SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

Total Cost Price (CP) of 20 quintals of rice = Rs 8,580

To earn a profit of 20%, the total Selling Price (SP) must be: Total SP = CP × (1 + Profit Percentage) Total SP = 8,580 × 1.20 = Rs 10,296

Since 22% of the rice is lost in transportation, the quantity of rice left for sale is: Remaining quantity = 20 quintals × (1 - 0.22) = 20 × 0.78 = 15.6 quintals

The selling rate per quintal to achieve the target SP is: Rate per quintal = Total SP / Remaining quantity Rate per quintal = 10,296 / 15.6 = Rs 660 per quintal

Explanation

The intercept form of the equation of a line with x-intercept a and y-intercept b is: x/a + y/b = 1

Substituting the given intercepts, a = 3/2 and b = -4/7: x/(3/2) + y/(-4/7) = 1

This simplifies to: 2x/3 - 7y/4 = 1

Multiplying the entire equation by the common denominator 12 to clear the fractions: 12 × (2x/3) - 12 × (7y/4) = 12 × 1 8x - 21y = 12

Explanation

Euler's polyhedral formula states that V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces. A tetrahedron consists of 4 vertices and 4 triangular faces. Substituting these values into the formula gives 4 - E + 4 = 2, which simplifies to 8 - E = 2, yielding E = 6.

Alternatively, a tetrahedron can be viewed as a shape where every vertex is connected to every other vertex. The number of edges is the number of ways to choose 2 vertices from the 4 available vertices, which is given by the combination formula 4C2 = (4 × 3) / 2 = 6.

Explanation

The circumference of a circle with radius r is given by the formula 2πr. Substituting the given radius r = 14 cm and using π = 22/7:

Circumference = 2 × (22/7) × 14 = 88 cm

The perimeter of an equilateral triangle with side length s is given by 3s. Since the circumference of the circle is equal to the perimeter of the equilateral triangle:

3s = 88

s = 88/3 cm

Explanation

The angle 5π/3 lies in the fourth quadrant (between 3π/2 and 2π), where the cosecant function is negative.

We can rewrite the angle as: 5π/3 = 2π - π/3

Using the trigonometric identity cosec(2π - θ) = -cosec θ, we get: cosec(5π/3) = cosec(2π - π/3) = -cosec(π/3)

Since sin(π/3) = √3/2, its reciprocal cosec(π/3) is 2//√3. Substituting this value gives: cosec(5π/3) = -2/√3

Explanation

Using the standard trigonometric expansion formulas:

sin(A+B) = sinA cosB + cosA sinB sin(A-B) = sinA cosB - cosA sinB

Subtracting the second identity from the first:

sin(A+B) - sin(A-B) = (sinA cosB + cosA sinB) - (sinA cosB - cosA sinB) = sinA cosB + cosA sinB - sinA cosB + cosA sinB = 2cosA sinB

Explanation

Using the algebraic identity (x + y)² = x² + 2xy + y² and the fundamental trigonometric identity sin²A + cos²A = 1, expand the term:

(cosA + sinA)² = cos²A + 2 sinA cosA + sin²A = 1 + 2 sinA cosA

Substitute this value back into the original expression:

2 - (cosA + sinA)² = 2 - (1 + 2 sinA cosA) = 1 - 2 sinA cosA

Expand (cosA - sinA)² using the algebraic identity (x - y)² = x² - 2xy + y²:

(cosA - sinA)² = cos²A - 2 sinA cosA + sin²A = 1 - 2 sinA cosA

Both expressions simplify to 1 - 2 sinA cosA, establishing their equivalence.

Explanation

To find the total value of the stock, we calculate the sum of the products of quantity and average cost for each item category.

