SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

Let the total electricity bill be x.

If 25% of the bill is deducted, the remaining portion to be paid is 100% - 25% = 75%.

According to the given information, this 75% is equal to Rs 1788. We can write the relation as:

75% of x = 1788

0.75 × x = 1788

x = 1788 / 0.75

x = 1788 × 4/3

x = 596 × 4

x = 2384

Explanation

Four years ago, the sum of their ages was: 60 - 4 - 4 = 52 years

Since the ratio of their ages four years ago was 7:6, the total ratio parts equal 7 + 6 = 13.

The value of each ratio part is: 52 / 13 = 4 years

The age of the husband four years ago was: 7 × 4 = 28 years

The present age of the husband is: 28 + 4 = 32 years

Explanation

Solve the first inequality: 5x - 1 < 3x + 2 2x < 3 x < 3/2

Solve the second inequality: 5x + 5 > 6 - 2x 7x > 1 x > 1/7

Combining the two inequalities gives the range for x: 1/7 < x < 3/2

The value 1 satisfies this condition since 1/7 < 1 < 1.5.

Explanation

The formula for the n-th term of an arithmetic progression is aₙ = a + (n - 1)d, where a is the first term and d is the common difference.

Using the given terms: a₅ = a + 4d = 9 a₁₂ = a + 11d = -26

Subtracting the first equation from the second equation: 7d = -35 d = -5

Substituting d = -5 into the equation for the 5th term: a + 4(-5) = 9 a = 29

To find which term corresponds to -6, we use the n-th term formula: aₙ = a + (n - 1)d = -6 29 + (n - 1)(-5) = -6 (n - 1)(-5) = -35 n - 1 = 7 n = 8

Explanation

The distance d between two points (x₁, y₁) and (x₂, y₂) in a Cartesian plane is given by the standard distance formula:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Substituting the coordinates (11, -5) and (-1, 0) into the formula:

d = √((-1 - 11)² + (0 - (-5))²) d = √((-12)² + 5²) d = √(144 + 25) d = √169 d = 13 units

Explanation

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

y - y₁ = m(x - x₁)

Substituting the slope m = 4/3 and the point (x₁, y₁) = (-7, 2):

y - 2 = (4/3)(x - (-7))

y - 2 = (4/3)(x + 7)

Multiplying both sides by 3 to eliminate the fraction:

3(y - 2) = 4(x + 7)

3y - 6 = 4x + 28

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

4x - 3y = -34

Explanation

In triangle PQR, ST is parallel to QR with S on PQ and T on PR. This makes triangle PST similar to triangle PQR because angle PST is equal to angle PQR and angle PTS is equal to angle PRQ as they are corresponding angles, while angle P is common to both triangles.

For similar triangles, the ratio of corresponding sides is equal, which gives: ST/QR = PS/PQ

The point S divides the side PQ in the ratio 1:5, which means: PS/SQ = 1/5

The total length of PQ is the sum of PS and SQ. Therefore, the ratio of PS to the full side PQ is: PS/PQ = PS/(PS + SQ) = 1/(1 + 5) = 1/6

Substituting this ratio into the similarity relation: ST/QR = 1/6

Using the given length of QR which is 15 cm, we find: ST/15 = 1/6 ST = 15/6 ST = 2.5 cm

Explanation

In compound interest, the amount accumulated at the end of any year earns interest for the subsequent year. Therefore, the relation between the amounts at the end of the 2nd year (A₂) and the 3rd year (A₃) with an annual interest rate r is:

A₃ = A₂ × (1 + r/100)

Given A₂ = Rs 16854 and A₃ = Rs 17865.24:

17865.24 = 16854 × (1 + r/100)

The interest earned in the 3rd year is the difference between these two amounts:

Interest = 17865.24 - 16854 = 1011.24

Using this, the rate of interest is computed as:

r = (1011.24 / 16854) × 100

r = 101124 / 16854

r = 6

Explanation

Let Meghnad's efficiency be E. Since he completes the work in 45 hours, the total work is represented as 45E.

Jayshri is 50% more efficient than Meghnad, which means Jayshri's efficiency is E + 0.5E = 1.5E.

