SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The area of a regular hexagon with side length a is given by the formula: Area = (3√3 / 2) × a²

Given that the area of the hexagon is 54√3 sq cm, we can set up the equation: (3√3 / 2) × a² = 54√3

Dividing both sides of the equation by √3 simplifies it to: (3/2) × a² = 54

Solving for a²: a² = 54 × (2/3) a² = 36

Taking the positive square root of both sides gives the length of the side: a = 6 cm

Explanation

The given equation is x = cot(π/3) - tan(π/4), where the angles are measured in radians.

Using the standard values of trigonometric functions: cot(π/3) = 1/√3 tan(π/4) = 1

Substituting these values into the equation gives: x = 1/v3 - 1

By taking the common denominator, we get: x = (1 - √3)/√3

Explanation

The effective annual rate of interest is calculated using the formula:

R = (1 + r/n)ⁿ - 1

where r is the nominal annual rate of interest and n is the number of compounding periods per year.

For a nominal interest rate of 19% per annum compounded half-yearly: r = 19% = 0.19 n = 2

Substituting these values into the formula:

R = (1 + 0.19/2)² - 1 R = (1 + 0.095)² - 1 R = (1.095)² - 1 R = 1.199025 - 1 R = 0.199025

Converting the decimal value to a percentage:

R = 0.199025 × 100% = 19.9025% ≈ 19.9%

Explanation

Runs scored in the first 38 overs = 38 × 4.5 = 171 runs

Remaining runs needed to reach the target of 288 runs = 288 - 171 = 117 runs

Required run rate in the remaining 12 overs = 117 / 12 = 9.75 runs/over

Explanation

Let the present ages of the husband and the wife be x and y.

We are given that the sum of their present ages is 56: x + y = 56

Ten years ago, their ages were x - 10 and y - 10. The product of their ages at that time was 320: (x - 10)(y - 10) = 320

Substituting y = 56 - x into the product equation gives: (x - 10)(56 - x - 10) = 320 (x - 10)(46 - x) = 320 46x - x² - 460 + 10x = 320 x² - 56x + 780 = 0

Factoring this quadratic equation: x² - 30x - 26x + 780 = 0 x(x - 30) - 26(x - 30) = 0 (x - 30)(x - 26) = 0

Solving for x yields x = 30 or x = 26. If one is 30, the other is 26. Checking these values, their present sum is 30 + 26 = 56, and ten years ago their ages were 20 and 16, which multiply to 20 × 16 = 320.

Explanation

Let the total electricity bill be B.

If 27% of the bill is deducted, the remaining portion of the bill to be paid is: Remaining percentage = 100% - 27% = 73%

The remaining amount to be paid is given as Rs 1825. Setting up the relation: 73% of B = 1825 0.73 × B = 1825 B = 1825 / 0.73 B = 2500

Explanation

The fundamental trigonometric identity relating secant and tangent is: sec²A - tan²A = 1

Rearranging this equation to express sec²A in terms of tan²A yields: sec²A = 1 + tan²A

Taking the square root on both sides gives: secA = √(1 + tan²A)

Given that secA = x, substituting x results in: x = √(1 + tan²A)

Explanation

The total cost of purchasing daily tickets for 29 days is calculated by multiplying the daily ticket price by the number of travel days: 29 × ₹130 = ₹3770

The cost of the Monthly Pass is ₹3091.40.

The amount saved by purchasing the Monthly Pass instead of individual daily tickets is: ₹3770 - ₹3091.40 = ₹678.60

The savings percentage is calculated by dividing the total savings by the total cost of the daily tickets, then multiplying by 100: (₹678.60 / ₹3770) × 100 = 18%

Explanation

To find the overall profit or loss percentage, we can equalize the quantity of lemons bought and sold.

The vendor buys lemons at the rate of 11 for Rs 10, and sells them at the rate of 4 for Rs 5. The least common multiple of the quantities 11 and 4 is 44.

