SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

To find the time taken when A, B, and C work together, we use the work-rate formula where the combined rate is the sum of individual rates: 1/T = 1/T_A + 1/T_B + 1/T_C. Substituting the given values, the combined rate is 1/6 + 1/9 + 1/11. The Least Common Multiple (LCM) of 6, 9, and 11 is 198, which we use as the total units of work.

The individual efficiencies are calculated by dividing the total work by each person's time: A produces 198 / 6 = 33 units per day, B produces 198 / 9 = 22 units per day, and C produces 198 / 11 = 18 units per day. The total efficiency of all three working together is 33 + 22 + 18 = 73 units per day.

The time required to complete the work is the total work divided by the total efficiency: T = 198 / 73. Dividing 198 by 73 gives a quotient of 2 and a remainder of 52. This can be written as the mixed fraction 2 ⁵²⁄₇₃. Match this result to the provided options where the mixed fraction is represented as 2 52 73.

Explanation

The term breakeven price implies that the selling price of the mixture is equal to the weighted average cost of its individual components. The relationship between the costs of the components and the mixture is given by the weighted average formula M = (n₁ c₁ + n₂ c₂)/(n₁ + n₂), where M is the mean price, n₁ and n₂ are the units in the mixing ratio, and c₁ and c₂ are the unit prices of the individual colors.

Substituting the given values into the formula: 650 = (2 × 500 + 3 × c₂)/(2 + 3) 650 × 5 = 1000 + 3c₂ 3250 = 1000 + 3c₂ 2250 = 3c₂ c₂ = ²²⁵⁰⁄₃ = 750

Explanation

Let the cost price of the painting be CP. When the painting is sold at Rs.1,300, the loss incurred is CP - 1300. When it is sold at Rs.1,498, the gain is 1498 - CP. According to the problem, the gain is 25% more than the loss, which means Gain = Loss + 0.25 × Loss = 1.25 × Loss.

Substituting the values into the equation: 1498 - CP = 1.25 × (CP - 1300) 1498 - CP = ⁵⁄₄ × (CP - 1300) Multiplying both sides by 4 to clear the fraction: 4 × (1498 - CP) = 5 × (CP - 1300) 5992 - 4CP = 5CP - 6500 5992 + 6500 = 9CP 12492 = 9CP CP = ¹²⁴⁹²⁄₉ = 1388

The cost price of the painting is Rs.1,388. To achieve a gain of 25%, the selling price must be 125% of the cost price: Selling Price = 1388 × 1.25 Selling Price = 1388 × ⁵⁄₄ Selling Price = 347 × 5 = 1735

Explanation

The net effect of successive discounts can be calculated using the successive discount formula x + y - ˣʸ⁄₁₀₀ or by calculating the final multiplier.

Using the successive discount formula, we first combine the first two discounts of 15% and 20%: Net discount for the first two = 15 + 20 - (15 × 20)/(100) = 35 - 3 = 32%

Now, we combine this 32% result with the third discount of 25%: Total net discount = 32 + 25 - (32 × 25)/(100) = 57 - ⁸⁰⁰⁄₁₀₀ = 57 - 8 = 49%

Alternatively, let the initial price be 100. Price after 15% discount = 100 × 0.85 = 85 Price after 20% discount = 85 × 0.80 = 68 Price after 25% discount = 68 × 0.75 = 51 The final price is 51% of the original price, which means the total discount is 100% - 51% = 49%.

Explanation

The time taken by two pipes working together to fill a tank is calculated using the formula T = (xy)/(x + y), where x and y are the individual times taken by each pipe.

Substituting the given values, where x = 18 hours and y = 24 hours:

T = (18 × 24)/(18 + 24)

T = ⁴³²⁄₄₂

To simplify the fraction, divide both the numerator and the denominator by their greatest common divisor, which is 6:

T = ⁷²⁄₇

Converting this improper fraction into a mixed number:

72 ÷ 7 = 10 with a remainder of 2.

Thus, the time taken is 10²⁄₇ hours.

Explanation

In the initial mixture of 55 litres, the ratio of fruit juice to water is 4:1. The total parts in the mixture are 4 + 1 = 5. The volume of one part is calculated as 55 / 5 = 11 litres. Thus, the quantity of fruit juice is 4 × 11 = 44 litres and the quantity of water is 1 × 11 = 11 litres.

