SSC CHSL Tier 1 2012 Quantitative Aptitude Questions with Solutions

Explanation

Let the initial ratio of A and B be 7:5. After removing 9 litres of the mixture, the ratio of A and B in the remaining mixture remains 7:5.

When 9 litres of liquid B are added, the new ratio of A and B becomes 7:9. Since only liquid B is added, the quantity of A remains constant, and the ratio parts for A in both states are already equal at 7 parts.

The change in liquid B is 9 - 5 = 4 parts, which corresponds to the 9 litres of B added.

4 parts = 9 litres 1 part = 9/4 litres

The total volume of the final mixture is 7 + 9 = 16 parts. Total final volume = 16 × 9/4 = 36 litres.

Since the total volume after replacing 9 litres remains equal to the initial volume, the initial total volume of the mixture was also 36 litres.

Initial quantity of liquid A = 36 × 7/(7 + 5) = 36 × 7/12 = 21 litres.

Explanation

By the rule of alligation, the ratio of the number of students in class A to class B is determined by comparing the differences between their individual averages and the overall average:

Ratio of Class A to Class B = (Average of Class B - Overall Average) / (Overall Average - Average of Class A)

Substituting the given values into the formula:

Ratio = (40 - 30) / (30 - 25) = 10 / 5 = 2 / 1

This yields a ratio of 2:1.

Explanation

Let the total capital be 12 units. This convenient value is chosen because 12 is the least common multiple of the denominators 4 and 3.

The first portion of the capital is one-fourth of 12, which is 3 units, yielding 3% interest. The second portion is two-thirds of 12, which is 8 units, yielding 5% interest. The remaining portion is 12 - (3 + 8) = 1 unit, yielding 11% interest.

The total interest earned from the whole capital is calculated by adding the interest from each portion: Total interest = (3 × 3%) + (8 × 5%) + (1 × 11%) Total interest = 9% + 40% + 11% = 60%

To find the percentage received on the whole capital of 12 units, we divide the total interest by the total capital: Overall percentage = 60% / 12 = 5%

Explanation

The first five odd multiples of 3 are obtained by multiplying 3 by the first five odd natural numbers (1, 3, 5, 7, and 9). These multiples are: 3 × 1 = 3 3 × 3 = 9 3 × 5 = 15 3 × 7 = 21 3 × 9 = 27

Since these terms form an arithmetic progression with a constant common difference of 6, the average of the sequence is equal to its middle term. For a sequence of five terms, the middle (third) term is 15.

Alternatively, the average of an arithmetic progression can be calculated using the formula: Average = (First term + Last term) / 2 Average = (3 + 27) / 2 = 15

Explanation

Let CP₁ and CP₂ be the cost prices of the first and second fans, respectively. Since there is neither gain nor loss in the entire transaction, the profit earned on the first fan is equal to the loss incurred on the second fan.

15% of CP₁ = 9% of CP₂

CP₁ / CP₂ = 9 / 15 = 3 / 5

The ratio of their cost prices is 3 : 5.

Using this ratio to divide the total cost price of 2,160:

CP₁ = (3 / 8) × 2160 = 810

CP₂ = (5 / 8) × 2160 = 1350

Explanation

Let the Cost Price of the handcart be CP.

When sold at a loss of 25%, the Selling Price is 75% of the Cost Price. CP × 75% = 720 CP × 3/4 = 720 CP = 720 × 4/3 = 960

To obtain a gain of 25%, the new Selling Price must be 125% of the Cost Price. New Selling Price = CP × 125% New Selling Price = 960 × 5/4 = 1200

Alternatively, the ratio of the two selling prices can be calculated directly. New Selling Price = 720 × (100 + 25) / (100 - 25) New Selling Price = 720 × 125/75 New Selling Price = 720 × 5/3 = 1200

Explanation

Let the price A paid for the cycle be x.

The successive selling price formula is: Final Price = Initial Price × (1 + P₁/100) × (1 + P₂/100)

Here, the final price paid by C is 264, the first profit percentage P₁ is 10%, and the second profit percentage P₂ is 20%.

