SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

The compound interest formula for compounding half-yearly is A = P(1 + R/200)²ᵀ, where A is the amount, P is the principal sum, R is the annual rate of interest, and T is the time in years.

Here, the given values are: Amount A = 9,826 Annual Rate R = 2½% = 2.5% Time T = 18 months = 1.5 years Number of half-yearly periods n = 2T = 3

The half-yearly interest rate is R/2 = 1.25% per half-year, which is equivalent to 1.25/100 = 1/80 in fraction form.

Substituting these values into the formula: 9,826 = P(1 + 1/80)³ 9,826 = P(81/80)³ P = 9,826 × (80/81)³ P = 9,826 × 512,000 / 531,441 P = 5,030,912,000 / 531,441 P ≈ 9,466.55

Explanation

The total surface area of a right circular cone is given by the formula TSA = πr(l + r), where r is the radius and l is the slant height of the cone.

Substituting the given values: Radius (r) = 21 cm Slant height (l) = 28 cm π = 22/7

TSA = (22/7) × 21 × (28 + 21) TSA = 22 × 3 × 49 TSA = 66 × 49 TSA = 3234 cm²

Explanation

The side lengths of the triangle are 8 cm, 15 cm, and 17 cm. Since 8² + 15² = 64 + 225 = 289 = 17², these lengths form a Pythagorean triple. This confirms that the triangle is a right-angled triangle, where the base and height are the two shorter sides of length 8 cm and 15 cm, and the hypotenuse is 17 cm.

The area of a right-angled triangle is calculated using the formula: Area = ½ × base × height

Substituting the values into the formula: Area = ½ × 8 × 15 = 60 cm²

Explanation

The weight of the teacher can be calculated directly using the average weight formula:

Teacher's weight = Old Average + (New Total Number of People × Increase in Average)

Here, the old average is 55 kg, the number of students is 45, and the increase in average weight is 500 grams, which is equal to 0.5 kg.

When the teacher is included, the total number of people becomes 45 + 1 = 46.

Substituting these values into the formula:

Teacher's weight = 55 + 46 × 0.5 Teacher's weight = 55 + 23 Teacher's weight = 78 kg

Explanation

To find the total number of students in all the subjects, we sum the number of students listed for each individual subject:

Total number of students = 110 + 70 + 50 + 40 + 80 + 90 = 440

Explanation

To solve the expression (0.125 × 0.01) ÷ (0.5 × 0.005), convert each decimal into fractions to simplify the calculation.

First, calculate the numerator: 0.125 × 0.01 = (1/8) × (1/100) = 1/800

Next, calculate the denominator: 0.5 × 0.005 = (1/2) × (5/1000) = 5/2000 = 1/400

Now, divide the numerator by the denominator: (1/800) ÷ (1/400) = (1/800) × 400 = 400/800 = 1/2 = 0.5

Explanation

The length of the common chord of two equal circles of radius r, where each circle passes through the center of the other, is given by the standard relation:

L = r√3

Substituting the given radius r = 6 cm into the formula:

L = 6√3 cm

This relation is derived by considering the two centers O₁ and O₂. Since each circle passes through the center of the other, the distance between the centers O₁O₂ is equal to the radius r = 6 cm. The common chord AB intersects the line of centers O₁O₂ perpendicularly at its midpoint M. Therefore, O₁M = r/2 = 3 cm.

In the right-angled triangle O₁MA, the hypotenuse O₁A is the radius of the circle, which is 6 cm. Using the Pythagorean theorem:

AM = √(O₁A² - O₁M²) AM = √(6² - 3²) AM = √(36 - 9) AM = √27 = 3√3 cm

The total length of the common chord AB is twice the length of AM:

AB = 2 × AM = 2 × 3√3 = 6√3 cm

Explanation

Correct Answer: B

For any two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides.

Perimeter of triangle GST / Perimeter of triangle FAX = GS / FA

Substituting the given values into the formula:

48 / 36 = GS / 12

Simplifying the fraction on the left side:

4/3 = GS / 12

Solving for GS:

GS = (4 × 12) / 3 = 16 cm

Explanation

Let the cost of one gold ring be r and the cost of one gold bangle be b.

