SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

Using the algebraic identity (a + b)² = a² + b² + 2ab, let a = tan θ and b = cot θ. Given the expression tan θ + cot θ = 12, we square both sides:

(tan θ + cot θ)² = 12²

tan² θ + cot² θ + 2 tan θ cot θ = 144

Since tan θ and cot θ are reciprocal trigonometric functions, their product tan θ · cot θ = 1. Substituting this identity into the equation yields:

tan² θ + cot² θ + 2(1) = 144

Subtracting 2 from both sides to isolate the required terms:

tan² θ + cot² θ = 144 - 2

tan² θ + cot² θ = 142

Explanation

The production of Type B two-wheelers peaked in 1995 at 25 thousand units. By examining the data for Type B across the years—20 (1993), 22 (1994), 25 (1995), 23 (1996), 19.5 (1997), and 18 (1998)—it is clear that the highest production occurred in 1995. Additionally, the year 1995 is the only year among the options where the total production of all types of two-wheelers was exactly 251.5 thousand units. Furthermore, the increase in total production from 1994 to 1995 was 14.5 thousand units (251.5 - 237), which represents the lowest annual increase in total production across the period.

Explanation

Statement I follows from the algebraic identity (x + ¹⁄ₓ)² = x² + (1)/(x²⁾ + 2. Substituting the given value K + (1)/(K) = 12 into the identity, we get 12² = K² + (1)/(K²⁾ + 2. This simplifies to 144 = K² + (1)/(K²⁾ + 2, and by subtracting 2 from both sides, we find K² + (1)/(K²⁾ = 142.

Statement II involves the repeated application of the difference of squares formula (a - b)(a + b) = a² - b². Rearranging the terms of the expression, we first multiply (K - (1)/(K))(K + (1)/(K)) to obtain K² - (1)/(K²⁾. Multiplying this result by the next factor, (K² + (1)/(K²⁾), gives K⁴ - (1)/(K⁴⁾. Finally, multiplying by the last factor, (K⁴ + (1)/(K⁴⁾), results in K⁸ - (1)/(K⁸⁾. Since the resulting expression K⁸ - (1)/(K⁸⁾ is not equal to K¹⁶ - (1)/(K¹⁶⁾, Statement II is incorrect.

Explanation

The maximum marks for the English subject is 200. Based on the table provided, the percentage of marks obtained by Shyam and Sushil in English (the fifth subject listed) are 65% and 75% respectively.

To find the actual marks obtained: Marks obtained by Shyam = 65% of 200 = ⁶⁵⁄₁₀₀ × 200 = 130 Marks obtained by Sushil = 75% of 200 = ⁷⁵⁄₁₀₀ × 200 = 150

To determine the difference: Difference = 150 - 130 = 20

Comparing the two results, Sushil obtained 150 marks while Shyam obtained 130 marks. Therefore, the marks obtained by Sushil in English are 20 more than the marks obtained by Shyam.

Explanation

Let the total votes be 100%. The candidate in third place received 20% of the total votes. The remaining 80% of the votes were shared between the winner (W) and the first runner-up (R1). Thus, W + R1 = 80%.

The difference between the winner and the first runner-up is given as 20% of the total votes, so W - R1 = 20%. Solving these two linear equations by adding them gives 2W = 100%, which means W = 50%. Substituting W back into the first equation, we find R1 = 80% - 50% = 30%.

The difference between the first runner-up and the second runner-up is R1 - R2 = 30% - 20% = 10%. According to the problem, this 10% difference corresponds to 37,000 votes. Therefore, 10% = 37,000.

To find the votes received by the winner, we calculate 50% of the total votes. Since 10% equals 37,000, then 50% is 5 × 37,000 = 185,000.

Explanation

To calculate the overall percentage gain or loss, we use the effective multiplier method by combining the price factor and the weight factor.

The price factor is given by the ratio of the stated selling price to the cost price, which is ³⁶⁄₄₀ = ⁹⁄₁₀.

The weight factor arises from the use of a false weight. Since Ramesh gives 800 grams while charging for 1000 grams, the profit ratio for weight is ¹⁰⁰⁰⁄₈₀₀ = ⁵⁄₄.