Mobile Phones value = 25 × 15000 = 375000 rupees Cameras value = 80 × 16000 = 1280000 rupees TVs value = 54 × 55000 = 2970000 rupees Refrigerators value = 41 × 27000 = 1107000 rupees ACs value = 64 × 24000 = 1536000 rupees

Total value = 375000 + 1280000 + 2970000 + 1107000 + 1536000 = 7268000 rupees

Since 1 lakh is equal to 100000, we convert the total value into lakh rupees: Total value in lakh rupees = 7268000 ÷ 100000 = 72.68

Explanation

The ratio of Import to Export is given by: Ratio = Import / Export

This can be rearranged to find exports: Export = Import / Ratio

For the year 2012: Ratio = 0.9 Import = 700 crores Export in 2012 = 700 / 0.9 = 777.78 crores

Given that the total exports for 2012 and 2013 combined is 4000 crores: Export in 2012 + Export in 2013 = 4000 777.78 + Export in 2013 = 4000 Export in 2013 = 4000 - 777.78 = 3222.22 crores

For the year 2013: Ratio = 1 Import in 2013 / Export in 2013 = 1 Import in 2013 = Export in 2013 = 3222.22 crores (approximately 3222 crores)

Explanation

According to the given table, the height of the child on their 7th birthday was 125 cms, and their height on their 16th birthday was 175 cms.

To find the increase in height between these two birthdays, subtract the height on the 7th birthday from the height on the 16th birthday:

Increase in height = Height on 16th birthday - Height on 7th birthday Increase in height = 175 cms - 125 cms = 50 cms

Explanation

The total duration from 10 pm to 6 am is 8 hours.

Summing the values from the data table gives 5 + 30 + 5 + 20 + 40 = 100, which indicates that the recorded data represents percentages of the total time.

The combined percentage of time the user spent in Deep Sleep or was Awake is: 5% + 40% = 45%

To find this duration in hours, we calculate 45% of the total 8 hours: 8 hours × 0.45 = 3.6 hours

Explanation

To find the product of the algebraic expression (3x + 4y)(4x - 3y), we expand it using the distributive property:

(3x + 4y)(4x - 3y) = 3x(4x - 3y) + 4y(4x - 3y)

Expanding both terms: 3x(4x - 3y) = 12x² - 9xy 4y(4x - 3y) = 16xy - 12y²

Combining these results: 12x² - 9xy + 16xy - 12y²

Combining the like terms -9xy and 16xy: -9xy + 16xy = 7xy

This simplifies to: 12x² + 7xy - 12y²

Explanation

The given series is an arithmetic progression (AP) where the first term a = 81, the last term l = 130, and the common difference d = 1.

The number of terms n is calculated as: n = l - a + 1 n = 130 - 81 + 1 = 50

The sum of an arithmetic progression is given by the formula: Sₙ = n/2 × (a + l)

Substituting the values into the formula: S₅₀ = 50/2 × (81 + 130) S₅₀ = 25 × 211 S₅₀ = 5275

Explanation

Let the present ages of the husband and the wife be H and W respectively. The sum of their present ages is given as: H + W = 100

Ten years ago, the sum of their ages was: (H - 10) + (W - 10) = H + W - 20 = 100 - 20 = 80

Since the ratio of their ages ten years ago was 9:7, we can represent their ages ten years ago as 9x and 7x. 9x + 7x = 80 16x = 80 x = 5

The husband's age ten years ago was: 9 × 5 = 45 years

Therefore, the present age of the husband is: 45 + 10 = 55 years

Explanation

Solving the first inequality: 3x - 1 < 5x + 2 Subtracting 3x and 2 from both sides: -3 < 2x x > -1.5

Solving the second inequality: 3x + 5 < 6 - 4x Adding 4x and subtracting 5 from both sides: 7x < 1 x < 1/7

Combining both conditions gives the range of x: -1.5 < x < 1/7

The value x = -1 satisfies this combined inequality since -1.5 < -1 < 1/7.

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.

Using the given values: a₃ = a + 2d = 13 a₁₃ = a + 12d = -47

Subtracting the first equation from the second equation: 10d = -60 d = -6

Substituting d = -6 into the equation for the third term: a + 2(-6) = 13 a = 25

To find which term has the value -17, substitute a = 25 and d = -6 into the general term formula: -17 = 25 + (n - 1)(-6) -42 = -6(n - 1) n - 1 = 7 n = 8

Explanation

Let the first number be 6k.