According to the given information, the total cost of 4 rings and 2 bangles is: 4r + 2b = 40000

Dividing this equation by 2 gives the cost relation for 2 rings and 1 bangle: 2r + b = 20000

We need to find the cost of 10 rings and 5 bangles, which can be factored as: 10r + 5b = 5 × (2r + b)

Substituting the value of (2r + b) into this equation: 5 × 20000 = 100000

Explanation

Rebate per item = 25% of ₹32 = (25/100) × 32 = ¼ × 32 = ₹8

To get a total rebate of ₹40: Number of items = Total rebate required ÷ Rebate per item Number of items = 40 ÷ 8 = 5

Explanation

By drawing a horizontal reference line at the level of Suhas's eyes (1.8 m above the ground), we can analyze the problem using similar triangles formed by the line of sight.

The height of the top of the tree above this reference line is: 3.15 - 1.8 = 1.35 m

The horizontal distance from Suhas to the tree is 7.5 m, and the total horizontal distance from Suhas to the building is 7.5 + 45 = 52.5 m.

Let H be the height of the building. The height of the top of the building above the reference line is H - 1.8.

Using the property of similar triangles: Relative height of tree / Distance to tree = Relative height of building / Total distance to building

1.35 / 7.5 = (H - 1.8) / 52.5

0.18 = (H - 1.8) / 52.5

H - 1.8 = 0.18 × 52.5

H - 1.8 = 9.45

H = 11.25 m

Explanation

From the given equation, 3cot A = 4, we get cot A = 4/3. In a right-angled triangle, cot A is the ratio of the adjacent side to the opposite side. Thus, we can associate:

Adjacent side = 4 Opposite side = 3

Using the Pythagorean theorem, the hypotenuse of the triangle is:

Hypotenuse = √(Adjacent² + Opposite²) = √(4² + 3²) = √25 = 5

The sec A function is defined as the ratio of the hypotenuse to the adjacent side:

sec A = Hypotenuse / Adjacent side = 5/4

Explanation

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For the number 76050, the last two digits form the number 50. Dividing 50 by 4 yields 12 with a remainder of 2. Since 50 is not divisible by 4, the entire number 76050 is not divisible by 4.

Explanation

The relationship between Cost Price (CP), Marked Price (MP), profit percentage (p), and discount percentage (d) is given by the formula:

MP / CP = (100 + p) / (100 - d)

Given that the merchant marks his goods 20% above the cost price, the ratio of Marked Price to Cost Price is:

MP / CP = 120 / 100 = 1.2

The profit percentage is p = 8%. Substituting these values into the formula:

1.2 = (100 + 8) / (100 - d)

1.2 = 108 / (100 - d)

100 - d = 108 / 1.2

100 - d = 90

d = 10%

Explanation

Correct Answer: B

The formula for the circumference of a circle is C = 2πr, where r is the radius. This relation shows that the circumference is directly proportional to the radius (C ∝ r).

If the radius increases by 20%, the new radius becomes 1.2r. The new circumference C' is: C' = 2π(1.2r) = 1.2(2πr) = 1.2C

The percentage increase in the circumference is: ((1.2C - C) / C) × 100% = 0.2 × 100% = 20%

Explanation

Using the algebraic identity: (a³ - b³) / (a² + ab + b²) = a - b

Let a = 0.83 and b = 0.1.

The terms in the expression are: a² = (0.83)² ab = 0.83 × 0.1 = 0.083 b² = (0.1)² = 0.01

Substituting these values: a - b = 0.83 - 0.1 = 0.73

Explanation

From the given equation x + 1/x = 2, we can deduce that x = 1. Substituting x = 1 into the expression x⁵ + 1/x⁵ gives 1⁵ + 1/1⁵ = 1 + 1 = 2, which matches Option D.

Explanation

Let the common tangent to the two circles at P intersect the direct common tangent AB at T.

Since the lengths of tangents drawn from an external point to a circle are equal, for the first circle, TA = TP. Similarly, for the second circle, TB = TP.

Combining these gives TA = TB = TP. Since T is the midpoint of AB and is equidistant from A, B, and P, the point P lies on a circle with AB as the diameter. The angle subtended by a diameter at any point on the circumference is a right angle, which means ∠APB = 90°.

In triangle APB, the sum of all interior angles is 180°: ∠PAB + ∠ABP + ∠APB = 180°

Substituting the given values: 22° + ∠ABP + 90° = 180° ∠ABP = 180° - 112° = 68°

Explanation

A perpendicular drawn from the center of a circle to a chord bisects the chord. Therefore, the length of the half-chord is 24 / 2 = 12 cm.

