SSC CHSL Tier 1 2012 Quantitative Aptitude Questions with Solutions

Explanation

Let the three dimensions of the rectangular parallelepiped be a = 1 cm, b = 2 cm, and c.

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

Substituting the given total surface area of 22 square cm and the known side lengths: 22 = 2(1×2 + 2c + 1c) 11 = 2 + 3c 3c = 9 c = 3 cm

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

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

Explanation

The upstream speed of a swimmer is the difference between the speed of the swimmer in still water and the speed of the water stream.

Upstream speed = Speed in still water - Speed of water stream 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 / Upstream speed

Substituting the given values:

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

Explanation

Let C be the capital of the person.

Since the tax rate is 4 paise per rupee, the tax rate is 4% of the income. The net income is therefore 96% of the gross income.

The interest rate falls from 4% to 3.75%, making the decrease in the interest rate: 4% - 3.75% = 0.25%

The net decrease in yearly income is calculated using the relation: Net decrease = Capital × Decrease in interest rate × (100% - Tax rate)

Substituting the given values: 48 = C × (0.25 / 100) × (1 - 0.04) 48 = C × 0.0025 × 0.96 48 = C × 0.0024 C = 48 / 0.0024 C = 20,000

Explanation

Using the population depreciation formula, the present population (P) is calculated from the population n years ago (P₀) at a annual decrease rate of r% as:

P = P₀ × (1 - r/100)ⁿ

Substituting the given values, where P₀ = 10000, r = 20, and n = 2:

P = 10000 × (1 - 20/100)² P = 10000 × (4/5)² P = 10000 × 16/25 P = 400 × 16 P = 6400

Explanation

The ratio of the sides of the triangle is 3:4:5. Because 3² + 4² = 5², this is a right-angled triangle.

Let the sides of the triangle be 3x, 4x, and 5x.

The area of a right-angled triangle is given by the formula: Area = ½ × base × height Area = ½ × 3x × 4x = 6x²

Since the area of the triangle is 72 square units: 6x² = 72 x² = 12 x = √12 = 2√3

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

The perimeter of the equilateral triangle is the same as this perimeter. Let the side of the equilateral triangle be a: 3a = 24√3 a = 8√3

The area of an equilateral triangle is given by the formula: Area = (√3/4) × a² Area = (√3/4) × (8√3)² Area = (√3/4) × 192 Area = 48√3 square units

Explanation

The distance between the centres of the two touching circles is AB = AC + BC = 2 + 2 = 4.

Since the third circle with centre C and radius 2 meets the first circle (centre A, radius 2) at D, we have AC = CD = AD = 2. Thus, triangle ACD is an equilateral triangle of side 2.

Similarly, the third circle meets the second circle (centre B, radius 2) at E, so BC = CE = BE = 2. Thus, triangle BCE is an equilateral triangle of side 2.

Since A, C, and B are collinear, the angle ∠ACB = 180°. The angles in the equilateral triangles are ∠DCA = 60° and ∠ECB = 60°. Therefore, the angle ∠DCE is: ∠DCE = 180° - 60° - 60° = 60°

In triangle CDE, we have CD = CE = 2 and ∠DCE = 60°, which makes triangle CDE an equilateral triangle of side 2 as well.

The quadrilateral ABDE is composed of these three non-overlapping equilateral triangles: ACD, CDE, and BCE.

Using the formula for the area of an equilateral triangle of side s: Area = (√3/4) × s²

Substituting s = 2: Area of one triangle = (√3/4) × 2² = √3

Since there are three such triangles, the total area of the quadrilateral ABDE is: Total Area = 3 × √3 = 3√3 sq. units

Explanation

Let r₁ be the radius of the sphere and r₂ be the radius of the hemisphere.

The volume of a sphere is given by the formula: V₁ = (4/3) π r₁³

The volume of a hemisphere is given by the formula: V₂ = (2/3) π r₂³

Since their volumes are equal, we equate the two formulas: (4/3) π r₁³ = (2/3) π r₂³

Dividing both sides by (2/3) π: 2 r₁³ = r₂³

Taking the cube root on both sides: ∛2 r₁ = r₂

Rearranging the terms to find the ratio of their radii (r₁ : r₂): r₁/r₂ = 1 / ∛2

Thus, the ratio of their radii is 1 : ∛2.