When working together, their combined efficiency is the sum of their individual efficiencies: E + 1.5E = 2.5E

The time taken by both of them to complete the work together is calculated by dividing the total work by their combined efficiency: 45E / 2.5E = 45 / 2.5 = 18 hours

Explanation

The revenue of the sixth year is counted in both the first six years and the last six years. Therefore, its value can be calculated by subtracting the total revenue of all eleven years from the sum of the revenues of the first six years and the last six years.

Total revenue of 11 years = 11 × 77 = 847 lakhs Total revenue of first 6 years = 6 × 72 = 432 lakhs Total revenue of last 6 years = 6 × 84 = 504 lakhs

Revenue of the sixth year = (Total revenue of first 6 years + Total revenue of last 6 years) - Total revenue of 11 years Revenue of the sixth year = (432 + 504) - 847 Revenue of the sixth year = 936 - 847 = 89 lakhs

Explanation

In triangle ABC, AD is the median to the side CB. Since D is the midpoint of CB and CB = 16 cm, we have:

BD = CD = CB / 2 = 16 / 2 = 8 cm

We are given that the length of the median AD is 8 cm. Therefore:

AD = BD = CD = 8 cm

In triangle ABD, since AD = BD, the opposite angles are equal: angle BAD = angle ABD = x

In triangle ACD, since AD = CD, the opposite angles are equal: angle CAD = angle ACD = y

The sum of the angles in triangle ABC is: angle ABC + angle ACB + angle CAB = 180° x + y + (x + y) = 180° 2(x + y) = 180° x + y = 90°

Since angle CAB = x + y, the measure of angle CAB is 90°.

Explanation

The third proportional c of two numbers a and b is defined by the relation a/b = b/c, which gives: c = b²/a

Substituting the given values, a = 12 and b = 48: c = 48² / 12 c = (48 × 48) / 12 c = 4 × 48 c = 192

Explanation

The curved surface area of a cylinder is given by the formula: Curved Surface Area = 2 × π × r × h

where r is the radius and h is the height of the cylinder. Using the approximation π = 22/7 and substituting the given values: 616 = 2 × (22/7) × r × 14

Simplifying the equation: 616 = 2 × 22 × r × 2 616 = 88 × r

Solving for the radius r: r = 616 / 88 r = 7 cm

Explanation

Let the marks obtained by the two students be x and y, where x is the higher mark.

From the given conditions: x - y = 22

Also, the higher mark is 55% of the sum of their marks: x = 0.55(x + y) x = 0.55x + 0.55y 0.45x = 0.55y x/y = 55/45 = 11/9

Let the marks of the students be 11k and 9k. Using the difference equation: 11k - 9k = 22 2k = 22 k = 11

Substituting the value of k: x = 11 × 11 = 121 y = 9 × 11 = 99

Explanation

Using the trigonometric identity cot(2π - θ) = -cot θ, the given expression can be written as:

cot 11π/6 = cot(2π - π/6) = -cot(π/6)

Since cot(π/6) = √3, the expression evaluates to -√3.

Explanation

By using the standard trigonometric sum-to-product identity for the difference of two sine functions, we have:

sin C - sin D = 2 cos((C + D)/2) sin((C - D)/2)

This relationship can be derived using the angle addition and subtraction formulas for sine:

sin(x + y) = sin x cos y + cos x sin y sin(x - y) = sin x cos y - cos x sin y

Subtracting the second equation from the first gives:

sin(x + y) - sin(x - y) = 2 cos x sin y

By setting C = x + y and D = x - y, we solve for x and y to get x = (C + D)/2 and y = (C - D)/2. Substituting these values back into the equation yields the expression 2 cos((C + D)/2) sin((C - D)/2) which is equal to sin C - sin D.

Explanation

Using the fundamental trigonometric identity sin²A = 1 - cos²A, the expression tan²Asin²A can be rewritten by substituting sin²A:

tan²Asin²A = tan²A (1 - cos²A)

Expanding the terms gives:

tan²Asin²A = tan²A - tan²A cos²A

Since tan A = sin A / cos A, squaring both sides gives tan²A = sin²A / cos²A. Substituting this into the second term of the expression:

tan²Asin²A = tan²A - (sin²A / cos²A) × cos²A

Simplifying the second term by cancelling cos²A yields:

tan²Asin²A = tan²A - sin²A

Explanation

The average bonus is calculated by dividing the total bonus paid to all employees by the total number of employees.