Cost Price (CP) of 44 lemons: Since 11 lemons cost Rs 10, then 44 lemons cost: 10 × 4 = Rs 40

Selling Price (SP) of 44 lemons: Since 4 lemons are sold for Rs 5, then 44 lemons are sold for: 5 × 11 = Rs 55

Since the Selling Price is greater than the Cost Price, there is a gain. Gain = SP - CP = 55 - 40 = Rs 15

Gain percentage = (Gain / CP) × 100 Substituting the values: Gain percentage = (15 / 40) × 100 Gain percentage = (3 / 8) × 100 Gain percentage = 37.5%

Explanation

We are given the system of linear equations: 7x - 5y = 13 2x - y = 5

From the second equation, we can express y in terms of x: y = 2x - 5

Substituting this expression for y into the first equation: 7x - 5(2x - 5) = 13 7x - 10x + 25 = 13 -3x = -12 x = 4

Now, substituting x = 4 into the expression for y: y = 2(4) - 5 = 3

Thus, we find the value of x - y: x - y = 4 - 3 = 1

Explanation

Let the side of the square be a. The diagonal of the square is given by a√2. Since the diagonal is 9 cm, we have: a√2 = 9 a = 9/√2 cm

The perimeter of the square is 4a, which equals: 4 × (9/√2) = 18√2 cm

Since the perimeter of the equilateral triangle is equal to the perimeter of the square, the perimeter of the equilateral triangle is also 18√2 cm. Let the side of the equilateral triangle be s. The perimeter is 3s, so: 3s = 18√2 s = 6√2 cm

The area of an equilateral triangle with side s is given by the formula (√3/4)s². Substituting the value of s: Area = (√3/4) × (6√2)² Area = (√3/4) × 72 Area = 18√3 sq cm

Explanation

The ratio of the speed of the policeman to that of the thief is 12:7. Since the duration of travel is the same for both until the catch, the distance covered by each is directly proportional to their speeds. Thus, the ratio of the distance covered by the policeman to that covered by the thief is also 12:7.

Let the distance covered by the policeman be 12x and the distance covered by the thief be 7x.

The difference in their distances is equal to the initial gap between them: 12x - 7x = 350 5x = 350 x = 70

The distance run by the thief before being overtaken is: 7x = 7 × 70 = 490 metres.

Explanation

Let the rates of work of painters A, B, and C be represented by 1/A, 1/B, and 1/C respectively. The combined rate of all three painters working together is 1/T, where T is the total time taken.

The relation is given by the formula: 1/A + 1/B + 1/C = 1/T

Substitute the given values A = 10, B = 20, and T = 5: 1/10 + 1/20 + 1/C = 1/5

Find a common denominator of 20 for the fractions: 2/20 + 1/20 + 1/C = 4/20

3/20 + 1/C = 4/20

Subtract 3/20 from both sides: 1/C = 4/20 - 3/20 1/C = 1/20 C = 20

Explanation

The slope (m) 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 (-4, -6) and (2, -3) into the formula:

m = (-3 - (-6)) / (2 - (-4)) m = (-3 + 6) / (2 + 4) m = 3/6 = ½

Since parallel lines have equal slopes, the slope of the line parallel to the line passing through these points is also ½.

Explanation

The total surface area of a cuboid is given by the formula: Total Surface Area = 2(lb + bh + hl)

Here, length l = 10 cm and breadth b = 8 cm. Substituting these values along with the given total surface area of 376 sq cm: 2(10 × 8 + 8 × h + h × 10) = 376 2(80 + 18h) = 376 80 + 18h = 188 18h = 108 h = 6 cm

The volume of a cuboid is given by the formula: Volume = l × b × h

Substituting the dimensions: Volume = 10 × 8 × 6 = 480 cubic cm

Explanation

The value of the total stock is calculated by summing up the product of the quantity and the average cost for each item.

Value of Mobile Phones = 47 × 11,000 = 517,000 Rs Value of Cameras = 60 × 9,000 = 540,000 Rs Value of TVs = 21 × 43,000 = 903,000 Rs Value of Refrigerators = 31 × 60,000 = 1,860,000 Rs Value of ACs = 82 × 22,000 = 1,804,000 Rs

Total Value of Stock = 517,000 + 540,000 + 903,000 + 1,860,000 + 1,804,000 = 5,624,000 Rs

To find the value in lakh rupees, divide the total value by 100,000: Value in lakh rupees = 5,624,000 / 100,000 = 56.24

Explanation

Imports in 2012 (I₁₂) = 1500 crores Ratio of Import to Export in 2012 = 1.1

Therefore, exports in 2012 (E₁₂) = I₁₂ / 1.1 = 1500 / 1.1 ≈ 1363.64 crores

Total exports in 2012 and 2013 (E₁₂ + E₁₃) = 2000 crores

Exports in 2013 (E₁₃) = 2000 - 1363.64 = 636.36 crores

Ratio of Import to Export in 2013 = 0.9

Imports in 2013 (I₁₃) = E₁₃ × 0.9 = 636.36 × 0.9 = 572.72 crores, which is approximately 573 crores.

