SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

Average speed is calculated using the formula v = ᵈ⁄ₜ, where d is the distance covered and t is the time taken. Substituting the given values: v = (90 km)/(2.5 hours) = 36 km/h

To express the speed in meters per second (m/s), multiply the value by the conversion factor ⁵⁄₁₈, which is derived from (1000 meters)/(3600 seconds): v = 36 × ⁵⁄₁₈ = 10 m/s

Explanation

The simple interest rate is determined by the percentage increase in the principal over the given time. Since the principal increases by 18% in 3 years at simple interest, the annual interest rate R is calculated as R = (18%)/(3) = 6% per annum.

To find the compound interest on Rs. 25,000 for 3 years at 6% per annum, we use the formula for the effective compound interest rate over 3 years: Rₑ_f_f = x + y + z + (xy + yz + zx)/(100) + ˣʸᶻ⁄₁₀₀₀₀. For x = y = z = 6, the effective rate is 6 + 6 + 6 + (36 + 36 + 36)/(100) + ²¹⁶⁄₁₀₀₀₀ = 18 + 1.08 + 0.0216 = 19.1016%.

The compound interest is calculated as CI = (19.1016)/(100) × 25000. Simplifying the calculation, CI = 19.1016 × 250 = 4775.4. Alternatively, the amount A = 25000 × (1.06)³ = 25000 × 1.191016 = 29775.4. Subtracting the principal, CI = 29775.4 - 25000 = 4775.4.

Explanation

The given linear equation is 5x + 6(3 - 2x) = 4. Expanding the brackets by distributing the multiplier 6 results in 5x + 18 - 12x = 4. Combining the like terms involving x yields -7x + 18 = 4. Subtracting 18 from both sides of the equation gives -7x = 4 - 18, which simplifies to -7x = -14. Dividing both sides by -7 results in x = ⁻¹⁴⁄₋₇ = 2.

Explanation

The value of a³ + b³ is calculated using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). This formula is derived from the expansion of the cube of a binomial, (a + b)³ = a³ + b³ + 3ab(a + b).

Given the values a + b = 1 and ab = -6, substitute them directly into the identity:

a³ + b³ = (1)³ - 3(-6)(1) a³ + b³ = 1 - (-18) a³ + b³ = 1 + 18 a³ + b³ = 19

The identity provides the most direct path to the solution by utilizing the known sum and product of the variables.

Explanation

Let the non-zero number be x. According to the problem statement, the sum of the number and twenty times its reciprocal is equal to 9. This relationship is represented by the equation x + ²⁰⁄ₓ = 9. To eliminate the denominator, multiply every term by x, which results in the quadratic equation x² + 20 = 9x. By rearranging the terms into the standard form ax² + bx + c = 0, we get x² - 9x + 20 = 0. This quadratic can be solved by factoring the middle term; we seek two numbers that multiply to 20 and add up to -9. Those numbers are -4 and -5. Thus, the equation becomes (x - 4)(x - 5) = 0. This gives two possible solutions for x, which are 4 and 5. Since 5 is the value provided in the options, it is the correct number.

Explanation

The n-th term of an Arithmetic Progression (AP) is given by the formula aₙ = a + (n - 1)d, where a is the first term and d is the common difference. We are given the 3rd term a₃ = 13 and the 5th term a₅ = 21.

The common difference d can be calculated by the change in values divided by the change in the number of terms: d = (a₅ - a₃)/(5 - 3) = (21 - 13)/(2) = ⁸⁄₂ = 4

To find the 13th term a₁₃, we can start from the 5th term and add the common difference for the remaining (13 - 5) = 8 steps: a₁₃ = a₅ + (13 - 5)d a₁₃ = 21 + 8 × 4 a₁₃ = 21 + 32 = 53

Alternatively, using the 3rd term: a₁₃ = a₃ + (13 - 3)d a₁₃ = 13 + 10 × 4 a₁₃ = 13 + 40 = 53

Explanation

The reflection of a point (x, y) in the origin results in the point (-x, -y). For the point (3, -5), the coordinates after changing the sign of both values are (-3, 5).