When water is added to the mixture, the amount of fruit juice remains constant at 44 litres. The required new ratio of juice to water is 2:1. In this new ratio, 2 parts represent the 44 litres of juice. Therefore, 1 part of water in the new mixture must be 44 / 2 = 22 litres. Since the initial quantity of water was 11 litres, the amount of water that must be added is 22 - 11 = 11 litres.

Explanation

The volume V of a cylindrical flask is determined by the formula V = π r² h, where r is the radius and h is the height. Given a diameter of 12 cm, the radius r = ¹²⁄₂ = 6 cm. The height h is given as 21 cm. Using π = ²²⁄₇, the calculation is:

V = ²²⁄₇ × 6² × 21

V = ²²⁄₇ × 36 × 21

To simplify the calculation, divide 21 by 7:

V = 22 × 36 × 3

V = 22 × 108

V = 2376 cm³

Explanation

The time taken by the policeman to catch the thief and the time the thief runs are equal. Since time is constant, distance is directly proportional to speed. The ratio of the speeds of the policeman and the thief is 10:8, which simplifies to 5:4. Consequently, the ratio of the distances covered by the policeman and the thief is also 5:4.

The difference in the distance covered represents the initial gap of 1250 meters, which is 1.25 km. In terms of the ratio, the difference is 5 - 4 = 1 unit. Therefore, 1 unit corresponds to 1.25 km. To find the distance run by the thief, we multiply the thief's ratio part by this value: 4 × 1.25 = 5 km.

Explanation

The fundamental trigonometric identity states that sec² θ - tan² θ = 1. This can be factored as (sec θ + tan θ)(sec θ - tan θ) = 1. Given that sec θ + tan θ = 5, we can substitute this value to find the conjugate: sec θ - tan θ = ⅕

To find sec θ, we add the two equations: (sec θ + tan θ) + (sec θ - tan θ) = 5 + ⅕ 2 sec θ = 5 + ⅕

Dividing both sides by 2 yields: sec θ = ½(5 + ⅕)

Explanation

In a circle, a perpendicular line drawn from the center to a chord bisects that chord. Since the total length of the chord is 14 cm, the distance from the midpoint of the chord to one of its ends is 7 cm. This perpendicular distance from the center, the half-length of the chord, and the radius of the circle form a right-angled triangle.

Let the radius be r, the perpendicular distance be d = 24 cm, and the half-chord length be l = 7 cm. According to the Pythagoras theorem:

r² = d² + l² r² = 24² + 7² r² = 576 + 49 r² = 625 r = √(625) = 25 cm

The calculation follows the standard Pythagorean triplet (7, 24, 25), where the hypotenuse represents the radius of the circle.

Explanation

The downstream time t_d is 15 minutes. This is given as 25% less than the upstream time tᵤ, meaning t_d = 0.75 × tᵤ. By substituting the value of t_d, we find 15 = 0.75 × tᵤ, which gives tᵤ = 20 minutes.

To find the speeds in km/h, we convert the times into hours: t_d = 15/60 = 1/4 hour and tᵤ = 20/60 = 1/3 hour. The downstream speed v_d is distance / t_d = 16 / (1/4) = 64 km/h. The upstream speed vᵤ is distance / tᵤ = 16 / (1/3) = 48 km/h.

The speed of the man in still water u is the average of the downstream and upstream speeds: u = v_d + vᵤ{2} = (64 + 48)/(2) = ¹¹²⁄₂ = 56 km/h. In one hour, the man can row 56 km in still water.

Explanation

The simple interest formula is SI = (P × R × T)/(100). Since the principal (P) and rate (R) are the same for both cases, the difference in interest arises from the difference in time (T).

Let the principal be P. The difference in simple interest over the two periods is given by: (P × 4 × 11)/(100) - (P × 4 × 6)/(100) = 7500

Factoring out the common terms: (P × 4)/(100) × (11 - 6) = 7500 (P × 4 × 5)/(100) = 7500 (20P)/(100) = 7500 (P)/(5) = 7500 P = 7500 × 5 P = 37500

Alternatively, the difference in interest corresponds to the interest for the difference in duration. The difference in time is 5 years. At a rate of 4% per annum, the total interest difference is 20% of the principal. Since 20% of the principal equals ₹7,500, the total principal (100%) is 7500 × 5 = 37500.

Explanation

The list price (Marked Price) is calculated using the formula Selling Price = List Price × (1 - (Discount%)/(100)). Given that the selling price is ₹54,000 and the discount is 20%, the equation becomes 54,000 = List Price × (1 - 0.20). Simplifying this, we get 54,000 = List Price × 0.80.

