SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

The final amount A in a fixed deposit scheme with different annual interest rates r1 and r2 is calculated using the formula A = x(1 + (r₁)/(100))(1 + (r₂)/(100)). In this case, the principal x is invested at r₁ = 10% for the first year and r₂ = 15% for the second year to yield a total amount of Rs 20,240.

Substituting the given values into the formula: 20,240 = x(1 + ¹⁰⁄₁₀₀)(1 + ¹⁵⁄₁₀₀) 20,240 = x × ¹¹⁰⁄₁₀₀ × ¹¹⁵⁄₁₀₀ 20,240 = x × ¹¹⁄₁₀ × ²³⁄₂₀ 20,240 = x × ²⁵³⁄₂₀₀

Rearranging to solve for x: x = (20,240 × 200)/(253)

By dividing 20,240 by 253, we find that 253 × 8 = 2024, which means 253 × 80 = 20,240. x = 80 × 200 x = 16,000

Explanation

Distribute the ⁵⁄₂ term and simplify the equation: ¹⁰ˣ⁄₃ + 5 - ⁵ˣ⁄₆ = ⁷⁄₂

Multiply the entire equation by 6 to eliminate the denominators: 20x + 30 - 5x = 21

Combine the x terms and move the constant term to the other side: 15x = 21 - 30 15x = -9

Solve for x by dividing by 15: x = -⁹⁄₁₅

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 3: x = -⅗

Explanation

The algebraic identity for the difference of two cubes is a³ - b³ = (a - b)³ + 3ab(a - b). By substituting the given values a - b = 2 and ab = 15 into this formula, the expression simplifies as follows:

a³ - b³ = (2)³ + 3(15)(2) a³ - b³ = 8 + 90 a³ - b³ = 98

Alternatively, one can find the individual values of a and b by solving the system of equations. Two numbers that have a product of 15 and a difference of 2 are 5 and 3. Substituting a = 5 and b = 3 into the expression: a³ - b³ = 5³ - 3³ a³ - b³ = 125 - 27 = 98

Explanation

Let the fraction be represented by x. The problem states that the sum of the fraction and four times its reciprocal is 13/3. This is represented by the equation x + 4(1/x) = 13/3. To solve for x, we first simplify the left side to get (x² + 4)/x = 13/3. Cross-multiplying the terms results in 3(x² + 4) = 13x, which simplifies to the quadratic equation 3x² - 13x + 12 = 0.

This quadratic equation can be solved by factoring the middle term: 3x² - 9x - 4x + 12 = 0. Grouping the terms, we get 3x(x - 3) - 4(x - 3) = 0, which factors into (3x - 4)(x - 3) = 0. Solving for x gives two possible values: x = 4/3 or x = 3. Testing the fraction 4/3 in the original equation, we have 4/3 + 4(3/4) = 4/3 + 3 = (4 + 9)/3 = 13/3, which satisfies the given condition.

Explanation

The nᵗʰ term of an arithmetic progression is given by the formula aₙ = a + (n-1)d, where a is the first term and d is the common difference. Given the 3rd term a₃ = -14 and the 8th term a₈ = 1, the difference between these terms can be expressed as a₈ - a₃ = (a + 7d) - (a + 2d) = 5d. Substituting the given values, we get 1 - (-14) = 15, so 5d = 15, which gives d = 3. The 11th term a₁₁ can be calculated directly from the 8th term using the relation a₁₁ = a₈ + (11 - 8)d. Substituting the known values results in a₁₁ = 1 + 3(3) = 1 + 9 = 10.

Explanation

The distance between two points (x₁, y₁) and (x₂, y₂) is given by the distance formula d = √((x₂ - x₁)² + (y₂ - y₁)²). Substituting the given points (4, 8) and (k, -4) along with the distance d = 13 into this formula results in:

13 = √((k - 4)² + (-4 - 8)²)

Squaring both sides of the equation to remove the square root:

13² = (k - 4)² + (-12)² 169 = (k - 4)² + 144

Subtracting 144 from both sides:

25 = (k - 4)²

Taking the square root of both sides gives two possible values for k - 4:

k - 4 = 5 or k - 4 = -5

Solving for k in both cases:

k = 9 or k = -1

Among the provided options, the value k = -1 matches.

