SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The given compound inequality can be split into two separate inequalities: 2x - 2(x - 2) < 5 - x and 5 - x > -2x + 2

Solving the first inequality: 2x - 2x + 4 < 5 - x 4 < 5 - x x < 1

Solving the second inequality: 5 - x > -2x + 2 x > 2 - 5 x > -3

Combining the two solutions yields the interval: -3 < x < 1

Only the value 0 lies within this open interval.

Explanation

The effective discount on purchasing 7 items can be calculated directly using the weighted average formula, as the marked price of each item is constant:

Effective discount percentage = (n₁ × d₁ + n₂ × d₂) / (n₁ + n₂)

Here, Raksha buys 7 items, which is a combination of 2 items with a 16% discount and 5 items with a 44% discount.

n₁ = 2, d₁ = 16% n₂ = 5, d₂ = 44%

Substituting these values into the formula:

Effective discount percentage = (2 × 16 + 5 × 44) / (2 + 5) Effective discount percentage = (32 + 220) / 7 Effective discount percentage = 252 / 7 = 36%

Explanation

Given the linear equation: 6x - 5 = 4x + 7

We can solve for x by grouping the like terms: 6x - 4x = 7 + 5 2x = 12 x = 6

Substituting this value of x into the expression (x + 1)³: (6 + 1)³ = 7³ = 343

Explanation

Collinear points lie on the same straight line, which means the slope of the line segment AB is equal to the slope of the line segment BC.

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)

For points A(3, -2) and B(1, 4): Slope of AB = (4 - (-2)) / (1 - 3) = 6 / -2 = -3

For points B(1, 4) and C(-2, x): Slope of BC = (x - 4) / (-2 - 1) = (x - 4) / -3

Equating the two slopes because the points are collinear: -3 = (x - 4) / -3

Multiplying both sides by -3: 9 = x - 4

Solving for x: x = 13

Explanation

Let the number of selected and unselected candidates initially be 5x and x respectively. Thus, the total number of applied candidates is 5x + x = 6x.

According to the given conditions, if 100 less candidates had applied, the new total of applied candidates would be 6x - 100. If 20 less candidates were selected, the new number of selected candidates would be 5x - 20.

The new number of unselected candidates is the difference between the new total applied and the new selected candidates: New unselected = (6x - 100) - (5x - 20) = x - 80

The new ratio of selected to unselected candidates is given as 6:1: (5x - 20) / (x - 80) = 6 / 1 5x - 20 = 6(x - 80) 5x - 20 = 6x - 480 x = 460

Therefore, the total number of candidates who had applied for the process is: 6x = 6 × 460 = 2760

Explanation

Using the algebraic identity for the square of a binomial:

(x + y)² = x² + y² + 2xy

Substitute the given values x² + y² = 100 and xy = 48 into the equation:

(x + y)² = 100 + 2 × 48 (x + y)² = 100 + 96 (x + y)² = 196

Taking the square root on both sides:

x + y = √196 x + y = 14

Explanation

Two angles are supplementary when their sum is 180°. Let the measure of the angle be x. This means its supplementary angle is 180° - x.

According to the given condition, the angle is 1/3 of its supplementary angle: x = (180° - x) / 3

Multiplying both sides by 3 gives: 3x = 180° - x

Adding x to both sides yields: 4x = 180°

Dividing by 4 gives: x = 45°

Explanation

The given equation is (5/9 of x) - (2/5 of 9/4) = -4/5.

First, calculate the term (2/5 of 9/4): (2/5) × (9/4) = 18/20 = 9/10

Substitute this value back into the equation: (5/9) × x - 9/10 = -4/5

Add 9/10 to both sides of the equation to isolate the term containing x: (5/9) × x = -4/5 + 9/10

Convert -4/5 to a fraction with a denominator of 10: -4/5 = -8/10

Now substitute and add the terms: (5/9) × x = -8/10 + 9/10 = 1/10

Isolate x by multiplying both sides by 9/5: x = (1/10) × (9/5) = 9/50

To convert the fraction 9/50 to a decimal, multiply both the numerator and the denominator by 2: x = 18/100 = 0.18

Explanation

An isosceles trapezium is a quadrilateral in which the non-parallel sides are equal in length, and the base angles are equal. In an isosceles trapezium ABCD where AB is parallel to CD and AD = BC, the triangles ACD and BDC are congruent by the Side-Angle-Side (SAS) congruence criterion. This is because AD = BC, the base angles ∠ADC and ∠BCD are equal, and the side CD is common to both triangles. Since triangles ACD and BDC are congruent, their corresponding parts are equal, which means the diagonals AC and BD are equal in length (congruent).