To find the list price, we divide the selling price by the remaining percentage: List Price = (54,000)/(0.8) = (540,000)/(8). Performing the division, 540,000 ÷ 8 equals 67,500. Since a 20% discount implies the item was sold at 80% of its original value, calculating 54,000 × ⁵⁄₄ also yields 67,500.

Explanation

The expression x⁴ - (1)/(x⁴⁾ can be factored using the identity for the difference of squares as (x² - (1)/(x²⁾)(x² + (1)/(x²⁾). We are given that x² - (1)/(x²⁾ = 4√2. To find the value of x² + (1)/(x²⁾, we use the algebraic relationship (a + b)² = (a - b)² + 4ab.

Let a = x² and b = (1)/(x²⁾. Substituting the values into the identity gives (x² + (1)/(x²⁾)² = (x² - (1)/(x²⁾)² + 4(x²)((1)/(x²⁾). This simplifies to (x² + (1)/(x²⁾)² = (4√2)² + 4. Calculating the square, we get 32 + 4 = 36. Taking the square root, x² + (1)/(x²⁾ = √(36) = 6.

Finally, multiplying the two components together, x⁴ - (1)/(x⁴⁾ = (4√2)(6) = 24√2.

Explanation

The amount of work done by a person in one day is the reciprocal of the total time taken to complete the entire task. Therefore, the daily work rates for P, Q, and R are ⅑ , ¹⁄₁₂ , and ¹⁄₁₈ , respectively.

To find the total work completed by all three working together in one day, we sum their individual rates:

Total daily work = ⅑ + ¹⁄₁₂ + ¹⁄₁₈

The Least Common Multiple (LCM) of 9, 12, and 18 is 36. Converting the fractions to have a common denominator:

Total daily work = ⁴⁄₃₆ + ³⁄₃₆ + ²⁄₃₆ Total daily work = (4 + 3 + 2)/(36) Total daily work = ⁹⁄₃₆

Simplifying the fraction ⁹⁄₃₆ results in ¼ .

Explanation

Let the total work be the Least Common Multiple (LCM) of the given time periods: 12, 15, and 10, which is 60 units.

The daily efficiency of Ram and Ramesh is 60 / 12 = 5 units. The daily efficiency of Ramesh and Somesh is 60 / 15 = 4 units. The daily efficiency of Somesh and Ram is 60 / 10 = 6 units.

Adding these efficiencies, we get twice the combined efficiency of Ram, Ramesh, and Somesh: 2 × (Ram + Ramesh + Somesh) = 5 + 4 + 6 = 15 units per day.

The combined efficiency of Ram, Ramesh, and Somesh working together is 15 / 2 = 7.5 units per day.

The time taken for all three to complete the work together is Total Work / Combined Efficiency = 60 / 7.5 = 8 days.

Explanation

Let the distance covered by the first person be d and the time taken be t. Using the speed formula s = ᵈ⁄ₜ, the speed of the first person is s₁ = ᵈ⁄ₜ.

The second person covers 25% of the distance, which is ᵈ⁄₄, in triple the time taken by the first person, which is 3t. The speed of the second person is calculated as s₂ = (Distance)/(Time) = (d)/(4){3t} = ᵈ⁄₁₂ₜ.

To find the ratio of the speed of the first person to that of the second person, we divide s₁ by s₂:

Ratio = (s₁)/(s₂) = (d)/(t){ᵈ⁄₁₂ₜ}

Simplifying the expression by multiplying by the reciprocal:

Ratio = ᵈ⁄ₜ × (12t)/(d) = 12

This gives a ratio of 12:1.

Explanation

The overall profit or loss percentage is determined by the combined effect of the stated loss and the weight discrepancy. Let the cost price be C per unit weight. The dealer claims to sell 25g at a 15% loss. The selling price for this quantity is 25 × C × (1 - 0.15) = 21.25C.

However, the dealer only provides 20g of the article. The actual cost to the dealer for the quantity sold is 20 × C = 20C. To find the net profit or loss, compare the actual selling price to the actual cost price: Profit = Selling Price - Actual Cost = 21.25C - 20C = 1.25C

The profit percentage is calculated on the actual cost incurred by the dealer: Profit % = (Profit)/(Actual Cost) × 100 = (1.25C)/(20C) × 100 Profit % = (1.25)/(20) × 100 = 0.0625 × 100 = 6.25%

Using the multiplier method, the price factor is 0.85 (representing a 15% loss) and the weight factor is ²⁵⁄₂₀ = 1.25 (representing the ratio of true weight to false weight). The effective multiplier is 0.85 × 1.25 = 1.0625. A multiplier greater than 1 indicates a profit, specifically (1.0625 - 1) × 100 = 6.25%.