Explanation

The midpoint formula states that if P(x, y) is the midpoint of a line segment with endpoints A(x₁, y₁) and B(x₂, y₂), then x = (x₁ + x₂)/(2) and y = (y₁ + y₂)/(2). Substituting the known coordinates of midpoint P(5, -1) and endpoint A(2, -4) into these equations allows us to solve for the coordinates of B(x₂, y₂).

For the x-coordinate: 5 = (2 + x₂)/(2) 10 = 2 + x₂ x₂ = 8

For the y-coordinate: -1 = (-4 + y₂)/(2) -2 = -4 + y₂ y₂ = 2

The coordinates of point B are therefore (8, 2).

Explanation

The slope m₁ of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula m = y₂ - y₁{x₂ - x₁}. Substituting the given points (-2, 3) and (2, 0) into the formula:

m₁ = (0 - 3)/(2 - (-2)) = ⁻³⁄₄

For two lines to be perpendicular, the product of their slopes must be -1, expressed as m₁ · m₂ = -1. To find the slope of the perpendicular line m₂, we take the negative reciprocal of m₁:

m₂ = -(1)/(m₁₎ = -(1)/(-3/4) = ⁴⁄₃

Explanation

For any two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides. Given that Δ ABC ∼ Δ PQR, the relationship between their perimeters and corresponding sides AB and PQ is:

(Perimeter(Δ ABC))/(Perimeter(Δ PQR)) = (AB)/(PQ)

The problem states that the ratio of the perimeters is 1:2 and the length of side PQ is 10 cm. Substituting these values into the equation:

½ = (AB)/(10)

To find the length of AB, we solve for the unknown:

AB = 10 × ½ = 5 cm

Explanation

The trigonometric values for the standard angle 30° are sin 30° = ½ and cos 30° = √3{2}. To find the value of the given expression, we substitute these values and perform the addition:

sin 30° + cos 30° = ½ + √3{2}

Because the fractions share a common denominator of 2, the numerators can be combined directly over the same denominator:

1 + √3{2}

Explanation

In a triangle, the sum of all internal angles is 180°. Given that △ ABC is right-angled at B, we have ∠ B = 90°. Since the sum of the angles in any triangle is 180°, the two acute angles A and C must be complementary, meaning ∠ A + ∠ C = 90°. Given that ∠ A = 30°, the value of angle C is calculated as ∠ C = 90° - 30° = 60°.

The trigonometric function sec C is defined as the reciprocal of cos C. Using the standard trigonometric ratio for 60°, we know that cos 60° = 1/2. Therefore:

sec C = sec 60° = (1)/(cos 60°) = (1)/(1/2) = 2

Explanation

In a right-angled triangle, the sine of an angle is the ratio of the perpendicular to the hypotenuse, so sin θ = (P)/(H) = ¹²⁄₁₃. Using the Pythagorean theorem, H² = P² + B², we calculate the base B = √(13² - 12²) = √(169 - 144) = √(25) = 5. The cotangent of an angle is the ratio of the base to the perpendicular, defined as cot θ = (B)/(P). Substituting the values, cot θ = ⁵⁄₁₂.

Explanation

The Highest Common Factor (HCF) of 6345 and 2160 is calculated using the Euclidean division algorithm, which involves repeated division until the remainder is zero.

Step 1: Divide 6345 by 2160. 6345 = 2 × 2160 + 2025

Step 2: Divide the previous divisor (2160) by the remainder (2025). 2160 = 1 × 2025 + 135

Step 3: Divide the previous divisor (2025) by the new remainder (135). 2025 = 15 × 135 + 0

Since the remainder is zero, the last divisor used, 135, is the HCF. This result can also be verified through prime factorization. The number 2160 is 2⁴ × 3³ × 5 and 6345 is 3³ × 5 × 47. The common prime factors are 3³ × 5, which results in 27 × 5 = 135.