The net ratio of the transaction is the product of these two factors: Net Ratio = ⁹⁄₁₀ × ⁵⁄₄ = ⁴⁵⁄₄₀ = ⁹⁄₈

A net ratio of ⁹⁄₈ indicates that for every 8 units of cost, there are 9 units of revenue. Since the numerator is greater than the denominator, it results in a gain.

Gain Percentage = (⁹⁄₈ - 1) × 100 = ⅛ × 100 = 12.5%

Explanation

The amount Jasmine needs to pay is the selling price after deducting the discount from the marked price. This is calculated using the formula SP = MP × (1 - (D)/(100)), where MP is the marked price and D is the discount percentage. Substituting the given values, we get 800 × (1 - ³⁰⁄₁₀₀) = 800 × 0.70 = 560. Alternatively, the discount amount can be found by calculating 30% of 800, which is 0.3 × 800 = 240. Subtracting this discount from the original price results in 800 - 240 = 560.

Explanation

The mean proportional x between two numbers a and b is defined by the relation a:x = x:b, which simplifies to the geometric mean formula x = √(a × b). Substituting the given values a = 27 and b = 300 into the formula:

x = √(27 × 300) x = √(8100)

Since 90 × 90 = 8100, the square root is 90. One can also arrive at this by factoring the terms into perfect squares: √(27 × 300) = √((9 × 3) × (3 × 100)) = √(9 × 9 × 100) = 3 × 3 × 10 = 90.

Explanation

The interest rate is given as 5 paisa per rupee per month. Since ₹1 equals 100 paisa, a rate of 5 paisa per rupee is equivalent to 5/100 or 5% per month. The principal amount P is ₹2,700 and the time duration T is 8 months.

Since the rate is already specified per month, we can apply the simple interest formula SI = P × R × T directly using the monthly rate and the number of months:

SI = 2,700 × ⁵⁄₁₀₀ × 8

SI = 27 × 5 × 8

SI = 27 × 40

SI = 1,080

The total simple interest earned over 8 months is ₹1,080.

Explanation

Let n be the original number of apples. Using the concept of total weight, the sum of weights before and after adding the new apples must be consistent. The initial total weight is 50n kg. When 6 apples with an average weight of 60 kg are added, the total weight becomes 50n + (6 × 60). The new average weight of the basket is given as 50 + 5 = 55 kg for a total of n + 6 apples.

Equating the two expressions for total weight: 50n + 360 = 55(n + 6) 50n + 360 = 55n + 330 55n - 50n = 360 - 330 5n = 30 n = 6

Another efficient approach is the deviation method. The new average of the basket is 55 kg. The 6 added apples have an average weight 5 kg higher than this new average (60 - 55 = 5), providing a total surplus of 6 × 5 = 30 kg. This surplus is what increased the average weight of the original n apples from 50 kg to the new average of 55 kg. Since each of the n apples saw an increase of 5 kg, the equation is 5n = 30, which results in n = 6.

Explanation

The Highest Common Factor (HCF) of two or more numbers is always a factor of their Least Common Multiple (LCM). This relationship exists because the HCF is composed of the common prime factors in their lowest powers, while the LCM includes all prime factors in their highest powers. Consequently, the LCM must be exactly divisible by the HCF, yielding an integer as the quotient. Given that the HCF is 8, any possible LCM must be a multiple of 8. Since 42 ÷ 8 = 5.25, which is not an integer, 42 cannot be the LCM of two numbers that have 8 as their HCF.

Explanation

The area of an equilateral triangle is calculated using the formula Area = √3{4} a², where a represents the length of the side. Given the side a = 8 cm, substituting this value into the formula gives Area = √3{4} × (8)².

Calculating the square of the side, we get 8² = 64. Substituting this back into the equation yields Area = √3{4} × 64, which simplifies to 16√3. Using the approximation √3 ≈ 1.73205, the calculation becomes 16 × 1.73205 = 27.7128. When rounded to two decimal places, the area of the triangle is 27.71 cm².

