SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

The total selling price (SP) is calculated by adding the profit to the cost price (CP). Since Aditya bought the mangoes for Rs. 1,100 and made a profit of Rs. 150, the total SP is 1,100 + 150 = Rs. 1,250.

The number of mangoes sold is determined by dividing the total SP by the selling price per mango. Thus, the number of mangoes sold is 1,250 / 5 = 250 mangoes.

The number of mangoes thrown away is the difference between the total number of mangoes purchased and the number of mangoes sold, which is 300 - 250 = 50 mangoes.

To find the percentage of mangoes thrown away, use the formula: Percentage = (Mangoes thrown away / Total mangoes purchased) × 100 Percentage = (50 / 300) × 100 = (1 / 6) × 100 = 50 / 3 = 16 ⅔%

Explanation

To find the value of N, we use the work-rate formula M₁ D₁ H₁{W₁} = M₂ D₂ H₂{W₂}, where M is the number of persons, D is the number of days, H is the number of hours per day, and W is the units of work completed.

In the first scenario, M₁ = N+15, D₁ = 8, H₁ = 9, and W₁ = 36. In the second scenario, M₂ = N+9, D₂ = 20, H₂ = 7, and W₂ = 64 (since the remaining work is 100% - 36% = 64%).

Substituting these values into the formula:

((N+15) × 8 × 9)/(36) = ((N+9) × 20 × 7)/(64)

Simplifying the left side: ((N+15) × 72)/(36) = 2(N+15)

Simplifying the right side: ((N+9) × 140)/(64) = ((N+9) × 35)/(16)

Equating both sides: 2(N+15) = (35(N+9))/(16) 32(N+15) = 35(N+9) 32N + 480 = 35N + 315 35N - 32N = 480 - 315 3N = 165 N = 55

Explanation

According to the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is half the length of the third side. In △ ABC, since X and Y are the midpoints of AB and AC, we have the relation XY = ½ BC.

Substituting this relation into the given sum BC + XY = 18: BC + ½ BC = 18 ³⁄₂ BC = 18 BC = (18 × 2)/(3) = 12 units.

Using the value of BC, we find XY = ½ × 12 = 6 units. The difference BC - XY is calculated as 12 - 6 = 6 units.

Explanation

Given the equation sin θ - cos θ = 0, we can rearrange it to find that sin θ = cos θ. The expression to be evaluated is sin³ θ - cos³ θ. Using the algebraic identity for the difference of cubes, a³ - b³ = (a - b)(a² + ab + b²), we substitute a = sin θ and b = cos θ:

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

Since it is given that sin θ - cos θ = 0, the entire product becomes:

0 · (sin² θ + sin θ cos θ + cos² θ) = 0

Alternatively, because sin θ = cos θ, it follows that their cubes must also be equal, sin³ θ = cos³ θ. Therefore, subtracting one from the other yields zero.

Explanation

Let the cost price of the article be 100x. The dealer marks the article 60% above the cost price, so the marked price is 100x + 60x = 160x.

The selling price is determined by applying two successive discounts of 10% and 20% on the marked price. The formula for the selling price after discounts is SP = MP × (1 - d₁) × (1 - d₂). SP = 160x × (1 - ¹⁰⁄₁₀₀) × (1 - ²⁰⁄₁₀₀) SP = 160x × 0.9 × 0.8 SP = 160x × 0.72 = 115.2x

The gain is the difference between the selling price and the cost price: Gain = SP - CP = 115.2x - 100x = 15.2x

According to the problem, the gain is ₹1,064. We can set up the equation: 15.2x = 1064 x = (1064)/(15.2) = ¹⁰⁶⁴⁰⁄₁₅₂ = 70

The cost price of the article is 100x = 100 × 70 = 7000.

Explanation

In a chase scenario where both parties start and stop at the same time, the distance traveled is directly proportional to their speeds since d = v × t. The ratio of the policeman's speed to the thief's speed is given as 12 : 10, which simplifies to 6 : 5. Consequently, the ratio of the distance covered by the policeman to the distance covered by the thief is also 6 : 5. The difference between these distances in ratio units is 6 - 5 = 1 unit. This difference represents the initial gap of 300 m or 0.3 km that the policeman must close. Since 1 unit equals 0.3 km, the total distance run by the thief, which corresponds to 5 units, is calculated as 5 × 0.3 km = 1.5 km.