Explanation

To find the fraction of work left, first determine the total work using the Least Common Multiple (LCM) of the time taken by A and B. The LCM of 24 and 40 is 120, so let the total work be 120 units.

The efficiency (work per day) of A is 120 / 24 = 5 units per day, and the efficiency of B is 120 / 40 = 3 units per day. Working together, their combined efficiency is 5 + 3 = 8 units per day.

In 10 days, the amount of work completed by A and B together is 10 × 8 = 80 units. The remaining work is the difference between the total work and the completed work: 120 - 80 = 40 units. To find the fraction of work left, divide the remaining work by the total work: 40 / 120 = 1/3.

Explanation

The area of a circle is given by the formula A = π r². Substituting the given area of 616 sq cm into the equation, we have 616 = ²²⁄₇ × r². To find r², we calculate 616 × ⁷⁄₂₂, which results in 28 × 7 = 196. Therefore, the radius r = √(196) = 14 cm.

The circumference of a circle is calculated using the formula C = 2π r. By substituting the radius of 14 cm into this formula, we get C = 2 × ²²⁄₇ × 14. Simplifying this expression gives 2 × 22 × 2, which equals 88 cm.

Explanation

Let the Marked Price (MP) of the goods be 100 units. The retailer offers a discount of 20% on the marked price, so the Selling Price (SP) is determined as: SP = MP × (1 - Discount Rate) SP = 100 × (1 - 0.20) = 80

The problem states that by selling at this price, the retailer ends up selling at the Cost Price (CP). Therefore, SP = CP = 80 units.

The markup is the difference between the Marked Price and the Cost Price. The formula for the markup amount is: Markup Amount = MP - CP Markup Amount = 100 - 80 = 20

The percentage markup is always calculated with respect to the Cost Price. Using the standard formula for markup percentage: Percentage Markup = ( (MP - CP)/(CP) ) × 100 Percentage Markup = ( ²⁰⁄₈₀ ) × 100 = ¼ × 100 = 25%

This can also be expressed through the relationship CP = MP × 0.8. Rearranging for MP gives MP = CP / 0.8, which simplifies to MP = 1.25 × CP. A multiplier of 1.25 corresponds to a 25% increase over the original cost price.

Explanation

Initial ratio of selected to unselected candidates is 11:2, which means for every 13 applicants, 11 were selected and 2 were unselected. Let the initial selected candidates be 11x and the unselected candidates be 2x. The total number of initial applicants is 13x.

In the second scenario, 40 fewer candidates applied and 20 fewer were selected. Thus, the new total number of applicants is 13x - 40 and the new number of selected candidates is 11x - 20. The new number of unselected candidates is the difference between these two values: (13x - 40) - (11x - 20) = 2x - 20.

The new ratio of selected to unselected is 10:1. Setting up the equation: (11x - 20)/(2x - 20) = ¹⁰⁄₁ 11x - 20 = 10(2x - 20) 11x - 20 = 20x - 200 9x = 180 x = 20

The total number of candidates who applied is 13x = 13 × 20 = 260. A quicker method for competitive exams is to observe that the total number of applicants must be a multiple of 11 + 2 = 13. Among the choices provided, only 260 is exactly divisible by 13.

Explanation

The set of natural numbers from 21 to 39 forms an arithmetic progression because the difference between any two consecutive terms is constant. For any sequence in an arithmetic progression, the average is calculated using the formula Average = (a + l)/(2), where a is the first term and l is the last term.

Substituting the given values into the formula, where a = 21 and l = 39:

Average = (21 + 39)/(2) = ⁶⁰⁄₂ = 30

Alternatively, since the total number of terms n = (39 - 21) + 1 = 19 is an odd number, the average is exactly equal to the middle term of the sequence. The position of the middle term is (n + 1)/(2) = (19 + 1)/(2) = 10. The 10th term in the sequence starting from 21 is 21 + (10 - 1) = 30.

Explanation

Let the cost price of the sheep be S and the cost price of the goat be G. According to the problem, the total cost price is S + G = 3500. Since the selling prices of both animals are equal, we can equate the expressions for their selling prices based on the given profit and loss percentages.