The perpendicular distance from the center, the half-chord, and the radius of the circle form a right-angled triangle where the radius is the hypotenuse. Applying Pythagoras' theorem:

Radius² = Perpendicular² + Half-chord² Radius² = 9² + 12² Radius² = 81 + 144 Radius² = 225 Radius = √225 = 15 cm.

Explanation

Let P be the principal sum borrowed.

The total duration is 7 years. For the first 3 years, the simple interest rate is 10% per annum: I₁ = P × 10% × 3 = 0.30P

For the remaining 4 years (7 - 3 = 4), the simple interest rate is 15% per annum: I₂ = P × 15% × 4 = 0.60P

The total simple interest paid is the sum of interest from both periods: Total Simple Interest = 0.30P + 0.60P = 0.90P

Using the total interest value of ₹2,884.50: 0.90P = 2884.50 P = 2884.50 / 0.90 = 3205

Explanation

Using the trigonometric co-function identities where cos θ = sin(90° - θ) and sin θ = cos(90° - θ), the terms in the expression can be rewritten.

For the second term: cos 43° = sin(90° - 43°) = sin 47° sin 57° = cos(90° - 57°) = cos 33°

Substituting these conversions back into the second term of the expression: cos 43° sin 57° = sin 47° cos 33°

Now, substituting this into the original expression: sin 47° cos 33° - cos 43° sin 57° = sin 47° cos 33° - sin 47° cos 33° = 0

Explanation

The given expression is of the algebraic form (a³ - b³) / (a² + ab + b²), where a = 4.2 and b = 0.2.

This is because: b³ = (0.2)³ = 0.008 ab = 4.2 × 0.2 = 0.84 b² = (0.2)² = 0.04

Using the algebraic identity for the difference of cubes: a³ - b³ = (a - b)(a² + ab + b²)

Dividing both sides by (a² + ab + b²) gives: (a³ - b³) / (a² + ab + b²) = a - b

Substituting the values of a and b: 4.2 - 0.2 = 4

Explanation

Correct Answer: D

Using the weighted average formula for mixtures:

Cost of mixture per kg = (w₁ × c₁ + w₂ × c₂) / (w₁ + w₂)

Here, the ratios of the two qualities of fruit are w₁ = 5 and w₂ = 4, with their respective costs being c₁ = ₹20 per kg and c₂ = ₹38 per kg.

Substituting these values into the formula:

Cost of mixture per kg = (5 × 20 + 4 × 38) / (5 + 4) Cost of mixture per kg = (100 + 152) / 9 Cost of mixture per kg = 252 / 9 = 28

Explanation

The total surface area of a cylinder is given by the formula:

Total Surface Area = 2πr(r + h)

Given: Radius r = 6 cm Height h = 8 cm π = 22/7

Substitute these values into the formula:

Total Surface Area = 2 × (22/7) × 6 × (6 + 8) Total Surface Area = 2 × (22/7) × 6 × 14 Total Surface Area = 2 × 22 × 6 × 2 Total Surface Area = 44 × 12 Total Surface Area = 528 cm²

Explanation

Let the total number of votes polled be T. Since 20% of the votes are invalid, the valid votes constitute 80% of the total votes, which is 0.8T.

The winning candidate obtained 73% of the valid votes. This means the losing candidate received 27% of the valid votes (100% - 73% = 27%).

The winner's majority is the difference between the winner's and the loser's shares of the valid votes: 73% - 27% = 46% of the valid votes

We are given that this majority is 1426 votes: 46% of valid votes = 1426 Valid votes = 1426 ÷ 0.46 = 3100

Since valid votes represent 80% of the total votes polled: 0.8 × T = 3100 T = 3100 ÷ 0.8 = 3875

Explanation

Let the total quantity of goods be 100 units and the cost price of each unit be ₹1. Total Cost Price = 100 × ₹1 = ₹100.

Due to a 10% loss during transportation, the quantity of goods remaining for sale is 90 units.

The shopkeeper sells his goods at a 15% profit, making the selling price of each unit ₹1.15. Total Selling Price = 90 × ₹1.15 = ₹103.5.

Overall Profit = Total Selling Price - Total Cost Price = ₹103.5 - ₹100 = ₹3.5. Overall Profit Percentage = (3.5 / 100) × 100 = 3.5%.

Explanation

Total work can be assumed as the Least Common Multiple (LCM) of 25 and 15, which is 75 units.

Efficiency of A = 75 ÷ 25 = 3 units per day. Efficiency of B = 75 ÷ 15 = 5 units per day.