Explanation

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

A 40% increase in the principal amount P over 10 years at simple interest means the interest earned is 0.40 × P. Substituting these values: 0.40 × P = (P × R × 10) / 100 0.40 = R / 10 R = 4%

The formula for compound interest CI is: CI = P × ((1 + R/100)ᵀ - 1)

For a principal of Rs. 30000, a rate of 4%, and a time period of 3 years: CI = 30000 × ((1 + 4/100)³ - 1) CI = 30000 × (1.04³ - 1) CI = 30000 × (1.124864 - 1) CI = 30000 × 0.124864 CI = 3745.92

Explanation

Let the number be x. According to the given condition, the difference between the number and thrice its reciprocal is 26/3. This can be written as the algebraic equation:

x - 3/x = 26/3

To find x, substitute the value 9 into the equation:

9 - 3/9 = 9 - 1/3 = 26/3

Since this satisfies the equation, the number is 9.

Explanation

The cross-section of the canal is a trapezium with parallel sides of length 4 m and 5 m, and a depth of 2 m.

The area of this trapezoidal cross-section is given by the formula: Area = ½ × (sum of parallel sides) × depth Area = ½ × (4 + 5) × 2 = 9 square meters

The maximum capacity of the canal is its volume, which is the product of its cross-sectional area and its length: Capacity = Area × length Capacity = 9 × 120 = 1080 cubic meters

Explanation

The GDP of the country at the beginning of 2015 is equivalent to the GDP of the country at the end of 2014.

To find this value, we only need to account for the GDP growth rate during the year 2014, which is given as 4% in the data table.

GDP at the beginning of 2015 = GDP at the end of 2013 × (1 + Growth rate in 2014 / 100)

Substituting the given values:

GDP at the beginning of 2015 = 5 × (1 + 4/100) = 5 × 1.04 = 5.2 trillion

Explanation

The coordinates of the midpoint of a line segment joining two points (x₁, y₁) and (x₂, y₂) are given by the formula ((x₁ + x₂) / 2, (y₁ + y₂) / 2).

Given that Q(-2, b) is the midpoint of segment EF with endpoints E(-7, -6) and F(a, 0), we can equate the corresponding coordinates to find the values of a and b.

For the x-coordinate: -2 = (-7 + a) / 2 -4 = -7 + a a = 3

For the y-coordinate: b = (-6 + 0) / 2 b = -3

This yields a = 3 and b = -3.

Explanation

The given equation is: 693.5 - (47.23 - x) - 82.48 = 8.172

Simplify the terms on the left side by expanding the parenthesis: 693.5 - 47.23 + x - 82.48 = 8.172

Combine the numerical constants: (693.5 - 47.23 - 82.48) + x = 8.172 (646.27 - 82.48) + x = 8.172 563.79 + x = 8.172

Isolate x by subtracting 563.79 from both sides of the equation: x = 8.172 - 563.79 x = -555.618

Explanation

Expanding the first two terms of the expression: (x + 7)(5 - 3x) = 5x - 3x² + 35 - 21x = -3x² - 16x + 35

Multiplying this quadratic expression by the third term: (-3x² - 16x + 35)(3x - 6)

To find the coefficient of x², we select the products from the multiplication that result in an x² term: (-3x² × -6) + (-16x × 3x) = 18x² - 48x² = -30x²

The coefficient of x² is -30.

Explanation

Let the time spent traveling by rail be t hours. Since the man spends 8 hours more on the steamer than on the rail, the time spent on the steamer is t + 8 hours.