Explanation

In any triangle, an altitude is the perpendicular distance from a vertex to the opposite side. In a right-angled triangle, the hypotenuse is the longest side because it is opposite the 90° angle. When altitude AL is drawn to side BC, it creates a right-angled triangle ALB where AB is the hypotenuse. Therefore, AL < AB. Similarly, altitude BM drawn to side AC creates right-angled triangle BMC where BC is the hypotenuse, meaning BM < BC. Finally, altitude CN drawn to side AB creates right-angled triangle CNA where AC is the hypotenuse, meaning CN < AC. Summing these three individual inequalities, we get AL + BM + CN < AB + BC + CA.

Explanation

Since Bharat starts the work and they work on alternate days for exactly 8 days, Bharat and Ajay each work for exactly 4 days. Ajay can finish the work alone in 32 days, so his daily efficiency is 1/32 of the work per day. In 4 days, Ajay completes 4 × ¹⁄₃₂ = ⅛ of the total work.

The remaining work, 1 - ⅛ = ⅞, was completed by Bharat in his 4 working days. Bharat's rate of work is therefore (7/8)/(4) = ⁷⁄₃₂ of the work per day. To find the time Bharat takes to complete 7 times the same work, we divide the total required work by Bharat's efficiency:

Time = (7 × Work)/(Bharat's Rate) = (7)/(7){32} = 7 × ³²⁄₇ = 32 days

Explanation

Given 3x = sec A and 3/x = tan A, squaring both equations gives 9x² = sec² A and 9/x² = tan² A. Subtracting the second from the first yields 9(x² - 1/x²) = sec² A - tan² A. Since sec² A - tan² A = 1, it follows that 9(x² - 1/x²) = 1, which means x² - 1/x² = 1/9.

Explanation

The largest chord of a circle is its diameter. According to the inscribed angle theorem, 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 circle. Since a diameter divides the circle into two semicircles, it subtends a straight angle of 180° at the center. Consequently, the angle subtended by the diameter at any point on the circumference is half of the central angle, which is (180°)/(2) = 90°. This geometric principle is also known as Thales's Theorem, which states that any triangle inscribed in a semicircle with the diameter as one of its sides must be a right-angled triangle.

Explanation

The given expression follows the algebraic identity for the difference of two cubes: (a³ - b³)/(a² + ab + b²) = a - b

By assigning a = 120 and b = 100, the expression can be rewritten as: (|a³ - b³|)/(|a² + ab + b²|)

Since a > b, both the numerator and the denominator are positive, allowing the expression to simplify directly using the identity: ((a - b)(a² + ab + b²))/(a² + ab + b²) = a - b

Substituting the values: 120 - 100 = 20

Explanation

To find the total time, we first determine the total work by taking the Least Common Multiple (LCM) of 20, 30, and 60, which is 60 units. Based on this, the individual efficiencies (work per day) are: Efficiency of A = 60 / 20 = 3 units/day Efficiency of B = 60 / 30 = 2 units/day Efficiency of C = 60 / 60 = 1 unit/day

The work follows a three-day cycle: Day 1: A works alone, completing 3 units. Day 2: A works alone, completing 3 units. Day 3: A, B, and C work together, completing 3 + 2 + 1 = 6 units.

Total work completed in one 3-day cycle = 3 + 3 + 6 = 12 units. To complete the total work of 60 units, the number of cycles required is 60 / 12 = 5 cycles. Total time taken = 5 × 3 = 15 days.

Explanation

The relationship between income, expenditure, and savings is given by the formula Income = Expenditure + Savings. According to the ratio 5:3, if the expenditure is 5x, the savings are 3x, and the total initial income is 5x + 3x = 8x.

The problem states that the initial savings are ₹4,500. Equating this to the savings part of the ratio, we get: 3x = 4500 x = 1500

The initial income is 8x = 8 × 1500 = 12000.

The income increases by 25%. The new income is calculated by adding the increment to the initial income: New Income = Initial Income × (1 + ²⁵⁄₁₀₀) New Income = 12000 × 1.25 = 15000

Explanation

The algebraic identity for the sum of cubes is a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - (ab + bc + ca)). To solve this using the given values, we can rewrite the term (ab + bc + ca) using the identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca), which gives ab + bc + ca = ((a + b + c)² - (a² + b² + c²))/(2).

