SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

To find the value of x² - (1)/(x²⁾, we use the algebraic identity a² - b² = (a - b)(a + b). Applying this to the expression, we have x² - (1)/(x²⁾ = (x - ¹⁄ₓ)(x + ¹⁄ₓ).

The value of (x - ¹⁄ₓ) is given as 6. To find the value of (x + ¹⁄ₓ), we use the standard identity (x + ¹⁄ₓ)² = (x - ¹⁄ₓ)² + 4. Substituting the given value into this formula: (x + ¹⁄ₓ)² = 6² + 4 (x + ¹⁄ₓ)² = 36 + 4 = 40

Since x > 0, the sum x + ¹⁄ₓ must be positive. Taking the square root of both sides: x + ¹⁄ₓ = √(40) = √(4 × 10) = 2√(10)

Now, we multiply the two terms to find the final result: x² - (1)/(x²⁾ = (x - ¹⁄ₓ)(x + ¹⁄ₓ) = 6 × 2√(10) = 12√(10)

Explanation

The expression is solved using the fundamental Pythagorean identity of trigonometry, which states that 1 + tan²A = sec²A. This identity is derived by dividing the primary identity sin²A + cos²A = 1 by cos²A. By rearranging the terms of 1 + tan²A = sec²A and subtracting tan²A from both sides, we directly obtain sec²A - tan²A = 1. This relation holds for all angles A where sec A and tan A are defined.

Explanation

Let the monthly incomes of the father, mother, and son be F, M, and S respectively. Based on the averages provided, the total income for each pair is calculated by multiplying the average by two: F + M = 2 × 5000 = 10000 M + S = 2 × 6000 = 12000 F + S = 2 × 10000 = 20000

By summing these three equations, we get: 2(F + M + S) = 10000 + 12000 + 20000 = 42000 F + M + S = 21000

To find the father's income, subtract the combined income of the mother and son from the total sum of all three: F = (F + M + S) - (M + S) F = 21000 - 12000 = 9000

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. For △ ABC ∼ △ PQR, this relationship is expressed as:

(ar(△ ABC))/(ar(△ PQR)) = ((BC)/(QR))²

Substituting the given values into the formula:

¹⁶⁄₂₅ = ((20)/(QR))²

Taking the square root on both sides:

⅘ = (20)/(QR)

Solving for QR:

4 × QR = 20 × 5

QR = ¹⁰⁰⁄₄

QR = 25 cm

Explanation

The total earnings of an individual are determined by the product of the number of days worked and their daily wage rate. Given the daily wage ratio is 4:5:3, we can represent their daily wages as 4k, 5k, and 3k respectively. The total earnings for X, Y, and Z are calculated by multiplying their respective working days by these wage rates: X earns 5 × 4k = 20k, Y earns 7 × 5k = 35k, and Z earns 10 × 3k = 30k.

The sum of these individual earnings equals the total cost of the work, which is ₹3,400. Thus, 20k + 35k + 30k = 3400, which simplifies to 85k = 3400. Solving for k gives k = 3400 / 85 = 40. To find the amount received by X, we substitute the value of k back into X's total earnings expression: 20 × 40 = 800. Alternatively, using the simplified ratio of their total shares 20:35:30 as 4:7:6, the share of X is (4 / (4 + 7 + 6)) × 3400 = (4 / 17) × 3400 = 800.

Explanation

The total time taken is the sum of the travel time and the total stoppage time.

First, calculate the travel time using the formula Time = Distance / Speed. Travel time = 84 km / 36 km/h = 7/3 hours. Converting this to minutes: 7/3 × 60 = 140 minutes.

Next, find the number of stops. Ram stops at the end of every 10 km. For a distance of 84 km, the number of stops is the integer part of 84 / 10, excluding a stop at the final destination. Number of stops = 84 // 10 = 8. Total stoppage time = 8 stops × 5 minutes/stop = 40 minutes.

Total time taken = 140 minutes + 40 minutes = 180 minutes.

Explanation

In a circle, a perpendicular drawn from the center to a chord bisects that chord. Given the chord length is 40 cm, the length of the segment from the midpoint to an endpoint is 20 cm. This segment and the perpendicular distance from the center (15 cm) form the legs of a right-angled triangle where the hypotenuse is the radius r of the circle.