Using the relationship distance = velocity × time: Distance by rail = 65 × t Distance by steamer = 35 × (t + 8)

The total distance traveled is 420 km, which gives the equation: 65t + 35(t + 8) = 420 65t + 35t + 280 = 420 100t = 140 t = 1.4 hours

The time spent on the steamer is: 1.4 + 8 = 9.4 hours

The distance covered by the steamer is: 35 × 9.4 = 329 km

Explanation

Using the trigonometric identity cos(-θ) = cos(θ), the given expression simplifies to:

cos(-300°) = cos(300°)

Since 300° lies in the fourth quadrant where the cosine function is positive, it can be written as:

cos(300°) = cos(360° - 60°) = cos(60°)

The standard trigonometric value for cos(60°) is ½. Therefore, the value of x is ½.

Explanation

To find the slope of the line 4x - 8y = 3, we can rewrite the equation in the slope-intercept form, y = mx + c, where m represents the slope of the line.

Rearranging the terms of the given equation: -8y = -4x + 3

Dividing the entire equation by -8: y = (-4/-8)x - 3/8 y = ½x - 3/8

Comparing this equation with y = mx + c, the slope m is ½.

Explanation

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

When the machine is sold for ₹53 lakhs, the loss incurred is: Loss = CP - 53

When the machine is sold for ₹64 lakhs, the gain obtained is: Gain = 64 - CP

According to the given condition, the gain is 10 times the former loss: Gain = 10 × Loss 64 - CP = 10 × (CP - 53)

Expanding and solving the linear equation: 64 - CP = 10CP - 530 11CP = 594 CP = 594 ÷ 11 CP = 54 lakhs

Explanation

Let the present ages of Rambha and Sarvesh be 8x and 5x respectively.

According to the given condition, after 7 years, the ratio of their ages becomes 5:4. This can be expressed as:

(8x + 7) / (5x + 7) = 5 / 4

By cross-multiplying to solve for x:

4(8x + 7) = 5(5x + 7)

32x + 28 = 25x + 35

7x = 7

x = 1

Rambha's present age is 8x, which is 8 × 1 = 8 years.

Explanation

In right-angled triangle PQR, right-angled at Q, with altitude QS on the hypotenuse PR, the triangles PQS and PRQ are similar. This similarity gives the projection theorem relation:

PQ² = PS × PR

Substituting the given values PQ = 4√29 cm and PS = 8 cm into the equation:

(4√29)² = 8 × PR

16 × 29 = 8 × PR

464 = 8 × PR

PR = 58 cm

The hypotenuse PR is composed of the segments PS and SR, such that PR = PS + SR. Substituting the known lengths yields:

58 = 8 + SR

SR = 50 cm

Explanation

A rectangle is defined as a quadrilateral with four right angles, meaning each of its four interior angles measures exactly 90°. Consecutive angles are any two adjacent angles that share a common side. Since all four interior angles in a rectangle have an equal measure of 90°, any pair of consecutive angles is equal in measure and thus congruent. Furthermore, the sum of these two consecutive angles is 90° + 90° = 180°, which by definition makes them supplementary.

Explanation

The fundamental trigonometric identity states that cosec²A - cot²A = 1. Rearranging this relation gives cosec²A - 1 = cot²A. Taking the square root on both sides yields √(cosec²A - 1) = cotA. Given that x = √(cosec²A - 1), substituting the identity directly results in x = cotA.

Explanation

Let the total work be the Least Common Multiple (LCM) of 15 and 20, which is 60 units.

The daily efficiency of A is 60 ÷ 15 = 4 units. The daily efficiency of B is 60 ÷ 20 = 3 units.

The combined efficiency of A and B working together is 4 + 3 = 7 units per day.

In 5 days, the total work completed by both is 7 × 5 = 35 units.

The remaining work is 60 - 35 = 25 units.

The fraction of the work left is 25/60, which simplifies to 5/12.

Explanation

Let the volume of milk purchased be 5 litres. The cost price of 5 litres of milk at Rs 30 per litre is 5 × 30 = Rs 150.

The milkman adds 1/5 of water to this milk. Quantity of water added = 1/5 × 5 = 1 litre. Total quantity of the mixture = 5 + 1 = 6 litres.