Explanation

The slope m₁ of the given line 2x + 3y = -6 is determined by converting it into the slope-intercept form y = mx + c. Rearranging the equation gives 3y = -2x - 6, which simplifies to y = -⅔x - 2. Therefore, the slope m₁ is -⅔.

Two lines are perpendicular if the product of their slopes is -1. Let m₂ be the slope of the required line. Using the relation m₁ · m₂ = -1, we have (-⅔) · m₂ = -1, which yields m₂ = ³⁄₂. The problem also states that the line has a y-intercept of 3, meaning c = 3.

Substituting these values into the slope-intercept equation y = m₂x + c, we obtain y = ³⁄₂x + 3. To convert this into the general form, we multiply every term by 2 to get 2y = 3x + 6. Rearranging the variables and constants to one side results in 3x - 2y = -6.

Explanation

In Δ ABC, since DE is parallel to BC, Δ ADE is similar to Δ ABC by the Angle-Angle similarity criterion. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

The length of side AB can be expressed as AD + DB. Given the ratio AD:DB = 2:3, the ratio of the side of the smaller triangle to the larger triangle is:

(AD)/(AB) = (AD)/(AD + DB) = (2)/(2 + 3) = ⅖

The ratio of the areas of Δ ADE and Δ ABC is the square of this side ratio:

(Area(Δ ADE))/(Area(Δ ABC)) = ((AD)/(AB))² = (⅖)² = ⁴⁄₂₅

Given that the area of Δ ADE is 4 sq cm, we can set up the equation:

(4)/(Area(Δ ABC)) = ⁴⁄₂₅

Solving for the area of Δ ABC gives Area(Δ ABC) = 25 sq cm. The area of the quadrilateral BDEC is the difference between the area of the larger triangle and the smaller triangle:

Area(BDEC) = Area(Δ ABC) - Area(Δ ADE) = 25 - 4 = 21 sq cm

Explanation

The value of the trigonometric function sin 30° is ½ and the value of csc 45°, which is the reciprocal of sin 45° = (1)/(√(2)), is √2. Substituting these values into the expression sin 30° - csc 45° gives:

½ - √2

To simplify this subtraction, find a common denominator of 2:

½ - 2√2{2} = 1 - 2√2{2}

Explanation

In right-angled triangle LMN, given that ∠ M = 90° and ∠ N = 45°, the third angle ∠ L must also be 45° since the sum of angles in a triangle is 180°. This identifies Δ LMN as an isosceles right-angled triangle where the legs MN and ML are equal in length.

Using the trigonometric ratio for cosine at ∠ N: cos(N) = (MN)/(NL) cos(45°) = (MN)/(9√(2))

Substituting the value cos(45°) = (1)/(√(2)): (1)/(√(2)) = (MN)/(9√(2)) MN = 9√2{√2} MN = 9 cm

Alternatively, in a 45°-45°-90° triangle, the relationship between the hypotenuse h and a leg s is defined by the formula h = s√2. Here, NL is the hypotenuse and MN is a leg. Thus, MN = (NL)/(√(2)) = 9√2{√2} = 9 cm.

Explanation

In a right-angled triangle, cos θ is defined as the ratio of the base to the hypotenuse. Given cos θ = 35/37, we can let the base B = 35 and the hypotenuse H = 37. To find the value of cot θ, which is the ratio of the base to the perpendicular P, we first determine P using the Pythagorean theorem: H² = B² + P².

Substituting the given values, we have 37² = 35² + P². Solving for P gives P = √(37² - 35²). Using the identity a² - b² = (a + b)(a - b), we get P = √((37 + 35)(37 - 35)) = √(72 × 2) = √144 = 12.

The value of cot θ is the ratio (Base)/(Perpendicular). Substituting B = 35 and P = 12, we obtain cot θ = 35/12.