Explanation

To find the increase in the height of the child from the 7th Birthday to the 13th Birthday, we retrieve the height values for these specific ages from the given data table.

At the 7th Birthday, the child's height is 125 cms. At the 13th Birthday, the child's height is 170 cms.

The increase in height is calculated by subtracting the height at the 7th Birthday from the height at the 13th Birthday: 170 cms - 125 cms = 45 cms

Explanation

The total duration of time between 10 PM and 6 AM is 8 hours.

The values given in the table represent the percentages of this total duration, which sum to 100% (10 + 25 + 10 + 10 + 45 = 100).

The percentage of time spent in Deep Sleep is 10%, and the percentage of time spent in Extremely light sleep is 10%.

To find the combined duration of these two states, we add their percentages: 10% + 10% = 20%

Now, we calculate 20% of the total 8 hours: 8 × 0.20 = 1.6 hours

Explanation

To find the value of x, we solve the system of inequalities by splitting it into two parts.

First inequality: x + 5(x - 2) < 4 - x x + 5x - 10 < 4 - x 6x - 10 < 4 - x 7x < 14 x < 2

Second inequality: 4 - x < 4x - 1 5 < 5x x > 1

Combining these two inequalities gives the range: 1 < x < 2

The value 3/2 (which is 1.5) lies within this interval.

Explanation

When a point (x, y) is reflected in the Y-axis, its x-coordinate changes sign while the y-coordinate remains unchanged. Therefore, reflecting the point Q(a, b) in the Y-axis gives the coordinates of Q₁ as (-a, b).

When a point (x, y) is subsequently reflected in the X-axis, its y-coordinate changes sign while the x-coordinate remains unchanged. Therefore, reflecting Q₁(-a, b) in the X-axis results in the coordinates (-a, -b).

We are given that this final reflected point is (-4, 5). Equating the coordinates: -a = -4, which gives a = 4 -b = 5, which gives b = -5

Thus, the coordinates of the original point Q(a, b) are (4, -5).

Explanation

Using the algebraic identity (a - b)(a + b) = a² - b²:

Let a = 100000 and b = 7.

The given expression can be written as: 99993 × 100007 = (100000 - 7)(100000 + 7)

Applying the formula: = 100000² - 7² = 10000000000 - 49 = 9999999951

Explanation

Using the standard algebraic identity for the square of a binomial difference: (a - b)² = a² - 2ab + b²

By substituting a = 5x and b = y into this formula: (5x - y)² = (5x)² - 2(5x)(y) + y² (5x - y)² = 25x² - 10xy + y²

This algebraic simplification directly yields the correct relation.

Explanation

The trigonometric function sec θ is defined as the reciprocal of cos θ.

The angle 225° lies in the third quadrant, where the cosine function is negative. Using the trigonometric reduction formula cos(180° + θ) = -cos θ, we can write:

cos 225° = cos(180° + 45°) = -cos 45°

Since the standard value of cos 45° is 1/√2:

cos 225° = -1/√2

Taking the reciprocal to find the secant value:

sec 225° = 1/cos 225° = 1/(-1/√2) = -√2

Thus, the value of x is -√2.

Explanation

For the quadratic equation 15x² - 2x - 8 = 0, we can find the roots by splitting the middle term. We look for two numbers whose sum is -2 and whose product is 15 × (-8) = -120. These numbers are -12 and 10.

Rewriting the middle term: 15x² - 12x + 10x - 8 = 0

Factoring by grouping: 3x(5x - 4) + 2(5x - 4) = 0 (3x + 2)(5x - 4) = 0

Setting each factor to zero: 3x + 2 = 0 which gives x = -2/3 5x - 4 = 0 which gives x = 4/5

Explanation

Total Cost Price of 26 quintals of rice is Rs 1872. To earn a 10% profit, the desired total Selling Price is: Total Selling Price = 1.10 × 1872 = Rs 2059.2

Since 28% of the rice is lost in transportation, the remaining quantity of rice is: Remaining quantity = 26 × (1 - 0.28) = 26 × 0.72 = 18.72 quintals

The rate per quintal at which the remaining rice should be sold is: Selling rate = Total Selling Price / Remaining quantity = 2059.2 / 18.72 = Rs 110 per quintal