First, determine the total number of employees: Total employees = 1 + 6 + 2 = 9

Next, calculate the total bonus for each job category in lakhs: Bonus for Managers = 1 × 36 lakhs × 60% = 21.6 lakhs Bonus for Executives = 6 × 8 lakhs × 30% = 14.4 lakhs Bonus for Trainees = 2 × 2 lakhs × 20% = 0.8 lakhs

Summing these amounts gives the total bonus for all employees: Total bonus = 21.6 + 14.4 + 0.8 = 36.8 lakhs Converting lakhs to rupees, 36.8 lakhs equals 3,680,000 rupees.

Now, divide the total bonus by the total number of employees to find the average bonus: Average bonus = 3,680,000 / 9 = 408,888.89 rupees

Rounding to the nearest whole rupee yields 408,889 rupees.

Explanation

The total value of exports is computed by adding the values of all listed categories:

Total exports = 950 + 525 + 725 + 525 + 950 = 3675 million USThe export value of Electronics is 950 million US.

The percentage of Electronics exports relative to the total is calculated using the formula:

Percentage = (Electronics value / Total value) × 100

Percentage = (950 / 3675) × 100 ≈ 25.85%

Explanation

The total number of machines purchased by the factory is 8. The number of Machine-C bought is 8 - (3 + 4) = 1.

To find the average cost, first calculate the total cost of all the machines: Total Cost = (3 × 45000) + (4 × 25000) + (1 × 35000) Total Cost = 135000 + 100000 + 35000 = 270000

The formula for the average cost of the machines is: Average Cost = Total Cost / Total number of machines Average Cost = 270000 / 8 = 33750

Explanation

Using the ratio method, the present ratio of the ages of Rangitha and Shaheen is 6:5, and after 12 years, the ratio becomes 7:6.

Since the difference between the ratio terms in both cases is the same (6 - 5 = 1 and 7 - 6 = 1), the increase in ratio units can be directly equated to the number of years.

The change in units for each person is: 7 - 6 = 1 unit

This 1 unit change corresponds directly to the 12 years that have passed: 1 unit = 12 years

Therefore, Rangitha's present age, which is represented by 6 units, is: 6 × 12 = 72 years

Explanation

The cosecant function is the reciprocal of the sine function, defined as csc θ = 1 / sin θ.

For θ = 240°: csc 240° = 1 / sin 240°

The angle 240° lies in the third quadrant, where the sine function is negative. Using the trigonometric identity sin(180° + θ) = -sin θ: sin 240° = sin(180° + 60°) = -sin 60°

The value of sin 60° is √3/2, which gives: sin 240° = -√3/2

Substituting this back into the cosecant relation: x = 1 / (-√3/2) = -2/√3

Explanation

Following the BODMAS rule, the expression is evaluated step-by-step:

First, simplify the innermost bracket: 1.3 + 3.7 = 5

Next, simplify the expression within the square brackets: -0.3 × 5 = -1.5

Then, evaluate the "of" operation: -1.5 of 0.8 = -1.5 × 0.8 = -1.2

Finally, perform the outer multiplication: 2 × (-1.2) = -2.4

Explanation

Let the speed of the goods train be S km/hr.

The goods train passes the station and travels for 9 hours before the second train reaches the station. After that, the second train takes 3 hours to overtake the goods train. This means the goods train travels for a total of 9 + 3 = 12 hours from the station to the point of overtake.

Since the second train travels for 3 hours at a speed of 72 km/hr to reach the same overtake point, the distance covered by both trains from the station is equal.

Using the formula Distance = Speed × Time:

Distance covered by the goods train = Distance covered by the second train S × 12 = 72 × 3 S = (72 × 3) / 12 S = 6 × 3 S = 18 km/hr

Explanation

The selling price (SP) is directly proportional to its corresponding percentage of the cost price (CP).