Explanation

The number of employees in each department as per the data provided in the graph for this examination is as follows: Department A: 350 Department B: 700 Department C: 450 Department D: 600 Department E: 650 Department F: 400

By comparing these values, we can determine the ranking of the departments by their employee counts: Department B: 700 (Highest) Department E: 650 (Second Highest) Department D: 600 Department C: 450 Department F: 400 Department A: 350

Department E has 650 employees, which is the second highest value among all the departments.

Explanation

Based on the data for the given departments: Department A = 17 employees Department B = 6 employees Department C = 10 employees Department D = 28 employees Department E = 15 employees Department F = 4 employees

If 5 employees are transferred from department A to department F: New number of employees in department A = 17 - 5 = 12 New number of employees in department F = 4 + 5 = 9

The new count for department F (9) falls between the employee counts of department B (6) and department C (10), as 6 < 9 < 10.

Explanation

The employee distribution for the departments is A = 20%, B = 6%, C = 12%, D = 15%, E = 22%, and F = 25%. When departments F and D are merged to create department G, the new department accounts for 25% + 15% = 40% of the total employees. The remaining departments to consider are A (20%), B (6%), C (12%), and E (22%). Comparing the shares of all departments in the new configuration—A (20%), B (6%), C (12%), E (22%), and G (40%)—it is clear that department B has the lowest percentage share. Since the absolute number of employees is calculated as a direct percentage of the constant total workforce, the department with the lowest percentage share necessarily contains the least number of employees.

Explanation

The total number of employees in the company, as specified in the data set for this exam shift, is 64. To find the total expense, we use the formula:

Total Expense = Total Employees × Average Salary

Given that the average salary of these employees is Rs 8,000, we substitute the values into the formula:

Total Expense = 64 × 8,000 = 5,12,000

To convert the amount from Rupees to Lakhs, we divide the total by 1,00,000:

Expense in Lakhs = 5,12,000 / 1,00,000 = 5.12

The total expenditure incurred by the company for salaries is 5.12 Lakhs.

Explanation

To convert the binary number 10110 to its decimal equivalent, each digit is multiplied by 2ⁿ, where n is the position of the digit starting from 0 on the right. The conversion follows the formula:

Decimal = (dₙ × 2ⁿ) + (dₙ₋₁ × 2ⁿ⁻¹) + … + (d₀ × 2⁰)

For the binary value 10110: The bit at position 4 is 1, so 1 × 2⁴ = 16. The bit at position 3 is 0, so 0 × 2³ = 0. The bit at position 2 is 1, so 1 × 2² = 4. The bit at position 1 is 1, so 1 × 2¹ = 2. The bit at position 0 is 0, so 0 × 2⁰ = 0.

Summing these individual values gives 16 + 0 + 4 + 2 + 0, which equals 22. This summation of powers of two effectively translates the base-2 representation into the base-10 decimal system.

Explanation

To find the time taken when A, B, and C work together, we calculate the total work using the Least Common Multiple (LCM) of their individual completion times. The LCM of 12, 16, and 24 is 48, so we assume the total work is 48 units.

Next, we determine the efficiency (work done per day) for each individual: Efficiency of A = 48 / 12 = 4 units/day Efficiency of B = 48 / 16 = 3 units/day Efficiency of C = 48 / 24 = 2 units/day

The combined efficiency of A, B, and C working together is the sum of their individual efficiencies: Combined Efficiency = 4 + 3 + 2 = 9 units/day

The time required to complete the job together is the total work divided by the combined efficiency: Time = 48 / 9 days

Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor, 3, yields 16 / 3 days.

Explanation

In a square with side length a, the diagonal d represents the hypotenuse of a right-angled triangle formed by two adjacent sides. According to the Pythagorean theorem, the relationship between the side and the diagonal is given by the formula d = a√2. Substituting the given diagonal length of 10 cm into this equation, we have 10 = a√2. To find the side length a, we divide both sides by √2, resulting in a = (10)/(√(2)). To simplify this expression, we rationalize the denominator by multiplying the numerator and denominator by √2, which gives a = 10√2{2}. This simplifies to 5√2 cm.