Using the Pythagorean theorem: r = √(15² + 20²) r = √(225 + 400) r = √(625) = 25 cm

This calculation follows the 3-4-5 Pythagorean triplet scaled by a factor of 5, where 15:20:25 corresponds to 3(5):4(5):5(5). The diameter of the circle is twice the radius: Diameter = 2 × 25 = 50 cm

Explanation

The expression x⁸ - (1)/(x⁸) can be factored using the identity a² - b² = (a+b)(a-b) iteratively: x⁸ - (1)/(x⁸) = (x⁴ + (1)/(x⁴))(x⁴ - (1)/(x⁴)) = (x⁴ + (1)/(x⁴))(x² + (1)/(x²))(x² - (1)/(x²)) = (x⁴ + (1)/(x⁴))(x² + (1)/(x²))(x + ¹⁄ₓ)(x - ¹⁄ₓ)

Given x + ¹⁄ₓ = 3√2, we calculate the individual components:

To find x² + (1)/(x²), use (x + ¹⁄ₓ)² - 2 = (3√2)² - 2 = 18 - 2 = 16.

To find x⁴ + (1)/(x⁴), use (x² + (1)/(x²))² - 2 = 16² - 2 = 256 - 2 = 254.

To find x - ¹⁄ₓ, use the relation (x - ¹⁄ₓ)² = (x + ¹⁄ₓ)² - 4. (x - ¹⁄ₓ)² = 18 - 4 = 14. Since x > 1, x - ¹⁄ₓ = √(14).

Now, substitute these values into the factored expression: x⁸ - (1)/(x⁸) = 254 × 16 × 3√2 × √(14) x⁸ - (1)/(x⁸) = 4064 × 3√(28) x⁸ - (1)/(x⁸) = 12192 × 2√7 x⁸ - (1)/(x⁸) = 24384√7

Explanation

Since DE ∥ BC, △ ADE ∼ △ ABC by the AA similarity criterion. The property of similar triangles states that the ratio of their areas is equal to the square of the ratio of their corresponding sides.

(Area(△ ADE))/(Area(△ ABC)) = ((AD)/(AB))²

Given that DE divides △ ABC into two equal parts, the area of △ ADE is half the area of △ ABC. Therefore:

(Area(△ ADE))/(Area(△ ABC)) = ½

Equating the two expressions for the area ratio:

((AD)/(AB))² = ½ ⟹ (AD)/(AB) = (1)/(√(2))

We are required to find the ratio DB:AB. Since D lies on AB, we have AB = AD + DB, which implies DB = AB - AD. Dividing both sides by AB:

(DB)/(AB) = (AB - AD)/(AB) = 1 - (AD)/(AB)

Substituting the value of (AD)/(AB):

(DB)/(AB) = 1 - (1)/(√(2)) = √2 - 1{√2}

This yields the ratio DB:AB = √2 - 1 : √2.

Explanation

To simplify the expression [0.08 - {3.5 - 4.9 - (12.5 - 7.8 - 4.6)}], we apply the BODMAS rule, which dictates solving the innermost brackets first.

First, calculate the value inside the parentheses (): 12.5 - 7.8 - 4.6 = 12.5 - 12.4 = 0.1

Next, substitute this value into the curly braces {}: {3.5 - 4.9 - 0.1} = 3.5 - 5.0 = -1.5

Finally, solve the remaining expression by substituting the value from the curly braces: 0.08 - (-1.5) = 0.08 + 1.5 = 1.58

Explanation

In a right-angled triangle, the relationship between the sides is governed by the Pythagorean theorem: p² + b² = h², where p is the perpendicular, b is the base, and h is the hypotenuse. Given the hypotenuse h = 5 cm and the base b = 3 cm, the perpendicular side is calculated as: p² + 3² = 5² p² + 9 = 25 p² = 16 p = 4 cm The trigonometric ratio tan θ is defined as the ratio of the side opposite to the angle θ (perpendicular) to the side adjacent to the angle (base). Taking θ as the angle opposite to the side of length 4 cm, the value is: tan θ = (Perpendicular)/(Base) = ⁴⁄₃

Explanation

The perimeter P of a rectangle is determined by the formula P = 2(L + W), where L is the length and W is the width. Given that the width is 2 m less than the length, the relationship is W = L - 2. By substituting the given perimeter of 68 m into the equation:

68 = 2(L + (L - 2)) 68 = 2(2L - 2)

Dividing both sides by 2 results in:

34 = 2L - 2 36 = 2L L = 18

Explanation

Let the total capacity of the tank be the Least Common Multiple (LCM) of the individual times, which is LCM(12, 18, 36) = 36 units. The filling efficiency of Pipe A is 36 / 12 = 3 units per minute, and for Pipe B, it is 36 / 18 = 2 units per minute. Since Pipe C empties the tank, its efficiency is taken as negative, which is 36 / 36 = -1 unit per minute.

