SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

Let the efficiency of Mayur be 1 unit per hour. Since Mayur takes 27 hours to complete the work, the total work is 27 units.

Jayantika is 100% more efficient than Mayur, which means her efficiency is 1 + 1 = 2 units per hour.

When they work together, their combined efficiency is 1 + 2 = 3 units per hour.

The time taken by both of them together to complete the total work is: Total work / Combined efficiency = 27 / 3 = 9 hours.

Explanation

The reciprocal of 3/8 is 8/3, and the reciprocal of 8/11 is 11/8.

The sum of these reciprocals is: 8/3 + 11/8 = (8 × 8 + 11 × 3) / (3 × 8) = (64 + 33) / 24 = 97/24

The reciprocal of this sum is: 1 / (97/24) = 24/97

Explanation

To find the average bonus per employee, first calculate the total bonus earned by each category of employees. Since 1 lakh equals 100,000 rupees, the annual salary of a manager is 4,800,000 rupees, an executive is 2,000,000 rupees, and a trainee is 400,000 rupees.

For the managers, the bonus per employee is 60% of 4,800,000 rupees, which is 2,880,000 rupees. With 2 managers, the total bonus is 5,760,000 rupees.

For the executives, the bonus per employee is 20% of 2,000,000 rupees, which is 400,000 rupees. With 9 executives, the total bonus is 3,600,000 rupees.

For the trainees, the bonus per employee is 10% of 400,000 rupees, which is 40,000 rupees. With 2 trainees, the total bonus is 80,000 rupees.

The total bonus for all employees is the sum of these values: 5,760,000 + 3,600,000 + 80,000 = 9,440,000 rupees.

The total number of employees is 2 + 9 + 2 = 13.

Dividing the total bonus by the total number of employees gives 9,440,000 ÷ 13 ≈ 726153.85 rupees, which rounds to 726154 rupees.

Explanation

The total exports from the three companies together for the years 2012 and 2014 are calculated as follows:

For the year 2012: Total exports = Export of Company A + Export of Company B + Export of Company C Total exports = 3000 + 4000 + 1000 = 8000 lakh rupees

For the year 2014: Total exports = Export of Company A + Export of Company B + Export of Company C Total exports = 3000 + 4000 + 1000 = 8000 lakh rupees

Since both years have a total export value of 8000 lakh rupees, the total exports for the pair 2014 and 2012 are equal.

Explanation

The total profit or loss over the 5-year period is calculated by taking the algebraic sum of the profit or loss values for each year:

Total = (-15) + (-5) + (-20) + 15 + 20

Grouping the positive and negative terms makes calculation simple:

Total = (-15 + 15) + (-20 + 20) - 5 = -5

A negative result indicates a net loss. Therefore, the company incurred a total loss of Rs 5 crores over the last 5 years.

Explanation

The percentage of jewellery exports is calculated by dividing the value of jewellery exports by the total value of all exports and then multiplying by 100.

First, determine the total value of India's exports in 2015 by adding the export values of all categories: Total exports = 1000 + 900 + 575 + 925 + 950 = 4350 million USNext, calculate the percentage of jewellery exports using the formula: Percentage of jewellery = (Value of Jewellery / Total Exports) × 100 Percentage of jewellery = (1000 / 4350) × 100 Percentage of jewellery = 2000 / 87 ≈ 22.99%

Explanation

A quadratic equation ax² + bx + c = 0 has real roots if its discriminant D = b² - 4ac is greater than or equal to zero (D ≥ 0).

For the quadratic equation 2x² - 7x + 5 = 0, the coefficients are a = 2, b = -7, and c = 5.

Calculating the discriminant: D = (-7)² - 4 × 2 × 5 D = 49 - 40 D = 9

Since the discriminant is 9, which is greater than zero, the equation has real and distinct roots.

Explanation

To find the possible values of x, solve the two inequalities separately and find their intersection.