Explanation

The given equation is x + ¹⁄ₓ = 2. Multiplying the entire equation by x yields x² + 1 = 2x. By rearranging the terms to one side, we get the quadratic equation x² - 2x + 1 = 0. This is equivalent to the algebraic identity (x - 1)² = 0. Taking the square root of both sides, we find x - 1 = 0, which means x = 1. Substituting this value into the expression x⁷ + (1)/(x¹¹⁷⁾, we obtain 1⁷ + (1)/(1¹¹⁷⁾. Since 1 raised to any real power is 1, the expression becomes 1 + ¹⁄₁ = 1 + 1 = 2.

Explanation

According to the properties of modular arithmetic, if a number m leaves a remainder r when divided by n, then k × m will leave the same remainder as k × r when divided by n. Given that m divided by 7 leaves a remainder of 5, we can write m ≡ 5 7.

To find the remainder when 3m is divided by 7, we multiply the individual remainder by 3: 3 × 5 = 15

Now, we divide 15 by 7 to find the final remainder: 15 = (2 × 7) + 1

By representing m as 7k + 5 for some integer k, we can also see that: 3m = 3(7k + 5) = 21k + 15 3m = 21k + 14 + 1 3m = 7(3k + 2) + 1

Since 7(3k + 2) is exactly divisible by 7, the remaining term 1 is the remainder.

Explanation

The total work is constant and can be calculated using the relationship M₁ D₁ = M₂ D₂, where M represents the number of men and D represents the number of days.

Substituting the initial conditions where M₁ = 12 and D₁ = 10: Total Work = 12 × 10 = 120 man-days

To find the total number of men (M₂) needed to finish the same work in D₂ = 6 days: 120 = M₂ × 6 M₂ = ¹²⁰⁄₆ = 20 men

The question asks for the number of additional men required. This is the difference between the new total and the original group: Additional men = M₂ - M₁ = 20 - 12 = 8 men

Explanation

To find the time taken when working alternately, first determine the total work by taking the Least Common Multiple (LCM) of the individual times. The LCM of 9 and 36 is 36, so let the total work be 36 units.

The efficiency of X is 36 / 9 = 4 units per day, and the efficiency of Y is 36 / 36 = 1 unit per day. Since they work alternately starting with X, in one cycle of 2 days, the total work completed is 4 + 1 = 5 units.

To complete the 36 units of work, determine how many full 2-day cycles are completed by dividing the total work by the work per cycle: 36 / 5 = 7 full cycles with a remainder of 1 unit. These 7 cycles take 7 × 2 = 14 days and account for 7 × 5 = 35 units of work.

The remaining 1 unit of work (36 - 35 = 1) is to be done by X on the next day. Since X can do 4 units in a full day, X will take 1 / 4 of a day to complete the remaining unit. Therefore, the total time required is 14 + 1/4 = 14¼ days.

Explanation

The algebraic identity for the given expression is a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - (ab + bc + ca)). To find the value of (ab + bc + ca), we utilize the formula for the square of a trinomial: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca).

Substituting the given values a + b + c = 20 and a² + b² + c² = 152 into the trinomial formula, we get 20² = 152 + 2(ab + bc + ca). This simplifies to 400 = 152 + 2(ab + bc + ca), leading to 2(ab + bc + ca) = 248, which gives ab + bc + ca = 124.

Finally, substitute the values into the target identity: a³ + b³ + c³ - 3abc = 20 × (152 - 124). Calculating the difference inside the parentheses gives 152 - 124 = 28. Thus, 20 × 28 = 560.

Explanation

The surface area of a cube is given by the formula 6a², where a is the side length. By substituting the given value, we can solve for a:

6a² = 5046 a² = 841 a = √(841) = 29 cm

The volume of a cube is determined by the formula V = a³. Substituting the calculated side length of 29 cm into this formula:

V = 29³ V = 24389 cm³

The derivation follows from first finding the linear dimension of the cube from its two-dimensional surface measurement and then raising that dimension to the third power to find the three-dimensional space it occupies.

Explanation

The length of a direct common tangent to two circles is calculated using the formula L = √(d²⁾⁻⁽^R⁻ʳ⁾^^^², where d represents the distance between the centers of the circles, and R and r are the radii of the two circles. In this problem, the radii are R = 6 cm and r = 2 cm, and the distance between the centers is d = 5 cm.