When all three pipes are opened simultaneously, the combined net filling rate is the sum of their efficiencies: 3 + 2 - 1 = 4 units per minute. The time required to fill the 36-unit tank at this combined rate is calculated as total capacity divided by the net rate: 36 / 4 = 9 minutes.

Explanation

At the same time, the sun's rays fall at the same angle of elevation, meaning the height of an object and the length of its shadow are in direct proportion. This relationship can be expressed by the formula (h₁)/(s₁) = (h₂)/(s₂), where h represents the height of the pole and s represents the length of its shadow.

Given: h₁ = 28 m s₁ = 19.2 m h₂ = 52.5 m

Substitute these values into the proportion: (28)/(19.2) = (52.5)/(s₂)

Solving for s₂: s₂ = (52.5 × 19.2)/(28)

Divide 19.2 and 28 by their common factor 4: s₂ = (52.5 × 4.8)/(7)

Divide 52.5 by 7: s₂ = 7.5 × 4.8

Calculating the product: s₂ = 36 m

Explanation

The given expression can be evaluated using the trigonometric identity for complementary angles, which states that cos θ = sin(90° - θ) and sin θ = cos(90° - θ).

For the first term, (cos 41°)/(sin 49°), we observe that 41° and 49° are complementary because 41° + 49° = 90°. Substituting the identity, cos 41° = sin(90° - 41°) = sin 49°. Thus, the fraction becomes (sin 49°)/(sin 49°) = 1.

For the second term, (sin 51°)/(cos 39°), we see that 51° and 39° are also complementary since 51° + 39° = 90°. Using the identity, sin 51° = cos(90° - 51°) = cos 39°. Consequently, this fraction becomes (cos 39°)/(cos 39°) = 1.

Adding the values of the two terms together results in 1 + 1 = 2.

Explanation

The cost price of 1000 g of rice is given as ₹x. When the machine is changed, it shows a reading of 1000 g for an actual weight of 950 g. Therefore, the cost incurred by the seller for the quantity actually sold is the cost price of 950 g of rice.

Cost Price (CP) of 950 g = ˣ⁄₁₀₀₀ × 950 = 0.95x

To earn a profit of 15%, the selling price (SP) must be 115% of the actual cost price incurred by the seller. The seller charges the customer based on the machine reading of 1000 g. Let the required selling price per 1000 g be SP.

SP = Actual CP × (1 + (Profit %)/(100)) SP = 0.95x × (1 + ¹⁵⁄₁₀₀) SP = 0.95x × 1.15

Calculating the product: 0.95 × 1.15 = 0.95 × (1 + 0.15) = 0.95 + 0.1425 = 1.0925

Thus, the selling price should be 1.0925x.

Explanation

Let the original length, breadth, and height of the cuboid be l, b, and h respectively. The original volume is V = l × b × h.

When a value increases by x%, the new value becomes (1 + x/100) times the original. Following the given percentage increases: The new length l' = l × (1 + 200/100) = 3l The new breadth b' = b × (1 + 300/100) = 4b The new height h' = h × (1 + 300/100) = 4h

The new volume V' is calculated as: V' = l' × b' × h' = (3l) × (4b) × (4h) = 48(l × b × h) = 48V

The increase in volume is the difference between the new volume and the original volume: Increase in volume = V' - V = 48V - V = 47V

Therefore, the increase in the volume of the cuboid is 47 times the original volume.

Explanation

The speed of the boat upstream is the difference between the speed of the boat in still water and the speed of the river flow, calculated as sᵤ = 45 - 15 = 30 km/h. Conversely, the speed downstream is the sum of these two speeds, s_d = 45 + 15 = 60 km/h.

Let the distance to be covered be D. The time taken for the upstream journey is tᵤ = (D)/(sᵤ₎ = (D)/(30), and the time taken for the downstream journey is t_d = (D)/(s_d₎ = (D)/(60).