Since the first number is twice the second, the second number is 3k. Since the first number is thrice the third, the third number is 2k.

The average of the three numbers is given as 506: (6k + 3k + 2k) ÷ 3 = 506 11k ÷ 3 = 506 k = (506 × 3) ÷ 11 k = 46 × 3 = 138

The difference between the first and the third number is: 6k - 2k = 4k 4 × 138 = 552

Explanation

The distance d between two points (x₁, y₁) and (x₂, y₂) is calculated using the distance formula:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Here, the given points are (4, -8) and (-2, 0). Substituting these values into the formula:

d = √((-2 - 4)² + (0 - (-8))²)

d = √((-6)² + 8²)

d = √(36 + 64)

d = √100

d = 10 units

Explanation

The slope of the base line QR is calculated using the slope formula: m_QR = (5 - 3) / (-4 - 0) = 2 / -4 = -½

Since the altitude PS is perpendicular to QR, the product of their slopes is -1: m_PS × m_QR = -1 m_PS × (-½) = -1 m_PS = 2

Using the point-slope form for the line PS passing through P(6, 2) with a slope of 2: y - 2 = 2(x - 6) y - 2 = 2x - 12 2x - y = 10

Explanation

Since ST is parallel to QR in ΔPQR, ΔPST is similar to ΔPQR by Angle-Angle similarity. For similar triangles, the ratio of corresponding sides is equal:

ST / QR = PS / PQ

S divides PQ in the ratio 4:3, which means PS : SQ = 4 : 3. Therefore, the ratio of PS to the entire side PQ is:

PS / PQ = PS / (PS + SQ) = 4 / (4 + 3) = 4/7

Substituting this and the value of QR into the similarity relation:

ST / 4.9 = 4/7

ST = 4.9 × 4 / 7

ST = 0.7 × 4 = 2.8 cm

Explanation

If Samarjit's daily work rate is S, then Makarand's daily work rate is 2S since he works 2 times as fast.

Since Samarjit completes the job alone in 27 days, the total work can be represented as 27S.

When working together, their combined daily work rate is 2S + S = 3S.

The number of days they take to complete the job together is the total work divided by their combined daily work rate: 27S ÷ 3S = 9 days.

Explanation

Let the present ages of Ramya and Saurabh be 8x and 7x respectively.

After 10 years, their ages will be 8x + 10 and 7x + 10. The ratio of their ages at that time is 12:11, which gives the equation:

(8x + 10) / (7x + 10) = 12 / 11

By cross-multiplying: 11 × (8x + 10) = 12 × (7x + 10) 88x + 110 = 84x + 120 4x = 10 x = 2.5

Ramya's present age is 8x: 8 × 2.5 = 20 years

Explanation

First, calculate the compound interest on Rs 12000 for 2 years at 10% per annum. The effective compound interest rate for 2 years at 10% per annum is 10 + 10 + (10 × 10)/100 = 21%.

Compound Interest = 12000 × 21% = 12000 × 21/100 = 2520.

The simple interest is half of this compound interest: Simple Interest = 2520 / 2 = 1260.

Now, use the simple interest formula to find the principal sum, P: Simple Interest = (P × R × T) / 100

Substituting the given values: 1260 = (P × 16 × 3) / 100 1260 = 48 × P / 100 P = 126000 / 48 = 2625.

Explanation

Let the maximum marks be M.

The student secured 56 marks and failed by 34 marks. Therefore, the minimum marks required to pass are: Pass Marks = 56 + 34 = 90

Since the passing marks are 25% of the maximum marks: 25% of M = 90 0.25 × M = 90 M = 90 ÷ 0.25 M = 360

Explanation

The effective discount for two successive discount percentages is given by the formula:

Net Discount = d₁ + d₂ - (d₁ × d₂)/100

Here, the initial discount d₁ is 30% and the subsequent cashback d₂ is 20%. Substituting these values into the formula:

Net Discount = 30 + 20 - (30 × 20)/100 Net Discount = 50 - 6 = 44%

Explanation

The curved surface area of a cylinder is given by the formula: CSA = 2πrh

Substituting the given values, CSA = 1320 sq cm and h = 10 cm, using π ≈ 22/7: 1320 = 2 × (22/7) × r × 10 1320 = (440/7) × r

Solving for the radius r: r = (1320 × 7) / 440 r = 3 × 7 r = 21 cm

Explanation

Let the cost price of the watch for the wholesaler be CP.