Substituting these values into the formula: L = √(5²⁾⁻⁽⁶⁻²⁾^^^² L = √(25 - (4)²⁾ L = √(25 - 16) L = √9 L = 3 cm

Explanation

To keep the expenditure constant when the price of a commodity increases, the percentage reduction in consumption is calculated using the formula (r)/(100 + r) × 100%, where r is the percentage increase in price. Given that the price of petrol increases by 7%, substituting r = 7 into the formula gives (7)/(100 + 7) × 100% = ⁷⁰⁰⁄₁₀₇%. Dividing 700 by 107 results in a quotient of 6 and a remainder of 58. This improper fraction can be written as the mixed fraction 6 ⁵⁸⁄₁₀₇%.

Explanation

In simple interest, the interest earned each year is constant. The difference between the amount after 4 years and 3 years gives the interest for exactly one year: I = 1708 - 1630 = 78. To find the principal sum P, subtract the total interest for 3 years from the amount accumulated at the end of 3 years. This is expressed as P = 1630 - (3 × 78). Multiplying 78 by 3 equals 234. Subtracting 234 from 1630 results in 1396.

Explanation

The actual profit percentage is determined by the ratio of the total selling price to the actual cost incurred for the quantity delivered. Let the cost price of 1000 gm be 1000 units. The shopkeeper delivers only 940 gm, so his actual cost price (CP) is 940 units. He sells this quantity at a 4% profit over the price of 1000 gm, making the selling price (SP) equal to 1000 × (1 + 0.04) = 1040 units.

The actual profit is calculated as SP - CP = 1040 - 940 = 100 units. The actual profit percentage is given by the formula:

Actual Profit % = (Actual Profit)/(Actual CP) × 100

Substituting the values, we get ¹⁰⁰⁄₉₄₀ × 100 = ¹⁰⁰⁰⁄₉₄ ≈ 10.6382%. When rounded off to two decimal places, the actual profit percentage is 10.64%.

Explanation

In a right-angled triangle, the midpoint of the hypotenuse is the circumcenter of the triangle. For △ ABC with ∠ A = 90°, the side BC is the hypotenuse. The circumcenter of any triangle is equidistant from all its vertices. Since D is the midpoint of the hypotenuse BC, it acts as the circumcenter, meaning the distances from D to the vertices A, B, and C are equal. Mathematically, this is expressed as AD = BD = CD. Because D is the midpoint of BC, the lengths BD and CD are both equal to (BC)/(2). Consequently, the length of the median AD is also equal to (BC)/(2).

Explanation

Using the rule of alligation, where the cost price of water is assumed to be ₹0 per litre, the ratio of the quantities mixed is determined by:

(Quantity of Water)/(Quantity of Milk) = (Cost Price of Milk - Mean Price)/(Mean Price - Cost Price of Water)

Substituting the given values into the formula: (Quantity of Water)/(Quantity of Milk) = (32 - 28)/(28 - 0) (Quantity of Water)/(Quantity of Milk) = ⁴⁄₂₈ (Quantity of Water)/(Quantity of Milk) = ⅐

Explanation

Let W and G represent the daily earnings of one woman and one girl, respectively. Based on the given information, we can establish the daily earning equations:

For the first case: 5W + 9G = ¹⁸⁷²⁰⁄₉ = 2080 (Equation 1) For the second case: 9W + 16G = ⁵²⁰⁸⁰⁄₁₄ = 3720 (Equation 2)

To solve for W and G, multiply Equation 1 by 9 and Equation 2 by 5 to equate the W coefficients: 45W + 81G = 18720 45W + 80G = 18600

Subtracting the second derived equation from the first gives G = 120. Substituting G = 120 into Equation 1: 5W + 9(120) = 2080 5W + 1080 = 2080 5W = 1000 ⟹ W = 200

The total earnings for 12 women and 7 girls in 13 days is calculated as follows: Total = 13 × (12W + 7G) Total = 13 × (12 × 200 + 7 × 120) Total = 13 × (2400 + 840) Total = 13 × 3240 = 42120

Explanation

The net selling price is calculated by applying the successive discounts to the marked price and then subtracting the cash value of the reward points.