According to the problem, the time difference between the upstream and downstream journeys is 9 hours. This gives the equation: tᵤ - t_d = 9 (D)/(30) - (D)/(60) = 9

To solve for D, find a common denominator: (2D - D)/(60) = 9 (D)/(60) = 9 D = 9 × 60 = 540 km

Alternatively, using the ratio method, the ratio of upstream speed to downstream speed is 30:60, which simplifies to 1:2. Since time is inversely proportional to speed for a constant distance, the ratio of time taken upstream to time taken downstream is 2:1. The difference of 1 unit in the ratio corresponds to the 9 hours given in the question. Therefore, the time taken upstream is 2 × 9 = 18 hours. The distance is calculated as speed multiplied by time: 30 km/h × 18 h = 540 km.

Explanation

The relationship between Marked Price (MP) and Cost Price (CP) when profit and discount percentages are known is: MP = CP × (100 + Gain%)/(100 - Discount%)

Given CP = 2000, Gain% = 20, and Discount% = 20, substituting these values into the formula yields: MP = 2000 × (100 + 20)/(100 - 20) MP = 2000 × ¹²⁰⁄₈₀ MP = 2000 × ³⁄₂ MP = 3000

This formula accounts for the fact that the selling price must simultaneously represent a 20 percent increase over the cost price and an 80 percent value of the marked price. By simplifying the ratio of the profit-adjusted cost to the discount-adjusted price, the marked price is determined to be 3,000.

Explanation

For a number to be divisible by 33, it must be simultaneously divisible by both 3 and 11, since 33 = 3 × 11. A number is divisible by 3 if the sum of its digits is a multiple of 3. For 35508, the sum of the digits is 3 + 5 + 5 + 0 + 8 = 21. Because 21 is divisible by 3, the number 35508 is divisible by 3.

A number is divisible by 11 if the difference between the sum of the digits at odd positions and the sum of the digits at even positions is 0 or a multiple of 11. In the case of 35508, the sum of digits at odd positions is 3 + 5 + 8 = 16, and the sum of digits at even positions is 5 + 0 = 5. The difference is 16 - 5 = 11. Since 11 is divisible by 11, the number 35508 satisfies the condition for divisibility by 11. Having met both criteria, 35508 is divisible by 33.

Explanation

Let the speed of the motorboat in still water be u = 31x and the speed of the current be v = 6x. The downstream speed is S_d = u + v = 31x + 6x = 37x, and the upstream speed is Sᵤ = u - v = 31x - 6x = 25x.

The time taken to travel downstream is T_d = 4 h 10 min, which is equal to 250 minutes. Since the distance covered is the same for both the onward and return journeys, the time taken is inversely proportional to the speed. This relationship is expressed as Tᵤ / T_d = S_d / Sᵤ.

Substituting the known values into the ratio gives Tᵤ = T_d × (S_d / Sᵤ) = 250 × (37x / 25x). Simplifying this calculation, we get Tᵤ = 10 × 37 = 370 minutes. To convert 370 minutes back into hours and minutes, we divide by 60, resulting in 6 hours and 10 minutes.

Explanation

The simple interest (SI) earned is the difference between the final amount and the initial principal sum. SI = Amount - Principal = 38400 - 30000 = 8400

Using the standard simple interest formula SI = (P × R × T)/(100), where P is the principal, R is the annual interest rate, and T is the time in years: 8400 = (30000 × 8 × T)/(100) 8400 = 300 × 8 × T 8400 = 2400 × T T = ⁸⁴⁰⁰⁄₂₄₀₀ T = ⁸⁴⁄₂₄ T = 3.5 years

Explanation

The total number of students in all five colleges is 2080. Summing the students in IIT Delhi, IIT Kanpur, and IIT Guwahati gives 430 + 350 + 400 = 1180. Therefore, the combined number of students in IIT Bombay (B) and IIT Madras (M) is 2080 - 1180 = 900.

According to the problem, the number of students in IIT Bombay is 20% less than in IIT Madras, which gives the relation B = M - 0.20M = 0.8M. Substituting this into the combined total: 0.8M + M = 900 1.8M = 900 ⟹ M = 500 B = 900 - 500 = 400

Next, calculate the number of students in the Applied and Civil departments for these two colleges: Students in Applied department in IIT Bombay = 32% of 400 = 0.32 × 400 = 128 Students in Applied department in IIT Madras = 35% of 500 = 0.35 × 500 = 175 Total students in Applied department = 128 + 175 = 303

Students in Civil department in IIT Bombay = 18% of 400 = 0.18 × 400 = 72 Students in Civil department in IIT Madras = 25% of 500 = 0.25 × 500 = 125 Total students in Civil department = 72 + 125 = 197

The difference between the total students in the Applied department and the Civil department in these two colleges is 303 - 197 = 106.