Explanation

The average foreign trade by country P is calculated using the formula:

Average = (Total foreign trade of P)/(Total number of years)

First, calculate the sum of foreign trade for country P over the 5 years: Sum = 220 + 200 + 125 + 115 + 85 = 745

Next, divide the total sum by the number of years, which is 5: Average = ⁷⁴⁵⁄₅ = 149

Explanation

The trigonometric double angle identity for cosine provides a direct relationship between cos 2Y and sin Y through the formula: cos 2Y = 1 - 2sin² Y

Given the equation sin Y = x, we can substitute x directly into this identity. Replacing sin Y with x results in the following expression: cos 2Y = 1 - 2(x)² cos 2Y = 1 - 2x²

This identity is valid for any angle Y, satisfying the given range 0 ≤ Y ≤ 90°.

Explanation

Using the standard algebraic expansions, (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b². By substituting these expansions into the given expression, we get:

(a + b)² - (a - b)² = (a² + 2ab + b²) - (a² - 2ab + b²)

When subtracting the second polynomial, the signs of all its terms are reversed:

a² + 2ab + b² - a² + 2ab - b²

The a² and -a² terms cancel each other, and the b² and -b² terms also cancel each other. This simplifies the expression to:

2ab + 2ab = 4ab

This result is a standard algebraic identity which shows that the difference between the square of the sum and the square of the difference of two numbers is always four times their product.

Explanation

The ratio of the number of monuments in city B to the number of monuments in city D is determined by comparing their respective values from the table. City B has 30 monuments and city D has 150 monuments.

Ratio = (Number of monuments in B)/(Number of monuments in D)

Substituting the given values:

Ratio = ³⁰⁄₁₅₀

To simplify this fraction, divide both the numerator and the denominator by their greatest common divisor, which is 30:

Ratio = (30 ÷ 30)/(150 ÷ 30) = ⅕

Expressing this as a ratio gives 1:5.

Explanation

The percentage increase in production is calculated using the formula: Percentage Increase = (Final Production - Initial Production)/(Initial Production) × 100.

From the provided table, the total production of all two-wheelers in 1996 is 268 and the total production in 1998 is 334.

Increase in production = 334 - 268 = 66.

Percentage increase = ⁶⁶⁄₂₆₈ × 100 ≈ 24.62%.

Rounding off to the nearest integer, we get 25%.

Explanation

The length of a string tightly tied around n identical circles of radius r arranged in a compact formation is calculated by summing the lengths of the straight segments and the circular arcs. For three circles touching each other, their centers form an equilateral triangle with side length 2r. The string consists of three straight sections that are parallel to the sides of this triangle, each having a length equal to the distance between the centers, which is 2r = 14 cm. Therefore, the total length of the straight sections is 3 × 14 = 42 cm.

The remaining parts of the string are arcs that wrap around the outer edges of the circles. At each of the three circles, the string turns through an angle of 120° (or 2π/3 radians). The sum of these three arcs equals the circumference of one full circle, which is 2π r = 2π(7) = 14π cm. Representing π as p as per the options, the total length of the string is the sum of the straight and curved parts: 42 + 14p cm.

Explanation

Let the cost price of the goods be 100. The dealer marks the goods at 15% above the cost price, so the marked price becomes 115. A discount of 5% is then offered on this marked price for cash payment. The discount amount is calculated as 5% of 115, which is 115 × ⁵⁄₁₀₀ = 5.75. The selling price is the marked price minus the discount, which equals 115 - 5.75 = 109.25. Since the cost price is 100 and the selling price is 109.25, the profit earned is 109.25 - 100 = 9.25, resulting in a profit percentage of 9.25%.

Alternatively, the net profit percentage can be calculated using the formula for successive percentage changes: x + y + ˣʸ⁄₁₀₀. Here, the markup x = 15% and the discount y = -5%. Substituting these values gives 15 - 5 + (15 × (-5))/(100) = 10 - ⁷⁵⁄₁₀₀ = 10 - 0.75 = 9.25%.

Explanation

The curved surface area of a right circular cone is calculated using the formula A = π r l, where r is the radius of the base and l is the slant height. In a right circular cone, the radius, height h, and slant height form a right-angled triangle, satisfying the Pythagorean theorem r² + h² = l². Given the height h = 8m and slant height l = 10m, the radius is found by r = √(10²⁾⁻⁸^^^² = √(100 - 64) = √(36) = 6m. Substituting the radius r = 6m and slant height l = 10m into the area formula yields A = π × 6 × 10 = 60π m².