Explanation

The initial discount of 40% is applied to the marked price of Rs 1000. Price after first discount = 1000 × (1 - ⁴⁰⁄₁₀₀) = 1000 × 0.60 = 600. The customer actually pays a final sale price of Rs 510. The additional discount amount = 600 - 510 = 90. To find the additional discount percentage, we calculate the ratio of this discount to the price after the first discount: Additional discount % = (⁹⁰⁄₆₀₀) × 100 = ⁹⁰⁄₆ = 15%.

Explanation

The third proportional to two numbers a and b is a value c such that a:b = b:c. This relationship is represented by the formula c = b²{a}.

Given a = 9 and b = 45, the third proportional is calculated as:

c = 45²{9}

c = (45 × 45)/(9)

c = 5 × 45

c = 225

Explanation

The number of boys in the class is calculated as 39 - 26 = 13. The ratio of the number of girls to the number of boys is 26 : 13, which simplifies to 2 : 1. Let the average weight of the boys be x. Applying the weighted average formula:

Average weight of class = (n_gᵢᵣₗₛ × Avg_gᵢᵣₗₛ) + (n_bₒ_yₛ × Avg_bₒ_yₛ){n_gᵢᵣₗₛ + n_bₒ_yₛ}

Using the simplified ratio of 2 girls to 1 boy:

48 = (2 × 42 + 1 × x)/(2 + 1)

48 = (84 + x)/(3)

144 = 84 + x

x = 144 - 84 = 60

Explanation

The initial selling price of Rs 2880 results in a loss of 10%, which means this price represents 90% of the cost price (100% - 10%). To achieve a profit of 20%, the target selling price must represent 120% of the cost price (100% + 20%).

The required selling price can be calculated directly using the ratio of the percentages: SPₙₑ_w = SPₒₗ_d × (100 + Profit%)/(100 - Loss%) SPₙₑ_w = 2880 × ¹²⁰⁄₉₀ SPₙₑ_w = 2880 × ⁴⁄₃ SPₙₑ_w = 960 × 4 = 3840

Alternatively, using the unitary method, if 90% corresponds to 2880, then 1% corresponds to 2880 / 90 = 32. Therefore, 120% corresponds to 32 × 120 = 3840.

Explanation

Let the marks obtained in History be H and the marks in Geography be G. We are given H = G + 9 and H = 0.56(H + G). Substituting H into the second equation gives G + 9 = 0.56(G + 9 + G), which simplifies to G + 9 = 0.56(2G + 9). Expanding the right side results in G + 9 = 1.12G + 5.04. Rearranging the equation to solve for G, we get 9 - 5.04 = 1.12G - G, which leads to 3.96 = 0.12G. Dividing 3.96 by 0.12 results in G = 33. Alternatively, since History is 56% of the total, Geography must be 100% - 56% = 44% of the total. The difference between History and Geography is 56% - 44% = 12% of the total marks. We know this difference is 9 marks. Therefore, 12% of the total marks T is 9, so 0.12T = 9, which gives T = 75. Geography marks are 44% of 75, calculated as 0.44 × 75 = 33.

Explanation

Let the time taken by the Express train and the Duronto train to travel 720 km be T_E and T_D respectively. From the first condition, the Express train takes 6 hours more than the Duronto, so T_E - T_D = 6.

If the speed of the Express train is doubled, its time taken is halved because time is inversely proportional to speed for a constant distance. The new time for the Express train is T_E / 2. According to the second condition, this new time is 2 hours less than the Duronto's time, so T_D - T_E / 2 = 2.

Adding these two equations eliminates T_D: (T_E - T_D) + (T_D - T_E / 2) = 6 + 2 T_E / 2 = 8 T_E = 16 hours

Substituting T_E = 16 into the first equation: 16 - T_D = 6 T_D = 10 hours

The speed of the Duronto train is calculated as Distance / Time: Speed_Dᵤᵣₒₙₜₒ = 720 / 10 = 72 km/hr

Explanation

For yearly compounding at 10% per annum, the interest for 1 year is calculated simply as 1000 × 10/100 = 100.