Explanation

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

The volume of the sphere is 4/3 × π × r³ and the volume of the cylinder is π × r² × h. Since their volumes are equal: 4/3 × π × r³ = π × r² × h h = 4/3 × r

The total surface area of the cylinder is: A_cylinder = 2 × π × r × h + 2 × π × r² A_cylinder = 2 × π × r × (4/3 × r) + 2 × π × r² A_cylinder = 8/3 × π × r² + 2 × π × r² A_cylinder = 14/3 × π × r²

The surface area of the sphere is: A_sphere = 4 × π × r²

The ratio of the surface area of the cylinder to the surface area of the sphere is: A_cylinder / A_sphere = (14/3 × π × r²) / (4 × π × r²) A_cylinder / A_sphere = (14/3) / 4 A_cylinder / A_sphere = 14 / 12 A_cylinder / A_sphere = 7 / 6

Explanation

Using the algebraic identity for the cube of a sum: (a + b)³ = a³ + b³ + 3ab(a + b)

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

Since k = a³ + b³ + 3ab, it directly follows that k = 1.

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

Explanation

We square both of the given equations and then add them together:

(ax + by)² = 3² = 9 (bx - ay)² = 4² = 16

Adding these two squared equations:

(ax + by)² + (bx - ay)² = 9 + 16

Expanding the terms on the left side:

a²x² + b²y² + 2abxy + b²x² + a²y² - 2abxy = 25

The term 2abxy cancels out with -2abxy:

a²x² + b²x² + a²y² + b²y² = 25

Grouping the terms by factoring:

(a² + b²)x² + (a² + b²)y² = 25 (a² + b²)(x² + y²) = 25

Given that x² + y² = 1, substituting this into the equation yields:

(a² + b²)(1) = 25 a² + b² = 25

Explanation

The circumcentre of a triangle is the point where the perpendicular bisectors of its sides intersect. For an obtuse-angled triangle, where one angle is greater than 90°, the circumcentre always lies outside the boundary of the triangle. This occurs because the perpendicular bisectors of the two sides forming the obtuse angle extend away from the interior of the triangle to meet at a point in the exterior.

Explanation

The perimeter of a semicircle of radius r is given by the sum of its curved boundary and its diameter: Perimeter = πr + 2r = r(π + 2)

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

The area of a semicircle is given by the formula: Area = ½ π r²

Substituting r = 7 m and π ≈ 22/7 into the formula: Area = ½ × (22/7) × 7² Area = 11 × 7 Area = 77 sq.m

Explanation

To find the point where the graphs of the equations 25x + 75y = 225 and x = 9 meet, we substitute the value of x = 9 into the first equation:

25(9) + 75y = 225

225 + 75y = 225

Subtracting 225 from both sides gives:

75y = 0

y = 0

Therefore, the coordinates of the point of intersection are (9, 0).

Explanation

Using the standard algebraic identity: a² + b² + c² - ab - bc - ca = ½ [(a - b)² + (b - c)² + (c - a)²]

We first calculate the differences between the given variables: a - b = (x + y) - (x - y) = 2y b - c = (x - y) - (x + 2y) = -3y c - a = (x + 2y) - (x + y) = y

Substituting these values into the identity yields: ½ [(2y)² + (-3y)² + (y)²] = ½ [4y² + 9y² + y²] = ½ [14y²] = 7y²

Explanation

The given equation is cos θ cosec 23° = 1.

Using the reciprocal trigonometric identity, cosec A = 1 / sin A, the equation can be rewritten as: cos θ × (1 / sin 23°) = 1 cos θ = sin 23°

Applying the co-function identity, sin A = cos(90° - A): cos θ = cos(90° - 23°) cos θ = cos 67°

Comparing both sides gives: θ = 67°

Explanation

The given series is sin²5° + sin²10° + sin²15° + ... + sin²85° + sin²90°.

The angles form an arithmetic progression with a common difference of 5°, ranging from 5° to 90°. Since 90 ÷ 5 = 18, there are 18 terms in the series.

Using the trigonometric identity sin(90° - θ) = cos θ, we get sin²(90° - θ) = cos²θ. Therefore, we can pair terms using the relation sin²θ + sin²(90° - θ) = sin²θ + cos²θ = 1.