Explanation

Total production of Product Z is calculated by summing its production values across all four months: Total_Z = 15 + 18 + 12 + 19 = 64 units.

Total production of Product Y is calculated similarly: Total_Y = 18 + 13 + 17 + 8 = 56 units.

To find Y's total production as a percentage of Z's total production, we use the following relation: Percentage = (Total_Y)/(Total_Z) × 100

Substituting the calculated totals into the formula: Percentage = ⁵⁶⁄₆₄ × 100

The fraction ⁵⁶⁄₆₄ simplifies to ⅞ by dividing both the numerator and the denominator by 8. Using the standard conversion where ⅛ equals 12.5%, we multiply by 7: 7 × 12.5% = 87.5%

Explanation

In △ ABC, since PQ ∥ AC, the smaller triangle △ BPQ is similar to the larger triangle △ BAC by the AA similarity criterion. The ratio of their corresponding sides must be equal, which gives the relation (BP)/(BA) = (PQ)/(AC). Substituting the given values into this equation, we have (BP)/(26) = ⁷⁄₁₀.

Solving for BP, we get BP = (7 × 26)/(10) = ¹⁸²⁄₁₀ = 18.2. Because the point P lies on the line segment AB, the length of AP is calculated by subtracting BP from the total length AB. Thus, AP = AB - BP = 26 - 18.2 = 7.8.

Explanation

The volume of the second solution can be determined using the weighted average formula for mixtures: C₁ V₁ + C₂ V₂ = Cₘ (V₁ + V₂), where C represents the concentration and V represents the volume.

Given that V₁ = 100 ml, C₁ = 20%, C₂ = 50%, V₂ = x, and the final concentration Cₘ = 30%, the equation is set up as follows:

20 × 100 + 50 × x = 30 × (100 + x) 2000 + 50x = 3000 + 30x

By isolating the variable x: 50x - 30x = 3000 - 2000 20x = 1000 x = 50 ml

Alternatively, using the rule of alligation, the ratio of the volume of the 20% solution to the 50% solution is calculated by the difference between the concentrations: Ratio V₁ : V₂ = (50 - 30) : (30 - 20) = 20 : 10 = 2 : 1. Since V₁ = 100 ml corresponds to 2 units, 1 unit corresponds to 100 / 2 = 50 ml.

Explanation

Total work is calculated as the product of the number of men and the time required, which gives 10 × 40 = 400 man-days. Let x be the number of days after which 4 men left the job. For the first x days, all 10 men worked, completing 10x man-days of work. After x days, only 10 - 4 = 6 men remained to complete the job. Since the total time taken for completion was 50 days, these 6 men worked for (50 - x) days, completing 6(50 - x) man-days of work. The sum of the work done in these two periods must equal the total work required: 10x + 6(50 - x) = 400. Expanding the equation results in 10x + 300 - 6x = 400. This simplifies to 4x = 400 - 300, which means 4x = 100. Solving for x gives x = 25.

Explanation

Given the relationship A + B = C, we take the tangent of both sides to get tan(A + B) = tan C. Applying the trigonometric identity for the tangent of a sum, we have:

(tan A + tan B)/(1 - tan A tan B) = tan C

Cross-multiplying to eliminate the denominator results in:

tan A + tan B = tan C(1 - tan A tan B)

Distributing tan C on the right side yields:

tan A + tan B = tan C - tan A tan B tan C

To find the value of tan A tan B tan C, we rearrange the equation by moving the product term to the left side and the terms tan A and tan B to the right side:

tan A tan B tan C = tan C - tan A - tan B

Explanation

To find the value of 3m + n, we use the algebraic identity (a + b)² = a² + b² + 2ab. Let a = 3m and b = n. Expanding the square of the required expression gives:

(3m + n)² = (3m)² + (n)² + 2(3m)(n) (3m + n)² = 9m² + n² + 6mn

Substituting the given values 9m² + n² = 40 and mn = 4 into the equation:

(3m + n)² = 40 + 6(4) (3m + n)² = 40 + 24 (3m + n)² = 64

Taking the square root of both sides, we get 3m + n = √(64). Since m and n are specified as positive real numbers, the sum 3m + n must also be positive. Therefore, 3m + n = 8.