1.20 × S = 0.90 × G

By simplifying the ratio of the cost prices: (S)/(G) = (0.90)/(1.20) = ¾

The cost prices of the sheep and the goat are in the ratio 3:4. The total ratio is 3 + 4 = 7 units. We can find the value of one unit by dividing the total cost price by the total number of units: Value per unit = ³⁵⁰⁰⁄₇ = 500

The cost price of the sheep is 3 × 500 = 1500, and the cost price of the goat is 4 × 500 = 2000. Comparing these two values, the cost price of the cheaper animal is 1500.

Explanation

Let the third number be 100. According to the problem, the first number is 10% less than the third number, so the first number is 100 - (10% of 100) = 90. The second number is 20% less than the third number, so the second number is 100 - (20% of 100) = 80.

To make the second number (80) equal to the first number (90), the second number must be increased by the difference, which is 90 - 80 = 10. The percentage enhancement required is calculated relative to the second number using the formula Percentage Increase = (Difference)/(Base Value) × 100.

Substituting the values into the formula gives ¹⁰⁄₈₀ × 100. Simplifying the fraction ¹⁰⁄₈₀ yields ⅛. Since ⅛ is equivalent to 0.125, multiplying by 100 results in 12.5%.

Explanation

When the distance d between two points is constant, the relationship between speeds v₁, v₂ and the difference in time taken Δ t is given by the formula:

d = (v₁ × v₂)/(|v₁ - v₂|) × Δ t

Substituting the given values, where v₁ = 35 km/hr, v₂ = 45 km/hr, and Δ t = 2 hours:

d = (35 × 45)/(45 - 35) × 2

d = ¹⁵⁷⁵⁄₁₀ × 2

d = 157.5 × 2 = 315 km

Alternatively, using the ratio method, the ratio of speeds of car A and car B is 35:45, which simplifies to 7:9. Since distance is constant, the ratio of time taken is the inverse of the speed ratio, which is 9:7. The difference in the time ratio parts is 9 - 7 = 2 units. Given that the actual time difference is 2 hours, 1 unit equals 1 hour. Therefore, car A takes 9 hours to complete the journey. The distance is calculated as speed multiplied by time: 35 km/hr × 9 hours = 315 km.

Explanation

The relationship between the principal P, the amount A, the rate r, and the time n in compound interest is given by the formula A = P(1 + ʳ⁄₁₀₀)ⁿ. Substituting the given values A = 18,150, P = 15,000, and n = 2 into the formula:

18,150 = 15,000(1 + ʳ⁄₁₀₀)²

Dividing both sides by 15,000:

(18,150)/(15,000) = (1 + ʳ⁄₁₀₀)²

(1,815)/(1,500) = (1 + ʳ⁄₁₀₀)²

Simplifying the fraction by dividing both the numerator and the denominator by 15:

¹²¹⁄₁₀₀ = (1 + ʳ⁄₁₀₀)²

Taking the square root of both sides:

√(¹²¹⁄₁₀₀) = 1 + ʳ⁄₁₀₀

¹¹⁄₁₀ = 1 + ʳ⁄₁₀₀

1.1 = 1 + ʳ⁄₁₀₀

Subtracting 1 from both sides:

0.1 = ʳ⁄₁₀₀

r = 0.1 × 100

r = 10

Explanation

To find the value of x, simplify the given linear equation 9x - (5(2x + 1))/(2) = ⁹⁄₂. Multiplying every term by 2 to eliminate the denominators results in the following expression: 2(9x) - 5(2x + 1) = 9. Expanding the brackets on the left side gives 18x - 10x - 5 = 9. Combining the x-terms leads to 8x - 5 = 9. Adding 5 to both sides results in 8x = 14. Solving for x gives x = ¹⁴⁄₈, which simplifies to ⁷⁄₄ after dividing the numerator and denominator by their greatest common divisor, 2.