We can group the first 16 terms into 8 pairs: (sin²5° + sin²85°) + (sin²10° + sin²80°) + ... + (sin²40° + sin²50°)

Each of these 8 pairs sums to 1, contributing a value of 8.

The remaining two terms that are not part of these pairs are sin²45° and sin²90°. Evaluating them individually: sin²45° = (1/√2)² = ½ sin²90° = 1² = 1

Summing all the components: Total Sum = (8 × 1) + ½ + 1 = 9½

Explanation

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

This configuration forms a right-angled triangle with the ground, where the vertical side is h, the hypotenuse is 15 - h, and the angle of elevation from the ground is 30°.

Using the sine trigonometric ratio: sin 30° = height / hypotenuse ½ = h / (15 - h)

Multiplying both sides gives: 15 - h = 2h 3h = 15 h = 5 ft

Explanation

In a right-angled triangle ABC with ∠BAC = 90° and AD perpendicular to BC, the altitude AD to the hypotenuse BC satisfies the geometric mean theorem:

AD² = BD × CD

Substituting the given values AD = 6 cm and BD = 4 cm:

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 of a circle at the center is double the angle subtended by it at any point on the remaining part of the circle.

For the arc CD, the angle subtended at the center O is ∠COD = 140°. Therefore, the angle subtended by the same arc at the point A on the circumference is ∠CAD = ½ × ∠COD = ½ × 140° = 70°.

The total angle at vertex A is ∠BAD = ∠BAC + ∠CAD. Substituting the given value of ∠BAC = 40°, we get ∠BAD = 40° + 70° = 110°.

In a cyclic quadrilateral ABCD, the sum of opposite angles is 180°. Since ∠BAD and ∠BCD are opposite angles, we have ∠BAD + ∠BCD = 180°.

Substituting the value of ∠BAD gives 110° + ∠BCD = 180°, which yields ∠BCD = 70°.

Explanation

According to the properties 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(ΔABC) / Area(ΔDEF) = (AB / DE)²

Substituting the given values AB = 10 cm and DE = 8 cm into the relation:

Area(ΔABC) / Area(ΔDEF) = (10 / 8)² = (5 / 4)² = 25 / 16

Explanation

Let the total work be the Least Common Multiple of the given number of days, which is the LCM of 36, 60, and 45.

LCM(36, 60, 45) = 180 units

Now, determine the daily work efficiency of each pair: Efficiency of A and B = 180 / 36 = 5 units per day Efficiency of B and C = 180 / 60 = 3 units per day Efficiency of A and C = 180 / 45 = 4 units per day

Adding these three equations gives the combined efficiency of all three working together twice: 2 × (Efficiency of A + B + C) = 5 + 3 + 4 = 12 units per day Efficiency of A + B + C = 12 / 2 = 6 units per day

To find C's individual efficiency, subtract the efficiency of A and B from the total efficiency: Efficiency of C = (Efficiency of A + B + C) - (Efficiency of A + B) = 6 - 5 = 1 unit per day

The time taken by C alone to complete the work is: Time = Total work / Efficiency of C = 180 / 1 = 180 days

Explanation

Total work can be assumed as the Least Common Multiple of individual days 16, 24, and 8, which is 48 units.

Efficiency of the first man = 48 ÷ 16 = 3 units per day. Efficiency of the second man = 48 ÷ 24 = 2 units per day. Combined efficiency of all three men working together = 48 ÷ 8 = 6 units per day.

Efficiency of the third man = Combined efficiency - (First man's efficiency + Second man's efficiency) = 6 - (3 + 2) = 1 unit per day.

Since all of them work together for the same duration of 8 days, the total wages are shared in the ratio of their daily efficiencies.

Ratio of shares of the three men = 3 : 2 : 1 Sum of the ratio terms = 3 + 2 + 1 = 6

Share of the third man = (1/6) × 960 = 160

Explanation

Let x and y be two natural numbers such that x² - y² = 45. Using the algebraic identity for the difference of squares, this equation can be factored as (x - y)(x + y) = 45.

Since x and y are natural numbers, both (x - y) and (x + y) must be integers. Because x + y must be positive and their product is 45, both factors must be positive integers, with x + y > x - y.

Let a = x - y and b = x + y. Thus, a and b are positive factors of 45 such that ab = 45 and a < b.