Substituting the values into the formula: 264 = x × (1 + 10/100) × (1 + 20/100) 264 = x × 1.1 × 1.2 264 = x × 1.32 x = 264 / 1.32 x = 200

Explanation

The net percentage change when a quantity is successively increased by a% and then decreased by b% is given by the formula:

Net percentage change = a - b - (a × b)/100

Substituting the given values of a = 10 and b = 10 into the formula:

Net percentage change = 10 - 10 - (10 × 10)/100 = -1%

The negative sign represents a net decrease, meaning the salary decreased by 1%.

Explanation

Let Kishan's total monthly salary be S.

The total percentage of salary spent on food and the donation is 30% + 3% = 33%.

According to the problem, this combined expenditure is 2,310.

Thus, 33% of S = 2,310.

Solving for S: S = (2,310 × 100) / 33 S = 70 × 100 S = 7,000

Explanation

To find the length of the train, we use the relation:

Distance = Speed × Time

First, the speed of the train must be converted from kilometres per hour to metres per second by multiplying by 5/18:

Speed = 90 × 5/18 = 25 m/s

When a train crosses a stationary point object like a signal, the total distance travelled by the train is equal to the length of the train. Using the formula:

Length of the train = 25 m/s × 10 s = 250 metres

Explanation

The speed of a swimmer upstream is the difference between the swimmer's speed in still water and the speed of the water current.

Upstream speed = Speed in still water - Speed of water current Upstream speed = 4 km/hr - 2 km/hr = 2 km/hr

The time taken to cover a certain distance is calculated using the formula:

Time = Distance / Speed

Substituting the given values:

Time = 10 km / 2 km/hr = 5 hours

Explanation

Let C be the capital of the person.

The income tax rate is 4 paise per rupee, which is equal to 4%. Therefore, the net income after tax is 96% of the gross income.

The interest rate falls from 4% to 3.75%, representing a decrease of 0.25% in the gross interest rate.

The decrease in net yearly income is given by the formula: Decrease in Net Income = Capital × Decrease in Gross Rate × (1 - Tax Rate)

Substituting the given values into the formula: 48 = C × 0.25% × 96% 48 = C × 0.0025 × 0.96 48 = C × 0.0024

Solving for C: C = 48 / 0.0024 C = 480000 / 24 C = 20000

Explanation

The present population when a population decreases at a constant annual rate is given by the formula:

Present population = P × (1 - r/100)ⁿ

Where P is the population 2 years ago, r is the annual rate of decrease, and n is the number of years.

Substituting the given values into the formula:

P = 10000 r = 20 n = 2

Present population = 10000 × (1 - 20/100)² Present population = 10000 × (1 - 0.2)² Present population = 10000 × (0.8)² Present population = 10000 × 0.64 Present population = 6400

Explanation

Let the three dimensions of the rectangular parallelepiped be a, b, and c. We are given two adjacent sides as a = 1 cm and b = 2 cm.

The formula for the total surface area (A) of a rectangular parallelepiped is: A = 2(ab + bc + ca)

Substituting the given values into the formula: 22 = 2(1×2 + 2×c + c×1) 11 = 2 + 3c 3c = 9 c = 3 cm

The formula for the length of the diagonal (d) of a rectangular parallelepiped is: d = √(a² + b² + c²)

Substituting the dimensions a = 1 cm, b = 2 cm, and c = 3 cm: d = √(1² + 2² + 3²) d = √(1 + 4 + 9) d = √14 cm

Explanation

Let the line of centers of the first two circles be the x-axis, with the point of contact C at the origin (0, 0).

Since the circles with centers A and B have radius 2 and touch externally at C, their coordinates are: A = (-2, 0) and B = (2, 0) Consequently, the base AB = 4.

The third circle with center C (0, 0) and radius 2 has the equation: x² + y² = 4

The circle with center A (-2, 0) and radius 2 has the equation: (x + 2)² + y² = 4

Solving these equations for the intersection point D in the upper half-plane (y > 0): (x + 2)² - x² = 0 4x + 4 = 0 x = -1 y = √3 Thus, D = (-1, √3).