Explanation

The sum of cubes a³ + b³ is derived from the algebraic identity for the cube of a binomial: (a + b)³ = a³ + b³ + 3ab(a + b). This can be rearranged to isolate the sum of cubes as a³ + b³ = (a + b)³ - 3ab(a + b).

By substituting the given values a + b = 4 and ab = -21 into this formula: a³ + b³ = (4)³ - 3(-21)(4) a³ + b³ = 64 - (-252) a³ + b³ = 64 + 252 a³ + b³ = 316

Alternatively, identifying the numbers a and b whose sum is 4 and product is -21 leads to 7 and -3. Calculating the sum of their cubes gives 7³ + (-3)³ = 343 - 27 = 316.

Explanation

Let the fraction be x. According to the problem, the sum of the fraction and 10 times its reciprocal is 37/4. This can be expressed as the equation x + 10 · ¹⁄ₓ = ³⁷⁄₄. To solve for x, multiply the entire equation by 4x to eliminate denominators: 4x² + 40 = 37x. Rearranging this into a standard quadratic equation gives 4x² - 37x + 40 = 0.

We solve this by splitting the middle term. We look for two numbers that multiply to 4 × 40 = 160 and add to -37. These numbers are -32 and -5. The equation becomes 4x² - 32x - 5x + 40 = 0. Factoring by grouping, we get 4x(x - 8) - 5(x - 8) = 0, which simplifies to (4x - 5)(x - 8) = 0. Setting each factor to zero gives x = ⁵⁄₄ or x = 8. Testing x = ⁵⁄₄ in the original condition: ⁵⁄₄ + 10 · ⅘ = ⁵⁄₄ + 8 = (5 + 32)/(4) = ³⁷⁄₄. This matches the given sum.

Explanation

The n-th 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. According to the problem, the 2nd term a₂ = 8 and the 6th term a₆ = 20.

Representing these terms using the formula: a + d = 8 a + 5d = 20

Subtracting the first equation from the second allows for the calculation of the common difference: (a + 5d) - (a + d) = 20 - 8 4d = 12 d = 3

Substituting d = 3 into the first equation gives a + 3 = 8, which means the first term a = 5. The 20th term is then calculated by substituting these values into the n-th term formula: a₂₀ = a + (20 - 1)d a₂₀ = 5 + 19(3) a₂₀ = 5 + 57 a₂₀ = 62

Explanation

When a point (x, y) is reflected in a horizontal line y = k, the x-coordinate remains unchanged because the movement is strictly vertical. The line y = k acts as the perpendicular bisector of the segment joining the original point and its image. Thus, the new y-coordinate, y', satisfies the midpoint formula (y + y') / 2 = k, which simplifies to y' = 2k - y.

For the point (3, 2) and the line y = -2, we have x = 3, y = 2, and k = -2. The x-coordinate of the reflection remains 3. The y-coordinate of the reflection is 2(-2) - 2 = -4 - 2 = -6. Combining these coordinates yields the point (3, -6).

Explanation

The distance d between two points (x₁, y₁) and (x₂, y₂) is calculated using the distance formula derived from the Pythagorean theorem: d = √((x₂ - x₁)² + (y₂ - y₁)²).

Substituting the given coordinates (3, 6) and (-2, -6) into the formula: x₁ = 3, y₁ = 6 x₂ = -2, y₂ = -6

d = √((-2 - 3)² + (-6 - 6)²) d = √((-5)² + (-12)²) d = √(25 + 144) d = √(169) d = 13

The horizontal difference is 5 units and the vertical difference is 12 units. This forms a Pythagorean triple (5, 12, 13), where the hypotenuse represents the straight-line distance between the two points.

Explanation

The equation of a line in intercept form is given by the formula ˣ⁄ₐ + (y)/(b) = 1, where a is the x-intercept and b is the y-intercept. Substituting the given values a = 3 and b = -5 into this equation:

ˣ⁄₃ + ʸ⁄₋₅ = 1

ˣ⁄₃ - ʸ⁄₅ = 1

To convert this into the standard linear form, multiply the entire equation by the least common multiple of the denominators, which is 15:

15 · (ˣ⁄₃ - ʸ⁄₅) = 15 · 1

5x - 3y = 15

This equation correctly represents the line that crosses the x-axis at 3 and the y-axis at -5.