He sells the mixture at Rs 32 per litre. The selling price of 6 litres of mixture is 6 × 32 = Rs 192.

Total profit earned = Selling Price - Cost Price = 192 - 150 = Rs 42. Gain percentage = (Profit / Cost Price) × 100 = (42 / 150) × 100 = 28%

Explanation

The area of a circle is directly proportional to the square of its radius, given by the formula A = πr².

When the radius of the circle is increased by 21%, the new radius becomes 1.21 times the original radius.

Using the successive percentage change formula, the net percentage increase in the area is calculated as:

Net percentage increase = x + y + (x × y)/100

Substituting x = 21 and y = 21:

Net percentage increase = 21 + 21 + (21 × 21)/100 = 42 + 441/100 = 42 + 4.41 = 46.41%

Alternatively, if the original area is A = πr², the new area with a radius of 1.21r is π(1.21r)² = 1.4641πr². The fractional increase in the area is 1.4641 - 1 = 0.4641, which corresponds to an increase of 46.41%.

Explanation

The total length of the capsule is 40 m and its radius is 6 m. Since the capsule has hemispherical ends of the same radius, the length of the cylindrical part (h) is calculated by subtracting the radius of both hemispherical ends from the total length: h = 40 - 2 × 6 = 28 m

The total surface area of the capsule is the sum of the curved surface area of the cylinder and the surface areas of the two hemispherical ends. Two hemispherical ends of radius r combine to form a sphere of radius r. Thus, the total surface area (A) is: A = 2 × π × r × h + 4 × π × r² A = 2 × π × r × (h + 2 × r)

Substituting the values r = 6 m, h = 28 m, and π = 22/7: A = 2 × (22/7) × 6 × (28 + 2 × 6) A = 2 × (22/7) × 6 × 40 A = 10560 / 7 A ≈ 1508.57 sq mts

Explanation

Using the algebraic identity for the difference of squares, a² - b² = (a - b)(a + b), we can rewrite the expression cos⁴A - sin⁴A as:

cos⁴A - sin⁴A = (cos²A - sin²A)(cos²A + sin²A)

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

cos⁴A - sin⁴A = cos²A - sin²A

Substituting sin²A = 1 - cos²A into the expression yields:

cos²A - (1 - cos²A) = 2cos²A - 1

Since we are given x = 2cos²A - 1, it follows that x = cos⁴A - sin⁴A.

Explanation

Since the total number of students in the class is 56 and there are 28 girls, the number of boys is 56 - 28 = 28. Because the number of boys and girls is equal, the average weight of the entire class is the simple arithmetic mean of the average weight of the girls and the average weight of the boys.

Using the formula for the weighted average where the group sizes are equal:

Average weight of class = (Average weight of girls + Average weight of boys) / 2

Substituting the given values:

51 = (46 + Average weight of boys) / 2

Multiplying both sides by 2:

102 = 46 + Average weight of boys

Subtracting 46 from both sides gives:

Average weight of boys = 102 - 46 = 56 kg

Explanation

Let the present ages of the father and the son be 7x and 2x respectively.

According to the given condition, five years ago, the product of their ages was 150: (7x - 5)(2x - 5) = 150

Expanding the equation: 14x² - 35x - 10x + 25 = 150 14x² - 45x - 125 = 0

Solving this quadratic equation by factoring: 14x² - 70x + 25x - 125 = 0 14x(x - 5) + 25(x - 5) = 0 (14x + 25)(x - 5) = 0

Since age cannot be negative, x must be a positive integer, which gives: x = 5

Substituting this value, the present age of the father is 7x = 7 × 5 = 35 years.

Explanation

Let the three consecutive natural numbers be n, n+1, and n+2, where n represents the smallest number.

The square of the greatest number is (n+2)² and the product of the other two numbers is n(n+1).

According to the given condition: (n+2)² - n(n+1) = 19

Expanding and simplifying the algebraic expression: n² + 4n + 4 - n² - n = 19 3n + 4 = 19 3n = 15 n = 5

Explanation

The simple interest is calculated using the formula: Simple Interest = (Principal × Rate × Time) / 100.