For the first inequality: 5x + 5 > 2 + 2x 5x - 2x > 2 - 5 3x > -3 x > -1

For the second inequality: 5x + 3 ≤ 4x + 5 5x - 4x ≤ 5 - 3 x ≤ 2

Combining both inequalities yields: -1 < x ≤ 2

The value 1 satisfies this combined inequality since -1 < 1 ≤ 2.

Explanation

An arithmetic progression term aₙ is given by the formula aₙ = a + (n - 1)d, where a is the first term and d is the common difference.

The relation between any two terms of an arithmetic progression can be written as aₙ = a_m + (n - m)d.

Given the 4th term a₄ = 15 and the 15th term a₁₅ = -29, the common difference d can be calculated as: d = (a₁₅ - a₄) / (15 - 4) d = (-29 - 15) / 11 d = -44 / 11 d = -4

Using the common difference d = -4, the 10th term a₁₀ can be found directly from the 4th term: a₁₀ = a₄ + (10 - 4)d a₁₀ = 15 + 6(-4) a₁₀ = 15 - 24 a₁₀ = -9

Explanation

The ratio k:1 in which the line segment joining the points (x₁, y₁) and (x₂, y₂) is divided by the x-axis is given by the formula:

k = -y₁ / y₂

For the given points (-1, -12) and (3, 4), we have y₁ = -12 and y₂ = 4. Substituting these coordinates into the formula yields:

k = -(-12) / 4 = 12 / 4 = 3

This means the line segment is divided internally in the ratio 3:1.

Explanation

Let the total work be 100 units, which is the least common multiple of 20 and 50.

The daily efficiency of A is 100 ÷ 20 = 5 units per day. The daily efficiency of B is 100 ÷ 50 = 2 units per day.

When working together, their combined efficiency is 5 + 2 = 7 units per day.

In 5 days, the total work completed by A and B together is 7 × 5 = 35 units.

The remaining work is 100 - 35 = 65 units.

The fraction of work left is 65/100, which simplifies to 13/20.

Explanation

To find where the line intercepts the y-axis, set x = 0 in the given equation of the line.

Substituting x = 0 into 4x - 3y = -6 gives:

4(0) - 3y = -6 -3y = -6 y = 2

Therefore, the point of intersection on the y-axis is (0, 2).

Explanation

By the Basic Proportionality Theorem, a line drawn parallel to one side of a triangle divides the other two sides in the same ratio. Since GH is parallel to EF in ΔDEF, we have:

DG/GE = DH/HF

Given that G divides DE in the ratio 1:3, the ratio DG/GE is equal to 1/3. Substituting this and HF = 7.2 cm into the equation gives:

1/3 = DH/7.2

Solving for DH:

DH = 7.2 / 3 = 2.4 cm

The total length of side DF is the sum of DH and HF:

DF = DH + HF = 2.4 + 7.2 = 9.6 cm

Explanation

To find the combined mean of the three divisions, we use the formula for the weighted mean:

x̄ = (n₁x̄₁ + n₂x̄₂ + n₃x̄₃) / (n₁ + n₂ + n₃)

Here, the given values are: Division A: n₁ = 65, x̄₁ = 54 Division B: n₂ = 30, x̄₂ = 50 Division C: n₃ = 55, x̄₃ = 48

Substituting these values into the formula: Total sum of marks = (65 × 54) + (30 × 50) + (55 × 48) = 3510 + 1500 + 2640 = 7650

Total number of students = 65 + 30 + 55 = 150

Combined mean = 7650 / 150 = 51

Explanation

The average cost of the machines is calculated by dividing the total cost of all machines by the total number of machines.

The total number of machines purchased is 10. This consists of: 1 of Machine-A at Rs 90000 each 4 of Machine-B at Rs 40000 each 5 of Machine-C (calculated as 10 - 1 - 4) at Rs 40000 each

The total cost of these machines is: (1 × 90000) + (4 × 40000) + (5 × 40000) = 90000 + 160000 + 200000 = 450000

The average cost is the total cost divided by the total number of machines: 450000 ÷ 10 = 45000

Explanation

The reciprocal of 8/7 is 7/8, and the reciprocal of 5/6 is 6/5.