Explanation

Let the required number be x. According to the given condition, the difference between the number and twice its reciprocal is equal to 49/5. This relation can be written in the form of an equation:

x - 2/x = 49/5

To clear the fractions, multiply the entire equation by 5x:

5x² - 10 = 49x

Rearranging this into standard quadratic form:

5x² - 49x - 10 = 0

Factorizing the quadratic expression by splitting the middle term:

5x² - 50x + x - 10 = 0

5x(x - 10) + 1(x - 10) = 0

(5x + 1)(x - 10) = 0

Solving for x gives:

x = 10 or x = -1/5

Evaluating x = 10 in the original equation:

10 - 2/10 = 10 - 1/5 = 49/5

This confirms that the number is 10.

Explanation

Let CP be the Cost Price and MP be the Marked (List) Price of the item.

When the merchant offers a 10% discount on the marked price, the selling price SP₁ is: SP₁ = 0.9 × MP

Since this results in a 16% loss, the selling price is also: SP₁ = 0.84 × CP

Equating the two expressions for SP₁: 0.9 × MP = 0.84 × CP MP / CP = 0.84 / 0.9 = 14 / 15

When a discount of 25% is offered on the marked price, the new selling price SP₂ is: SP₂ = 0.75 × MP

To find the new profit or loss percentage, we calculate the ratio of SP₂ to CP: SP₂ / CP = (0.75 × MP) / CP = 0.75 × (14 / 15) = (3/4) × (14 / 15) = 42 / 60 = 7 / 10 = 0.7

This means the new selling price is 70% of the cost price, representing a loss of 100% - 70% = 30%.

Explanation

To find the required value, we use the percentage formula:

Part = (Percentage / 100) × Whole

Substituting 31 for the part and 0.8 for the percentage:

31 = (0.8 / 100) × x

Rearranging the equation to solve for x:

x = (31 × 100) / 0.8

x = 3100 / 0.8

Multiplying both the numerator and the denominator by 10 to simplify the calculation:

x = 31000 / 8

x = 3875

Explanation

The cross-section of the canal is an isosceles trapezium. The area of a trapezium is given by the formula:

Area = ½ × (sum of parallel sides) × height

Here, the parallel sides are the bottom width of 2 m and the top width of 3 m, and the height is the depth of 1 m. Substituting these values into the formula:

Area = ½ × (2 + 3) × 1 = 2.5 square meters

The maximum capacity of the canal is its total volume, which is the cross-sectional area multiplied by the length of the canal:

Capacity = Area × Length

Capacity = 2.5 × 100 = 250 cubic meters

Explanation

For three points A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) to be collinear, the slope of the line segment AB must equal the slope of the line segment BC.

The formula for the slope of a line passing through two points is: m = (y₂ - y₁) / (x₂ - x₁)

Using the coordinates of A(2, -3) and B(-2, 0): Slope of AB = (0 - (-3)) / (-2 - 2) = 3 / -4 = -3/4

Using the coordinates of B(-2, 0) and C(6, x): Slope of BC = (x - 0) / (6 - (-2)) = x / 8

Equating the two slopes for collinearity: -3/4 = x/8

Solving for x: x = 8 × (-3/4) x = -6

Explanation

Let the original fare and the new fare be represented by the ratio 9:11.

Given that the original fare is Rs. 675, we can set up the proportion: 9 units = Rs. 675

To find the value of 1 unit, we divide: 1 unit = 675 / 9 = Rs. 75

The increase in the fare is the difference between the new fare and the original fare: Increase in units = 11 units - 9 units = 2 units

Substituting the value of 1 unit, we find the increase in the fare: Increase in fare = 2 × Rs. 75 = Rs. 150

Explanation

Let the first, second, and third numbers be represented as N₁, N₂, and N₃ respectively.

According to the given conditions: N₁ = 2 × N₂ → N₂ = N₁ / 2 N₁ = 3 × N₃ → N₃ = N₁ / 3

The average of the three numbers is given as 638: (N₁ + N₂ + N₃) / 3 = 638 N₁ + N₂ + N₃ = 638 × 3 = 1914

Substituting the values of N₂ and N₃ in terms of N₁: N₁ + N₁ / 2 + N₁ / 3 = 1914 N₁ × (1 + 1/2 + 1/3) = 1914 N₁ × (11/6) = 1914 N₁ = (1914 × 6) / 11 N₁ = 174 × 6 = 1044

The third number N₃ is: N₃ = 1044 / 3 = 348

The difference between the first and the third number is: N₁ - N₃ = 1044 - 348 = 696

