = Upstream time + Downstream time = 3 + 3 = 6 hours

Explanation

The expression is simplified by finding a common denominator:

√((cosec A + 1)/(cosec A - 1)) + √((cosec A - 1)/(cosec A + 1))

Combining the terms under a single common denominator:

((cosec A + 1) + (cosec A - 1)) / √(cosec² A - 1)

Simplifying the numerator: cosec A + 1 + cosec A - 1 = 2 cosec A

Using the trigonometric identity cosec² A - 1 = cot² A, the denominator simplifies to: √(cot² A) = cot A

Substituting these simplified terms back: 2 cosec A / cot A

Expressing the trigonometric functions in terms of sine and cosine: (2 / sin A) / (cos A / sin A) = 2 / cos A = 2 sec A

Explanation

The average of a set of consecutive integers (which forms an arithmetic progression) is equal to the average of its first and last terms. Let the 10 consecutive integers be n, n+1, ..., n+9.

The average of these 10 consecutive integers is given by: (n + (n + 9)) / 2 = 33/2 2n + 9 = 33 2n = 24 n = 12

The first three integers in this sequence are 12, 13, and 14.

The average of these first three consecutive integers is the middle term: (12 + 13 + 14) / 3 = 13

Explanation

Since SX is the median on TU, X is the midpoint of TU, which gives TX = UX.

We are given that SX = TX. Combining these relations, we get SX = TX = UX.

In ΔSTX, since SX = TX, the opposite angles are equal. Let ∠XST = ∠XTS = α.

In ΔSUX, since SX = UX, the opposite angles are equal. Let ∠XSU = ∠XUS = β.

The sum of angles in ΔSTU is ∠T + ∠U + ∠TSU = 180°.

Substituting the angle variables gives: α + β + (α + β) = 180° 2(α + β) = 180° α + β = 90°

Since ∠TSU = α + β, the value of ∠TSU is 90°.

Explanation

In an equilateral triangle, the median is also the altitude. The relation between the altitude h and the side length a of an equilateral triangle is given by the formula h = (√3 / 2) × a.

Substituting the given median value into the formula: 15√3 = (√3 / 2) × a

Dividing both sides by √3: 15 = a / 2

Solving for a: a = 15 × 2 = 30 cm

Explanation

Simple interest is calculated using the formula SI = (P × R × T) / 100, where P is the principal, R is the rate of interest per annum, and T is the time in years.

Substituting the given values: 1200 = (P × 8 × 3) / 100 1200 = 24P / 100 P = (1200 × 100) / 24 P = 5000

The new principal is twice the original principal: P' = 2 × 5000 = 10000

For compound interest compounding annually at 10% per annum for 2 years, the effective rate of interest can be calculated using the successive percentage formula: Effective rate = 10 + 10 + (10 × 10) / 100 = 21%

Now, calculate the compound interest accrued on the new principal: Compound Interest = 10000 × 21% = 10000 × 21/100 = 2100

Explanation

The centroid G of a triangle divides each median in the ratio 2:1 from the vertex. For the median AD, the centroid G divides it such that AG : GD = 2 : 1.

This means that the length of AG is two-thirds of the total length of the median AD:

AG = (2/3) × AD

Substituting the given value of AG = 14 cm into the relation:

14 = (2/3) × AD

AD = 14 × 3/2 = 21 cm

Explanation

The expression 3 sin A cos A can be written in terms of tan A by dividing and multiplying by cos² A, which is equivalent to dividing the expression by sin² A + cos² A:

3 sin A cos A = 3 sin A cos A / (sin² A + cos² A)

Dividing the numerator and the denominator by cos² A gives:

3 sin A cos A = 3 tan A / (tan² A + 1)

Substituting the given value tan A = 3 into the formula:

3 sin A cos A = 3 × 3 / (3² + 1) = 9 / (9 + 1) = 9/10

Explanation

Let the original price be P and the original consumption be C.

The relationship between expenditure, price, and consumption is given by: Expenditure = Price × Consumption

Since the expenditure remains constant after the price decrease of 25% and consumption increase of 20 kg: P × C = 0.75P × (C + 20)

Dividing both sides by P: C = 0.75(C + 20) C = 0.75C + 15 0.25C = 15 C = 60 kg

Explanation

The given algebraic expression can be simplified using the standard algebraic identity:

(x + y + z)(x² + y² + z² - xy - yz - zx) = x³ + y³ + z³ - 3xyz

By comparing the given expression (a + b + 2c)(a² + b² + 4c² - ab - 2bc - 2ca) with the identity, we can substitute: x = a y = b z = 2c

Substituting these values into the RHS of the identity:

x³ + y³ + z³ - 3xyz = a³ + b³ + (2c)³ - 3(a)(b)(2c) = a³ + b³ + 8c³ - 6abc

Explanation

Correct Answer: D

The length of the direct common tangent (L) to two circles is given by the formula:

L = √(d² - (R - r)²)

where d is the distance between the centres of the two circles, R is the radius of the larger circle, and r is the radius of the smaller circle.