Work completed by B in 10 days = 10 × 5 = 50 units. Remaining work = 75 - 50 = 25 units.

Time taken by A alone to finish the remaining work = Remaining work ÷ Efficiency of A = 25 ÷ 3 = 8⅓ days.

Explanation

The fundamental trigonometric identity states that sec²θ - tan²θ = 1.

By applying the difference of squares formula, a² - b² = (a - b)(a + b), this identity can be written as: (secθ - tanθ)(secθ + tanθ) = 1

Given that secθ + tanθ = 4, substituting this value into the equation yields: (secθ - tanθ) × 4 = 1

Solving for secθ - tanθ gives: secθ - tanθ = ¼

Explanation

The manufacturing numbers (in thousands) for aerated drink B from 2020 to 2022 are: In 2020: 230 In 2021: 210 In 2022: 175

Comparing these values year-on-year shows that 230 > 210 > 175, indicating a continuous decrease in the manufacturing of aerated drink B over this period.

Explanation

When a journey covers equal distances at two different speeds, say u and v, the average speed for the entire journey is given by the harmonic mean formula:

Average Speed = 2uv / (u + v)

Substituting the given values, where u = 110 km/h and v = 76 km/h:

Average Speed = 2 × 110 × 76 / (110 + 76) Average Speed = 16720 / 186 Average Speed = 8360 / 93

Converting this improper fraction into a mixed fraction:

8360 ÷ 93 = 89 with a remainder of 83

Average Speed = 89 83/93 km/h

Explanation

The third proportional c of two numbers a and b is defined by the relation a/b = b/c, which gives the formula:

c = b² / a

Given the two numbers: a = 2√3 b = 6√5

Substituting these values into the formula:

c = (6√5)² / (2√3)

c = (36 × 5) / (2√3)

c = 180 / (2√3)

c = 90 / √3

To rationalize the denominator, multiply the numerator and the denominator by √3:

c = (90 × √3) / (√3 × √3)

c = 90√3 / 3

c = 30√3

Explanation

The selling price after a discount is calculated using the formula:

Price after discount = Marked Price × (100 - Discount Percentage) / 100

Substituting the given values into the formula:

Marked Price = £250 Discount Percentage = 18%

Price after discount = 250 × (100 - 18) / 100 Price after discount = 250 × 82 / 100 Price after discount = 2.5 × 82 Price after discount = 205

Explanation

To find the time when they meet for the first time on the circular track, we use the relative speed formula.

First, convert the speeds from kmph to m/s by multiplying by 5/18: Speed of A = 27 × 5/18 = 7.5 m/s Speed of B = 36 × 5/18 = 10 m/s

Since both persons are running in the same direction, their relative speed is the difference of their individual speeds: Relative Speed = Speed of B - Speed of A = 10 - 7.5 = 2.5 m/s

The time taken for them to meet for the first time on the track is the time it takes for the faster runner to cover a relative distance of one full lap (1200 m) with respect to the slower runner: Time to meet = Track Length / Relative Speed = 1200 / 2.5 = 480 seconds

Explanation

Using the algebraic identity for the square of a sum: (p + q)² = p² + 2pq + q²

Rearranging this gives: p² + q² = (p + q)² - 2pq

We need to find the value of p² - pq + q². Substituting the above expression: p² - pq + q² = (p + q)² - 3pq

Substitute the given values p + q = 12 and pq = 14 into the equation: 12² - 3 × 14 = 144 - 42 = 102

Explanation

To find the number of votes Abha got, we first calculate the total number of valid votes and then determine Abha's percentage share of those valid votes.

First, calculate the total number of valid votes: Total votes = 40,000 Invalid votes = 10% of 40,000 = 4,000 Valid votes = 40,000 - 4,000 = 36,000

Next, find the percentage of valid votes received by Abha: Total percentage of valid votes = 100% Rajat's share = 36% Pooja's share = 40% Abha's share = 100% - (36% + 40%) = 24%

Finally, calculate the number of votes Abha received: Abha's votes = 24% of 36,000 = 0.24 × 36,000 = 8,640

Explanation

The formula for the length of a transverse common tangent (T) to two circles is:

T = √(d² - (r₁ + r₂)²)

where d is the distance between the centres of the two circles, and r₁ and r₂ are the radii of the two circles.