First, calculate the price after the two successive discounts of 20% and 25%. Using the successive discount formula, the effective discount is dₑ_f_f = 20 + 25 - (20 × 25)/(100) = 40%. Alternatively, the price after discounts can be found using the multiplier method: Price_dᵢₛ_cₒᵤₙₜₑ_d = 72000 × (1 - ²⁰⁄₁₀₀) × (1 - ²⁵⁄₁₀₀) Price_dᵢₛ_cₒᵤₙₜₑ_d = 72000 × ⅘ × ¾ = 72000 × ⅗ = 43200

Next, determine the cash value of the 2850 reward points. Since each point is worth ₹0.10: Reward_Value = 2850 × 0.10 = 285

Finally, subtract the reward value from the discounted price to find the net selling price: Net_SP = 43200 - 285 = 42915

Explanation

The algebraic identity for the sum of cubes of three variables is given by a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca). When the sum of the variables is zero, i.e., (a + b + c) = 0, the product on the right-hand side of the identity becomes zero regardless of the value of the second factor. This leads to the simplified equation a³ + b³ + c³ - 3abc = 0. By moving 3abc to the right side of the equation, we obtain a³ + b³ + c³ = 3abc.

Explanation

Using the trigonometric identity cos(x + y)cos(x - y) = cos² x - sin² y, we can expand both terms of the expression.

For the first term, cos(36° + A)cos(36° - A) = cos² 36° - sin² A.

For the second term, cos(54° + A)cos(54° - A) = cos² 54° - sin² A.

Adding these two results gives cos² 36° + cos² 54° - 2sin² A.

Since 54° and 36° are complementary angles (54° = 90° - 36°), we have cos 54° = sin 36°. Substituting this into the sum, we get cos² 36° + sin² 36° - 2sin² A.

By the Pythagorean identity cos² θ + sin² θ = 1, the expression simplifies to 1 - 2sin² A.

Applying the double angle identity cos 2A = 1 - 2sin² A, the value of the expression is cos 2A.

Explanation

The cost price (CP) of the headphones is determined using the formula CP = (SP × 100)/(100 + Gain%). Substituting the given values SP = 5520 and Gain = 20%: CP = (5520 × 100)/(120) = (5520)/(1.2) = 4600.

To find the gain or loss percentage for the new selling price of ₹4,370, we compare it with the CP. Since the new selling price is less than the cost price (4370 < 4600), there is a loss. Loss = CP - New SP = 4600 - 4370 = 230.

The loss percentage is calculated as Loss% = (Loss)/(CP) × 100: Loss% = ²³⁰⁄₄₆₀₀ × 100 = ²³⁰⁄₄₆ = 5%.

Explanation

The total increase in population is 19200 - 18000 = 1200. To find the ratio of males to females, we can use the method of alligation by first calculating the overall percentage increase: (1200 / 18000) × 100 = 20/3%.

By alligating the growth rate of males (5%) and females (7%) against the average growth rate (20/3%): The male component ratio is 7 - 20/3 = 1/3. The female component ratio is 20/3 - 5 = 5/3. The ratio of males to females is 1/3 : 5/3, which simplifies to 1:5.

Alternatively, using equations, let M be the number of males and F be the number of females. Given M + F = 18000 and 0.05M + 0.07F = 1200. Multiplying the first equation by 0.05 gives 0.05M + 0.05F = 900. Subtracting this from the second equation yields 0.02F = 300, so F = 15000. Consequently, M = 3000, giving a ratio M:F = 3000:15000 = 1:5.

Explanation

The efficiency of a single inlet pipe is 1/120 of the tank per hour, while the efficiency of a single outlet pipe is 1/54 of the tank per hour. When 8 inlet pipes and 3 outlet pipes are opened simultaneously, the net rate of filling is calculated by subtracting the total drainage rate from the total filling rate.

Net Efficiency = 8 × (¹⁄₁₂₀) - 3 × (¹⁄₅₄)

Simplifying the fractions: Net Efficiency = ¹⁄₁₅ - ¹⁄₁₈

To solve this, find a common denominator for 15 and 18, which is 90: Net Efficiency = ⁶⁄₉₀ - ⁵⁄₉₀ = ¹⁄₉₀

Since the net efficiency is 1/90 of the tank per hour, the time required to fill the entire tank is the reciprocal of this rate: Time = (1)/(1/90) = 90 hours.