Substituting this into the primary identity yields a direct relation between the given terms: a³ + b³ + c³ - 3abc = (a + b + c)/(2) [3(a² + b² + c²) - (a + b + c)²]

Plugging in the provided values a + b + c = 14 and a³ + b³ + c³ - 3abc = 98: 98 = ¹⁴⁄₂ [3(a² + b² + c²) - 14²] 98 = 7 [3(a² + b² + c²) - 196] 14 = 3(a² + b² + c²) - 196 210 = 3(a² + b² + c²) a² + b² + c² = ²¹⁰⁄₃ = 70

Explanation

In triangle ABC, point D lies on side BC. Given the lengths BC = 16 cm and BD = 11 cm, the length of the segment CD is calculated as CD = BC - BD = 16 - 11 = 5 cm.

Consider triangle ABC and triangle DAC. The problem states that ∠ BAC = ∠ ADC. Additionally, both triangles share the common angle at vertex C, so ∠ BCA = ∠ ACD. According to the AA (Angle-Angle) similarity criterion, triangle ABC is similar to triangle DAC (△ ABC ∼ △ DAC).

For similar triangles, the ratios of the lengths of corresponding sides are equal. Specifically, the ratio of the side opposite the given equal angles to the side opposite the common angle must be consistent: (BC)/(AC) = (AC)/(DC)

By cross-multiplying, we obtain the relation: AC² = BC × DC

Substituting the known values into the equation: AC² = 16 × 5 AC² = 80 AC = √(80) AC = √(16 × 5) AC = 4√5 cm

Explanation

Let the six consecutive natural numbers that form the digits of the number be n, n+1, n+2, n+3, n+4, and n+5. According to the divisibility rule for 3, a number is divisible by 3 if and only if the sum of its digits is divisible by 3.

To find the sum of these six digits, we use the formula for the sum of an arithmetic progression or simply add them: S = n + (n+1) + (n+2) + (n+3) + (n+4) + (n+5) S = 6n + 15

We can factor this expression by taking 3 as a common factor: S = 3(2n + 5)

Since the sum of the digits S is represented as a product of 3 and an integer (2n + 5), the sum is always a multiple of 3 for any natural number n. This holds true for all possible sets of consecutive digits such as {1, 2, 3, 4, 5, 6} where the sum is 21, {2, 3, 4, 5, 6, 7} where the sum is 27, {3, 4, 5, 6, 7, 8} where the sum is 33, and {4, 5, 6, 7, 8, 9} where the sum is 39. Because the sum of the digits is always divisible by 3, the 6-digit number itself is always divisible by 3.

Explanation

Total work is the Least Common Multiple (LCM) of the individual times taken by Benny, Chethan, and David, which is LCM(24, 36, 48) = 144 units. The efficiencies of the three workers are: Benny: 144 / 24 = 6 units/day Chethan: 144 / 36 = 4 units/day David: 144 / 48 = 3 units/day

Let the total work be completed in T days. Since David worked for the entire duration, Benny left 4 days before completion, and Chethan left 10 days before completion, the total work done is the sum of their individual contributions: 6(T - 4) + 4(T - 10) + 3(T) = 144

Simplifying the equation: 6T - 24 + 4T - 40 + 3T = 144 13T - 64 = 144 13T = 208 T = 208 / 13 = 16 days

Explanation

Since the police and the thief start moving at the same time and the chase ends when the police catches the thief, the duration of travel is the same for both. When time is constant, the distance covered is directly proportional to the speed (d ∝ v).

The ratio of the speed of the police (vₚ) to the speed of the thief (vₜ) is 11:6. Consequently, the ratio of the distances they cover is also 11:6. Let the distance covered by the police be 11x and the distance covered by the thief be 6x. The difference between these distances must equal the initial gap between them:

11x - 6x = 1150 m 5x = 1150 m x = ¹¹⁵⁰⁄₅ = 230 m

The distance run by the thief before being caught corresponds to 6x: dₜ = 6 × 230 = 1380 m

Explanation

To find the value of sec² θ + cos² θ, use the algebraic identity (a + b)² = a² + b² + 2ab. Let a = sec θ and b = cos θ. Because secant and cosine are reciprocal functions, their product ab = sec θ · cos θ = 1.