Similarly, the circle with center B (2, 0) and radius 2 intersects the third circle at point E in the upper half-plane: (x - 2)² - x² = 0 -4x + 4 = 0 x = 1 y = √3 Thus, E = (1, √3).

The quadrilateral ABDE is an isosceles trapezoid with parallel sides AB and DE. The lengths of the parallel sides are: AB = 4 DE = 1 - (-1) = 2

The height of the trapezoid is the y-coordinate of D and E: h = √3

Using the formula for the area of a trapezoid: Area = ½ × (AB + DE) × h Area = ½ × (4 + 2) × √3 = 3√3 sq. units

Explanation

Let r₁ and r₂ be the radii of the sphere and the hemisphere respectively.

The volume of a sphere of radius r₁ is given by: V₁ = (4/3)πr₁³

The volume of a hemisphere of radius r₂ is given by: V₂ = (2/3)πr₂³

Since the sphere and the hemisphere have the same volume, we set V₁ = V₂: (4/3)πr₁³ = (2/3)πr₂³

Dividing both sides by (2/3)π simplifies the equation to: 2r₁³ = r₂³

We can rewrite this to find the ratio of their cubed radii: r₁³ / r₂³ = ½

Taking the cube root on both sides gives the ratio of their radii: r₁ / r₂ = 1 / ∛2

Explanation

Let the radius of both the cylinder and the sphere be r, and the height of the cylinder be h.

Since the cylinder and the sphere have the same volume: πr²h = 4/3 πr³

Solving for h gives: h = 4/3 r

The total surface area of the cylinder (A_c) is: A_c = 2πrh + 2πr²

Substituting h = 4/3 r into the equation: A_c = 2πr(4/3 r) + 2πr² A_c = 8/3 πr² + 2πr² A_c = 14/3 πr²

The surface area of the sphere (A_s) is: A_s = 4πr²

The ratio of their surface areas is: A_c / A_s = (14/3 πr²) / (4πr²) A_c / A_s = 14 / (3 × 4) A_c / A_s = 14 / 12 A_c / A_s = 7/6

Explanation

The sides of the given triangle are in the ratio 3:4:5, which represents a right-angled triangle. Let the sides of this triangle be 3x, 4x, and 5x.

The area of this right-angled triangle is: Area = ½ × base × height 72 = ½ × 3x × 4x 72 = 6x² x² = 12 x = 2√3

The perimeter of this triangle is: Perimeter = 3x + 4x + 5x = 12x Perimeter = 12 × 2√3 = 24√3

An equilateral triangle has the same perimeter, so if s is the side length of the equilateral triangle: 3s = 24√3 s = 8√3

The area of this equilateral triangle is: Area = (√3 / 4) × s² Area = (√3 / 4) × (8√3)² Area = (√3 / 4) × 192 Area = 48√3 square units

Explanation

The perimeter of a semicircular path of radius r is given by the formula: Perimeter = πr + 2r = r(π + 2)

Given that the perimeter is 36 m, we substitute the value of π as 22/7: r(22/7 + 2) = 36 r(36/7) = 36 r = 7 m

The area of the semicircular path is: Area = ½πr² Area = ½ × (22/7) × 7² Area = 11 × 7 = 77 sq. m

Explanation

To find the intersection point of the two graphs, substitute the value of x from the second equation into the first equation.

Given equations: 25x + 75y = 225 x = 9

Substituting x = 9 into the first equation: 25(9) + 75y = 225 225 + 75y = 225

Subtracting 225 from both sides: 75y = 0 y = 0

This yields the coordinates (9, 0) for the point where the two lines meet.