Using the successive percentage change relation for successive profit and loss: CP × (1 + Gain%/100) × (1 - Loss%/100) = Selling Price to Customer

Substituting the given values into the equation: CP × 1.35 × 0.75 = 2379.375

Expressing the decimals as fractions to simplify the division: CP × 135/100 × 75/100 = 2379.375 CP × 27/20 × 3/4 = 2379.375 CP × 81/80 = 2379.375

Solving for CP: CP = (2379.375 × 80) / 81 CP = 190350 / 81 CP = 2350

Explanation

Using the trigonometric identity for angles in the fourth quadrant, the angle 7π/4 can be written as:

7π/4 = 2π - π/4

Since the tangent function is negative in the fourth quadrant, the relation is:

tan(2π - θ) = -tan(θ)

By substituting θ = π/4 into the formula, we get:

tan(7π/4) = tan(2π - π/4) = -tan(π/4)

Since the standard value of tan(π/4) is 1, the expression simplifies directly to:

tan(7π/4) = -1

Explanation

Let t be the time spent by the man on the steamer in hours. Since he spends 6 hours more on the steamer than on the rail, the time spent on the rail is t - 6 hours.

The standard formula for distance is: Distance = Speed × Time

The total distance is the sum of the distances covered by the steamer and the rail: Total Distance = (Speed of steamer × Time on steamer) + (Speed of rail × Time on rail) 401 = 21 × t + 56 × (t - 6) 401 = 21t + 56t - 336 737 = 77t t = 737 / 77 = 67/7 hours

The distance covered by the steamer is: Distance = Speed of steamer × Time on steamer Distance = 21 × (67/7) Distance = 3 × 67 = 201 km

Explanation

Using the standard sum-to-product formula in trigonometry, the sum of two sine functions is converted into a product of sine and cosine. This is derived from the angle sum and difference identities: sin(A + B) = sin A cos B + cos A sin B sin(A - B) = sin A cos B - cos A sin B

Adding these two equations yields: sin(A + B) + sin(A - B) = 2 sin A cos B

By defining C = A + B and D = A - B, we can solve for A and B as: A = (C + D)/2 B = (C - D)/2

Substituting these values back into the sum equation gives: sin C + sin D = 2 sin[(C + D)/2] cos[(C - D)/2]

Explanation

The total value of the stock is determined by the sum of the products of quantity and average cost for each item:

Total Value = ∑ (Quantity × Average Cost)

Calculating the value for each product: Mobile Phones: 100 × 18000 = 1800000 Rs Cameras: 87 × 6000 = 522000 Rs TVs: 80 × 56000 = 4480000 Rs Refrigerators: 38 × 28000 = 1064000 Rs ACs: 93 × 25000 = 2325000 Rs

Summing these individual values: Total Stock Value = 1800000 + 522000 + 4480000 + 1064000 + 2325000 = 10191000 Rs

Since 1 lakh equals 100000 Rs, the value in lakh rupees is: 10191000 / 100000 = 101.91 lakh Rs

Explanation

To find the increase in the height of the child from the 5th Birthday to the 16th Birthday, we extract the corresponding height values from the given data table.

Height of the child on the 5th Birthday = 110 cms Height of the child on the 16th Birthday = 190 cms

The increase in height is calculated by subtracting the initial height from the final height:

Increase in height = Height on the 16th Birthday - Height on the 5th Birthday Increase in height = 190 cms - 110 cms = 80 cms

Explanation

The total duration of time monitored by the fitness band between 10:00 PM and 6:00 AM is exactly 8 hours.