For half-yearly compounding, the annual rate is divided by 2, and the time period is multiplied by 2. Thus, the rate r = 5% per half-year and the number of periods n = 2. The compound interest is calculated using the formula CI = P[(1 + r/100)ⁿ - 1]. Substituting the values: CI = 1000 × [(1 + 5/100)² - 1] CI = 1000 × [(21/20)² - 1] CI = 1000 × [441/400 - 1] CI = 1000 × 41/400 = 102.5

Alternatively, the difference can be viewed as the interest earned on the first half-year's interest during the second half-year. The interest for the first 6 months is 1000 × 5% = 50. The extra interest generated in the second 6 months is 50 × 5% = 2.5.

The difference between the two interests is 102.5 - 100 = 2.5.

Explanation

To find the value of x, solve the linear equation 7x - ³⁄₂(4x - 9) = 6.5 by first expanding the terms within the parentheses. Multiply ³⁄₂ by each term inside the bracket: 7x - (³⁄₂ × 4x) + (³⁄₂ × 9) = 6.5 7x - 6x + 13.5 = 6.5

Combine the x terms on the left side: x + 13.5 = 6.5

Subtract 13.5 from both sides to isolate x: x = 6.5 - 13.5 x = -7

Explanation

The value of the sum of cubes a³ + b³ can be calculated directly using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). Given the values a + b = 8 and ab = 15, substitute them into the formula:

a³ + b³ = (8)³ - 3(15)(8) a³ + b³ = 512 - 360 a³ + b³ = 152

Alternatively, by inspecting the given conditions, the two numbers whose sum is 8 and product is 15 are 3 and 5. Calculating their cubes gives 3³ = 27 and 5³ = 125. Their sum is 27 + 125 = 152.

Explanation

Let the number be x. The equation formed from the given condition is x + ⁴⁄ₓ = ¹⁷⁄₂. Substituting x = 8 into the expression: 8 + ⁴⁄₈ = 8 + ½ = (16 + 1)/(2) = ¹⁷⁄₂ Alternatively, solving the quadratic equation 2x² - 17x + 8 = 0 by splitting the middle term: 2x² - 16x - x + 8 = 0 2x(x - 8) - 1(x - 8) = 0 (2x - 1)(x - 8) = 0 The solutions are x = 8 and x = 0.5.

Explanation

In an arithmetic progression, the nᵗʰ term is given by the formula aₙ = a + (n-1)d, where a is the first term and d is the common difference. The common difference can be calculated using any two terms aₘ and aₙ as d = (aₘ - aₙ)/(m - n).

Given a₅ = 7 and a₉ = 13, the common difference is: d = (13 - 7)/(9 - 5) = ⁶⁄₄ = 1.5

To find the 15th term, we can use the 9th term as a reference point: a₁₅ = a₉ + (15 - 9)d a₁₅ = 13 + 6(1.5) a₁₅ = 13 + 9 = 22

Explanation

When a point (x, y) is reflected in a horizontal line y = k, the x-coordinate of the image remains the same as that of the original point, while the y-coordinate changes. The line y = k acts as the midpoint between the y-coordinate of the original point and the y-coordinate of its image.

The formula for the reflected point (x', y') is x' = x and y' = 2k - y. Given the point (6, -3) and the line of reflection y = 2, we substitute x = 6, y = -3, and k = 2 into the formula.

x' = 6 y' = 2(2) - (-3) = 4 + 3 = 7

Therefore, the reflection of the point (6, -3) in the line y = 2 is (6, 7).

Explanation

The section formula states that if a point (x, y) divides the line segment joining (x₁, y₁) and (x₂, y₂) in the ratio m:n internally, the coordinates are given by x = (mx₂ + nx₁)/(m+n) and y = (my₂ + ny₁)/(m+n). Given point A(4, 2) divides the segment BC where B is (2, 6) and C is (7, y) in the ratio 2:5, we substitute the y-coordinates into the formula. Here, the y-coordinate of A is 2, m = 2, n = 5, y₁ = 6, and y₂ = y.