Substituting the given values, d = 18 cm, r₁ = 8.5 cm, and r₂ = 5.5 cm:

T = √(18² - (8.5 + 5.5)²) T = √(18² - 14²)

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

T = √((18 - 14) × (18 + 14)) T = √(4 × 32) T = √128 T = 8√2 cm

Explanation

Using the direct relation for the change in selling price: New Selling Price = Initial Selling Price × (100 + Profit%) / (100 - Loss%)

Given: Initial Selling Price = ₹402 Loss% = 33% Profit% = 37%

Substitute the values into the formula: New Selling Price = 402 × (100 + 37) / (100 - 33) New Selling Price = 402 × 137 / 67

Since 402 ÷ 67 = 6: New Selling Price = 6 × 137 = 822

The required selling price is ₹822.

Explanation

The mean proportional of two numbers a and b is given by the formula √(a × b).

Given the numbers a = 8.41 and b = 16.81, the mean proportional is: Mean proportional = √(8.41 × 16.81)

Using the algebraic property √(x × y) = √x × √y: Mean proportional = √8.41 × √16.81

We know that 2.9² = 8.41 and 4.1² = 16.81, which gives: √8.41 = 2.9 √16.81 = 4.1

Multiplying these two values: 2.9 × 4.1 = 11.89

Explanation

For a number to be divisible by 88, it must be divisible by both 8 and 11.

For divisibility by 8, the last three digits of the 10-digit number 620x976y52, which are y52, must be divisible by 8. Testing the possible single-digit values of y from 0 to 9, y52 is divisible by 8 when y is 1, 3, 5, 7, or 9.

For divisibility by 11, the difference between the sum of the digits at the odd positions and the sum of the digits at the even positions must be 0 or a multiple of 11. Sum of digits at odd positions = 6 + 0 + 9 + 6 + 5 = 26 Sum of digits at even positions = 2 + x + 7 + y + 2 = 11 + x + y Difference = 26 - (11 + x + y) = 15 - (x + y)

For this difference to be a multiple of 11, 15 - (x + y) can be 11 or 0, which gives: x + y = 4 or x + y = 15

Since we need the least value of x² + y², we test the possible values of y in both cases.

From Case 1 where x + y = 4: If y = 1, then x = 3. This gives x² + y² = 3² + 1² = 10. If y = 3, then x = 1. This gives x² + y² = 1² + 3² = 10.

From Case 2 where x + y = 15: If y = 7, then x = 8. This gives x² + y² = 8² + 7² = 113. If y = 9, then x = 6. This gives x² + y² = 6² + 9² = 117.

Comparing all possible values, the least value of x² + y² is 10.

Explanation

Simple interest is calculated using the formula SI = P × R × T / 100, where P is the principal, R is the rate of interest per annum, and T is the time in years.

For the first 10 years: P₁ = 2000 R = 5% T₁ = 10 years SI₁ = 2000 × 5 × 10 / 100 = 1000

After 10 years, this interest is added to the principal to form the new principal: P₂ = 2000 + 1000 = 3000

To reach the target amount of 4000, the additional interest required in the second period is: SI₂ = 4000 - 3000 = 1000

Using the simple interest formula for the second period with P₂ = 3000: 1000 = 3000 × 5 × T₂ / 100 1000 = 150 × T₂ T₂ = 1000 / 150 = 20/3 = 6 ⅔ years

The total time required for the investment to amount to 4000 is the sum of both periods: Total Time = T₁ + T₂ = 10 + 6 ⅔ = 16 ⅔ years

Explanation

The selling price of an item after successive discounts is calculated using the formula:

Selling Price = Marked Price × (1 - d₁/100) × (1 - d₂/100)

Here, the marked price is £2,000, the first discount is 15%, and the second discount is 5%.

Substituting these values into the formula:

Selling Price = 2000 × (1 - 15/100) × (1 - 5/100) Selling Price = 2000 × 0.85 × 0.95 Selling Price = 1700 × 0.95 Selling Price = 1615

Explanation

The quantity of water in the mixture after n replacement operations can be calculated using the dilution formula:

Wₙ = W₀ × (1 - y/V)ⁿ

where: W₀ is the initial quantity of water = 10 - 8 = 2 litres y is the volume of solution replaced in each step = 2 litres V is the total volume of the solution = 10 litres n is the number of operations = 3 (the initial process plus two more repetitions)

Substituting these values into the formula:

W₃ = 2 × (1 - 2/10)³ W₃ = 2 × (0.8)³ W₃ = 2 × 0.512 W₃ = 1.024 litres

Since the total volume of the mixture remains constant at 10 litres, the final quantity of milk is:

Milk = Total Volume - Final Water Volume Milk = 10 - 1.024 = 8.976 litres

Explanation

Since P beats S by 6 metres in a 120-metre race, P covers 120 metres in the same time S covers: 120 - 6 = 114 metres.