Explanation

The algebraic identity for the given expression is: a² + b² + c² - ab - bc - ca = ½ [ (a - b)² + (b - c)² + (c - a)² ]

Substitute the given values of a, b, and c to find the differences: a - b = (x + y) - (x - y) = 2y b - c = (x - y) - (x + 2y) = -3y c - a = (x + 2y) - (x + y) = y

Substituting these differences back into the identity: a² + b² + c² - ab - bc - ca = ½ [ (2y)² + (-3y)² + y² ] a² + b² + c² - ab - bc - ca = ½ [ 4y² + 9y² + y² ] a² + b² + c² - ab - bc - ca = ½ [ 14y² ] a² + b² + c² - ab - bc - ca = 7y²

Explanation

The algebraic identity for the cube of a sum of two numbers is: (a + b)³ = a³ + b³ + 3ab(a + b)

Substituting the given equation a + b = 1 into this identity: (1)³ = a³ + b³ + 3ab(1) 1 = a³ + b³ + 3ab

Comparing this result directly with the given expression a³ + b³ + 3ab = k, we get k = 1.

Alternatively, this can be solved quickly by choosing convenient values that satisfy the condition a + b = 1. If we let a = 1 and b = 0, then: k = 1³ + 0³ + 3(1)(0) k = 1

Explanation

Squaring and adding the two given linear equations:

(ax + by)² + (bx - ay)² = 3² + 4²

Expanding the terms on the left side:

(a²x² + b²y² + 2abxy) + (b²x² + a²y² - 2abxy) = 9 + 16

The middle terms +2abxy and -2abxy cancel each other out, leaving:

a²x² + b²y² + b²x² + a²y² = 25

Grouping the terms with a² and b²:

a²(x² + y²) + b²(x² + y²) = 25

Factoring out (x² + y²):

(a² + b²)(x² + y²) = 25

Substituting the given condition x² + y² = 1 into the equation:

(a² + b²)(1) = 25

a² + b² = 25

Explanation

The circumcentre of a triangle is the point where the perpendicular bisectors of its sides intersect, which also serves as the centre of its circumscribed circle. In an obtuse-angled triangle, one of the interior angles is greater than 90°. Because of this wide opening, the perpendicular bisectors of the sides diverge away from the interior of the triangle and intersect each other in the exterior region. Consequently, the circumcentre of an obtuse-angled triangle always lies outside the boundaries of the triangle.

Explanation

In a right-angled triangle ABC with ∠BAC = 90° and AD perpendicular to BC, the altitude AD to the hypotenuse BC divides the hypotenuse into two segments BD and CD. By the property of similar triangles ABD and CAD, the altitude AD is the geometric mean of BD and CD:

AD² = BD × CD

Given that AD = 6 cm and BD = 4 cm, we substitute these values into the equation:

6² = 4 × CD 36 = 4 × CD CD = 9 cm

The total length of the hypotenuse BC is the sum of the two segments BD and CD:

BC = BD + CD BC = 4 + 9 = 13 cm

Explanation

The angle subtended by an arc at the center of a circle is double the angle subtended by it at any point on the remaining part of the circumference. Since the angle subtended by arc CD at the center O is ∠COD = 140°, the angle subtended by the same arc CD at point A on the circumference is: ∠CAD = ½ × ∠COD = ½ × 140° = 70°

In the cyclic quadrilateral ABCD, the internal ray AC divides the angle ∠BAD into ∠BAC and ∠CAD. Therefore, the total angle at vertex A is: ∠BAD = ∠BAC + ∠CAD = 40° + 70° = 110°

In a cyclic quadrilateral, the sum of opposite angles is 180°. Thus, the opposite angles ∠BAD and ∠BCD satisfy: ∠BCD = 180° - ∠BAD = 180° - 110° = 70°

Explanation

According to the property of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Area(ABC) / Area(DEF) = (AB / DE)²

Substituting the given side lengths AB = 10 cm and DE = 8 cm:

Area(ABC) / Area(DEF) = (10 / 8)² = (5 / 4)² = 25 / 16

This gives the ratio of the areas as 25 : 16.