Explanation

Anuj's total travel time is 8.5 hours, which includes 3 halts of 10 minutes each. The total duration of Anuj's halts is 3 × 10 = 30 minutes, or 0.5 hours. His actual driving time is therefore 8.5 - 0.5 = 8 hours. Using the formula Distance = Speed × Time, the distance between place A and place B is 70 km/h × 8 hours = 560 km.

For Anup, the driving time required to cover this 560 km distance at a speed of 80 km/h is 560 / 80 = 7 hours. Anup takes 4 halts of 15 minutes each, which adds a total halt time of 4 × 15 = 60 minutes, or 1 hour. The total time taken by Anup to reach place B is the sum of his driving time and halt time, which is 7 hours + 1 hour = 8 hours.

Explanation

Using the trigonometric identity sec² A - tan² A = 1, we can express it as (sec A + tan A)(sec A - tan A) = 1. Given that sec A + tan A = 3, it follows that sec A - tan A = ⅓.

By adding these two equations: (sec A + tan A) + (sec A - tan A) = 3 + ⅓ 2 sec A = ¹⁰⁄₃ sec A = ⁵⁄₃

Alternatively, for any value x = sec A + tan A, the cosine value is given by the formula cos A = (2x)/(x² + 1). Substituting x = 3: cos A = (2(3))/(3² + 1) = ⁶⁄₁₀ = ⅗

Explanation

The algebraic identity for the square of a binomial sum is given by (x + y)² = x² + y² + 2xy. To find the value of x² + y², the formula can be rearranged as x² + y² = (x + y)² - 2xy.

Substituting the given values x + y = 7 and xy = 19 into this expression:

x² + y² = (7)² - 2(19) x² + y² = 49 - 38 x² + y² = 11

Explanation

The fundamental Pythagorean identity in trigonometry states that sin² θ + cos² θ = 1. The expression cos² θ - sin² θ represents the double angle formula for cosine, which is cos 2θ. Because cos 2θ varies with the value of θ and is not constantly equal to 1, the equation cos² θ - sin² θ ≡ 1 is not a valid trigonometric identity. An identity must hold true for all values of the variable within its domain, but this equality only occurs at specific points where sin θ = 0.

Explanation

The volume of the cuboid is calculated using the formula V_cᵤ_bₒᵢ_d = length × width × height. Substituting the given dimensions, V_cᵤ_bₒᵢ_d = 8 m × 8 m × 16 m = 1024 m³.

The problem states that the volume of the cuboid is twice the volume of the cube (V_cᵤ_bₑ). Therefore, V_cᵤ_bₑ = V_cᵤ_bₒᵢ_d / 2 = 1024 / 2 = 512 m³. If the side of the cube is a, then a³ = 512, which gives a = ³√(512) = 8 m.

The total surface area of a cube is calculated using the formula 6a². Substituting the value of a, the total surface area is 6 × (8 m)² = 6 × 64 m² = 384 m².

Explanation

The expression can be simplified by expanding each product term individually.

First term: (a - 2b)(b - 3a) = ab - 3a² - 2b² + 6ab = -3a² + 7ab - 2b²

Second term: (a + b)(a - 3b) = a² - 3ab + ab - 3b² = a² - 2ab - 3b²

Third term (including the negative sign): -(b - 3a)(4a - 5b) = -(4ab - 5b² - 12a² + 15ab) = -(-12a² + 19ab - 5b²) = 12a² - 19ab + 5b²

Now, combine the simplified components by grouping like terms: a² terms: -3a² + a² + 12a² = 10a² ab terms: 7ab - 2ab - 19ab = -14ab b² terms: -2b² - 3b² + 5b² = 0

Combining these results gives the simplified expression: 10a² - 14ab

Explanation

The expression a³ + b³ + c³ - 3abc can be calculated using the algebraic identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - (ab + bc + ca)). This identity can be rewritten in a more direct form using the given values of (a + b + c) and (ab + bc + ca) by substituting a² + b² + c² = (a + b + c)² - 2(ab + bc + ca).