Explanation

The expression tan x + cot x can be simplified using the fundamental trigonometric identities. Since tan x = (sin x)/(cos x) and cot x = (cos x)/(sin x), their sum is (sin x)/(cos x) + (cos x)/(sin x) = (sin² x + cos² x)/(sin x cos x). Applying the Pythagorean identity sin² x + cos² x = 1, the expression simplifies to (1)/(sin x cos x).

Given that sin x = ³⁄₁₀, the value of cos x is calculated using cos x = √(1 - sin² x) = √(1 - (³⁄₁₀)²) = √(1 - ⁹⁄₁₀₀) = √(91){10}. Substituting these values into the simplified expression results in:

(1)/((3){10)(√(91){10})} = (1)/(3√(91)){100} = (100)/(3√(91))

Explanation

The selling price after successive discounts is calculated using the formula SP = MP × (1 - d₁{100}) × (1 - d₂{100}), where MP is the marked price and d₁, d₂ are the discount percentages.

Given MP = 4500 and the first discount d₁ = 20%, the price after the first discount is 4500 × (1 - ²⁰⁄₁₀₀) = 4500 × ⅘ = 3600.

The second discount d₂ is applied to this intermediate price of 3600 to reach the final selling price of 2400. The amount of the second discount is 3600 - 2400 = 1200.

The second discount percentage is calculated as (Discount Amount)/(Price after first discount) × 100. Substituting the values, we get ¹²⁰⁰⁄₃₆₀₀ × 100 = ⅓ × 100 = 33⅓%.

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 either 0 or a multiple of 11. For the number 97174, the digits at odd positions are 9, 1, and 4, providing a sum of 9 + 1 + 4 = 14. The digits at even positions are 7 and 7, providing a sum of 7 + 7 = 14. The difference between these two sums is 14 - 14 = 0. Since the result is 0, the number satisfies the divisibility criteria for 11.

Explanation

When two trains run in the same direction, their relative speed is the difference between their individual speeds. Here, the relative speed is 50 km/h - 40 km/h = 10 km/h. To convert this speed into meters per second, multiply by ⁵⁄₁₈:

Relative Speed = 10 × ⁵⁄₁₈ = ²⁵⁄₉ m/s

The total distance to be covered for the trains to pass each other is the sum of their lengths.

Total Distance = 130 m + 120 m = 250 m

Using the formula Time = (Distance)/(Relative Speed):

Time = (250)/(25){9} = 250 × ⁹⁄₂₅ = 10 × 9 = 90 seconds

Explanation

Let the two natural numbers be a and b. We are given that (a + b)² = 784 and ab = 192. We need to find the positive difference between their squares, which is |a² - b²|.

Using the identity |a² - b²| = (a + b)|a - b|, we first determine the values of (a + b) and |a - b|.

From the given information: a + b = √(784) = 28

To find |a - b|, we use the algebraic relation: (a - b)² = (a + b)² - 4ab Substituting the known values: (a - b)² = 784 - 4(192) (a - b)² = 784 - 768 = 16 Taking the square root gives: |a - b| = √(16) = 4

Now, calculating the difference of the squares: |a² - b²| = (a + b) × |a - b| |a² - b²| = 28 × 4 = 112

Explanation

Using the complementary angle identity sec(90° - θ) = cscθ, the numerator sec²(90° - θ) - cot²θ transforms into csc²θ - cot²θ. According to the Pythagorean identity 1 + cot²θ = csc²θ, the value of csc²θ - cot²θ is 1.

For the denominator, we use the identity sin(90° - A) = cos A. Since 55° = 90° - 35°, we have sin 55° = cos 35°, and thus sin² 55° = cos² 35°. The expression inside the parentheses becomes sin² 35° + cos² 35°, which equals 1 based on the identity sin² A + cos² A = 1. The denominator is therefore 2(1) = 2. Substituting these values into the original expression gives ½, which equals 0.5.

Explanation

The total surface area of a cube is calculated using the formula 6a², where a is the side length of the cube. Given that the total surface area is 24 square units, the equation is 6a² = 24. Dividing both sides by 6 yields a² = 4, and taking the square root gives a side length of a = 2 units. The volume of a cube is determined by the formula V = a³. By substituting the side length, the volume is 2³, which equals 8 cubic units.