Explanation

Using the trigonometric identity cosec x = 1/sin x, the relation cos θ cosec 67° = 1 can be rewritten as:

cos θ / sin 67° = 1

cos θ = sin 67°

Using the complementary angle identity sin x = cos(90° - x):

cos θ = cos(90° - 67°)

cos θ = cos 23°

Comparing both sides, we get:

θ = 23°

Explanation

The given expression is a typographical representation of the trigonometric series: sin² 5° + sin² 10° + sin² 15° + ... + sin² 85° + sin² 90°

The angles in the series form an arithmetic progression starting from 5° to 90° with a common difference of 5°. The total number of terms is 90 ÷ 5 = 18 terms.

Using the identity sin(90° - θ) = cos θ, we have sin² (90° - θ) = cos² θ. This gives the relation: sin² θ + sin² (90° - θ) = sin² θ + cos² θ = 1

We can pair the terms from opposite ends of the series up to 40°: (sin² 5° + sin² 85°) + (sin² 10° + sin² 80°) + ... + (sin² 40° + sin² 50°) + sin² 45° + sin² 90°

There are 8 such pairs, with each pair summing to 1. The remaining unpaired terms are sin² 45° and sin² 90°.

Substituting the values: sin² 45° = (1/√2)² = ½ sin² 90° = 1² = 1

Summing these up: Total Sum = (8 × 1) + ½ + 1 = 9 ½

Explanation

Let h be the height in feet at which the post is broken. Since the total height of the post is 15 ft, the length of the upper broken part that bends to touch the ground is 15 - h.

This broken part forms the hypotenuse of a right-angled triangle, where the height from the ground to the break point, h, is the perpendicular side opposite to the 30° angle. Using the trigonometric sine ratio:

sin 30° = h / (15 - h)

Since sin 30° = ½, we can substitute this value into the equation:

½ = h / (15 - h)

Cross-multiplying to solve for h:

15 - h = 2h

3h = 15

h = 5 ft

Explanation

To compare the fractions p = 3/5, q = 7/9, and r = 5/7, we can observe the difference between the denominator and numerator of each fraction.

For p, the difference is 5 - 3 = 2. For q, the difference is 9 - 7 = 2. For r, the difference is 7 - 5 = 2.

Since the difference between the denominator and numerator is constant (equal to 2) for all three proper fractions, the fraction with the smaller numerator is smaller in value, and the fraction with the larger numerator is larger in value.

Comparing the numerators, we have 3 < 5 < 7.

Therefore, 3/5 < 5/7 < 7/9, which means p < r < q.

Explanation

To find the smallest possible integer value of n, solve the given inequality:

7n + 9 > 100

Subtract 9 from both sides of the inequality:

7n > 91

Divide both sides by 7:

n > 13

Since the inequality is strict (n must be strictly greater than 13) and n must be an integer, the smallest possible integer value that satisfies this condition is 14.

Explanation

To find the length of the train, use the distance formula:

Distance = Speed × Time

First, convert the speed of the train from kilometers per hour to meters per second by multiplying by the conversion factor 5/18:

Speed = 36 × (5/18) = 10 m/s

When a train crosses a pole, the total distance covered is equal to the length of the train itself. Substituting the calculated speed and the given time of 25 seconds into the formula:

Length of the train = 10 m/s × 25 s = 250 m

Explanation

The combined rate of work for multiple pipes is the sum of their individual rates of work. The rates at which pipes A, B, and C can fill the cistern are 1/10, 1/12, and 1/15 of the cistern per hour respectively.

Combined Rate = 1/10 + 1/12 + 1/15

To add these fractions, find the Least Common Multiple (LCM) of the denominators 10, 12, and 15, which is 60.

Combined Rate = (6 + 5 + 4) / 60 = 15 / 60 = 1/4 of the cistern per hour

The time taken to fill the entire cistern is the reciprocal of the combined rate:

Time = 1 / (1/4) = 4 hours

Explanation

Let M and B represent the daily work capacity of a man and a boy respectively.