Explanation

The expression x² + 4y² + 4y - 4xy - 2x - 8 can be rearranged to group related terms: (x² - 4xy + 4y²) - (2x - 4y) - 8

The first part of the expression, (x² - 4xy + 4y²), is a perfect square equivalent to (x - 2y)². By factoring out -2 from the terms -(2x - 4y), the expression becomes: (x - 2y)² - 2(x - 2y) - 8

Let z = (x - 2y). The expression now takes the form of a standard quadratic equation: z² - 2z - 8

Factoring this quadratic by splitting the middle term yields: z² - 4z + 2z - 8 = z(z - 4) + 2(z - 4) = (z - 4)(z + 2)

Substituting z = x - 2y back into these factors results in: (x - 2y - 4)(x - 2y + 2)

Explanation

The Highest Common Factor (HCF) is determined by finding the greatest divisor shared by all given numbers. Through prime factorization: 105 = 3 × 5 × 7 335 = 5 × 67 465 = 3 × 5 × 31

Comparing these factors, the only prime factor common to the sets of 105, 335, and 465 is 5. Because no other factor is present in all three factorizations, the HCF is 5. This is also evident from the divisibility rules, as all three numbers end in 5, confirming 5 as a divisor, while their differences 335 - 105 = 230 and 465 - 335 = 130 are both divisible by 5 but share no larger common factor that also divides the original odd numbers.

Explanation

The total number of accountants working in the organization in the year 2019 is the sum of all accountants who joined from the year of foundation (2015) to 2019, minus the sum of all accountants who left the organization during the same period.

Total Joined (2015-2019) = 35 + 26 + 15 + 20 + 14 = 110 Total Left (2015-2019) = 0 + 16 + 5 + 38 + 24 = 83

Total accountants working in 2019 = Σ Joined - Σ Left 110 - 83 = 27

Explanation

Let the total number of votes polled be V. If the winning candidate receives 76% of the total votes, then the losing candidate receives the remaining 100% - 76% = 24% of the total votes. The majority by which the winning candidate is elected is defined as the difference between the votes obtained by the winner and the loser.

The majority in terms of percentage is 76% - 24% = 52%. According to the problem, this majority corresponds to 1480 votes. We can express this relationship using the following formula:

Majority = (Percentage Difference)/(100) × V 1480 = ⁵²⁄₁₀₀ × V V = (1480 × 100)/(52) V = ¹⁴⁸⁰⁰⁰⁄₅₂ V = 2846.1538...

The question specifies to consider only the integral part of the result. Therefore, the total number of votes polled is 2846.

Explanation

From the given equations, we can isolate the trigonometric terms as ˣ⁄₈ = sin θ + cos θ and ʸ⁄₉ = sin θ - cos θ. To find the value of the required expression, we square both equations:

(x²)/(8²) = (sin θ + cos θ)² = sin² θ + cos² θ + 2 sin θ cos θ (y²)/(9²) = (sin θ - cos θ)² = sin² θ + cos² θ - 2 sin θ cos θ

Adding these two equations, the term 2 sin θ cos θ cancels out. Applying the fundamental identity sin² θ + cos² θ = 1, the calculation becomes:

(x²)/(8²) + (y²)/(9²) = (sin² θ + cos² θ) + (sin² θ + cos² θ) = 1 + 1 = 2

Explanation

The speed of an object is determined by the total distance covered in a specific time interval. Since the time interval is the same for both A and B, the ratio of their speeds is equal to the ratio of the distances they cover in that time.