The sum of these reciprocals is calculated by finding a common denominator: 7/8 + 6/5 = (7 × 5 + 6 × 8) / (8 × 5) = (35 + 48) / 40 = 83/40

Taking the reciprocal of this sum gives: 1 / (83/40) = 40/83

Explanation

To find the value of x, we split the compound inequality into two separate inequalities.

First inequality: 3x - 2(4 - 3x) < 2x - 5 3x - 8 + 6x < 2x - 5 9x - 8 < 2x - 5 7x < 3 x < 3/7

Second inequality: 2x - 5 < 3x + 3 -8 < x

Combining both inequalities, the range for x is: -8 < x < 3/7

Since 3/7 is approximately 0.43, the value of x must be between -8 and 0.43. The value 0 satisfies this condition since -8 < 0 < 0.43. Substituting x = 0 back into the original compound inequality yields -8 < -5 < 3, which is mathematically true.

Explanation

To simplify the given algebraic expression, apply the laws of exponents: xᵐ × xⁿ = xᵐ⁺ⁿ and xᵐ / xⁿ = xᵐ⁻ⁿ.

First, multiply the terms in the numerator: (b²x³a²z⁴) × (b²x⁴a⁴z³) = a²⁺⁴ × b²⁺² × x³⁺⁴ × z⁴⁺³ = a⁶b⁴x⁷z⁷

Next, divide this result by the denominator: (a⁶b⁴x⁷z⁷) / (a³b²z⁴) = a⁶⁻³ × b⁴⁻² × x⁷ × z⁷⁻⁴ = a³b²x⁷z³

Rearranging the variables to match the standard form gives: b²x⁷a³z³

Explanation

Correct Answer: D

The simple interest formula is: SI = (P × R × T) / 100

Substituting the given values where simple interest SI = 9648, rate R = 12%, and time T = 6 years: 9648 = (P × 12 × 6) / 100 9648 = 72P / 100 P = (9648 × 100) / 72 P = 13400

Explanation

In a right-angled triangle, the side opposite the right angle is the hypotenuse, which is given as 9√3 cm. To find the length of the side opposite the 30° angle, the trigonometric sine ratio can be used:

sin(θ) = opposite / hypotenuse

Substituting θ = 30° and hypotenuse = 9√3 cm:

sin(30°) = opposite / 9√3

Since sin(30°) = ½, the equation becomes:

½ = opposite / 9√3

Solving for the opposite side:

opposite = (9√3)/2 cm

Explanation

The cost price (CP) of the dried apricots is calculated using the formula: CP = SP₁ ÷ (1 - Loss%/100)

Substituting the first selling price (SP₁) = 1310 and Loss% = 13: CP = 1310 ÷ (1 - 0.13) CP = 1310 ÷ 0.87 CP ≈ 1505.75

When the shopkeeper decides to sell the apricots at the new selling price (SP₂) of 1441, the loss amount is: Loss = CP - SP₂ Loss = 1505.75 - 1441 Loss ≈ 64.75

The new loss percentage is: New Loss% = (Loss ÷ CP) × 100 New Loss% = (64.75 ÷ 1505.75) × 100 ≈ 4.3%

This results in a 4.3% loss.

Explanation

Using the trigonometric identity for angles in the third quadrant: cos(180° + θ) = -cos θ

Substituting θ = 60°: cos 240° = cos(180° + 60°) = -cos 60°

Since the value of cos 60° is ½, we get: x = -½

Explanation

Let G be the speed of the goods train in km/hr.

The goods train passes the station 3 hours before the second train, meaning it has a head start of 3 hours. Since the second train takes 8 hours to overtake the goods train after passing the station, the goods train travels for a total of 3 + 8 = 11 hours by the time it is overtaken.

The second train travels for 8 hours at a speed of 88 km/hr.