The positive factor pairs of 45 are: 1 × 45 = 45 3 × 15 = 45 5 × 9 = 45

Solving for x and y from the system of equations gives: x = (b + a)/2 y = (b - a)/2

For each of the three factor pairs, both a and b are odd, meaning their sum and difference are always even. This guarantees that x and y will be integers. Since b > a, y is at least 1, so both x and y are natural numbers.

For (1, 45): x = 23, y = 22 For (3, 15): x = 9, y = 6 For (5, 9): x = 7, y = 2

This yields exactly 3 pairs of natural numbers.

Explanation

The given series is an arithmetic progression with the first term a = 2, common difference d = 2, and the last term aₙ = 198.

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

Substituting the values to find the total number of terms, n: 198 = 2 + (n - 1) × 2 196 = (n - 1) × 2 98 = n - 1 n = 99

Since the total number of terms is 99, which is odd, there is only one middle term. The position of this middle term is: (n + 1)/2 = (99 + 1)/2 = 50th term

The value of the 50th term is: a₅₀ = a + (50 - 1)d a₅₀ = 2 + 49 × 2 a₅₀ = 2 + 98 a₅₀ = 100

Explanation

The total number of decimal places in the numerator is 4 + 4 + 3 = 11, and in the denominator is 3 + 4 + 4 = 11. Since the number of decimal places is equal, we can eliminate the decimal points entirely to simplify the expression:

(3555 × 5555 × 2025) / (225 × 17775 × 2222)

We can simplify this by grouping corresponding terms:

3555 / 17775 = 1/5

5555 / 2222 = 5/2

2025 / 225 = 9

Multiplying these simplified ratios together:

(1/5) × (5/2) × 9 = 9/2 = 4.5

Explanation

Total wages are calculated as the product of the number of workers and the individual wages per worker.

Initial total wages : New total wages = (Initial number of workers × Initial individual wages) : (New number of workers × New individual wages)

Substituting the given ratios: Initial total wages : New total wages = (15 × 22) : (11 × 25) Initial total wages : New total wages = 330 : 275

Dividing both terms by 55: 330 / 55 : 275 / 55 = 6 : 5

The total wages of the workers are decreased in the ratio 6 : 5.

Explanation

The equivalent discount rate of two successive discounts of d₁% and d₂% is given by the formula:

Equivalent Discount = d₁ + d₂ - (d₁ × d₂)/100

Substituting d₁ = 8 and d₂ = 8:

Equivalent Discount = 8 + 8 - (8 × 8)/100 = 16 - 0.64 = 15.36%

The shopkeeper compares this successive discount with a single discount of 16%. The difference in the discount rate is:

16% - 15.36% = 0.64%

The loss incurred by offering a single 16% discount instead of the two successive discounts is 0.64% of the marked price:

Loss = 0.64% of 400 = (0.64/100) × 400 = 2.56

Explanation

The single effective discount equivalent to two successive discounts of d₁% and d₂% is given by the formula:

Effective Discount = d₁ + d₂ - (d₁ × d₂) ÷ 100

Substituting the given successive discount values of d₁ = 30 and d₂ = 30:

Effective Discount = 30 + 30 - (30 × 30) ÷ 100 Effective Discount = 60 - 900 ÷ 100 Effective Discount = 60 - 9 Effective Discount = 51%

Explanation

Total cost price of 2000 copies is ₹70,000.

The number of books available for sale is the total printed copies minus the free specimen copies: 2000 - 400 = 1600 copies

The selling price of each book after a 30% discount on the printed price of ₹75 is: 75 × (1 - 0.30) = 75 × 0.70 = ₹52.5

The total selling price obtained from selling 1600 books is: 1600 × 52.5 = ₹84,000

Since the total selling price of ₹84,000 is greater than the total cost price of ₹70,000, there is a gain: Gain = 84,000 - 70,000 = ₹14,000

The gain percentage is calculated as: (Gain / Total Cost Price) × 100 = (14,000 / 70,000) × 100 = 20%

Explanation

Let the total capital be represented as 1.