The total work is equal in both scenarios: 10 × (2M + 3B) = 8 × (3M + 2B) 20M + 30B = 24M + 16B 4M = 14B 2M = 7B

Substitute 2M = 7B into the first equation to express the total work in terms of boys' daily work capacity: Total Work = 10 × (7B + 3B) = 100B

The time D required by 2 men and 1 boy to complete the work is given by: D × (2M + 1B) = 100B

Substitute 2M = 7B into the equation: D × (7B + 1B) = 100B 8B × D = 100B D = 100 / 8 = 12.5 days

Explanation

The total area of the rectangular paper ABCD is calculated by multiplying its length and breadth: Area of rectangle = AB × BC = 22 × 14 = 308 cm²

A semi-circular portion with BC as its diameter is removed. Since the diameter is BC = 14 cm, the radius r of this semi-circle is 7 cm.

The area of the semi-circular portion is given by the formula: Area of semi-circle = ½ × π × r²

Using π = 22/7: Area of semi-circle = ½ × (22/7) × 7² = 77 cm²

To find the remaining area of the paper, subtract the area of the semi-circular portion from the total area of the rectangle: Remaining area = Area of rectangle - Area of semi-circle = 308 - 77 = 231 cm²

Explanation

Let s be the side of the square and r be the radius of the circle.

The perimeter of a square is 4s, which is 4 × 22 = 88 cm.

The circumference of the circle is 2πr. Equating the circumference to the perimeter of the square: 2πr = 88

Using the standard approximation π = 22/7: 2 × (22/7) × r = 88 r = (88 × 7) / 44 r = 14 cm

The area of the circle is given by the formula πr²: Area = π × 14² = 196π cm²

Explanation

The selling price of an item after a discount is calculated using the formula:

Selling Price = Listed Price × (1 - Discount Percentage / 100)

Substituting the given values into the formula:

Listed Price = ₹4000 Discount Percentage = 5%

Selling Price = 4000 × (1 - 5 / 100) Selling Price = 4000 × (95 / 100) Selling Price = 40 × 95 Selling Price = ₹3800

Explanation

The net profit percentage when an item is marked up by m% and then sold at a discount of d% is given by the successive percentage formula:

Profit Percentage = m - d - (m × d) / 100

Given: Markup percentage (m) = 25% Discount percentage (d) = 9%

Substituting these values into the formula:

Profit Percentage = 25 - 9 - (25 × 9) / 100

Profit Percentage = 16 - 225 / 100

Profit Percentage = 16 - 2.25 = 13.75%

Explanation

The net selling price (SP) after successive discounts of d₁%, d₂%, and d₃% on a marked price (MP) is given by the formula:

SP = MP × (1 - d₁/100) × (1 - d₂/100) × (1 - d₃/100)

Substituting MP = ₹20,000, d₁ = 10, d₂ = 5, and d₃ = 2:

SP = 20000 × (1 - 10/100) × (1 - 5/100) × (1 - 2/100)

SP = 20000 × 0.90 × 0.95 × 0.98

SP = 18000 × 0.95 × 0.98

SP = 17100 × 0.98

SP = 16758

Explanation

We are given the relation: (1/3) A = 75% of B = 0.6 of C

Converting the percentage and decimal into their simplest fractional forms: 75% = 75/100 = 3/4 0.6 = 6/10 = 3/5

Substituting these fractions into the given relation: (1/3) A = (3/4) B = (3/5) C

Equating each term to a common constant k: (1/3) A = k → A = 3k (3/4) B = k → B = (4/3) k (3/5) C = k → C = (5/3) k

Expressing the ratio A : B : C: A : B : C = 3k : (4/3) k : (5/3) k A : B : C = 3 : 4/3 : 5/3

Multiplying each part of the ratio by 3 to eliminate the denominators: A : B : C = (3 × 3) : 4 : 5 A : B : C = 9 : 4 : 5

Explanation

Let the quantity of water in the mixture be 5 units. Since the quantity of milk is 3/5 that of water, the quantity of milk is 3/5 × 5 = 3 units.

The total quantity of the mixture is the sum of the quantity of milk and the quantity of water, which is 3 + 5 = 8 units.

The proportion of milk in the mixture is the ratio of the quantity of milk to the total quantity of the mixture, which is 3/8.