Since both trains cover the exact same distance from the station to the point where they meet, we can equate their distances using the formula Distance = Speed × Time:

Distance traveled by goods train = Distance traveled by second train G × 11 = 88 × 8 G = (88 × 8) / 11 G = 8 × 8 G = 64 km/hr

Explanation

Using the standard algebraic identity: (a + b)² = a² + b² + 2ab

Substitute the given values into the equation: 9² = 61 + 2ab 81 = 61 + 2ab

Subtract 61 from both sides: 2ab = 81 - 61 2ab = 20

Divide by 2: ab = 10

Explanation

The reflection of a point (x, y) in the x-axis changes the sign of its y-coordinate while keeping its x-coordinate unchanged, following the transformation rule (x, y) → (x, -y).

For the point (-2, 5): The x-coordinate remains -2. The y-coordinate changes from 5 to -5.

This yields the reflected point (-2, -5).

Explanation

Let the original price of the article be P. After a discount of 25%, the new price becomes: P' = P × (1 - 0.25) = 0.75P

To restore the price to its original value, the new price must be increased by the discounted amount, which is 0.25P. The percentage increase required on the new price P' is: Percentage Increase = (0.25P / 0.75P) × 100% = (1/3) × 100% = 33.33%

Explanation

Using the fundamental trigonometric identity: 1 + tan²A = sec²A

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

Substituting this into the given equation: x = 1/√(1 + tan²A) x = 1/secA

Since the reciprocal of secA is cosA: x = cosA

Explanation

The effective discount of two successive discounts of d₁% and d₂% can be calculated using the successive discount formula:

Effective discount = d₁ + d₂ - (d₁ × d₂)/100

Here, the initial discount d₁ is 20% and the cashback d₂ acts as a successive discount of 5%. Substituting these values into the formula:

Effective discount = 20 + 5 - (20 × 5)/100 = 25 - 1 = 24%

Alternatively, if the original price of the item is ₹100, a 20% discount reduces the price to ₹80. Applying the 5% cashback on this reduced price of ₹80 yields a cashback of ₹4, which is 5% of ₹80. This brings the final effective price down to ₹76. The total discount received is ₹24, which represents an effective discount of 24% on the original price of ₹100.

Explanation

The formula for the total surface area of a hollow hemisphere is: TSA = 2πR² + 2πr² + π(R² - r²) TSA = π(3R² + r²)

Here, the internal radius r is 8 cm and the thickness is 1 cm. Thus, the external radius R is 8 + 1 = 9 cm.

Substituting the values into the formula: TSA = π × (3 × 9² + 8²) TSA = π × (3 × 81 + 64) TSA = π × (243 + 64) TSA = π × 307

Using π ≈ 22/7: TSA = (22/7) × 307 TSA = 6754/7 TSA ≈ 964.85 sq cm

Explanation

To find the ratio A : B : C from the relation mA = nB = pC, we use the formula: A : B : C = 1/m : 1/n : 1/p

Substituting the given coefficients m = 6, n = 7, and p = 5: A : B : C = 1/6 : 1/7 : 1/5

To eliminate the fractions and express the ratio in simplest integer form, we multiply each term by the Least Common Multiple (LCM) of the denominators 6, 7, and 5.

LCM(6, 7, 5) = 6 × 7 × 5 = 210

Multiplying each term by 210: A = 210/6 = 35 B = 210/7 = 30 C = 210/5 = 42

This yields the ratio: A : B : C = 35 : 30 : 42

Explanation

The relation between distance, speed, and time difference is given by the formula:

d = (v₁ × v₂ × Δt) / (v₂ - v₁)

where d is the distance, v₁ and v₂ are the speeds of the two cars, and Δt is the difference in time taken.

Substituting the given values: v₁ = 30 km/hr v₂ = 44 km/hr Δt = 3.5 hours

d = (30 × 44 × 3.5) / (44 - 30) d = (1320 × 3.5) / 14 d = 4620 / 14 d = 330 km

Explanation

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

Sₙ = (n / 2) × (a + l)

Substituting the given values, n = 14, a = 23, and l = -42, into the formula:

S₁₄ = (14 / 2) × (23 + (-42))

S₁₄ = 7 × (-19)

S₁₄ = -133

Explanation

The equation of a line passing through a point (x₁, y₁) with a slope m is given by the point-slope form: y - y₁ = m(x - x₁)