At a 5% loss, the first selling price of ₹1,230 represents 95% of the CP.

Let P% be the percentage of the CP that the new selling price of ₹1,353 represents:

P% = (SP₂ / SP₁) × 95%

Substituting the values:

P% = (1353 / 1230) × 95%

Since 1353 / 1230 = 1.1, we can compute:

P% = 1.1 × 95% = 104.5%

Since the new selling price represents 104.5% of the CP, the net result is:

104.5% - 100% = 4.5% gain

Explanation

Using the fundamental trigonometric identity: sin²A + cos²A = 1

We can rewrite this to express cos²A as: 1 - sin²A = cos²A

Taking the square root on both sides gives: √(1 - sin²A) = cosA

Substituting this expression back into the given equation: x = 1/√(1 - sin²A) x = 1/cosA

Using the reciprocal identity: 1/cosA = secA

This yields: x = secA

Explanation

Total work can be assumed as the Least Common Multiple (LCM) of 15 and 20, which is 60 units.

The daily work rate of A is 60 / 15 = 4 units per day. The daily work rate of B is 60 / 20 = 3 units per day.

When working together, their combined daily work rate is 4 + 3 = 7 units per day.

In 6 days of working together, the work completed is 7 × 6 = 42 units.

The remaining work is 60 - 42 = 18 units.

The fraction of work left is 18 / 60, which simplifies to 3/10.

Explanation

Multiplying the numerator and the denominator of the given expression by (1 + cos a), we obtain:

((1 + cos a) × (1 + cos a)) / ((1 - cos a) × (1 + cos a)) = (1 + cos a)² / (1 - cos² a)

Using the trigonometric identity 1 - cos² a = sin² a, the expression becomes:

(1 + cos a)² / sin² a

This can be written as:

((1 + cos a) / sin a)²

Dividing each term in the numerator by sin a yields:

(1/sin a + cos a/sin a)²

Applying the reciprocal and quotient identities, where 1/sin a = cosec a and cos a/sin a = cot a, the expression simplifies to:

(cosec a + cot a)²

Explanation

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

M(x, y) = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Given the points C(5, -2) and D(-1, 8), substitute these coordinates into the formula:

x-coordinate = (5 + (-1)) / 2 = 4 / 2 = 2

y-coordinate = (-2 + 8) / 2 = 6 / 2 = 3

Thus, the coordinates of the midpoint are (2, 3).

Explanation

The slope of the first line passing through (-3,4) and (0,3) is -1/3. For the two lines to be perpendicular, the slope of the second line must be 3. Using the slope formula with points (5,7) and (4,x) gives (x - 7)/(4 - 5) = 3, which simplifies to x = 4.

Explanation

Solving the two inequalities, we get 2(3x - 4) - 2 < 4x - 2 which simplifies to x < 4, and 4x - 2 >= 2x - 4 which simplifies to x >= -1. Combining these gives the range -1 <= x < 4, and among the given options, only 2 (Option A) falls within this range.

Explanation

First, expand the squared term using the identity (a + b)² = a² + 2ab + b²: (3x + 2)² = 9x² + 12x + 4

Next, multiply this polynomial by (4x - 3): (9x² + 12x + 4)(4x - 3) = 9x²(4x - 3) + 12x(4x - 3) + 4(4x - 3) = 36x³ - 27x² + 48x² - 36x + 16x - 12 = 36x³ + 21x² - 20x - 12

Finally, subtract the second expression (2x³ - 9x² + 12x - 24) from the expanded term: (36x³ + 21x² - 20x - 12) - (2x³ - 9x² + 12x - 24) = 36x³ + 21x² - 20x - 12 - 2x³ + 9x² - 12x + 24

Combine the like terms: x³ terms: 36x³ - 2x³ = 34x³ x² terms: 21x² + 9x² = 30x² x terms: -20x - 12x = -32x Constant terms: -12 + 24 = 12

Combining these results yields: 34x³ + 30x² - 32x + 12

Explanation

Let the measure of the angle be x.

Two angles are supplementary when their sum is 180°. Therefore, the supplementary angle of x is 180° - x.