Given: d = 17 cm R = 11 cm r = 3 cm

Substituting these values into the formula: L = √(17² - (11 - 3)²) L = √(17² - 8²) L = √(289 - 64) L = √225 L = 15 cm

Explanation

The total salary of all the employees is the sum of the individual salaries of P, Q, R, S, T, and U.

Total Salary = 200 + 650 + 350 + 250 + 450 + 300 = 2200

Explanation

The given equation is x + 1/x = -2.

Multiplying the entire equation by x gives: x² + 1 = -2x x² + 2x + 1 = 0

This is a perfect square quadratic equation: (x + 1)² = 0 x = -1

Substituting x = -1 into the expression x⁷ + x⁻⁷ + x² + x⁻²: (-1)⁷ + (-1)⁻⁷ + (-1)² + (-1)⁻²

Since -1 raised to any odd power is -1, we have: (-1)⁷ = -1 (-1)⁻⁷ = 1/(-1)⁷ = -1

Since -1 raised to any even power is 1, we have: (-1)² = 1 (-1)⁻² = 1/(-1)² = 1

Adding these values together: -1 + (-1) + 1 + 1 = 0

Explanation

Let the time taken by the man to complete the work be 2x days and by the woman be 3x days.

The individual one-day work rates of the man and the woman are 1/(2x) and 1/(3x) respectively.

Together, they can complete the work in 24 days, which means their combined one-day work rate is 1/24.

We can set up the equation: 1/(2x) + 1/(3x) = 1/24

To solve for x, we find a common denominator on the left side: (3 + 2) / 6x = 1/24 5 / 6x = 1/24 6x = 5 × 24 6x = 120 x = 20

The time taken by the woman alone to complete the work is 3x days: 3 × 20 = 60 days.

Explanation

Selling price (SP) of each machine is Rs. 15000. Total selling price of both machines = 2 × 15000 = Rs. 30000.

The cost price (CP) of the first machine sold at 20% profit is: CP₁ = 15000 / (1 + 0.20) = 15000 / 1.2 = Rs. 12500.

The cost price (CP) of the second machine sold at 40% loss is: CP₂ = 15000 / (1 - 0.40) = 15000 / 0.6 = Rs. 25000.

Total cost price of both machines = 12500 + 25000 = Rs. 37500.

Since total CP is greater than total SP, there is a net loss: Net Loss = 37500 - 30000 = Rs. 7500.

Loss percentage = (Net Loss / Total CP) × 100 = (7500 / 37500) × 100 = 20%.

Explanation

By the principle of conservation of volume, the total volume of the spherical balls is equal to the volume of the melted hemisphere.

The volume of a hemisphere of radius R is: V_h = (2/3) π R³

The volume of a sphere of radius r is: V_s = (4/3) π r³

Let n be the number of spherical balls. The relation is: n × V_s = V_h n × (4/3) π r³ = (2/3) π R³

Simplifying the equation: n = ((2/3) / (4/3)) × (R / r)³ n = ½ × (R / r)³

Substituting the given values, R = 12 cm and r = 6 cm: n = ½ × (12 / 6)³ n = ½ × 2³ n = ½ × 8 n = 4

Explanation

Let the Cost Price of the article be CP and the Marked Price be MP.

When a discount of 20% is allowed on the MP, the Selling Price (SP) is: SP = MP × (1 - 20/100) = 0.80 × MP

Since there is a loss of 5% at this Selling Price: SP = CP × (1 - 5/100) = 0.95 × CP

Equating the two expressions for SP: 0.80 × MP = 0.95 × CP MP / CP = 0.95 / 0.80 = 95 / 80 = 19 / 16

If the article is sold at the marked price, the new Selling Price is equal to MP. Profit Percentage = ((MP - CP) / CP) × 100 Profit Percentage = ((19 - 16) / 16) × 100 Profit Percentage = (3 / 16) × 100 = 18.75%

Explanation

The length of an arc of a circle is given by the formula:

L = (θ / 360°) × 2πr

Given: Radius r = 42 cm Central angle θ = 150° π ≈ 22/7

Substituting these values into the formula:

L = (150 / 360) × 2 × (22 / 7) × 42

Simplifying the fraction and the terms:

L = (5 / 12) × 2 × 22 × 6

L = (5 / 12) × 264

L = 5 × 22

L = 110 cm

Explanation

The three-digit numbers divisible by 15 form an arithmetic progression. The first three-digit number divisible by 15 is 105, and the last is 990. The common difference is 15.

Using the formula for the n-th term of an arithmetic progression: l = a + (n - 1)d

Substituting the values: 990 = 105 + (n - 1) × 15 885 = (n - 1) × 15 n - 1 = 59 n = 60

Using the sum formula for an arithmetic progression: S = (n / 2) × (a + l)

Substituting the values: S = (60 / 2) × (105 + 990) S = 30 × 1095 S = 32850