Explanation

The prime factorization of 484 is determined by noting that 484 = 22². Since 22 = 2 × 11, we have 484 = (2 × 11)² = 2² × 11². For a divisor to be a perfect square, the exponents of its prime factors in its prime factorization must be even. In the expression 2² × 11², the available even exponents for the prime factor 2 are 0 and 2, and the available even exponents for the prime factor 11 are 0 and 2.

The sum of the divisors that are perfect squares can be calculated using the product of the sums of these even powers: Sum = (2⁰ + 2²) × (11⁰ + 11²) Sum = (1 + 4) × (1 + 121) Sum = 5 × 122 = 610

Alternatively, the individual divisors that are perfect squares are 2⁰ × 11⁰ = 1, 2² × 11⁰ = 4, 2⁰ × 11² = 121, and 2² × 11² = 484. Adding these values together yields 1 + 4 + 121 + 484 = 610.

Explanation

In concentric circles, a chord of the larger circle C1 that is tangent to the smaller circle C2 is bisected at the point of contact P. This follows from two geometric theorems. First, the radius OP of the smaller circle is perpendicular to the tangent AB at the point of contact P. Second, in the larger circle C1, OP is a line segment from the center perpendicular to the chord AB. Since a perpendicular from the center of a circle to a chord bisects the chord, P must be the midpoint of AB. Thus, the length of AP is calculated as 12 / 2 = 6 cm.

Explanation

Let n be the initial number of people in the group. The total age of the initial group is the average age multiplied by the number of people, which is 27n. When 4 new friends with an average age of 25 join, the total age of these new members is 4 × 25 = 100.

The total number of people in the group becomes n + 4, and their new average age is 26. We can set up the equation for the total sum of ages as follows: 27n + 100 = 26(n + 4) 27n + 100 = 26n + 104 27n - 26n = 104 - 100 n = 4

Using the method of allegation, the ratio of the number of initial friends to the new friends is the ratio of the differences between the averages. The difference between the initial average and the total average is 27 - 26 = 1. The difference between the total average and the new friends' average is 26 - 25 = 1. Thus, the ratio of the number of people is 1:1. Since there are 4 new friends, there were 4 friends initially.

Explanation

The algebraic identity for the difference of squares is defined as a² - b² = (a - b)(a + b). Applying this identity to the numerator of the expression, l² - m² can be factored into (l - m)(l + m). The original expression is then rewritten as: ((l - m)(l + m))/((l + m)²) Given that the denominator is non-zero, we can divide both the numerator and the denominator by the common factor (l + m). This cancellation simplifies the fraction to the form (l - m)/(l + m).

Explanation

The values of trigonometric functions for angles greater than 360° are determined by the periodicity formula f(n · 360° + θ) = f(θ), where n is an integer.

To solve the expression, each term is simplified by finding the equivalent angle within the standard range:

csc 2910° = csc(8 · 360° + 30°) = csc 30° = 2.

sec 4260° = sec(11 · 360° + 300°) = sec 300°. Since 300° is in the fourth quadrant where secant is positive, sec 300° = sec(360° - 60°) = sec 60° = 2.

tan 2565° = tan(7 · 360° + 45°) = tan 45° = 1.

cot 1755° = cot(4 · 360° + 315°) = cot 315°. Since cotangent is negative in the fourth quadrant, cot 315° = cot(360° - 45°) = -cot 45° = -1.

Substituting these individual values back into the original expression gives: 2 + 2 + 1 + (-1) = 4.

Explanation

The algebraic identity relating the given terms is a³ + b³ + c³ - 3abc = ½(a + b + c)[(a - b)² + (b - c)² + (c - a)²]. By substituting the known values into this equation, we get 405 = ½ × 15 × [(a - b)² + (b - c)² + (c - a)²]. This simplifies to 405 = 7.5 × [(a - b)² + (b - c)² + (c - a)²]. Dividing 405 by 7.5 results in 54, which is the value of the required expression.

Explanation

To find the ratio of the number of rabbits to the total number of pets, first calculate the total population of pets owned by Radhika. The total number of pets is the sum of the dogs, rabbits, and parrots, which is 2 + 3 + 4 = 9. The number of rabbits is given as 3. Therefore, the ratio of rabbits to the total number of pets is 3:9. This ratio can be simplified by dividing both terms by their greatest common divisor, 3, resulting in 1:3.