Let d_A be the distance covered by A in one jump and d_B be the distance covered by B in one jump. According to the problem, the distance covered by A in 7 jumps is equal to the distance covered by B in 5 jumps: 7 × d_A = 5 × d_B (d_A)/(d_B) = ⁵⁄₇

In the given time interval, A jumps 8 times and B jumps 6 times. The total distance covered by A is 8 × d_A and the total distance covered by B is 6 × d_B. The ratio of their speeds is: (Speed_A)/(Speed_B) = (8 × d_A)/(6 × d_B)

Substituting the value of (d_A)/(d_B) into the equation: (Speed_A)/(Speed_B) = ⁸⁄₆ × ⁵⁄₇ = ⁴⁰⁄₄₂

Reducing the fraction to its simplest form by dividing both the numerator and denominator by 2: ⁴⁰⁄₄₂ = ²⁰⁄₂₁

Explanation

In simple interest, if a sum becomes n times itself, the interest earned is (n-1) times the principal. For a fixed duration of time, the rate of interest r is directly proportional to the interest earned. This relationship is derived from the simple interest formula SI = (P × r × t)/(100), where SI = (n-1)P, leading to the proportionality (n₁ - 1) / r₁ = (n₂ - 1) / r₂.

Given that the sum becomes 2 times (n₁ = 2) at a rate r₁ = 5%, and we need to find the rate r₂ for the sum to become 5 times (n₂ = 5):

r₂ = r₁ × (n₂ - 1)/(n₁ - 1)

r₂ = 5 × (5 - 1)/(2 - 1)

r₂ = 5 × 4 = 20

The rate of interest required is 20 percent.

Explanation

Let the initial quantities of milk and water in the mixture be 5x and 3x respectively. When 7 litres of water are added to the mixture, the quantity of water becomes 3x + 7, while the quantity of milk remains 5x. The new ratio of milk to water is 1:2.

The relationship can be expressed as: (5x)/(3x + 7) = ½

Cross-multiplying to solve for x: 10x = 3x + 7 7x = 7 x = 1

The quantity of milk in the mixture is 5x. Substituting x = 1: Milk = 5 × 1 = 5 litres.

Using the ratio method, the initial ratio is 5:3. Since only water is added, the quantity of milk must remain constant. To make the milk parts equal in the final ratio of 1:2, multiply the entire ratio by 5, resulting in 5:10. The increase in the water component is 10 - 3 = 7 units. Given that 7 litres of water were added, 7 units correspond to 7 litres, meaning 1 unit equals 1 litre. The quantity of milk is represented by 5 units, which equals 5 litres.

Explanation

The difference in simple interest is due to the difference in the time period for which the sum was invested. Since the rate of interest and the principal amount remain constant, the difference in interest corresponds to the interest earned over the additional years.

Difference in time = 5 years - 3 years = 2 years. The rate of interest is 14% per annum. The total difference in interest as a percentage of the principal is 14% × 2 = 28%.

Given that this difference is ₹4,200, we can set up the equation: 28% of Principal = 4200 ²⁸⁄₁₀₀ × P = 4200 P = (4200 × 100)/(28)

Simplifying the calculation: P = (600 × 100)/(4) P = 150 × 100 P = 15,000

The principal sum is ₹15,000.

Explanation

The construction task follows the principle of work equivalence, where the total amount of work is constant and can be expressed using the formula M₁D₁ = M₂D₂. In this equation, M represents the number of men and D represents the number of days. Given M₁ = 450 men and D₁ = 20 days, the total work is 450 × 20 = 9000 man-days. To find the number of men (M₂) required to complete the same work in D₂ = 30 days, the equation is set up as 450 × 20 = M₂ × 30. Solving for M₂ involves dividing the total man-days by the new time frame: M₂ = ⁹⁰⁰⁰⁄₃₀. This calculation results in 300 men. Since the number of days increases, the number of men required decreases proportionally to maintain the same total work output.

Explanation

To find the value of a⁴ + b⁴, we first determine the value of a² + b² using the algebraic identity (a + b)² = a² + b² + 2ab. Substituting the given values a + b = 11 and ab = 35:

a² + b² = (a + b)² - 2ab a² + b² = 11² - 2(35) a² + b² = 121 - 70 = 51

Next, we use the identity (a² + b²)² = a⁴ + b⁴ + 2(ab)² to solve for a⁴ + b⁴:

a⁴ + b⁴ = (a² + b²)² - 2(ab)² a⁴ + b⁴ = 51² - 2(35²)

Calculating the squares: 51² = 2601 35² = 1225

Substituting these values into the equation: a⁴ + b⁴ = 2601 - 2(1225) a⁴ + b⁴ = 2601 - 2450 = 151

Explanation

When two numbers are in the ratio a:b and their Highest Common Factor (HCF) is h, the two numbers can be expressed as ah and bh. Since the ratio 6:5 consists of coprime numbers (numbers with no common factor other than 1), the Least Common Multiple (LCM) can be calculated directly using the formula LCM = a × b × h.