Substituting these values into the y-coordinate formula: 2 = (2(y) + 5(6))/(2+5) 2 = (2y + 30)/(7) 14 = 2y + 30 2y = 14 - 30 2y = -16 y = -8

Explanation

A line intersects the X-axis at a point where the y-coordinate is zero. To determine the coordinates of this intersection for the equation 2x - 3y = 6, we substitute y = 0 into the given linear equation:

2x - 3(0) = 6 2x = 6 x = ⁶⁄₂ x = 3

Substituting y = 0 yields a value of x = 3. In the Cartesian coordinate system, a point is represented as (x, y). Since the x-coordinate is 3 and the y-coordinate is 0, the line cuts the X-axis at the point (3, 0).

Explanation

In a right-angled triangle ΔABC where ∠B = 90° and BD is the altitude drawn to the hypotenuse AC, the Geometric Mean Theorem states that the altitude is the geometric mean of the two segments of the hypotenuse. The formula for the length of the altitude is BD² = AD × DC. Given the segments AD = 4 cm and DC = 9 cm, substituting these values into the formula yields BD² = 4 × 9 = 36. Taking the square root of both sides, we find BD = √36 = 6 cm.

Explanation

Using the standard trigonometric values for specific angles, we know that tan 45° = 1 and csc 60° = (1)/(sin 60°). Since sin 60° = √3{2}, it follows that csc 60° = (2)/(√(3)).

Substituting these values into the given expression: tan 45° + csc 60° = 1 + (2)/(√(3))

To simplify this, we find a common denominator: √3 + 2{√3}

To match the format of the provided options, we rationalize the denominator by multiplying both the numerator and the denominator by √3: (√3 + 2) · √3{√3 · √3} = (√3 · √3) + (2 · √3){3} = 3 + 2√3{3}

Explanation

In the right-angled triangle XYZ, the sum of all interior angles is 180°. Given that the triangle is right-angled at Y, we have ∠ Y = 90°. With ∠ Z = 60°, the measure of ∠ X can be found using the angle sum property: ∠ X = 180° - (∠ Y + ∠ Z) = 180° - (90° + 60°) = 30°. The cosecant of an angle is the reciprocal of its sine. Therefore, csc X = csc 30° = 1 / sin 30°. Since the value of sin 30° is 1/2, we find that csc 30° = 1 / (1/2) = 2.

Explanation

The trigonometric identity relating cosecant and cotangent is csc² θ - cot² θ = 1. Rearranging this formula to solve for cot θ, we get cot² θ = csc² θ - 1. Substituting the given value csc θ = ¹⁷⁄₈ into the equation, we have cot² θ = (¹⁷⁄₈)² - 1 = ²⁸⁹⁄₆₄ - 1. By taking the common denominator, the expression becomes cot² θ = (289 - 64)/(64) = ²²⁵⁄₆₄. Taking the square root of both sides, we find cot θ = ¹⁵⁄₈. This result can also be verified using the Pythagorean triplet (8, 15, 17), where if the hypotenuse is 17 and the opposite side is 8, the adjacent side must be 15. Since cot θ is the ratio of the adjacent side to the opposite side, it equals ¹⁵⁄₈.

Explanation

Based on the box office collection data for the six movies, the earnings are as follows: Movie E grossed 500 crores, Movie C grossed 450 crores, Movie A grossed 350 crores, Movie B grossed 250 crores, Movie F grossed 200 crores, and Movie D grossed 150 crores.

Comparing these values, Movie E has the highest collection, while Movie C, with 450 crores, has the second highest box office collection.

Explanation

Based on the movie collection data, the collection of movie B is 250 units and the collection of movie D is 150 units. The percentage by which the collection of movie D is lower than that of movie B is calculated using the formula for percentage decrease:

Percentage Decrease = (Collection of B - Collection of D)/(Collection of B) × 100

Substituting the respective values into the formula:

Percentage Decrease = (250 - 150)/(250) × 100

Percentage Decrease = ¹⁰⁰⁄₂₅₀ × 100

Percentage Decrease = ⅖ × 100 = 40%

Explanation

Based on the movie collection data, the collection of movie C is 225 crore. To identify which movies have a combined collection equal to this, we sum the individual collections of movies B, D, and F. The collection for movie B is 100 crore, movie D is 75 crore, and movie F is 50 crore.