Explanation

In geometry, a tangent to a circle is perpendicular to the radius at the point of tangency. This makes Δ OTP a right-angled triangle with the right angle at T. According to the Pythagorean theorem, the relationship between the sides is OT² + PT² = PO².

Given PT = 12 cm and PO = 13 cm, let the radius OT be r. r² + 12² = 13² r² + 144 = 169 r² = 169 - 144 r² = 25 r = √(25) = 5 cm

The result follows from the standard Pythagorean triplet (5, 12, 13), where 13 is the hypotenuse.

Explanation

To solve this, let the total work be the LCM of 18 and 30, which is 90 units. The efficiency of A is 90 / 18 = 5 units/day, and the efficiency of B is 90 / 30 = 3 units/day. A left 16⅔ days (which is ⁵⁰⁄₃ days) before the work was completed, meaning B worked alone for the final ⁵⁰⁄₃ days.

The work completed by B during these final days is 3 × ⁵⁰⁄₃ = 50 units. The remaining work, which A and B completed together at the start, is 90 - 50 = 40 units. The combined efficiency of A and B is 5 + 3 = 8 units/day. Therefore, the number of days A worked (the duration they worked together) is 40 / 8 = 5 days.

Explanation

Let the speed of the boat in still water be u km/h and the speed of the stream be v km/h. The downstream speed is D = u + v and the upstream speed is U = u - v. Based on the time taken for the two scenarios, we can set up the following equations:

(60)/(D) + (40)/(U) = 12.5

(84)/(D) + (63)/(U) = 18.9

In the second equation, 18 hours 54 minutes is converted to 18.9 hours because 54/60 = 0.9. To solve this system, let x = 1/D and y = 1/U. The equations become:

60x + 40y = 12.5 84x + 63y = 18.9

Multiplying the first equation by 1.4 gives 84x + 56y = 17.5. Subtracting this from the second equation:

(84x + 63y) - (84x + 56y) = 18.9 - 17.5 7y = 1.4 y = 0.2 = ⅕

Since 1/U = 1/5, we find U = 5. Substituting y = 0.2 back into 60x + 40(0.2) = 12.5:

60x + 8 = 12.5 60x = 4.5 x = (4.5)/(60) = ³⁄₄₀

Since 1/D = 3/40, we find D = ⁴⁰⁄₃. The speed of the boat in still water is the average of the downstream and upstream speeds:

u = (D + U)/(2) = (40)/(3) + 5{2} = (55)/(3){2} = ⁵⁵⁄₆ ≈ 9.17

Rounding to the nearest integer, the speed of the boat in still water is 9 km/h.

Explanation

The percentage of candidates that qualified from Goa over those that appeared is calculated using the formula P = (Qual)/(App) × 100. In 2015, the number of candidates who appeared from Goa was 7200 and the number who qualified was 850, resulting in a percentage of ⁸⁵⁰⁄₇₂₀₀ × 100 ≈ 11.81%.

Comparing this to the other years: for 2012, the percentage is ⁸¹⁰⁄₇₈₀₀ × 100 ≈ 10.38%; for 2013, it is ⁸⁰⁰⁄₇₅₀₀ × 100 ≈ 10.67%; for 2014, it is ⁹²⁰⁄₈₈₀₀ × 100 ≈ 10.45%; and for 2016, it is ⁵⁶⁰⁄₇₄₀₀ × 100 ≈ 7.57%. Since 11.81% is the highest value among these calculations, the qualification percentage for Goa peaked in 2015.

Explanation

Let the principal amount be P. The total amount A after 2 years is given as ⁸⁄₇P. The Simple Interest SI earned is the difference between the amount and the principal: SI = A - P = ⁸⁄₇P - P = ⅐P.

Using the formula for Simple Interest SI = (P · R · T)/(100), where R is the rate of interest per annum and T is the time in years, we substitute the values SI = ⅐P and T = 2:

⅐P = (P · R · 2)/(100)

Canceling P from both sides results in:

⅐ = (2R)/(100)

Rearranging the equation to solve for R:

2R = ¹⁰⁰⁄₇ R = ⁵⁰⁄₇

Converting the improper fraction ⁵⁰⁄₇ into a mixed fraction, we get 7⅐. Therefore, the rate of interest per annum is 7⅐%.