According to the given condition, the angle is smaller than its supplementary angle by 50°:

(180° - x) - x = 50°

180° - 2x = 50°

2x = 180° - 50°

2x = 130°

x = 65°

Explanation

The customer makes two deposits of Rs. 2000 each at different times of the year, with interest compounded half-yearly at a rate of 10% per half-year.

For the first deposit of Rs. 2000 made on 1st January, interest is earned for two half-years: Interest for the first half-year (Jan to July) = 10% of Rs. 2000 = Rs. 200 Amount at the end of the first half-year = Rs. 2000 + Rs. 200 = Rs. 2200

On 1st July, a second deposit of Rs. 2000 is made, which joins the accumulated amount of the first deposit. Total principal for the second half-year = Rs. 2200 + Rs. 2000 = Rs. 4200

Interest for the second half-year (July to Dec) = 10% of Rs. 4200 = Rs. 420

Total interest gained by the end of the year = Rs. 200 + Rs. 420 = Rs. 620

Explanation

The volume of water displaced by the completely immersed sphere is equal to the volume of the sphere itself.

Let the rise in the water level be h.

The volume of water displaced in the cylindrical vessel of radius R is given by: Volume of cylinder = πR²h

The volume of the sphere of radius r is given by: Volume of sphere = (4/3)πr³

Equating the two volumes: πR²h = (4/3)πr³ R²h = (4/3)r³

Substituting the given values, R = 7 cm and r = 6 cm: 7² × h = (4/3) × 6³ 49 × h = (4/3) × 216 49 × h = 4 × 72 49 × h = 288 h = 288 / 49 h ≈ 5.88 cm

Explanation

A rhombus is a parallelogram with all four sides of equal length. In any parallelogram, the consecutive interior angles lie between two parallel lines intersected by a transversal side. According to the properties of parallel lines, consecutive interior angles are supplementary, meaning their sum is always equal to 180°. Therefore, consecutive angles in a rhombus are supplementary.

Explanation

By using the algebraic identity: a³ - b³ = (a - b)³ + 3ab(a - b)

We substitute the given values, a - b = 3 and ab = 28, directly into the formula: a³ - b³ = 3³ + 3 × 28 × 3 a³ - b³ = 27 + 252 a³ - b³ = 279

Explanation

Cost of purchasing daily tickets for 26 days is calculated as: Cost of daily tickets = 26 × ₹30 = ₹780

Cost of the Monthly Pass is given as ₹670.8.

The savings obtained by purchasing the Monthly Pass instead of daily tickets is: Savings = Cost of daily tickets - Cost of Monthly Pass Savings = ₹780 - ₹670.8 = ₹109.2

The savings percentage is calculated based on the total cost of daily tickets: Savings percentage = (Savings ÷ Cost of daily tickets) × 100 Savings percentage = (109.2 ÷ 780) × 100 = 14%

Explanation

Let the number be x.

An increase of the number by 18 makes it 106% of itself, which means the increase of 18 corresponds to a 6% increase of the number.

This relation can be written as: 6% of x = 18 0.06 × x = 18 x = 18 / 0.06 x = 300

Explanation

The product of the two fractions 2/3 and -3/5 is: (2/3) × (-3/5) = -6/15 = -2/5

The sum of the two fractions is: 2/3 + (-3/5) = (2 × 5 - 3 × 3) / 15 = (10 - 9) / 15 = 1/15

Since both the product of -2/5 and the sum of 1/15 are satisfied, these two fractions are correct.

Explanation

To find the average bonus, we first calculate the total bonus paid to all employees. The total bonus is (2 * 60 lakhs * 50%) + (8 * 20 lakhs * 20%) + (4 * 4 lakhs * 10%) = 60 + 32 + 1.6 = 93.6 lakhs (or 9,360,000 rupees). Dividing this by the total number of employees (2 + 8 + 4 = 14) gives an average bonus of approximately 668,571 rupees.

Explanation

The total exports from the three companies together for each year are calculated by summing the export values of Company A, Company B, and Company C.

For the year 2013, the total exports are: 4000 + 3000 + 3000 = 10000 lakh rupees

For the year 2014, the total exports are: 4000 + 2000 + 4000 = 10000 lakh rupees

Both 2013 and 2014 have equal total exports of 10000 lakh rupees.