By squaring the given equation sec θ + cos θ = 32, we get: (sec θ + cos θ)² = 32² sec² θ + cos² θ + 2(sec θ cos θ) = 1024

Substituting the identity sec θ cos θ = 1 into the equation: sec² θ + cos² θ + 2(1) = 1024 sec² θ + cos² θ = 1024 - 2 sec² θ + cos² θ = 1022

Explanation

The expression is solved using the BODMAS rule, which dictates the order of operations as Division, Multiplication, and then Addition. First, convert the mixed fractions into improper fractions where 1 ¾ = ⁷⁄₄, 1 ⁵⁄₇ = ¹²⁄₇, and 2 ³⁄₇ = ¹⁷⁄₇. Substituting these into the expression gives ⁷⁄₄ + ¹²⁄₇ ÷ ¹⁷⁄₇ × ¹⁷⁄₇.

Following the order of operations, perform the division first: ¹²⁄₇ ÷ ¹⁷⁄₇ = ¹²⁄₇ × ⁷⁄₁₇ = ¹²⁄₁₇. Next, perform the multiplication: ¹²⁄₁₇ × ¹⁷⁄₇ = ¹²⁄₇. Finally, perform the addition by finding a common denominator for ⁷⁄₄ and ¹²⁄₇. The Least Common Multiple (LCM) of 4 and 7 is 28, so the sum becomes ⁴⁹⁄₂₈ + ⁴⁸⁄₂₈ = ⁹⁷⁄₂₈. Converting this improper fraction back into a mixed fraction, 97 ÷ 28 equals 3 with a remainder of 13, resulting in 3 ¹³⁄₂₈.

Explanation

The third proportional c of two quantities a and b is defined by the relation a:b = b:c, which gives the formula c = (b²)/(a). In this problem, we are given a = 3x², b = 4xy, and the third proportional c = 48.

Substituting these values into the formula: 48 = ((4xy)²)/(3x²) 48 = (16x² y²)/(3x²)

The x² terms in the numerator and denominator cancel out, simplifying the equation to: 48 = (16y²)/(3)

To solve for y², multiply both sides by 3 and divide by 16: y² = (48 × 3)/(16) y² = 3 × 3 y² = 9

Taking the square root to find the positive value: y = √9 = 3

Explanation

The length of the direct common tangent 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 values are d = 32 cm, R = 22 cm, and r = 18 cm.

Substituting these values into the formula: L = √(32²⁾⁻⁽²²⁻¹⁸⁾^^^² L = √(1024 - 4²⁾ L = √(1024 - 16) L = √(1008)

To simplify √(1008) to match the options, we look for square factors. Factoring 1008 as 4 × 252, we get: L = √(4 × 252) L = 2√(252) cm

Explanation

To find the required ratio, we calculate the total number of delivery partners who joined Clipkart and Tomato during the years 2019 and 2020 based on the provided table.

For Clipkart, the number of joining delivery partners was 5.6 thousand in 2019 and 6.4 thousand in 2020. Total Clipkart partners = 5.6 + 6.4 = 12.0 thousand

For Tomato, the number of joining delivery partners was 2.7 thousand in 2019 and 3.3 thousand in 2020. Total Tomato partners = 2.7 + 3.3 = 6.0 thousand

The ratio of the number of delivery partners who joined Clipkart to those who joined Tomato is: Ratio = (12.0)/(6.0) = ²⁄₁

This represents a ratio of 2:1.

Explanation

The sum of the areas of the floor and the flat roof is 2 × (L × W), while the area of the four walls is represented by the lateral surface area formula 2 × H × (L + W). Setting these two expressions as equal based on the given condition:

2LW = 2H(L + W)

Dividing both sides by 2 simplifies the relation to:

H = (LW)/(L + W)

Substituting the given dimensions L = 21 m and W = 14 m into the equation:

H = (21 × 14)/(21 + 14) H = ²⁹⁴⁄₃₅

Dividing both the numerator and the denominator by the common factor 7:

H = ⁴²⁄₅ = 8.4 m

Explanation

Using the trigonometric identity sec² A - tan² A = 1, we have (sec A + tan A)(sec A - tan A) = 1. Given sec A + tan A = 5, it follows that sec A - tan A = 1/5.

By adding these two equations: (sec A + tan A) + (sec A - tan A) = 5 + 1/5 2 sec A = 26/5 sec A = 13/5

By subtracting the second equation from the first: (sec A + tan A) - (sec A - tan A) = 5 - 1/5 2 tan A = 24/5 tan A = 12/5

Since sin A = (tan A)/(sec A), substituting the calculated values gives: sin A = (12/5)/(13/5) = 12/13

Explanation

Let the total work be the LCM of the individual days taken by A, B, and C, which is LCM(12, 15, 20) = 60 units.