Explanation

In right-angled Δ ABC with altitude BD drawn to the hypotenuse AC, Δ BDC is similar to Δ ABC by the AA similarity criterion. From this similarity, the ratio of corresponding sides is equal, leading to the relation BC/AC = DC/BC, which simplifies to the formula BC² = AC × DC. Substituting the given values AC = 25 cm and DC = 9 cm into the equation, we get BC² = 25 × 9 = 225. Taking the square root of both sides gives BC = √(225) = 15 cm.

Explanation

The standard trigonometric values for the angles in the expression are tan 45° = 1 and sec 60° = (1)/(cos 60°) = (1)/(1/2) = 2. Substituting these values into the given expression tan 45° - (1)/(√(3)) sec 60°, we have:

1 - (1)/(√(3))(2) = 1 - (2)/(√(3))

To simplify the result, we find a common denominator:

√3 - 2{√3}

Multiplying both the numerator and the denominator by √3 to rationalize the expression gives:

(√3 - 2) × √3{√3 × √3} = (√3)² - 2√3{3} = 3 - 2√3{3}

Explanation

In right-angled triangle DEF with ∠ E = 90°, the side EF is the side opposite to ∠ D and DE is the side adjacent to ∠ D. Using the trigonometric ratio for tangent:

tan D = (EF)/(DE)

Substituting the given values m∠ D = 30° and EF = 6√3 cm:

tan 30° = 6√3{DE}

The value of tan 30° is (1)/(√(3)). Substituting this into the equation gives:

(1)/(√(3)) = 6√3{DE}

By cross-multiplication:

DE = 6√3 × √3

DE = 6 × 3 = 18 cm.

Explanation

The trigonometric ratio sin θ is defined as the ratio of the perpendicular P to the hypotenuse H. Given sin θ = ²⁰⁄₂₉, we identify P = 20 and H = 29. According to the Pythagorean theorem H² = P² + B², the base B is calculated as:

B = √(H²⁾⁻^P^^^² B = √(29²⁾⁻²⁰^^^² = √(841 - 400) = √(441) = 21

The function sec θ is defined as the ratio of the hypotenuse to the base:

sec θ = (H)/(B) = ²⁹⁄₂₁

Explanation

This product can be solved efficiently using the algebraic identity (a - b)(a + b) = a² - b². The given numbers 9997 and 10003 can be rewritten as (10000 - 3) and (10000 + 3) respectively.

Applying the formula: 9997 × 10003 = (10000 - 3)(10000 + 3) = (10000)² - (3)² = 100,000,000 - 9 = 99,999,991

Explanation

When two individuals A and B can complete a task in x and y days respectively, the time taken by them to complete the work together is given by the formula (x × y)/(x + y). Substituting the given values where x = 36 and y = 12, the calculation is (36 × 12)/(36 + 12) = ⁴³²⁄₄₈ = 9 days.

Alternatively, using the efficiency method, let the total work be the Least Common Multiple (LCM) of 36 and 12, which is 36 units. The efficiency of A is 36 / 36 = 1 unit per day, and the efficiency of B is 36 / 12 = 3 units per day. Their combined efficiency is 1 + 3 = 4 units per day. Therefore, the time taken to complete the work together is Total Work divided by Total Efficiency, which is 36 / 4 = 9 days.

Explanation

The area of a rhombus is calculated using the formula A = ½ × d₁ × d₂, where d₁ and d₂ are the lengths of the diagonals. Given the diagonals are 12 cm and 14 cm, substituting these values into the formula gives A = ½ × 12 × 14. Simplifying this expression, we have 6 × 14, which results in an area of 84 sq cm. This relationship exists because the diagonals of a rhombus bisect each other at right angles, effectively dividing the rhombus into four congruent right-angled triangles.