Explanation

Let the current ages of the father and the son be 7x and 3x, respectively.

The product of their ages is given as: 7x × 3x = 756 21x² = 756 x² = 36 x = 6

Thus, their current ages are: Father's age = 7 × 6 = 42 years Son's age = 3 × 6 = 18 years

After 6 years, their ages will be: Father's age = 42 + 6 = 48 years Son's age = 18 + 6 = 24 years

The ratio of their ages after 6 years will be: 48 : 24 = 2 : 1

Explanation

Using the rule of alligation to find the ratio of men to women:

Average age of men: 45 Average age of women: 28 Combined average age: 36.5

The difference between the average age of men and the combined average is 45 - 36.5 = 8.5. The difference between the combined average and the average age of women is 36.5 - 28 = 8.5.

Therefore, the ratio of the number of men to the number of women is 8.5 : 8.5, which simplifies to 1 : 1.

With a total of 100 workers and a 1 : 1 ratio, the number of women in the factory is: 100 × ½ = 50

Explanation

The mean of n observations x₁, x₂, ..., xₙ is defined by the formula: x̄ = (x₁ + x₂ + ... + xₙ) / n

When each observation is divided by a constant a, the new set of observations becomes x₁/a, x₂/a, ..., xₙ/a. The mean of this new dataset is calculated as: Meanₙew = ((x₁/a) + (x₂/a) + ... + (xₙ/a)) / n Meanₙew = (1/a) × (x₁ + x₂ + ... + xₙ) / n Meanₙew = x̄/a

This demonstrates the scaling property of the arithmetic mean, which states that if every value in a dataset is divided by a constant, the mean of the dataset is also divided by that same constant.

Explanation

The relation between the selling price (SP), cost price (CP), and loss or gain percentage is given by: SP = CP × (100 ± Percentage) / 100

In the first case, selling the cow at a 15% loss yields: 136 = CP × (100 - 15) / 100 136 = CP × 85 / 100 CP = (136 × 100) / 85

To gain 15% in the second case, the new selling price (SP₂) is: SP₂ = CP × (100 + 15) / 100 SP₂ = CP × 115 / 100

Substituting the value of CP into this equation: SP₂ = ((136 × 100) / 85) × (115 / 100) SP₂ = 136 × 115 / 85

Simplifying the fraction by dividing both 115 and 85 by 5: SP₂ = 136 × 23 / 17

Dividing 136 by 17 gives 8: SP₂ = 8 × 23 SP₂ = 184

Explanation

The relationship between Cost Price (CP) and Market Price (MP) when a discount (D%) is offered and a profit (P%) is gained is given by the formula:

CP / MP = (100 - D) / (100 + P)

Given that the market price is ₹50, the discount is 10%, and the profit is 25%, substituting these values into the formula yields:

CP / 50 = (100 - 10) / (100 + 25)

CP / 50 = 90 / 125

CP = (90 / 125) × 50

CP = 90 × (2 / 5)

CP = 18 × 2

CP = 36

Explanation

Using the successive percentage increase formula:

Total Percentage Increase = x + y + xy/100

Here, A sells the article at a 20% profit (x = 20) and B sells it at a 10% profit (y = 10).

Substituting these values into the formula:

Total Percentage Increase = 20 + 10 + (20 × 10)/100 = 30 + 2 = 32%

This represents the net percentage increase in the price from what A paid to what C paid.

Explanation

Let the number be x.

According to the question: (3/5) × (2/3) × (3/7) × x = 150

Simplifying the fraction on the left side: (6/35) × x = 150

Solving for x: x = (150 × 35) / 6 x = 25 × 35 x = 875

To find 60% of this number: 60% of 875 = (60/100) × 875 = (3/5) × 875 = 3 × 175 = 525

Explanation

Let T represent the total number of people employed in the company.

Since the engineers constitute 40% of the total workers, the relationship is given by the formula:

Engineers = 40% of T

Substituting the given value of 80 engineers:

80 = 0.40 × T

Solving for T:

T = 80 ÷ 0.40 = 200