Explanation

To find the total profit or loss of the company over the last five years, we sum the values for each year from 2011 to 2015:

Total = 5 + (-15) + (-15) + 15 + (-10)

Combining the positive and negative values:

Total = (5 + 15) - (15 + 15 + 10) Total = 20 - 40 Total = -20

The negative sign represents a net loss, which equals a loss of Rs 20 crores.

Explanation

To find the percentage of steel exports relative to the total exports, the formula is:

Percentage of Steel = (Steel Exports / Total Exports) × 100

First, calculate the total value of exports by summing the individual values from the table:

Total Exports = 750 + 725 + 700 + 700 + 725 = 3600 million USThe export value of Steel is 700 million US.

Substitute these values into the formula:

Percentage of Steel = (700 / 3600) × 100 = 700 / 36 = 19.44%

Explanation

The standard trigonometric values are cos 45° = 1/√2 and sec 60° = 2.

Substituting these values into the given equation: x = cos 45° - sec 60° x = 1/|2 - 2

By taking the common denominator, this simplifies to: x = (1 - 2√2)/√2

Explanation

Let the time spent traveling by rail be t hours. Since the man spends 9 hours more on the steamer, the time spent traveling by steamer is t + 9 hours.

Using the relationship Distance = Speed × Time, we can express the distances covered by each mode of transport:

Distance by rail = 70 × t Distance by steamer = 30 × (t + 9)

The total distance traveled is 400 km. Setting up the equation:

70t + 30(t + 9) = 400 70t + 30t + 270 = 400 100t = 130 t = 1.3 hours

The time spent on the steamer is:

t + 9 = 1.3 + 9 = 10.3 hours

The distance covered by the steamer is:

30 × 10.3 = 309 km

Explanation

Let the cost price of the machine be C lakhs.

When the machine is sold for Rs 48 lakhs, the loss incurred is: Loss = C - 48

When the machine is sold for Rs 60 lakhs, the gain obtained is: Gain = 60 - C

According to the given condition, the gain is 5 times the former loss: Gain = 5 × Loss

Substituting the expressions into this relationship: 60 - C = 5 × (C - 48) 60 - C = 5C - 240 6C = 300 C = 50 lakhs

Explanation

Using the standard trigonometric double-angle formula for cosine: cos A = 2 cos²(A/2) - 1

By rearranging this equation to isolate the term cos²(A/2), we add 1 to both sides: 1 + cos A = 2 cos²(A/2)

Dividing both sides by 2 results in: (1 + cos A)/2 = cos²(A/2)

Given that x = (1 + cos A)/2, substituting this value gives: x = cos²(A/2)

Explanation

Let the three numbers be x, y, and z.

The sum of the three numbers whose average is 112 is: x + y + z = 112 × 3 = 336

The first number x is 3/13 times the sum of the other two numbers: x = (3/13) × (y + z)

This can be rewritten to express the sum of the other two numbers in terms of x: y + z = 13x / 3

Substituting this into the sum equation: x + 13x / 3 = 336 16x / 3 = 336 x = (336 × 3) / 16 x = 21 × 3 x = 63

Explanation

Let the list price of the article be L and the cost price be C.

When the discount is 20%, the selling price is: S₁ = L × (1 - 0.20) = 0.80 × L

Since there is a loss of 10% on the cost price: S₁ = C × (1 - 0.10) = 0.90 × C

Equating the two expressions for the first selling price: 0.80 × L = 0.90 × C L = (9/8) × C

When the discount offered is 5% of the list price, the new selling price S₂ is: S₂ = L × (1 - 0.05) = 0.95 × L

Substituting the value of L in terms of C into this equation: S₂ = 0.95 × (9/8) × C S₂ = (19/20) × (9/8) × C S₂ = (171/160) × C

Since 171/160 is greater than 1, it results in a profit. The profit percentage is calculated as: Profit % = (S₂ - C)/C × 100 Profit % = (171/160 - 1) × 100 Profit % = (11/160) × 100 Profit % = 110/16 = 6.875%