Substituting this into the original identity, we get: a³ + b³ + c³ - 3abc = (a + b + c)((a + b + c)² - 3(ab + bc + ca))

Now, substitute the given values a + b + c = 12 and ab + bc + ca = 47 into the formula: a³ + b³ + c³ - 3abc = 12 × (12² - 3 × 47) a³ + b³ + c³ - 3abc = 12 × (144 - 141) a³ + b³ + c³ - 3abc = 12 × 3 a³ + b³ + c³ - 3abc = 36

Explanation

The expression is solved using the VBODMAS rule, which dictates the order of operations: Vinculum, Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Step 1: Simplify the arithmetic operations outside the square brackets. 10 ÷ 5 × 1 + 3 = 2 × 1 + 3 = 2 + 3 = 5

Step 2: Solve the innermost parentheses. (7 - 7 - 9) = 0 - 9 = -9

Step 3: Simplify the expression within the square brackets. In these types of simplification problems, the subtractions are often nested to evaluate the inner relation first. 5 - (-9) = 14 8 - 14 = -6

Step 4: Combine the results from Step 1 and Step 3. 5 - (-6) = 5 + 6 = 11

Explanation

The trigonometric identity for complementary angles states that cot(90° - θ) = tanθ. Substituting this into the given expression, we have:

sec²θ - cot²(90° - θ) = sec²θ - tan²θ

Using the fundamental Pythagorean identity sec²θ - tan²θ = 1, the expression simplifies directly to 1.

Explanation

The time taken by B to complete the work alone can be determined using the work-rate formula 1/D_A₊_B = 1/D_A + 1/D_B, where D_A₊_B is the time taken by A and B together, and D_A and D_B are the times taken by A and B individually. Substituting the given values, we have 1/20 = 1/30 + 1/D_B. Rearranging to solve for 1/D_B gives 1/D_B = 1/20 - 1/30. Using a common denominator of 60, the calculation becomes 1/D_B = 3/60 - 2/60 = 1/60. Thus, D_B = 60 days.

Alternatively, using the unit work method, let the total work be the least common multiple of 20 and 30, which is 60 units. The combined efficiency of A and B is 60 / 20 = 3 units per day. The efficiency of A alone is 60 / 30 = 2 units per day. The efficiency of B is the difference between the combined efficiency and A's efficiency, which is 3 - 2 = 1 unit per day. Therefore, B alone will take 60 / 1 = 60 days to finish the work.

Explanation

The time taken to perform a task is directly proportional to the amount of work completed when the efficiency is constant. To find the time for a different portion of the work, we can set up a ratio based on the principle that the work done per day remains the same.

The worker completes ⅗ of the work in 12 days. First, determine the time taken to complete the entire work (1 unit): Full work time = 12 ÷ ⅗ = 12 × ⁵⁄₃ = 20 days.

Now, calculate the time required for ¾ of the work by applying this total time to the new fraction: Time for ¾ work = 20 × ¾ = 15 days.

This can be calculated in a single step using the direct variation formula (D₁)/(W₁) = (D₂)/(W₂) , where D is the number of days and W is the portion of work: (12)/(3/5) = (D₂)/(3/4) 12 × ⁵⁄₃ = D₂ × ⁴⁄₃ 20 = ⁴⁄₃ D₂ D₂ = (20 × 3)/(4) = 15 days.

Explanation

The average weight is calculated by dividing the sum of the individual weights by the total number of individuals. Using the formula for arithmetic mean:

Average = (Σ x)/(n)

Here, the weights are 49, 42, 61, 55, and 58, and the number of girls is 5.

Sum = 49 + 42 + 61 + 55 + 58 = 265

Average = ²⁶⁵⁄₅ = 53 kg

Explanation

In the first lag, the distance covered d₁ is calculated as speed multiplied by time: d₁ = 50 km/h × 5 hours = 250 km.

In the second lag, the speed v₂ is increased by 20%, so v₂ = 50 × 1.2 = 60 km/h. The distance d₂ is also 20% more than the first lag distance, so d₂ = 250 × 1.2 = 300 km. The time taken for the second lag t₂ is d₂ / v₂ = 300 / 60 = 5 hours.

For the third lag, the average speed v₃ of the first two lags is the total distance of the first two lags divided by their total time: v₃ = (250 + 300) / (5 + 5) = 550 / 10 = 55 km/h. The distance d₃ is 10% more than the second lag distance, so d₃ = 300 × 1.1 = 330 km. The time taken for the third lag t₃ is d₃ / v₃ = 330 / 55 = 6 hours.

The total time taken for all three lags is the sum of the time for each individual lag: T = t₁ + t₂ + t₃ = 5 + 5 + 6 = 16 hours.

Explanation

Initial monthly savings are calculated as the difference between income and expenditure: S = 12,000 - 8,000 = 4,000.

The increase in monthly income is 10% of 12,000, which is 1,200. The increase in monthly expenditure is 8% of 8,000, which is 640.