Summing these values: 100 + 75 + 50 = 225

Since the combined total of movies B, D, and F is 225 crore, it is exactly equal to the individual collection of movie C.

Explanation

The producer's share is calculated by taking the sum of the box office collections for movies A and E and applying the specified percentage. The box office collection for movie A is 100 crores and for movie E is 200 crores.

Total Box Office = 100 + 200 = 300 crores Producer Share = 300 × ⁶⁰⁄₁₀₀ = 180 crores

Explanation

To find the quotient when 1359 is divided by 48, we use long division. We start by dividing 135 by 48. Since 48 × 2 = 96 and 48 × 3 = 144, the divisor goes into the first three digits 2 times. Subtracting 96 from 135 yields 39. Bringing down the next digit, 9, gives the number 399. Next, we divide 399 by 48. Calculating multiples of 48, we have 48 × 8 = 384 and 48 × 9 = 432. Thus, 48 goes into 399 exactly 8 times with a remainder. Subtracting 384 from 399 leaves 15. The complete calculation shows 1359 = (48 × 28) + 15.

Explanation

The work rate of an individual is the reciprocal of the time taken to complete the job. Given the time taken by A is 10 days and B is 50 days, their respective work rates are 1/10 and 1/50 cupboards per day. The combined rate of A, B, and C is given as 1/6.25 cupboards per day.

Converting the decimal to a fraction, 6.25 = 25/4, so the combined rate is 4/25. To find the rate of C, we subtract the rates of A and B from the total combined rate:

Rate of C = ⁴⁄₂₅ - (⅒ + ¹⁄₅₀)

Using 50 as the common denominator:

Rate of C = ⁸⁄₅₀ - (⁵⁄₅₀ + ¹⁄₅₀) Rate of C = ⁸⁄₅₀ - ⁶⁄₅₀ = ²⁄₅₀ = ¹⁄₂₅

The time taken by C to complete the job alone is the reciprocal of its rate, which is 25 days.

Explanation

In a rectangle, the diagonal d and the sides a and b relate through the Pythagoras theorem: d² = a² + b². Given the diagonal d = 26 cm and one side a = 10 cm, the other side b is calculated as b = √(d² - a²). Substituting the values, b = √(26² - 10²) = √(676 - 100) = √576 = 24 cm. This can also be quickly identified using the Pythagorean triplet (5, 12, 13) scaled by a factor of 2, resulting in (10, 24, 26). The area of the rectangle is the product of its two sides: Area = a × b = 10 × 24 = 240 sq cm.

Explanation

The Selling Price (SP) of an article after a markup and a subsequent discount can be calculated directly using the formula:

SP = CP × (1 + (M)/(100)) × (1 - (D)/(100))

Where CP is the cost price, M is the markup percentage, and D is the discount percentage. Given CP = 450, M = 80, and D = 20, substitute the values into the formula:

SP = 450 × (1 + ⁸⁰⁄₁₀₀) × (1 - ²⁰⁄₁₀₀) SP = 450 × 1.8 × 0.8 SP = 450 × 1.44

To solve 450 × 1.44 quickly, treat it as 450 × ¹⁴⁴⁄₁₀₀, which simplifies to 4.5 × 144: 4 × 144 = 576 0.5 × 144 = 72 576 + 72 = 648

Alternatively, calculating step-by-step: Marked Price (MP) = 450 + 80% of 450 = 450 + 360 = 810. Selling Price (SP) = 810 - 20% of 810 = 810 - 162 = 648.