Given: Ratio a:b = 6:5 HCF h = 3

Substituting these values into the formula: LCM = 6 × 5 × 3 LCM = 30 × 3 = 90

Alternatively, the two numbers are 6 × 3 = 18 and 5 × 3 = 15. The LCM of 18 and 15 is the smallest number divisible by both, which is 90.

Explanation

The quantity of a liquid remaining in a container after n operations of replacing y units from a total of x units with another liquid is given by the formula: Remaining Quantity = x (1 - ʸ⁄ₓ)ⁿ

In this problem, the initial volume of milk x is 25 litres, the volume replaced in each step y is 5 litres, and the total number of operations n is 3 (one initial process plus two repetitions).

Substituting these values into the formula: Milk remaining = 25 × (1 - ⁵⁄₂₅)³ Milk remaining = 25 × (1 - ⅕)³ Milk remaining = 25 × (⅘)³ Milk remaining = 25 × ⁶⁴⁄₁₂₅ Milk remaining = ⁶⁴⁄₅ = 12.8 litres

Explanation

The Side-Angle-Side (SAS) congruence criterion states that two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle. In △ ABC and △ PQR, it is given that AB = PQ and ∠ B = ∠ Q. The angle ∠ B is the included angle between sides AB and BC in △ ABC, while ∠ Q is the included angle between sides PQ and QR in △ PQR. To satisfy the SAS condition, the side lengths forming the given equal angles must also be equal. Since AB = PQ is already established, the second pair of corresponding sides must be BC = QR.

Explanation

The annual rate of simple interest r when a principal amount becomes n times itself in t years can be determined using the formula r = ((n - 1) × 100)/(t). In this specific case, the principal triples, so n = 3, and the time duration t is given as 10 years. By substituting these values into the formula, the equation becomes r = ((3 - 1) × 100)/(10). Simplifying the numerator gives r = (2 × 100)/(10), which further reduces to r = 2 × 10 = 20. This formula is a direct derivation of the standard Simple Interest formula SI = (P × r × t)/(100). Since the principal P becomes 3P, the interest earned SI is 3P - P = 2P. Substituting SI = 2P and t = 10 into the standard equation results in 2P = (P × r × 10)/(100), which simplifies to 2 = ʳ⁄₁₀ and yields r = 20.

Explanation

When two objects move in opposite directions, their relative speed is the sum of their individual speeds. The relative speed of the two trains is 48 + 60 = 108 km/hr. Converting this speed into metres per second, we get 108 × ⁵⁄₁₈ = 30 m/s.

In the scenario where one train crosses a person sitting in another train, the distance covered is equal to the length of the train that is doing the crossing. Here, the slower train crosses a man in the faster train in 12 seconds. Using the formula Distance = Speed × Time, the length of the slower train is calculated as 30 m/s × 12 seconds = 360 metres.

Explanation

The selling price (SP) is obtained by applying a discount percentage to the marked price (MP). The mathematical relationship is expressed as SP = MP × (1 - (Discount %)/(100)).

In this case, the selling price is Rs. 1,445 and the discount is 15%. Plugging these values into the formula gives: 1445 = MP × (1 - ¹⁵⁄₁₀₀) 1445 = MP × ⁸⁵⁄₁₀₀ 1445 = MP × 0.85

To find the marked price, isolate MP: MP = (1445)/(0.85) MP = (1445 × 100)/(85)

Simplifying the fraction by dividing both the numerator and the denominator by 5: MP = (289 × 100)/(17)

Recognizing that 289 is the square of 17 (17 × 17 = 289): MP = 17 × 100 MP = 1700

Explanation

Let the total work be the Least Common Multiple (LCM) of the individual times taken by Ajay and Vijay. The LCM of 6 and 16 is 48 units.