Explanation

The discount percentage is determined by the formula: Discount % = (Marked Price - Selling Price)/(Marked Price) × 100. In this case, the marked price is Rs 5000 and the selling price is Rs 4400. The numerical discount offered by the shopkeeper is 5000 - 4400 = 600. By substituting the values into the percentage formula, we get ⁶⁰⁰⁄₅₀₀₀ × 100. This simplifies to ⁶⁰⁰⁄₅₀, resulting in a discount of 12%.

Explanation

The ratio of the original fare to the new fare is given as 9:11. This implies that if the original fare is represented by 9x, the new fare is 11x. The increase in fare is the difference between the two, which is 11x - 9x = 2x.

Given the original fare is Rs. 18,000, we can equate 9x = 18,000. Solving for x gives x = 18,000 / 9 = 2,000. Substituting the value of x into the expression for the increase, we get 2 × 2,000 = 4,000.

Explanation

Let the three numbers be x, y, and z. The sum of these numbers is calculated by multiplying the average by the count: x + y + z = 40 × 3 = 120. The problem states that the first number x is one-third the sum of the other two numbers, which is represented by the equation x = ⅓(y + z). Multiplying both sides by 3 gives 3x = y + z. Substituting this expression for the sum of the other two numbers back into the total sum equation results in x + 3x = 120. Combining like terms gives 4x = 120. Dividing both sides by 4, we find that x = 30.

Explanation

The total cost price for 500 kg of cotton is Rs 9,000. To earn a 10% profit, the trader needs a total selling price of 9,000 × (1 + 0.10) = 9,900 Rs.

The amount of cotton available for sale is reduced by 10% due to spoilage, which leaves 500 × (1 - 0.10) = 450 kg. To find the required selling rate per kilogram, divide the total target revenue by the weight of the remaining cotton:

Selling Rate = (9,900)/(450) = ⁹⁹⁰⁄₄₅ = 22 Rs/kg.

Explanation

To determine the maximum marks, first calculate the minimum passing marks by adding the student's score to the marks by which he failed. Since the student scored 67 and failed by 13, the passing marks are 67 + 13 = 80.

The problem states that the passing requirement is 40% of the total marks. Let the maximum marks be M. We can express this relationship as: 40% of M = 80 ⁴⁰⁄₁₀₀ × M = 80

Solving for M: M = (80 × 100)/(40) M = 2 × 100 = 200

Explanation

To determine the distance traveled in meters, the speed of the spaceship must be converted from kilometers per hour (km/h) to meters per second (m/s). The conversion factor is ⁵⁄₁₈ because 1 km/h = (1000 meters)/(3600 seconds) = ⁵⁄₁₈ m/s.

Speed in m/s = 1260 × ⁵⁄₁₈ = 70 × 5 = 350 m/s.

The distance D traveled in a given time t is calculated using the formula D = v × t. Given the time is ⅒th of a second: D = 350 × ⅒ = 35 meters.

Explanation

In compound interest, the amount at the end of the second year acts as the principal for the third year. The interest earned during the third year is simply the difference between the amount at the end of the third year (A₃) and the amount at the end of the second year (A₂).

Given that A₂ = 1800 and A₃ = 1926, the interest earned over that one-year period is: Interest = 1926 - 1800 = 126

The rate of interest (r) is calculated by determining what percentage this interest is of the principal for that year (A₂): r = (Interest)/(A₂) × 100 r = ¹²⁶⁄₁₈₀₀ × 100 r = ¹²⁶⁄₁₈ = 7%

This approach is derived from the standard compound interest formula Aₙ = P(1 + ʳ⁄₁₀₀)ⁿ. By taking the ratio of the amounts of two consecutive years, we get (A₃)/(A₂) = 1 + ʳ⁄₁₀₀, which simplifies directly to the calculation performed above.