For Jaipal: Principal = Rs. 8400, Time = 15 years, Rate = R% Simple Interest from Jaipal = (8400 × R × 15) / 100 = 1260 R

For Kareem: Principal = Rs. 5100, Time = 14 years, Rate = R% Simple Interest from Kareem = (5100 × R × 14) / 100 = 714 R

Total interest received is the sum of interest from both: Total Interest = 1260 R + 714 R = 1974 R

Given that the total interest is Rs. 17766: 1974 R = 17766 R = 17766 / 1974 R = 9

The rate of interest per annum is therefore 9 percent.

Explanation

The weight to height ratio is calculated using the formula: Ratio = Weight ÷ Height

Substituting the values from the table:

Amrita ratio = 53 ÷ 1.7 ≈ 31.18

Anagha ratio = 78 ÷ 1.55 ≈ 50.32

Anamika ratio = 52 ÷ 1.63 ≈ 31.90

Anandi ratio = 54 ÷ 1.62 ≈ 33.33

Comparing these computed ratios, Amrita possesses the lowest weight to height ratio of approximately 31.18.

Explanation

The least profit percentage from the given table is 5% in the year 2013.

The relationship between Revenue, Expenditure, and Profit percentage is given by the formula: Revenue = Expenditure × (1 + Profit % / 100)

Substituting the given values into the formula: Revenue = 400 × (1 + 5 / 100) Revenue = 400 × 1.05 Revenue = 420 crore

Explanation

The average number of children per house is calculated by dividing the total number of children by the total number of houses.

Total houses = 6 + 15 + 14 + 7 = 42

Total children = (0 × 6) + (1 × 15) + (2 × 14) + (3 × 7) = 0 + 15 + 28 + 21 = 64

Average = 64 / 42 ≈ 1.52

Explanation

The total number of shares issued by the company is six lakhs, which equals 6,00,000 shares.

Dhiraj currently holds a 30% share in the company.

The present number of shares held by Dhiraj is calculated as: 30% of 6,00,000 = 30/100 × 6,00,000 = 1,80,000 shares

When Chinmay sells 15,000 of his shares to Dhiraj, the number of shares held by Dhiraj increases by 15,000.

The new total of shares for Dhiraj is: 1,80,000 + 15,000 = 1,95,000 shares

Explanation

Using the rule of alligation, the ratio of wheat sold at 7% profit to the wheat sold at 17% profit is:

(17 - 11) : (11 - 7) = 6 : 4 = 3 : 2

The total quantity of wheat is 6 quintals. Since 1 quintal is equal to 100 kg, the total wheat is 600 kg.

The quantity of wheat sold at 17% profit is:

(2 / 5) × 600 kg = 240 kg

Explanation

The baking rate of Badki is 45 / 9 = 5 cakes per hour.

The combined baking rate of Badki and Chutki together is 80 / 10 = 8 cakes per hour.

Subtracting Badki's rate from the combined rate gives Chutki's baking rate: 8 - 5 = 3 cakes per hour.

In 40 hours, the number of cakes Chutki can bake is 3 × 40 = 120.

Explanation

The average of 5 numbers is 72, which means the sum of all 5 numbers is: 5 × 72 = 360

Let the first number be x and the sum of the other 4 numbers be S. Since the first number is 1/8 times the sum of the other 4 numbers: x = S / 8 S = 8 × x

The total sum of the 5 numbers is the sum of the first number and the remaining 4 numbers: x + S = 360

Substituting S = 8 × x into the equation: x + 8 × x = 360 9 × x = 360 x = 360 ÷ 9 = 40

Explanation

Let the number be x. According to the problem: (6/7) × (8/5) × x = 192

Simplifying the equation: (48/35) × x = 192

To find x, multiply both sides by the reciprocal: x = 192 × (35/48)

Since 192 divided by 48 is 4, we get: x = 4 × 35 x = 140

To find 3/4th of this number: (3/4) × 140 = 3 × 35 = 105

Explanation

By the geometric mean theorem (also known as the right triangle altitude theorem), the altitude drawn from the right-angle vertex of a right-angled triangle to the hypotenuse is the geometric mean of the two segments of the hypotenuse.