The fractions of the capital invested are: First part = 1/4 Second part = 2/3 The remainder = 1 - (1/4 + 2/3) = 1 - 11/12 = 1/12

The weighted average of the interest received on the entire capital is calculated by multiplying each fraction of the capital by its respective interest rate and summing them:

Average interest rate = (1/4 × 3%) + (2/3 × 5%) + (1/12 × 11%)

To add these terms, we find a common denominator of 12:

Average interest rate = (9/12)% + (40/12)% + (11/12)% Average interest rate = (9 + 40 + 11)/12 % Average interest rate = 60/12 % Average interest rate = 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. This gives the sequence 3, 9, 15, 21, and 27.

Since these numbers form an arithmetic progression with a constant common difference of 6, the average is equal to the middle term of the sequence. For a sequence of 5 terms, the middle term is the third term, which is 15.

Alternatively, the average of any arithmetic progression is given by the formula:

Average = (First term + Last term) / 2

Average = (3 + 27) / 2 = 15

Explanation

Let the quantities of alloy A and alloy B be equal.

In alloy A, the ratio of gold to copper is 7:2, giving a total of 7 + 2 = 9 parts. In alloy B, the ratio of gold to copper is 7:11, giving a total of 7 + 11 = 18 parts.

To equalize the quantities, we find the Least Common Multiple (LCM) of 9 and 18, which is 18.

Multiplying the ratio parts of alloy A by 2 gives: Gold in A : Copper in A = 14 : 4

For alloy B, the ratio remains: Gold in B : Copper in B = 7 : 11

When equal quantities of both alloys are melted together to form C, we add the corresponding quantities of gold and copper: Total Gold in C = 14 + 7 = 21 Total Copper in C = 4 + 11 = 15

The ratio of gold and copper in alloy C is 21 : 15, which simplifies to 7 : 5.

Explanation

Let the initial ratio of liquids A and B be 7:5. When 9 litres of mixture are drained off, the ratio of A and B in the remaining mixture remains 7:5.

When 9 litres of liquid B is added to this remaining mixture, the ratio of A and B becomes 7:9.

Since only liquid B is added, the quantity of liquid A remains constant. In the ratios 7:5 and 7:9, the ratio term representing A is indeed constant at 7.

The increase in liquid B is from 5 parts to 9 parts, which is a difference of 4 parts. This difference of 4 parts corresponds directly to the 9 litres of liquid B added: 4 parts = 9 litres

The total volume of the final mixture is 7 + 9 = 16 parts. Since the volume drained off is equal to the volume of B added, the total volume of the final mixture is equal to the initial total volume of the mixture: Total initial volume = (9 ÷ 4) × 16 = 36 litres

Initially, liquid A and B were in the ratio 7:5, which means liquid A constituted 7/12 of the total volume. Initial volume of liquid A = (7 ÷ 12) × 36 = 21 litres

Explanation

Using the allegation method, the ratio of the number of students in class A to class B is calculated as: (Average of Class B - Overall Average) : (Overall Average - Average of Class A)

Substituting the given values: (40 - 30) : (30 - 25) = 10 : 5 = 2 : 1

Explanation

The net percentage change for successive percentage changes of a% and b% is given by the formula: Net change % = a + b + (ab / 100)

Here, the salary is first increased by 10% (so a = 10) and then decreased by 10% (so b = -10). Substituting these values into the formula gives: Net change % = 10 - 10 + (10 × -10) / 100 = -1%

A negative value of 1% indicates a 1% decrease in the overall salary.

Explanation

Let Kishan's total monthly salary be S. The total percentage of salary spent on both food and donation is the sum of the individual percentages:

Total percentage spent = 30% + 3% = 33%

The value corresponding to this total percentage is given as ₹ 2,310. The relation to find the total salary S is:

33% of S = 2,310

(33 / 100) × S = 2,310

S = (2,310 × 100) / 33

S = 70 × 100 = 7,000

Explanation

To find the length of the train in metres, the speed must first be converted from km/hr to m/s.

To convert km/hr to m/s, multiply the speed by 5/18:

Speed = 90 × 5/18 = 25 m/s

When a train crosses a signal post, the distance it travels is exactly equal to its own length. Using the standard formula for distance:

Distance = Speed × Time

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

Explanation

Since there is no net profit or loss in the entire transaction, the profit earned on the first fan must equal the loss incurred on the second fan. Let CP₁ and CP₂ be the cost prices of the two fans.