The values given in the data table represent percentages, which sum up to a total of 100%: 5% + 30% + 20% + 10% + 35% = 100%

The combined percentage of time the user was either Dreaming or Awake is: 30% + 35% = 65%

To find the duration in hours, calculate 65% of the total 8 hours: 8 × 0.65 = 5.2 hours

Explanation

The effective discount for successive discounts of x% and y% is calculated using the formula:

Effective discount = x + y - xy/100

Here, the initial discount is x = 20 and the cashback percentage is y = 15. Substituting these values into the formula:

Effective discount = 20 + 15 - (20 × 15)/100 Effective discount = 35 - 300/100 Effective discount = 35 - 3 = 32%

Alternatively, if the marked price of the item is assumed to be ₹100, a 20% discount reduces the price to ₹80. Applying a 15% cashback on this price of ₹80 yields a benefit of 15% × 80 = ₹12. The final effective price paid by the customer is 80 - 12 = ₹68. Therefore, the total discount received is 100 - 68 = ₹32, which represents an effective discount of 32%.

Explanation

To find the Highest Common Factor (HCF) of 57 and 513, we can check if the larger number is divisible by the smaller number. Dividing 513 by 57 gives:

513 ÷ 57 = 9

Since 57 divides 513 exactly with a remainder of 0, 57 itself is the highest common factor of the two numbers.

Explanation

Let the efficiency of Murad be E. Since Joel is 100% more efficient than Murad, Joel's efficiency is E + 100% of E, which equals 2E.

The combined efficiency of Murad and Joel working together is E + 2E = 3E.

The total work is the product of Murad's efficiency and the time he takes to complete the work alone, which is E × 33 = 33E.

The time taken by both of them together to finish this work is the total work divided by their combined efficiency, which is 33E / 3E = 11 hours.

Explanation

To simplify the given algebraic expression, we divide the numerical coefficients and cancel out the common variables in the numerator and denominator:

Numerator = 437 × b × x × a × z = 437abxz Denominator = 23 × a × b = 23ab

First, dividing the numerical coefficients: 437 ÷ 23 = 19

Next, dividing the variable terms: abxz ÷ ab = xz

Combining these results gives: 19xz

Explanation

A quadratic equation ax² + bx + c = 0 has equal roots if its discriminant, D = b² - 4ac, is equal to zero.

For the equation x² + 14x + 49 = 0, the coefficients are a = 1, b = 14, and c = 49.

Calculating the discriminant: D = 14² - 4 × 1 × 49 D = 196 - 196 D = 0

Since the discriminant is zero, the equation has real and equal roots. Alternatively, the equation can be factored as a perfect square trinomial: (x + 7)² = 0 This yields two identical roots, x = -7.

Explanation

Solving the first inequality: 4 + 3x ≤ 6 + x Subtracting x from both sides: 4 + 2x ≤ 6 Subtracting 4 from both sides: 2x ≤ 2 Dividing by 2: x ≤ 1

Solving the second inequality: 3x + 5 > 2 + 2x Subtracting 2x from both sides: x + 5 > 2 Subtracting 5 from both sides: x > -3

Combining both results gives the solution interval: -3 < x ≤ 1

The value -2 is the only option that satisfies this condition.

Explanation

The sum S of an arithmetic progression with first term a, last term l, and number of terms n is given by the formula:

S = (n / 2) × (a + l)

Substituting the given values S = 114, a = 37, and l = -18:

114 = (n / 2) × (37 + (-18))

114 = (n / 2) × 19

Multiplying both sides by 2:

228 = 19 × n

n = 228 / 19

n = 12

Explanation

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

Perimeter of ΔABC : Perimeter of ΔDEF = AB : DE

Substituting the given values:

Perimeter of ΔABC : Perimeter of ΔDEF = 12 : 8 = 3 : 2

Explanation

The slope m₁ of a line passing through the points (x₁, y₁) and (x₂, y₂) is given by the formula: m₁ = (y₂ - y₁) / (x₂ - x₁)

For the points (-4, 5) and (3, -2), the slope is: m₁ = (-2 - 5) / (3 - (-4)) m₁ = -7 / 7 m₁ = -1

Two lines are perpendicular if the product of their slopes is -1. Therefore, the slope m₂ of the perpendicular line is: m₁ × m₂ = -1 -1 × m₂ = -1 m₂ = 1

Explanation

Let the two numbers be a and b.