The efficiency of each person is: Efficiency of A = ⁶⁰⁄₁₂ = 5 units/day Efficiency of B = ⁶⁰⁄₁₅ = 4 units/day Efficiency of C = ⁶⁰⁄₂₀ = 3 units/day

The combined efficiency of A, B, and C working together is 5 + 4 + 3 = 12 units/day. Over a period of 4 days, the amount of work completed is 12 × 4 = 48 units.

The remaining work is the difference between the total work and the completed work: 60 - 48 = 12 units. To find the fraction of the remaining work, we divide the remaining units by the total units: ¹²⁄₆₀ = ⅕ .

Explanation

To find the value of x⁵ + (1)/(x⁵), we first derive the values of lower powers of x from the given equation x⁴ + (1)/(x⁴) = 254. Using the algebraic identity a² + b² = (a+b)² - 2ab, we can write: (x² + (1)/(x²))² = x⁴ + (1)/(x⁴) + 2 = 254 + 2 = 256 Taking the square root (since x > 0), we get x² + (1)/(x²) = 16.

Repeating this process to find x + ¹⁄ₓ: (x + ¹⁄ₓ)² = x² + (1)/(x²) + 2 = 16 + 2 = 18 Thus, x + ¹⁄ₓ = √(18) = 3√2.

Next, we calculate x³ + (1)/(x³) using the identity a³ + b³ = (a+b)³ - 3ab(a+b): x³ + (1)/(x³) = (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ) = (3√2)³ - 3(3√2) x³ + (1)/(x³) = 54√2 - 9√2 = 45√2

The value of x⁵ + (1)/(x⁵) can be found using the relation: x⁵ + (1)/(x⁵) = (x² + (1)/(x²))(x³ + (1)/(x³)) - (x + ¹⁄ₓ) Substituting the calculated values: x⁵ + (1)/(x⁵) = (16)(45√2) - 3√2 x⁵ + (1)/(x⁵) = 720√2 - 3√2 = 717√2

Explanation

The shift from a 16% loss to a 20% gain represents a total change of 16% + 20% = 36% in the valuation of the article relative to its cost price (CP). This total percentage change is equivalent to the increase in the selling price, which is given as ₹324.

By using the relationship: 36% of CP = 324

We can solve for the cost price: CP = ³²⁴⁄₃₆ × 100

Dividing 324 by 36 gives 9. Therefore: CP = 9 × 100 = 900

Explanation

Using the algebraic identity for the sum of cubes, a³ + b³ = (a + b)(a² - ab + b²), we can expand the numerator by setting a = sin α and b = cos α.

sin³ α + cos³ α = (sin α + cos α)(sin² α - sin α cos α + cos² α)

Substituting this expansion back into the original expression:

((sin α + cos α)(sin² α - sin α cos α + cos² α))/(sin α + cos α)

The term (sin α + cos α) cancels out from the numerator and the denominator, leaving:

sin² α - sin α cos α + cos² α

By applying the Pythagorean identity sin² α + cos² α = 1, the expression simplifies to:

1 - sin α cos α

Explanation

The expression a³ - b³ - 3a²b + 3ab² is the standard algebraic expansion of the identity (a - b)³. By rearranging the terms to a³ - 3a²b + 3ab² - b³, it matches the binomial expansion formula exactly.

Given the values a = 17 and b = 13, the value of the expression is calculated by substituting these into the identity: (a - b)³ = (17 - 13)³ (a - b)³ = 4³ 4 × 4 × 4 = 64

Explanation

To simplify the expression 18 ÷ 3 of 2 × 5 + 72 ÷ 18 of 2 × 3 - 4 ÷ 8 × 2, we follow the BODMAS rule, which dictates the order of operations as Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, we calculate the of operations: 3 of 2 = 3 × 2 = 6 18 of 2 = 18 × 2 = 36

Substituting these values back into the expression, we get: 18 ÷ 6 × 5 + 72 ÷ 36 × 3 - 4 ÷ 8 × 2

Next, we perform the division operations from left to right: 18 ÷ 6 = 3 72 ÷ 36 = 2 4 ÷ 8 = 0.5

The expression now becomes: 3 × 5 + 2 × 3 - 0.5 × 2

Then, we perform the multiplication operations: 3 × 5 = 15 2 × 3 = 6 0.5 × 2 = 1

The expression is now: 15 + 6 - 1

Finally, performing addition and subtraction: 21 - 1 = 20

Explanation

Let the current ages of Gyanendra and Arbind be 6x and 5x respectively. According to the question, the sum of their current ages is 6x + 5x = 55, which simplifies to 11x = 55. Solving for x gives x = 5. Thus, Gyanendra's current age is 6 × 5 = 30 years and Arbind's current age is 5 × 5 = 25 years. To find their ages seven years from now, we add 7 to each of their current ages, resulting in 37 years for Gyanendra and 32 years for Arbind. Therefore, the ratio of their ages after seven years will be 37 : 32.