Explanation

To find the value of x, start with the given linear equation:

-½(x - 5) + 3 = -⁵⁄₂

First, subtract 3 from both sides of the equation. To simplify the right side, express 3 as a fraction with a common denominator of 2, which is ⁶⁄₂:

-½(x - 5) = -⁵⁄₂ - ⁶⁄₂

-½(x - 5) = -¹¹⁄₂

Next, multiply both sides of the equation by -2 to isolate the term (x - 5). This cancels out the negative sign and the fraction on the left side:

x - 5 = (-¹¹⁄₂) · (-2)

x - 5 = 11

Finally, isolate x by adding 5 to both sides:

x = 11 + 5 = 16

Explanation

The value of the expression a³ - b³ can be calculated using the standard algebraic identity a³ - b³ = (a - b)³ + 3ab(a - b). This formula is the most direct way to solve the problem as it utilizes the given values for the difference (a - b) and the product (ab) without requiring the determination of the individual values of a and b.

By substituting the provided values a - b = 1 and ab = 6 into the identity, the expression becomes (1)³ + 3(6)(1). Simplifying the terms, (1)³ equals 1 and the product 3 × 6 × 1 equals 18. Adding these two components results in 1 + 18, which yields 19. Alternatively, identifying two numbers whose difference is 1 and product is 6 leads to a = 3 and b = 2, and calculating 3³ - 2³ = 27 - 8 also confirms the result is 19.

Explanation

Let the required number be x. According to the problem, the number exceeds 58 times its reciprocal by ¾. This relationship can be expressed by the equation:

x - 58 · ¹⁄ₓ = ¾

To solve for x, multiply the entire equation by 4x to eliminate the denominators:

4x² - 232 = 3x

Rearranging this into a standard quadratic equation gives:

4x² - 3x - 232 = 0

We can solve this using the quadratic formula x = -b ± √(b² - 4ac){2a}, where a = 4, b = -3, and c = -232:

x = 3 ± √((-3)² - 4(4)(-232)){2(4)} x = 3 ± √(9 + 3712){8} x = 3 ± √(3721){8}

Since 61² = 3721, the equation becomes:

x = (3 ± 61)/(8)

The positive root is x = (3 + 61)/(8) = ⁶⁴⁄₈ = 8. Alternatively, testing the value 8 directly shows that 8 - ⁵⁸⁄₈ = 8 - 7.25 = 0.75, which equals ¾.

Explanation

The sum of the first n terms of an arithmetic progression can be calculated using the formula Sₙ = ⁿ⁄₂(a + l), where n is the number of terms, a is the first term, and l is the last term. This formula is derived from the fact that the sum of an arithmetic series is the product of the number of terms and the average of the first and last terms.

Given the parameters: Number of terms n = 13 First term a = -10 Last term l = 26

Substituting these values into the sum formula: S₁₃ = ¹³⁄₂ [ -10 + 26 ] S₁₃ = ¹³⁄₂ [ 16 ]

By simplifying the expression inside the brackets, we find the sum of the boundary terms is 16. Multiplying this by the ratio of the number of terms over two: S₁₃ = 13 × 8 S₁₃ = 104

This approach is the most direct way to solve the problem as it bypasses the need to find the common difference d of the progression, utilizing only the provided boundary values and the term count.

Explanation

When a point (x, y) is reflected in the x-axis, its horizontal position remains the same while its vertical position is mirrored across the axis. This means the x-coordinate remains unchanged and the y-coordinate changes its sign. The transformation rule is (x, y) → (x, -y). For the given point (-0.5, 6), the x-coordinate remains -0.5 and the y-coordinate 6 becomes -6, resulting in the reflected point (-0.5, -6).

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, the y-coordinate is given by y = (my₂ + ny₁)/(m+n). For the point T(x, 0) dividing the segment joining S(-4, -1) and U(1, 4), the y-coordinate is 0. Substituting the values y₁ = -1 and y₂ = 4 into the formula, we get 0 = (m(4) + n(-1))/(m+n). Multiplying both sides by m+n results in 4m - n = 0, which simplifies to 4m = n. Therefore, the ratio m/n is equal to 1/4, confirming the ratio is 1:4.