Ajay efficiency E_A = 48 / 6 = 8 units/hour. Vijay efficiency E_V = 48 / 16 = 3 units/hour.

Since they work on an alternate hour basis starting with Ajay, the work done in a 2-hour cycle is 8 + 3 = 11 units. In 4 such cycles, which equate to 8 hours, the total work completed is 11 × 4 = 44 units.

The remaining work is 48 - 44 = 4 units. After the 8th hour, it is Ajay turn again. Time taken by Ajay to finish the remaining 4 units is calculated as: Time = Work / Efficiency = 4 / 8 = 0.5 hours.

Converting 0.5 hours into minutes gives 0.5 × 60 = 30 minutes. Therefore, the total time taken to complete the work is 8 hours and 30 minutes.

Explanation

In △ DEF, the sum of the interior angles is 180°. Given that ∠ E = 85° and ∠ F = 40°, the measure of the third angle is ∠ D = 180° - (85° + 40°) = 55°.

Comparing △ ABC and △ DEF, it is given that AB = DE and AC = DF. Since ∠ A = 55° and we calculated ∠ D = 55°, the included angles between the respective equal sides are equal (∠ A = ∠ D). According to the Side-Angle-Side (SAS) congruence criterion, if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, the triangles are congruent. Therefore, △ ABC ≅ △ DEF by the SAS property.

Explanation

The average speed for the entire journey is calculated using the formula Average Speed = (Total Distance)/(Total Time). The total distance traveled is 250 km + 350 km = 600 km. To find the total time, we calculate the time taken for each segment separately using the relation Time = (Distance)/(Speed). For the first part, the time is t₁ = (250 km)/(50 km/h) = 5 hours. For the second part, the time is t₂ = (350 km)/(70 km/h) = 5 hours. Therefore, the total time taken is 5 hours + 5 hours = 10 hours. Substituting these values into the average speed formula gives (600 km)/(10 hours) = 60 km/h.

Explanation

Given y = 1 + √3 + √4, we simplify the expression for y as y = 1 + √3 + 2 = 3 + √3. Rearranging this gives y - 3 = √3. Squaring both sides to eliminate the square root, we get (y - 3)² = (√3)², which simplifies to y² - 6y + 9 = 3, or y² - 6y + 6 = 0.

To find the value of the polynomial 2y⁴ - 8y³ - 6y² + 28y - 84, we can divide it by the quadratic expression y² - 6y + 6. Using polynomial long division:

2y⁴ - 8y³ - 6y² + 28y - 84 = (2y² + 4y + 6)(y² - 6y + 6) + (40y - 120)

Since we established that y² - 6y + 6 = 0 for the given value of y, the entire first term becomes zero. The value of the original polynomial is therefore equal to the remainder:

40y - 120

Substituting the value y = 3 + √3 back into the remainder:

40(3 + √3) - 120 = 120 + 40√3 - 120 = 40√3

Explanation

If we assume the question contains a common typographical error and the intended expression is (x - 27)³ + (y - 9)³, we can let a = x - 27 and b = y - 9. Given x + y = 36, the sum of the bases is a + b = x + y - 36 = 36 - 36 = 0. Since a + b = 0, the sum of their cubes a³ + b³ is also 0.

Explanation

For two circles with radii r₁ and r₂ and distance between their centers d, the condition where the circles do not touch, intersect, or lie inside one another is represented by the inequality d > r₁ + r₂. In this geometric configuration, there are four common tangents in total. Two of these are direct common tangents (external tangents), which lie on the same side of the line segment joining the centers of the circles. The other two are transverse common tangents (internal tangents), which intersect the line segment joining the centers. Since the circles are completely separated, both pairs of tangents exist, resulting in 2 + 2 = 4 common tangents.

Explanation

To achieve a 25 percent gain on the whole transaction, the total profit needed is 100 (25 percent of 400). Since the first half sold at a 5 percent gain earned a profit of 10 (5 percent of 200), the second half must generate the remaining 90 in profit, which represents a 45 percent gain on the remaining cost of 200.