Explanation

Let the sides of the three small cubes be 3x, 4x, and 5x. When these cubes are melted to form a single large cube, the total volume remains conserved. Therefore, the sum of the volumes of the three smaller cubes equals the volume of the resulting large cube.

The diagonal of a cube with side A is given by the formula d = A√3. Given the diagonal is 18√3 cm, we can determine the side of the large cube: A√3 = 18√3 A = 18 cm

The volume of the large cube is A³ = 18³. The sum of the volumes of the three smaller cubes is: (3x)³ + (4x)³ + (5x)³ = 18³ 27x³ + 64x³ + 125x³ = 5832 216x³ = 5832

Solving for x: x³ = ⁵⁸³²⁄₂₁₆ = 27 x = ³√(27) = 3

Substituting the value of x back into the ratios, the sides of the three cubes are: First cube: 3 × 3 = 9 cm Second cube: 4 × 3 = 12 cm Third cube: 5 × 3 = 15 cm

Explanation

The efficiency relationship is established from the statement that 1 man or 2 women or 3 boys take the same time to complete the work. This means the rate of work for 1 man (M) is equal to the rate of 2 women (W) and 3 boys (B), expressed as 1M = 2W = 3B.

To find the time taken by 1 man, 1 woman, and 1 boy together, we use the formula for work equivalence:

Time = (Days)/(mₐₙ_d₎{mₒᵣ + wₐₙ_d{wₒᵣ} + bₐₙ_d{bₒᵣ}}

Substituting the given values into the formula:

Time = (121)/(1){1 + ½ + ⅓}

To simplify the denominator, find the least common multiple of 1, 2, and 3, which is 6:

¹⁄₁ + ½ + ⅓ = (6 + 3 + 2)/(6) = ¹¹⁄₆

Now, substitute this back into the time equation:

Time = (121)/(11){6} = (121 × 6)/(11)

Time = 11 × 6 = 66

Thus, one man, one woman, and one boy working together will complete the work in 66 days.

Explanation

Since triangles ABC and PQR are congruent (△ ABC ≅ △ PQR), their corresponding sides and angles are equal. Given that ∠ A = ∠ P = 90°, both are right-angled triangles. In △ ABC, the side BC is the hypotenuse because it is opposite the 90° angle. By congruence, the side AC corresponds to the side PR, which means AC = PR = 5 cm.

In the right-angled triangle ABC, we apply the Pythagoras theorem: AB² + AC² = BC² Substituting the known values: AB² + 5² = 13² AB² + 25 = 169 AB² = 169 - 25 AB² = 144 AB = √(144) = 12 cm.

The side AB is determined to be 12 cm, following the standard Pythagorean triplet (5, 12, 13).

Explanation

The mean proportion between two numbers a and b is calculated using the formula √(a × b). Given the values a = 7 and b = 112, the mean proportion is √(7 × 112).

To simplify the calculation, express 112 as a product of its factors: 112 = 7 × 16. Substituting this into the square root gives √(7 × 7 × 16). This can be rewritten as √(7²⁾^×⁴^^^², which simplifies to 7 × 4. The resulting value is 28.

Explanation

The total work is defined by the capacity of one man who completes it in 60 days. In standard work-rate problems, the efficiency of one boy is typically treated as equal to that of one man. Consequently, a group of 6 boys completes the work in 60 / 6 = 10 days. The problem states that 4 women or 6 boys finish the work in the same number of days, meaning 4 women also require 10 days to complete the task. To determine the time required for 5 women, we use the formula M₁ D₁ = M₂ D₂, where M is the number of workers and D is the number of days: 4 women × 10 days = 5 women × D₂ 40 = 5 × D₂ D₂ = 40 / 5 = 8

Explanation

The perpendicular drawn from the centre of a circle to a chord bisects the chord. Given the length of the chord is 40 cm, the length of each segment created by the perpendicular is 40 / 2 = 20 cm.

Consider the right-angled triangle formed by the radius (r), the perpendicular distance from the centre (15 cm), and the half-chord length (20 cm). According to the Pythagorean theorem:

r² = 15² + 20² r² = 225 + 400 r² = 625 r = √(625) = 25 cm

Alternatively, this follows the 3-4-5 Pythagorean triplet ratio scaled by a factor of 5, where 3 × 5 = 15 and 4 × 5 = 20, leading to a hypotenuse of 5 × 5 = 25 cm.