The relationship is: BD² = AD × DC

Substitute the given lengths AD = 8 cm and DC = 18 cm into the formula: BD² = 8 × 18 BD² = 144 BD = √144 = 12 cm

Explanation

The combined work rate of painters A, B, and C is the sum of their individual work rates. Let the number of days C takes to complete the job alone be x.

Work rate of A = 1/50 of the house per day Work rate of B = 1/25 of the house per day Work rate of C = 1/x of the house per day Combined work rate of A, B, and C = 1/10 of the house per day

Setting up the work rate equation: 1/50 + 1/25 + 1/x = 1/10

Solving for 1/x by using a common denominator of 50: 1/x = 1/10 - 1/50 - 1/25 1/x = 5/50 - 1/50 - 2/50 1/x = 2/50 1/x = 1/25

This gives x = 25 days.

Explanation

The first inequality is 4 + 2x ≤ 6 + x. Subtracting x and 4 from both sides yields: x ≤ 2

The second inequality is 2x + 5 < 2 + 4x. Subtracting 2x and 2 from both sides yields: 3 < 2x x > 1.5

Combining these two inequalities, the range for x is: 1.5 < x ≤ 2

Therefore, the only value that satisfies both inequalities is 2.

Explanation

Divide the quadratic equation 4x² + 6x - 18 = 0 by 2 to simplify it: 2x² + 3x - 9 = 0

To find the roots, factor the quadratic expression by splitting the middle term. The two numbers that multiply to 2 × (-9) = -18 and add up to 3 are 6 and -3.

2x² + 6x - 3x - 9 = 0 2x(x + 3) - 3(x + 3) = 0 (2x - 3)(x + 3) = 0

Solving for x by setting each factor to zero: 2x - 3 = 0 → x = 3/2 x + 3 = 0 → x = -3

These values of x are the roots of the equation.

Explanation

To find the percentage increase required to restore the original value after a decrease of d%, we use the percentage restoration formula:

Percentage Increase = (d / (100 - d)) × 100 %

Given that the price of the article is cut by 21%, we substitute d = 21 into the formula:

Percentage Increase = (21 / (100 - 21)) × 100 % = (21 / 79) × 100 % = 2100 / 79 % ≈ 26.58 %

This is the increase required on the reduced price to return to the original price.

Explanation

Let Haashim's salary be H.

Based on the given information, Gafur's salary (G) is (4/3)H and Satish's salary (S) is (5/4)H.

To find the ratio of Gafur's salary to Satish's salary, divide Gafur's salary by Satish's salary:

Ratio = G / S = ((4/3)H) / ((5/4)H) = (4/3) / (5/4) = (4/3) × (4/5) = 16/15

This simplifies to the ratio 16:15.

Explanation

To find the value of x for which the expressions are equal, we set them equal to each other:

5x + 17 = 17x - 5

Rearranging the terms to group the variables on one side and the constants on the other gives:

17 + 5 = 17x - 5x

22 = 12x

Solving for x:

x = 22/12

Simplifying the fraction by dividing the numerator and the denominator by 2 yields:

x = 11/6

Explanation

We can simplify the given expression by taking the least common multiple of the denominators:

x = cosecA / (cosecA - 1) + cosecA / (cosecA + 1)

x = [cosecA(cosecA + 1) + cosecA(cosecA - 1)] / [(cosecA - 1)(cosecA + 1)]

x = (cosec²A + cosecA + cosec²A - cosecA) / (cosec²A - 1)

x = 2cosec²A / (cosec²A - 1)

Using the trigonometric identity cosec²A - 1 = cot²A:

x = 2cosec²A / cot²A

Expressing cosecA and cotA in terms of sinA and cosA:

x = (2 / sin²A) / (cos²A / sin²A)

x = 2 / cos²A

x = 2sec²A

Explanation

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

Substituting a = 2x and b = y into this identity: (2x - y)² = (2x)² - 2(2x)(y) + y² (2x - y)² = 4x² - 4xy + y²