The slope m of a line making an angle θ with the positive x-axis is: m = tan(θ)

Given that θ = -45°, the slope is: m = tan(-45°) = -1

The line passes through the point (0, -4). Substituting x₁ = 0, y₁ = -4, and m = -1 into the point-slope equation: y - (-4) = -1(x - 0) y + 4 = -x x + y = -4

Explanation

In a rhombus ABCD, all sides are equal in length, which means AB = BC. This makes triangle ABC an isosceles triangle with AB = BC. In any isosceles triangle, the angles opposite to the equal sides are equal, which gives angle CAB = angle ACB = 30°.

The sum of angles in triangle ABC is 180°: angle CAB + angle ACB + angle ABC = 180° 30° + 30° + angle ABC = 180° 60° + angle ABC = 180° angle ABC = 120°

Alternatively, the diagonal AC of rhombus ABCD bisects angle DAB. Thus, angle DAB = 2 × angle CAB = 2 × 30° = 60°. Since consecutive angles in a rhombus are supplementary: angle DAB + angle ABC = 180° 60° + angle ABC = 180° angle ABC = 120°

Explanation

To simplify the expression √[(1 + sin A)/(1 - sin A)], we multiply both the numerator and the denominator inside the square root by (1 + sin A):

√[(1 + sin A)(1 + sin A) / (1 - sin A)(1 + sin A)]

= √[(1 + sin A)² / (1 - sin² A)]

Using the fundamental trigonometric identity sin² A + cos² A = 1, we can substitute 1 - sin² A = cos² A in the denominator:

= √[(1 + sin A)² / cos² A]

= (1 + sin A) / cos A

Dividing each term in the numerator by cos A gives:

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

= sec A + tan A

Explanation

The total number of students who chose Engineering is the sum of boys and girls in the Engineering row: 75 + 35 = 110

The number of girls who chose Engineering is 35.

The percentage of Engineering students who are girls is: (35 / 110) × 100 = 31.82%

Explanation

The total distance jogged during the week is calculated by summing the distances for each day:

Total distance = 0.5 + 3 + 0.5 + 2.5 + 2 + 3.5 + 1 = 13 km

The rate of burning calories is:

Rate = 400 calories ÷ 5 km = 80 calories per km

The total calories burned in the week is calculated as:

Total calories = Total distance × Rate = 13 km × 80 calories per km = 1040 calories

Explanation

First, calculate the total yearly expenses by summing up the costs of all individual items: Total Expenses = 9 + 7 + 4 + 3 + 3 = 26 Rs lakhs

Next, find the combined expense for Labour and Taxes: Labour and Taxes = 7 + 3 = 10 Rs lakhs

Finally, determine the percentage that this combined expense represents out of the total: Percentage = (10 / 26) × 100 = 38.46%

Explanation

Let CP be the cost price of the machine in lakhs.

The loss in the first case is given by the formula: Loss = CP - SP₁ Loss = CP - 55

The gain in the second case is given by the formula: Gain = SP₂ - CP Gain = 67 - CP

According to the problem, the gain is 5 times the loss: Gain = 5 × Loss 67 - CP = 5 × (CP - 55) 67 - CP = 5CP - 275 6CP = 342 CP = 57

Thus, the cost price of the machine is Rs 57 lakhs.

Explanation

The curved surface area of a cylinder is given by the formula CSA = 2 × π × r × h, where r is the radius of the base and h is the height.

The circumference of the base of a cylinder is given by the formula C = 2 × π × r.