The mean proportion of a and b is defined as √(ab). According to the problem: √(ab) = 12 ab = 144 a = 144/b

The third proportion of a and b is defined as b²/a. According to the problem: b²/a = 768 b² = 768a

Substituting the value of a from the first equation into the second equation: b² = 768 × (144/b) b³ = 768 × 144 b³ = 110592 b = 48

Now, substituting b = 48 back into the equation for a: a = 144/48 = 3

Thus, the two numbers are 3 and 48.

Explanation

The total runs scored in the first 43 overs is calculated as: Runs scored = 43 × 5 = 215 runs

The remaining runs required to reach the target of 285 runs is: Remaining runs = 285 - 215 = 70 runs

The required run rate for the remaining 7 overs is: Required run rate = 70 / 7 = 10 runs/over

Explanation

The sum of the interior angles of any polygon with n sides is calculated using the formula:

Sum = (n - 2) × 180°

A decagon is a polygon with 10 sides, so n = 10. Substituting this value into the formula gives:

Sum = (10 - 2) × 180° = 8 × 180° = 1440°

Explanation

The diagonal of a square is given by the formula d = a√2, where a is the side of the square. Substituting the given diagonal, we get 14√2 = a√2, which gives the side of the square a = 14 cm. Since the circle is inscribed in the square, its diameter is equal to the side length of the square. Thus, the radius of the circle r is a/2 = 14/2 = 7 cm. The area of the circle is calculated using the formula Area = πr². Substituting the values, we get Area = (22/7) × 7² = 22 × 7 = 154 sq cm.

Explanation

The total surface area of a hemisphere is given by the formula 3πr², where r represents the radius.

Using the given total surface area of 41.58 sq cm and taking π ≈ 22/7:

3 × (22/7) × r² = 41.58

(66/7) × r² = 41.58

r² = (41.58 × 7) / 66

r² = 0.63 × 7

r² = 4.41

r = √4.41 = 2.1 cm

Explanation

Let the distance covered by the steamer be d, which means the distance covered by the rail is 800 - d.

Using the relationship: Time = Distance / Speed

The time spent on the steamer is: t_s = d / 20

The time spent on the rail is: t_r = (800 - d) / 55

According to the given condition, the time spent on the steamer is 7 hours more than the time spent on the rail: t_s = t_r + 7

Substituting the expressions for t_s and t_r: d / 20 = (800 - d) / 55 + 7

To solve this equation, multiply all terms by the least common multiple of 20 and 55, which is 220: 11 × d = 4 × (800 - d) + 7 × 220

Expanding and simplifying the equation: 11 × d = 3200 - 4 × d + 1540 15 × d = 4740 d = 316 km

Explanation

In compound interest, the amount at the end of any year acts as the principal for the succeeding year. Therefore, the amount at the end of the third year (A₃) is obtained by applying the interest rate (R) for one year on the amount at the end of the second year (A₂).

The relation is given by the formula: A₃ = A₂ × (1 + R/100)

Substituting the given values into the formula: 29775.4 = 28090 × (1 + R/100)

Solving for R: 1 + R/100 = 29775.4 / 28090 R/100 = (29775.4 - 28090) / 28090 R/100 = 1685.4 / 28090 R = 168540 / 28090 R = 6 percent

Explanation

Using the standard half-angle trigonometric identities, we have: 1 - cos A = 2 sin²(A/2) 1 + cos A = 2 cos²(A/2)

Dividing the first identity by the second identity gives: (1 - cos A) / (1 + cos A) = 2 sin²(A/2) / 2 cos²(A/2)

Simplifying the right side of the equation yields: (1 - cos A) / (1 + cos A) = tan²(A/2)

Substituting x for tan(A/2) gives: x² = (1 - cos A) / (1 + cos A)

Taking the square root of both sides gives the value of x: x = √[1(1 - cos A) / (1 + cos A)]