The ratio of the speed of P to the speed of S is: Speed of P / Speed of S = 120 / 114 = 20 / 19.

In a 100-metre race where S has a head start of 7.5 metres, S needs to run: 100 - 7.5 = 92.5 metres to finish the race.

The distance P covers in the time S runs his remaining 92.5 metres to finish the race is: 92.5 × (20 / 19) = 1850 / 19 ≈ 97.37 metres.

Since P has only covered 97.37 metres when S reaches the 100-metre mark, S wins the race. The margin by which S wins is the distance P is short of the finish line: 100 - 97.37 = 2.63 metres.

Explanation

To simplify the given expression (1 + sin A) / cos A + cos A / (1 + sin A), we take the common denominator cos A (1 + sin A):

((1 + sin A)² + cos² A) / (cos A (1 + sin A))

Expanding the numerator gives:

(1 + 2 sin A + sin² A + cos² A) / (cos A (1 + sin A))

Using the fundamental trigonometric identity sin² A + cos² A = 1, the numerator becomes:

1 + 2 sin A + 1 = 2 + 2 sin A = 2(1 + sin A)

Substituting this back into the expression:

2(1 + sin A) / (cos A (1 + sin A))

Canceling the common term (1 + sin A) from the numerator and denominator:

2 / cos A = 2 sec A

Explanation

By the Tangent-Secant Theorem, if a tangent segment and a secant segment are drawn to a circle from an exterior point, the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external segment.

The formula is: CT² = CA × CB

Since the chord AB is extended to meet the tangent at C, the points lie in the order A, B, C on the secant line.

Given: AB = 3 cm BC = 2.4 cm

The entire secant segment length CA is: CA = AB + BC = 3 + 2.4 = 5.4 cm

Substituting these values into the formula: CT² = 5.4 × 2.4 CT² = 12.96 CT = √12.96 = 3.6 cm

Explanation

The algebraic identity for the square of a binomial sum is: (y + 1/y)² = y² + 2 + 1/y²

By rearranging the terms: (y + 1/y)² = (y² + 1/y²) + 2

Substitute the given value of y² + 1/y² = 167 into the equation: (y + 1/y)² = 167 + 2 (y + 1/y)² = 169

Since y > 0, the expression y + 1/y must also be positive. Taking the positive square root of both sides gives: y + 1/y = √169 = 13

Explanation

The surface area of a sphere is given by the formula: A = 4πr²

Substituting the given values, where the radius r is 3.5 cm (which can be written as 7/2 cm) and π is 22/7: A = 4 × (22/7) × (7/2)² A = 4 × (22/7) × (49/4) A = 154 cm²

Explanation

Using the algebraic identity (a+b)³ + (a-b)³ = 2a(a² + 3b²), we can simplify the numerator of the given expression by substituting a = x and b = 3:

(x+3)³ + (x-3)³ = 2x(x² + 3 × 3²) = 2x(x² + 27)

Substituting this simplified numerator back into the original expression gives:

[2x(x² + 27)] / (x² + 27)

Canceling the common factor (x² + 27) in both the numerator and denominator simplifies the expression to:

2x

Explanation

In any triangle PQR, the point of intersection of the angle bisectors is the incenter M. The angle formed at the incenter by the bisectors of angle P and angle R is related to the opposite angle Q by the standard geometric formula:

angle PMR = 90° + angle Q / 2

Substituting the given value of angle PMR = 127° into the formula:

127° = 90° + angle Q / 2

Subtracting 90° from both sides:

37° = angle Q / 2

Multiplying both sides by 2:

angle Q = 74°

Explanation

When two individuals can complete a work individually in A and B days, the time taken by them to complete the work together is given by the formula:

T = (A × B) / (A + B)

Here, the time taken by Jason is A = 23.4 days and the time taken by Rafique is B = 39 days.

Substituting these values into the formula:

T = (23.4 × 39) / (23.4 + 39)

T = 912.6 / 62.4

To simplify the calculation, write the decimal numbers as fractions:

T = (234/10 × 39) / (624/10)

T = (234 × 39) / 624

Since both 234 and 624 are divisible by 78 (234 = 78 × 3 and 624 = 78 × 8):

T = (3 × 39) / 8

T = 117 / 8

T = 14.625 days