The increase in savings is the difference between the increase in income and the increase in expenditure: 1,200 - 640 = 560.

To find the percentage increase in savings, we divide the increase in savings by the initial savings and multiply by 100: Percentage Increase = (560)/(4,000) × 100 = 14%

Alternatively, using the concept of weighted averages where income is the sum of expenditure and savings, we can represent the ratio of expenditure to savings as 8,000:4,000, which simplifies to 2:1. Let x be the percentage increase in savings. 10 = ((2 × 8) + (1 × x))/(2 + 1) 30 = 16 + x x = 14%

Explanation

The simple interest earned on the first sum is found by subtracting the principal from the amount: SI = 8,320 - 6,400 = 1,920. Using the formula SI = (P × R × T)/(100), we substitute the given values to determine the annual rate of interest: 1,920 = (6,400 × R × 3)/(100). This simplifies to 1,920 = 192R, resulting in R = 10% per annum.

To find what ₹7,200 will become in 5 years at the same rate, we calculate the new simple interest: SI = (7,200 × 10 × 5)/(100) = 3,600. The final amount is obtained by adding this interest to the principal: 7,200 + 3,600 = 10,800.

Explanation

The efficiency of A is 3.5 times that of B, which can be expressed as the ratio E_A : E_B = 3.5 : 1 = 7 : 2. Since time taken is inversely proportional to efficiency, the ratio of the time taken by A and B to complete the work is T_A : T_B = 2 : 7.

Let the time taken by A be 2x days and the time taken by B be 7x days. According to the problem, the difference in time is 45 days: 7x - 2x = 45 5x = 45 x = 9

Therefore, A takes 2 × 9 = 18 days and B takes 7 × 9 = 63 days to complete the work alone. To find the time taken when working together, we use the formula for combined work: Timeₜₒ_gₑₜₕₑᵣ = (T_A × T_B)/(T_A + T_B) Timeₜₒ_gₑₜₕₑᵣ = (18 × 63)/(18 + 63) = (18 × 63)/(81)

Simplifying the fraction by dividing both 63 and 81 by 9 gives: (18 × 7)/(9) = 2 × 7 = 14 days

Explanation

The expression 50 - [20 + (30 - (25 - 5))] is solved using the BODMAS rule, which dictates the order of operations by solving brackets from the innermost to the outermost.

First, the calculation inside the innermost parentheses is performed: 25 - 5 = 20

The expression now becomes: 50 - [20 + (30 - 20)]

Next, the remaining parentheses are simplified: 30 - 20 = 10

The expression is now: 50 - [20 + 10]

Then, the operation inside the square brackets is completed: 20 + 10 = 30

The final subtraction is: 50 - 30 = 20

Explanation

Let the cost price for A be 100. Since A sells the article to B at a 25% profit, the cost price for B is 100 × 1.25 = 125.

The cost price for C is given as 30% more than the cost price for A, so the cost price for C is 100 × 1.30 = 130.

The profit earned by B is the difference between the price at which B sells to C and the price at which B bought from A. Thus, B's profit is 130 - 125 = 5.

The profit percentage earned by B is calculated on B's cost price: Profit % of B = (Profit)/(CP of B) × 100 = ⁵⁄₁₂₅ × 100 = ¹⁄₂₅ × 100 = 4%

Explanation

To calculate the speed in km/h, first convert the distance from meters to kilometers and the time from minutes to hours. The distance is 3300 m = ³³⁰⁰⁄₁₀₀₀ = 3.3 km. The time is 45 minutes = ⁴⁵⁄₆₀ hours = 0.75 hours.

Using the standard speed formula: Speed = (Distance)/(Time) Speed = (3.3 km)/(0.75 h) Speed = ³³⁰⁄₇₅ = 4.4 km/h

Alternatively, speed can be calculated by first finding the speed in meters per minute and then converting to km/h: Speed in m/min = ³³⁰⁰⁄₄₅ = ²²⁰⁄₃ m/min To convert to km/h, multiply by ⁶⁰⁄₁₀₀₀: Speed = ²²⁰⁄₃ × ⁶⁰⁄₁₀₀₀ = 22 × 0.2 = 4.4 km/h

Explanation

The relationship between successive discounts and the final Selling Price (SP) is determined by the multiplier method. For a Marked Price (MP) of 100, a single equivalent discount of 31.6% results in an SP of 100 × (1 - 0.316) = 68.4.