Explanation

In a right-angled triangle, one angle is exactly 90°. According to the angle sum property of a triangle, the sum of all internal angles is 180°. Therefore, the sum of the two remaining acute angles must be 180° - 90° = 90°.

Let the measure of the smallest angle be x. The problem states that the other acute angle is 30° more than the smallest angle, so its measure is x + 30°. Setting up the equation for the sum of these two angles:

x + (x + 30°) = 90° 2x + 30° = 90° 2x = 60° x = 30°

The three angles of the triangle are 30°, 60°, and 90°. The smallest angle is 30°.

Explanation

A number is divisible by 3 if the sum of its digits is divisible by 3, and it is divisible by 9 if the sum of its digits is divisible by 9. To find numbers divisible by 3 but NOT by 9, we look for cases where the sum of digits S satisfies S ∈ {3, 6, 12, 15, 21, 24, 30, 33, …} but S ∉ {9, 18, 27, 36, …}.

For 5826: Sum = 5 + 8 + 2 + 6 = 21. Since 21 is divisible by 3 but not by 9, this number satisfies the condition. For 5964: Sum = 5 + 9 + 6 + 4 = 24. Since 24 is divisible by 3 but not by 9, this number satisfies the condition. For 6039: Sum = 6 + 0 + 3 + 9 = 18. Since 18 is divisible by 9, this number does not satisfy the condition. For 6336: Sum = 6 + 3 + 3 + 6 = 18. Since 18 is divisible by 9, this number does not satisfy the condition. For 6489: Sum = 6 + 4 + 8 + 9 = 27. Since 27 is divisible by 9, this number does not satisfy the condition. For 6564: Sum = 6 + 5 + 6 + 4 = 21. Since 21 is divisible by 3 but not by 9, this number satisfies the condition. For 6867: Sum = 6 + 8 + 6 + 7 = 27. Since 27 is divisible by 9, this number does not satisfy the condition. For 6960: Sum = 6 + 9 + 6 + 0 = 21. Since 21 is divisible by 3 but not by 9, this number satisfies the condition.

The numbers meeting the criteria are 5826, 5964, 6564, and 6960, making a total of 4 numbers.

Explanation

The two natural numbers are found by solving the equations a + b = 32 and a³ + b³ = 9728, which yields the numbers 20 and 12. Calculating their cubes gives 8000 and 1728 respectively, and their positive difference is 8000 - 1728 = 6272.

Explanation

The perimeter of a sector of a circle is calculated as the sum of its arc length and the lengths of the two radii that bound it. The formula for the perimeter P is P = 2r + l, where r is the radius and l is the arc length.

The arc length l for a central angle θ is given by l = (θ)/(360°) × 2π r. Given the radius r = 4 cm, the angle θ = 45°, and π = ²²⁄₇, the arc length is calculated as follows:

l = ⁴⁵⁄₃₆₀ × 2 × ²²⁄₇ × 4

l = ⅛ × 8 × ²²⁄₇

l = ²²⁄₇ cm

Substituting the values of r and l into the perimeter formula:

P = 2(4) + ²²⁄₇

P = 8 + ²²⁄₇

P = (56 + 22)/(7)

P = ⁷⁸⁄₇ cm

Explanation

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. For 11638, (1 + 6 + 8) - (1 + 3) = 15 - 4 = 11, which is divisible by 11. None of the other numbers satisfy this rule.

Explanation

The curved surface area (CSA) of a right circular cylinder is given by the formula CSA = π d h, where d is the diameter and h is the height. Given h = 56 cm and CSA = 1408 cm², substituting these values into the formula using π = ²²⁄₇ gives:

1408 = ²²⁄₇ × d × 56

1408 = 22 × d × 8

1408 = 176 × d

d = ¹⁴⁰⁸⁄₁₇₆ = 8 cm

To convert the diameter from centimeters to meters, divide by 100:

d = ⁸⁄₁₀₀ = 0.08 m