15% of CP₁ = 9% of CP₂ CP₁ / CP₂ = 9 / 15 = 3 / 5

The ratio of the cost prices of the two fans is 3 : 5, which means the sum of the ratio terms is 3 + 5 = 8.

Cost price of the first fan = (3 / 8) × 2160 = 810 Cost price of the second fan = (5 / 8) × 2160 = 1350

Explanation

The relationship between the Selling Price (SP), loss percentage, and gain percentage can be expressed using the formula:

New SP = Initial SP × (100 + Gain%) / (100 - Loss%)

By substituting the given values into the formula:

New SP = 720 × (100 + 25) / (100 - 25)

New SP = 720 × 125 / 75

Simplifying the fraction 125/75 to 5/3:

New SP = 720 × 5 / 3

New SP = 240 × 5

New SP = 1200

Explanation

Let the cost price of the cycle for A be x.

A sells the cycle to B at a profit of 10%, so the cost price for B is: x × (1 + 10/100) = 1.1x

B sells the cycle to C at a profit of 20%, so the cost price for C is: 1.1x × (1 + 20/100) = 1.1x × 1.2 = 1.32x

Given that C pays ₹ 264 for the cycle: 1.32x = 264 x = 264 / 1.32 = 200

Explanation

Let the two natural numbers be x and y with x > y. According to the problem: x² - y² = 45

Using the algebraic identity for the difference of squares: (x - y)(x + y) = 45

Since x and y are natural numbers, both (x - y) and (x + y) must be positive integers, with (x + y) > (x - y). We can express 45 as the product of two positive integers in the following ways: 45 = 1 × 45 45 = 3 × 15 45 = 5 × 9

We can solve for x and y for each of these factor pairs:

For the pair 1 and 45: x - y = 1 and x + y = 45 Adding these equations gives 2x = 46, which means x = 23. Substituting x gives y = 22. This yields the natural number pair (23, 22).

For the pair 3 and 15: x - y = 3 and x + y = 15 Adding these equations gives 2x = 18, which means x = 9. Substituting x gives y = 6. This yields the natural number pair (9, 6).

For the pair 5 and 9: x - y = 5 and x + y = 9 Adding these equations gives 2x = 14, which means x = 7. Substituting x gives y = 2. This yields the natural number pair (7, 2).

Thus, there are exactly 3 such pairs of natural numbers.

Explanation

The given series 2 + 4 + 6 + ... + 198 is an arithmetic progression where the first term a = 2 and the common difference d = 2.

The formula for the nth term of an arithmetic progression is: a_n = a + (n - 1)d

Substituting the last term to find the total number of terms: 198 = 2 + (n - 1) × 2 196 = (n - 1) × 2 98 = n - 1 n = 99

Since the total number of terms n is 99 (an odd number), there is only one middle term, which is the 50th term: (n + 1) / 2 = (99 + 1) / 2 = 50

We find the value of this 50th term using the nth term formula: a₅0 = a + (50 - 1)d a₅0 = 2 + 49 × 2 a₅0 = 2 + 98 a₅0 = 100

Explanation

To solve the expression, first count the total number of decimal places in the numerator and denominator.

In the numerator: 0.3555 has 4 decimal places. 0.5555 has 4 decimal places. 2.025 has 3 decimal places. Total decimal places in the numerator = 11.

In the denominator: 0.225 has 3 decimal places. 1.7775 has 4 decimal places. 0.2222 has 4 decimal places. Total decimal places in the denominator = 11.

Since both the numerator and the denominator have 11 decimal places, we can remove the decimal points: (3555 × 5555 × 2025) / (225 × 17775 × 2222)

Now, simplify the numbers step-by-step: First, divide 17775 by 3555: 17775 ÷ 3555 = 5

Next, divide both 5555 and 2222 by 1111: 5555 ÷ 1111 = 5 2222 ÷ 1111 = 2

Substitute these values back into the expression: (1 × 5 × 2025) / (225 × 5 × 2)

Cancel out the common factor of 5 from the numerator and denominator: 2025 / (225 × 2)

Divide 2025 by 225: 2025 ÷ 225 = 9

This leaves: 9 / 2 = 4.5

Explanation

Let the total work be the Least Common Multiple (LCM) of 36, 60, and 45. LCM(36, 60, 45) = 180 units.