Explanation

Using the fundamental trigonometric identity: sin²A + cos²A = 1

This can be rearranged as: 1 - cos²A = sin²A

Taking the square root on both sides: √(1 - cos²A) = sinA

Substituting this value into the given equation: x = 1/√(1 - cos²A) x = 1/sinA

Since the reciprocal of sinA is cosecA: x = cosecA

Explanation

When an object travels equal distances at two different speeds, v₁ and v₂, the average speed for the entire journey is given by the harmonic mean formula:

Average speed = 2v₁v₂ / (v₁ + v₂)

Substituting the given speeds v₁ = 30 km/h and v₂ = 45 km/h into the formula:

Average speed = 2 × 30 × 45 / (30 + 45)

Average speed = 2700 / 75

Average speed = 36 km/h

This calculation yields the average speed of the car for the entire round trip.

Explanation

We are given the expression: 1 / (sinA + cosA) + 1 / (sinA - cosA)

To simplify, we find a common denominator: = [(sinA - cosA) + (sinA + cosA)] / [(sinA + cosA)(sinA - cosA)]

Simplifying the numerator by combining like terms: (sinA - cosA) + (sinA + cosA) = 2sinA

Simplifying the denominator using the identity (x + y)(x - y) = x² - y²: (sinA + cosA)(sinA - cosA) = sin²A - cos²A

Using the fundamental trigonometric identity sin²A = 1 - cos²A, we can express the denominator purely in terms of cosA: sin²A - cos²A = (1 - cos²A) - cos²A = 1 - 2cos²A

Combining the simplified numerator and denominator yields the final expression: 2sinA / (1 - 2cos²A)

Explanation

First, determine the simple interest rate. An increase of 40% in the principal amount P over 8 years means the simple interest earned is 0.4P.

Simple Interest = (P × R × T) / 100 0.4P = (P × R × 8) / 100 R = (0.4 × 100) / 8 R = 5%

Next, calculate the compound interest on Rs. 20000 for 3 years at a 5% interest rate.

Amount = P × (1 + R / 100)ⁿ Amount = 20000 × (1 + 5 / 100)³ Amount = 20000 × (21 / 20)³ Amount = 20000 × (9261 / 8000) Amount = 2.5 × 9261 Amount = 23152.5

Compound Interest = Amount - Principal Compound Interest = 23152.5 - 20000 Compound Interest = 3152.5

Explanation

Two angles are supplementary when their sum is 180°. Let the supplementary angle be y. The angle of interest, x, is given as 4/5 of its supplementary angle:

x = (4/5)y

Since the sum of supplementary angles is 180°: x + y = 180°

Substituting the expression for x into the sum equation: (4/5)y + y = 180° (9/5)y = 180° y = 180° × 5 / 9 y = 100°

Solving for the angle of interest, x: x = (4/5) × 100° x = 80°

Explanation

An isosceles trapezium is a quadrilateral in which one pair of opposite sides (known as the bases) are parallel, and the other pair of opposite sides (known as the legs) are congruent, meaning they have the same length. This defining characteristic ensures that one pair of opposite sides is congruent.

Explanation

Let Mihir's work efficiency be 1 unit per hour. Since Mihir takes 30 hours to complete the work, the total work is 1 unit/hour × 30 hours = 30 units.

Jigya is 50% more efficient than Mihir, so Jigya's efficiency is 1.5 units per hour.

When working together, their combined efficiency is 1 + 1.5 = 2.5 units per hour.

The time taken by both of them to finish the work together is 30 units ÷ 2.5 units/hour = 12 hours.

Explanation

The columns of the data table represent the category of employee, the number of employees, the annual salary in lakhs, and the bonus as a percentage of the annual salary.

For Manager, the number of employees is 1, and the annual salary is 36 lakhs (3,600,000 rupees). The total bonus is 50% of 3,600,000, which equals 1,800,000 rupees.

For Executive, there are 5 employees, with an annual salary of 12 lakhs (1,200,000 rupees) each. The total bonus is 5 × (20% of 1,200,000) = 1,200,000 rupees.

For Trainee, there is 1 employee, with an annual salary of 2 lakhs (200,000 rupees). The total bonus is 20% of 200,000, which equals 40,000 rupees.

The total bonus distributed to all employees is 1,800,000 + 1,200,000 + 40,000 = 3,040,000 rupees.

The total number of employees is 1 + 5 + 1 = 7.