By substituting the circumference into the curved surface area formula, we get: CSA = C × h

Given that the curved surface area is 528 sq cm and the circumference of the base is 44 cm: 528 = 44 × h

Solving for height: h = 528 ÷ 44 h = 12 cm

Explanation

Using the triple-angle identity for cosine: cos 3A = 4 cos³A - 3 cos A

We can rewrite 3 cos A using the identity sin²A + cos²A = 1: 3 cos A = 3 (1 - sin²A) cos A 3 cos A = 3 cos A - 3 sin²A cos A

But let us express cos 3A + 3 sin²A cos A: cos 3A + 3 sin²A cos A = (4 cos³A - 3 cos A) + 3 (1 - cos²A) cos A cos 3A + 3 sin²A cos A = 4 cos³A - 3 cos A + 3 cos A - 3 cos³A cos 3A + 3 sin²A cos A = cos³A

Explanation

Applying the algebraic identity (a - b)² + (a + b)² = 2(a² + b²) to the expression (1 - tan A)² + (1 + tan A)² gives:

(1 - tan A)² + (1 + tan A)² = 2(1² + tan² A) = 2(1 + tan² A)

Using the fundamental trigonometric identity sec² A = 1 + tan² A, we can substitute this to get:

2(1 + tan² A) = 2 sec² A

Explanation

To find the weight to height ratio, we divide the weight in kilograms by the height in meters. For Anjalika, this ratio is:

Ratio = 51 / 1.72 ≈ 29.65

Since 29.65 is the smallest ratio among all the four individuals, Anjalika has the least weight to height ratio.

Explanation

The percentage profit is the least in the year 2011, which is 5%.

The standard formula for profit percentage is: Profit % = (Revenue - Expenditure) / Expenditure × 100

Substituting the given values into the formula: 5 = (Revenue - 450) / 450 × 100

5/100 = (Revenue - 450) / 450

0.05 × 450 = Revenue - 450

22.5 = Revenue - 450

Revenue = 450 + 22.5 = 472.5 crore

Explanation

The average number of children per house is calculated by dividing the total number of children across all houses by the total number of houses in the society.

The total number of houses is: 7 + 9 + 20 + 3 = 39

The total number of children is: (0 × 7) + (1 × 9) + (2 × 20) + (3 × 3) = 0 + 9 + 40 + 9 = 58

Dividing the total number of children by the total number of houses yields: 58 ÷ 39 ≈ 1.487

Rounding this result to two decimal places gives 1.49.

Explanation

To find the sum of the three algebraic expressions, we group and add the like terms for x, y, and z.

For the x-terms: 4x + x - 2x = 3x

For the y-terms: 3y - y - 3y = -y

For the z-terms: -7z - z + 4z = -4z

Combining these terms gives the total sum of 3x - y - 4z.

Explanation

The average speed of the car is calculated using the formula Speed = Distance / Time. Given a distance of 342 km and a time of 1.9 hours, the speed in kilometers per hour is 342 / 1.9 which equals 180 km/h. To convert this speed into meters per second, multiply the value by the conversion factor of 5/18. This yields 180 × 5/18 which equals 50 m/s.

Explanation

The total surface area of a cube is given by the formula 6a², where a represents the length of one edge of the cube.

Given that the total surface area is 486 sq cm, we can set up the equation:

6a² = 486

Dividing both sides by 6:

a² = 81

Taking the square root of both sides gives the edge length:

a = 9 cm

A cube has 12 edges in total. The sum of the lengths of all its edges is given by the formula 12a:

Sum of edges = 12 × 9 = 108 cm

Explanation

The total profit of ₹104000 is shared among Ashok, Bimala, and Chirag in the ratio 6:4:3.

The sum of the ratio parts is 6 + 4 + 3 = 13.

Chirag's share of the profit is represented by 3 parts out of the total 13 parts.

Chirag's share = 104000 × 3/13 = 8000 × 3 = 24000.

Explanation

Combining the fractions using a common denominator, the expression becomes:

x = ((1 + sinA)² + cos²A) / (cosA(1 + sinA))

Expanding the squared term in the numerator:

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

Using the fundamental trigonometric identity sin²A + cos²A = 1, the numerator simplifies to:

x = (1 + 2sinA + 1) / (cosA(1 + sinA))

x = (2 + 2sinA) / (cosA(1 + sinA))

Factoring out 2 in the numerator:

x = 2(1 + sinA) / (cosA(1 + sinA))

Canceling the common term (1 + sinA) from both the numerator and the denominator:

x = 2 / cosA

Since 1/cosA = secA:

x = 2secA