Explanation

Let the number to be added to each value be x. For the resulting numbers to be in proportion, the ratio of the first two must equal the ratio of the last two. The condition is expressed as:

(35 + x)/(115 + x) = (53 + x)/(165 + x)

By substituting the value x = 10 from the options into the equation, we get:

First ratio: (35 + 10)/(115 + 10) = ⁴⁵⁄₁₂₅ = (9 × 5)/(25 × 5) = ⁹⁄₂₅

Second ratio: (53 + 10)/(165 + 10) = ⁶³⁄₁₇₅ = (9 × 7)/(25 × 7) = ⁹⁄₂₅

Since both ratios simplify to ⁹⁄₂₅, the numbers 45, 125, 63, and 175 are in proportion. Alternatively, solving the linear equation formed by cross-multiplying (35 + x)(165 + x) = (115 + x)(53 + x) leads to 5775 + 200x + x² = 6095 + 168x + x². Simplifying this gives 32x = 320, which results in x = 10.

Explanation

To find the average after the 13th match, we use the relationship between the total runs and the average. Let the average after the 13th match be x. The average before this match (after 12 matches) was x - 4. The total runs after 13 matches is the sum of the runs after 12 matches and the runs from the 13th match. This gives the equation 13x = 12(x - 4) + 95. Solving for x yields 13x = 12x - 48 + 95, which simplifies to x = 47. A faster approach is to subtract the total increase applied to the previous matches from the current score: 95 - (12 × 4) = 95 - 48 = 47.

Explanation

To calculate the gain percentage, we find the cost price (CP) and selling price (SP) for a common quantity of bananas. The vendor buys 12 bananas for Rs 50 and sells 5 bananas for Rs 40. Taking the LCM of 12 and 5, which is 60, we can compare the prices for 60 units.

Cost Price of 60 bananas = ⁵⁰⁄₁₂ × 60 = 50 × 5 = Rs 250

Selling Price of 60 bananas = ⁴⁰⁄₅ × 60 = 40 × 12 = Rs 480

The total gain is calculated as SP - CP, which is 480 - 250 = Rs 230.

The gain percentage is determined using the formula: Gain % = (Total Gain)/(CP) × 100

Substituting the values: Gain % = ²³⁰⁄₂₅₀ × 100 = ²³⁄₂₅ × 100 = 23 × 4 = 92%

Explanation

Let the amount paid to labourer B be represented by x. According to the problem, labourer A is paid 150 percent of B's payment, which is 1.5x. The total daily payment for both labourers is Rs. 750, leading to the linear equation x + 1.5x = 750. This simplifies to 2.5x = 750. Solving for x by dividing 750 by 2.5, we get x = (750)/(2.5) = ⁷⁵⁰⁰⁄₂₅ = 300. Therefore, labourer B is paid Rs. 300.

Explanation

Let the time spent traveling on foot be t hours and the time spent on bicycle be (8 - t) hours. Given that the total distance covered is 50 km, we can establish the following equation based on the formula Distance = Speed × Time:

5t + 7(8 - t) = 50

Expanding the equation: 5t + 56 - 7t = 50 -2t = -6 t = 3

The time spent traveling on foot is 3 hours. To find the distance traveled on foot, multiply this time by the speed on foot: 5 km/hr × 3 hours = 15 km

Alternatively, using the method of alligation, the average speed for the entire journey is 50 / 8 = 6.25 km/hr. The ratio of time spent on foot to time spent on bicycle is (7 - 6.25) : (6.25 - 5) = 0.75 : 1.25, which simplifies to 3 : 5. Dividing the total time of 8 hours in the ratio 3 : 5 gives 3 hours for the foot journey and 5 hours for the bicycle journey. Thus, the distance on foot is 5 km/hr × 3 hours = 15 km.

Explanation

The effective rate of compound interest for two years can be calculated using the successive percentage formula x + y + ˣʸ⁄₁₀₀. For a rate of 8% per annum, the effective rate for two years is 8 + 8 + (8 × 8)/(100) = 16.64%.

According to the problem, the compound interest earned is Rs 4160. Therefore, 16.64% of the principal P is equal to 4160.

P × (16.64)/(100) = 4160

P = (4160 × 100)/(16.64)

P = (416000)/(16.64) = 25000

The sum invested is Rs 25000.