The average bonus is computed by dividing the total bonus by the total number of employees: Average bonus = 3,040,000 ÷ 7 ≈ 434285.71 rupees.

This rounds to 434286.

Explanation

The columns from left to right represent the years 2011, 2012, 2013, 2014, and 2015 respectively.

The total exports for the year 2012 are calculated as the sum of exports from companies A, B, and C in the second column: 5000 + 4000 + 5000 = 14000 lakh rupees

The total exports for the year 2014 are calculated as the sum of exports from companies A, B, and C in the fourth column: 4000 + 5000 + 5000 = 14000 lakh rupees

Both the years 2012 and 2014 have equal total exports of 14000 lakh rupees.

Explanation

The total profit or loss of the company over the last 5 years is calculated by finding the algebraic sum of the profit and loss values for each year from 2011 to 2015.

Using the given data: Total Profit/Loss = (-25) + 5 + (-10) + 10 + (-10)

Combining the positive and negative values: Total Profit/Loss = (5 + 10) - (25 + 10 + 10) Total Profit/Loss = 15 - 45 Total Profit/Loss = -30

A negative result indicates a net financial loss. Therefore, the company incurred a total loss of ₹30 crores over the 5-year period.

Explanation

To find the percentage of software exports relative to the total exports, first calculate the sum of all exports from the given table.

Total exports = 925 + 600 + 825 + 850 + 550 = 3750 million USThe export value of software is 600 million US.

Now, apply the percentage formula:

Percentage = (Software exports / Total exports) × 100 Percentage = (600 / 3750) × 100 Percentage = (4 / 25) × 100 Percentage = 4 × 4 = 16 percent

Explanation

The reflection of any point P(x, y) in the origin is given by the formula P'(-x, -y).

Applying this formula to the given point (4, -6): x-coordinate becomes -4 y-coordinate becomes -(-6) = 6

Therefore, the coordinates of the reflected point are (-4, 6).

Explanation

The reciprocal of a fraction x/y is y/x. For the given values 2/9 and 7/10, their respective reciprocals are 9/2 and 10/7.

The sum of these reciprocals is: 9/2 + 10/7 = (9 × 7 + 10 × 2) / (2 × 7) = (63 + 20) / 14 = 83/14

The reciprocal of this sum is: 1 / (83/14) = 14/83

Explanation

The equation of a straight line in slope-intercept form is y = mx + c, where m represents the slope and c represents the y-intercept of the line.

The slope m of a line making an angle θ with the positive x-axis is given by m = tan(θ). Since the line makes an angle of 45° with the positive x-axis, the slope is: m = tan(45°) = 1

The given y-intercept c is 4. Substituting m = 1 and c = 4 into the slope-intercept form equation gives: y = 1 · x + 4 y = x + 4

Rearranging the equation to standard form: x - y = -4

Explanation

The given expression can be simplified by factoring out the common trigonometric terms in both the numerator and the denominator:

Numerator: sin A - 2sin³ A = sin A (1 - 2sin² A) Denominator: 2cos³ A - cos A = cos A (2cos² A - 1)

Now, substituting these factored forms into the expression:

[sin A (1 - 2sin² A)] / [cos A (2cos² A - 1)]

By the double-angle identities for cosine: cos 2A = 1 - 2sin² A cos 2A = 2cos² A - 1

Replacing both terms in the expression with cos 2A:

[sin A × cos 2A] / [cos A × cos 2A]

Canceling out the common factor of cos 2A:

sin A / cos A = tan A

Explanation

The given compound inequality is: 5x - 5(3x - 2) < 4x - 1 < 2 + 2x

This can be split into two separate inequalities to find the range of x.

First inequality: 5x - 15x + 10 < 4x - 1 -10x + 10 < 4x - 1 11 < 14x x > 11/14

Second inequality: 4x - 1 < 2 + 2x 2x < 3 x < 3/2

Combining both inequalities gives: 11/14 < x < 3/2

Converting these fractions to decimals gives approximately 0.786 < x < 1.5. The value 1 falls within this interval.

Explanation

To simplify the algebraic fraction, we divide the numerical coefficients and cancel the common variable terms.

First, divide the coefficient of the numerator by the coefficient of the denominator: 533 ÷ 41 = 13

Next, simplify the algebraic variables by cancelling the common factors a and b present in both the numerator and the denominator: (b × a × x) / (a × b) = x

This leaves 13 as the simplified coefficient.