The final SP after successive discounts of 5%, 10%, and p% is expressed as: SP = MP × (1 - (d₁)/(100)) × (1 - (d₂)/(100)) × (1 - (d₃)/(100)) 68.4 = 100 × (1 - ⁵⁄₁₀₀) × (1 - ¹⁰⁄₁₀₀) × (1 - ᵖ⁄₁₀₀) 68.4 = 100 × 0.95 × 0.9 × (1 - ᵖ⁄₁₀₀) 68.4 = 85.5 × (1 - ᵖ⁄₁₀₀)

To find the value of the unknown discount factor: 1 - ᵖ⁄₁₀₀ = (68.4)/(85.5) 1 - ᵖ⁄₁₀₀ = ⁶⁸⁴⁄₈₅₅ Dividing both numerator and denominator by their common factors (first by 9, then by 19): 1 - ᵖ⁄₁₀₀ = ⁷⁶⁄₉₅ = ⅘ 1 - ᵖ⁄₁₀₀ = 0.8

Calculating p: ᵖ⁄₁₀₀ = 1 - 0.8 = 0.2 p = 20

Alternatively, using the successive discount formula x + y - ˣʸ⁄₁₀₀ for the first two discounts: 5 + 10 - (5 × 10)/(100) = 14.5% Using this combined value with p: 14.5 + p - (14.5 × p)/(100) = 31.6 0.855p = 31.6 - 14.5 0.855p = 17.1 p = (17.1)/(0.855) = 20

Explanation

The original speed is 72 km/h and the travel time is 7.5 hours, giving a distance of 540 km. Increasing the speed by 25% results in a new speed of 90 km/h, which covers the distance in 6 hours (540/90). The time saved is the difference between the initial 7.5 hours and the new 6 hours, which is 1.5 hours or 1 hour 30 minutes.

Explanation

The expression simplifies by following the BODMAS rule: first, the inner-most parenthesis (5/4 + 11/4) equals 4; then (38 - 4) equals 34; the curly braces {75 + 34} equal 109; the square brackets [142 - 109] equal 33; and finally, 264 - 33 equals 231.

Explanation

The weighted average formula is used: (1 * 126 + 1 * 135 + 2 * x) / (1 + 1 + 2) = 153. Solving the equation 126 + 135 + 2x = 612 gives 2x = 351, which results in x = 175.5.

Explanation

The quantity of flour required is directly proportional to the number of people. Using the unitary method, the amount of flour needed for one person is determined by dividing the initial weight of flour by the initial number of people:

Flour per person = (2.4 kg)/(15) = 0.16 kg

To calculate the total flour needed for 36 people, multiply the quantity required per person by the total number of people:

Total flour needed = 0.16 kg × 36 = 5.76 kg

This can also be solved using the direct proportion formula x₁{y₁} = x₂{y₂}:

(2.4)/(15) = ˣ⁄₃₆

x = (2.4 × 36)/(15)

x = (86.4)/(15) = 5.76 kg

Explanation

The amount returned A is ⁹⁄₅ of the principal P, which can be expressed as A = 1.8P. The simple interest I earned over the period is the difference between the amount and the principal, calculated as I = A - P = 1.8P - P = 0.8P. Using the simple interest formula I = (P × R × T)/(100), where T = 4 years, we substitute the values to find the rate R.

0.8P = (P × R × 4)/(100)

By cancelling P from both sides, the equation becomes 0.8 = (4R)/(100). Multiplying both sides by 100 gives 80 = 4R. Solving for R results in R = ⁸⁰⁄₄ = 20. The rate of interest is 20% per annum.

Explanation

Let the speed of the policeman be Vp and the speed of the thief be Vt. Given Vt = 0.8 Vp, the relative speed is Vp - Vt = 0.2 Vp. Using the formula distance = relative speed x time, we get 0.5 km = (0.2 Vp) x (12/60) hours, which simplifies to 0.5 = 0.04 Vp, giving Vp = 12.5 km/h. Thus, the speed of the thief is 0.8 x 12.5 = 10 km/h.

Explanation

The capacity of the tank is taken as the LCM of 9/2 and 36/5, which is 36 units. The inlet pipe fills 8 units per hour, and the outlet pipe drains 5 units per hour, resulting in a net fill of 3 units every 2 hours (one cycle). After 10 cycles (20 hours), the tank is filled with 30 units. In the 21st hour, the inlet pipe needs to fill the remaining 6 units (36 - 30 = 6), which takes 6/8 or 3/4 of an hour. Thus, the total time required is 20 3/4 hours.