The daily work rates (efficiencies) are calculated as: Efficiency of A and B = 180 / 36 = 5 units per day Efficiency of B and C = 180 / 60 = 3 units per day Efficiency of A and C = 180 / 45 = 4 units per day

Adding these efficiencies together gives: 2 × (Efficiency of A + B + C) = 5 + 3 + 4 = 12 units per day Efficiency of A + B + C = 12 / 2 = 6 units per day

To find the efficiency of C alone: Efficiency of C = (Efficiency of A + B + C) - (Efficiency of A + B) Efficiency of C = 6 - 5 = 1 unit per day

The number of days taken by C alone to complete the total work is: 180 units / 1 unit per day = 180 days.

Explanation

Let A and B be the two men, and C be the third man.

The number of days taken by A is 16, by B is 24, and by A, B, and C together is 8.

Let the total work be the LCM of 16, 24, and 8, which is 48 units.

Daily work of A = 48 / 16 = 3 units Daily work of B = 48 / 24 = 2 units Daily work of A, B, and C together = 48 / 8 = 6 units

Therefore, the daily work of C = 6 - 3 - 2 = 1 unit.

Since they all work together for the entire duration of 8 days, the wages are divided in the ratio of their daily work (efficiencies), which is A : B : C = 3 : 2 : 1.

The share of the third man C = (1 / 6) × 960 = 160.

Explanation

The formula for the net equivalent discount of two successive discounts d₁% and d₂% is: Net Discount = d₁ + d₂ - (d₁ × d₂)/100

Substituting the two successive discounts of 8% each: Net Discount = 8 + 8 - (8 × 8)/100 = 16 - 0.64 = 15.36%

If the shopkeeper offers a single discount of 16%, the difference between this single discount and the successive discounts is: Difference = 16% - 15.36% = 0.64%

The additional loss incurred by the shopkeeper on the marked price of 400 is: Loss = 400 × 0.64% = 400 × 0.0064 = 2.56

Explanation

The effective discount for two successive discounts of d₁% and d₂% is given by the formula:

Effective Discount = d₁ + d₂ - (d₁ × d₂) / 100

Substituting d₁ = 30 and d₂ = 30 into the formula:

Effective Discount = 30 + 30 - (30 × 30) / 100 Effective Discount = 60 - 900 / 100 Effective Discount = 60 - 9 Effective Discount = 51%

Explanation

The total cost of printing 2000 copies is ₹70,000.

Subtracting the 400 free specimen copies, the number of copies available for sale is: 2000 - 400 = 1600 copies

The selling price of each book after a 30% discount on the printed price of ₹75 is: 75 × (1 - 0.30) = ₹52.5

The total revenue from selling 1600 copies is: 1600 × 52.5 = ₹84,000

The profit gained from the sale is: 84,000 - 70,000 = ₹14,000

The profit percentage is: (14,000 / 70,000) × 100 = 20%

Explanation

The total wage bill of a factory is given by the relation: Total Wages = Number of Workers × Wage per Worker

Let the initial number of workers be 15x and the reduced number of workers be 11x. Similarly, let the initial wage per worker be 22y and the increased wage per worker be 25y.

Initial Total Wages = 15x × 22y = 330xy New Total Wages = 11x × 25y = 275xy

The ratio by which the total wages are decreased is the ratio of the initial total wages to the new total wages: 330xy : 275xy = 330 : 275

Dividing both terms by 55 simplifies the ratio to: 6 : 5

Explanation

To find the ratio of gold and copper in the new alloy C, we can take equal quantities of both alloys.

In alloy A, the ratio of gold to copper is 7:2, which gives a sum of 7 + 2 = 9 parts. In alloy B, the ratio of gold to copper is 7:11, which gives a sum of 7 + 11 = 18 parts.

To easily calculate with equal quantities, we find the least common multiple of 9 and 18, which is 18. Let us assume 18 units of each alloy are melted together.

For 18 units of alloy A: Gold = (7/9) × 18 = 14 units Copper = (2/9) × 18 = 4 units

For 18 units of alloy B: Gold = (7/18) × 18 = 7 units Copper = (11/18) × 18 = 11 units

In the new alloy C: Total gold = 14 + 7 = 21 units Total copper = 4 + 11 = 15 units

The ratio of gold to copper in alloy C is: 21 : 15 = 7 : 5