SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The normal cost of tickets without any discount is calculated by adding the individual costs of all adult and child tickets. Total normal cost = (25 × 1500) + (12 × 800) = 37500 + 9600 = 47100

According to the rule, 1 child goes free with 2 adults. With 25 adults, the maximum number of pairs of adults is the integer division of 25 by 2, which is 12 pairs. This means up to 12 children can go free. Since the group has exactly 12 children, all of them receive free entry.

The discount amount is the total cost of the 12 children's tickets that the group does not have to pay. Discount amount = 12 × 800 = 9600

The discount percentage is calculated as the ratio of the discount amount to the total normal cost, multiplied by 100. Discount percentage = (9600 / 47100) × 100 ≈ 20.38%

Explanation

Using the standard product-to-sum trigonometric identity:

2 cos A cos B = cos(A + B) + cos(A - B)

By setting A = (C + D)/2 and B = (C - D)/2, we find:

A + B = (C + D)/2 + (C - D)/2 = 2C/2 = C A - B = (C + D)/2 - (C - D)/2 = 2D/2 = D

Substituting these expressions back into the identity yields:

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

Explanation

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

cot²A cos²A = cot²A (1 - sin²A)

Multiplying cot²A into the terms inside the parentheses gives:

= cot²A - cot²A sin²A

Using the quotient identity cot A = cos A / sin A, we can write cot²A as cos²A / sin²A:

= cot²A - (cos²A / sin²A) sin²A

Canceling sin²A from the second term yields:

= cot²A - cos²A

Explanation

The value of the stock for each product category is calculated by multiplying its quantity by its average cost:

Value of Mobile Phones = 70 × 9000 = 630000 Rs Value of Cameras = 70 × 9000 = 630000 Rs Value of TVs = 82 × 49000 = 4018000 Rs Value of Refrigerators = 88 × 49000 = 4312000 Rs Value of ACs = 37 × 27000 = 999000 Rs

Summing these individual values gives the total stock value: Total Value = 630000 + 630000 + 4018000 + 4312000 + 999000 = 10589000 Rs

To convert the total amount from rupees to lakh rupees, we divide by 100000: Total Value in Lakhs = 10589000 ÷ 100000 = 105.89 lakh rupees

Explanation

For 2012, the ratio of import to export is 1.1: Import(2012) / Export(2012) = 1.1

Given that the imports in 2012 were Rs. 1000 crores: Export(2012) = 1000 / 1.1 = 909.09 crores

Since the total exports in 2012 and 2013 combined were Rs. 3200 crores, the exports in 2013 can be calculated as: Export(2013) = 3200 - Export(2012) Export(2013) = 3200 - 909.09 = 2290.91 crores

For 2013, the ratio of import to export is 0.9: Import(2013) / Export(2013) = 0.9

Substituting the value of Export(2013) yields: Import(2013) = 0.9 × 2290.91 = 2061.82 crores

Rounding to the nearest integer, the imports in 2013 were Rs. 2062 crores.

Explanation

The increase in height is calculated by subtracting the height of the child on the 5th birthday from the height on the 14th birthday.

From the table: Height on the 14th birthday = 175 cms Height on the 5th birthday = 105 cms

Increase in height = 175 - 105 = 70 cms

Explanation

The total duration monitored between 10:00 PM and 6:00 AM is 8 hours.

Based on the recorded data table: Time spent Dreaming = 10% Time spent in Extremely light sleep = 25%

Combining these two categories gives a total of: 10% + 25% = 35%

To find the duration in hours, we calculate 35% of the total 8 hours: 35% of 8 hours = (35/100) × 8 = 0.35 × 8 = 2.8 hours

Explanation

To find the value of x, simplify the given linear equation:

(9 - 3x) - (17x - 10) = -1

Remove the parentheses and distribute the negative sign:

9 - 3x - 17x + 10 = -1

Combine the like terms on the left side of the equation:

19 - 20x = -1

Subtract 19 from both sides of the equation:

-20x = -20

Divide both sides by -20:

x = 1

Explanation

The algebraic identity relating the given terms is: (a - b)² = a² + b² - 2ab

Substituting the given values, a - b = -7 and a² + b² = 85, into the identity yields: (-7)² = 85 - 2ab 49 = 85 - 2ab

Rearranging the terms to solve for ab: 2ab = 85 - 49 2ab = 36 ab = 18

Explanation

Since Prabodh completes ½ of the job in 30 days, he can complete the entire job alone in 30 × 2 = 60 days.

Similarly, Sapan completes the remaining ½ of the job in 45 days, which means Sapan can complete the entire job alone in 45 × 2 = 90 days.

Using the standard formula for the time taken by two individuals working together, the total days required is: (T₁ × T₂) / (T₁ + T₂) = (60 × 90) / (60 + 90) = 5400 / 150 = 36 days.

Explanation

Let the non-zero number be x. According to the given condition, the sum of the number and thrice its reciprocal is equal to 52/7. This can be represented by the equation:

x + 3/x = 52/7

Substituting x = 7 into the left side of the equation yields:

7 + 3/7 = (49 + 3)/7 = 52/7

Since the left-hand side matches the right-hand side, the value of the number is 7.

Explanation

To find the possible values of x, we solve the system of two inequalities.

First inequality: 2x - 2(3 + 4x) < -1 - 2x 2x - 6 - 8x < -1 - 2x -6 - 6x < -1 - 2x -5 < 4x x > -5/4 x > -1.25

Second inequality: -1 - 2x > -5/3 - x/3 Multiplying both sides by 3: -3 - 6x > -5 - x 2 > 5x x < 2/5 x < 0.4

Combining both conditions gives the interval -1.25 < x < 0.4. The value -1 lies within this range, making it a valid value for x.

Explanation

The weight to height ratio is calculated using the formula:

Ratio = Weight ÷ Height

Using the given data, the weight to height ratio for Aakriti is:

51 ÷ 1.76 ≈ 28.98 kg/m

Calculating the ratios for the others: Aabha: 68 ÷ 1.68 ≈ 40.48 kg/m Aahana: 77 ÷ 1.57 ≈ 49.04 kg/m Aakaanksha: 55 ÷ 1.8 ≈ 30.56 kg/m

Aakriti has the minimum value among all, which is approximately 28.98 kg/m.

Explanation

Profit percentage is calculated using the formula: Profit % = (Revenue - Expenditure) / Expenditure × 100

From the given data, the least profit percentage is 10% in the year 2011.

Using the given expenditure of Rs 225 crore: 10 = (Revenue - 225) / 225 × 100 0.1 = (Revenue - 225) / 225 22.5 = Revenue - 225 Revenue = 225 + 22.5 = 247.5

Explanation

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

The total number of houses is the sum of the houses in the frequency column: 4 + 10 + 19 + 3 = 36 houses.

The total number of children is the sum of the product of the number of children and the number of houses: (0 × 4) + (1 × 10) + (2 × 19) + (3 × 3) = 0 + 10 + 38 + 9 = 57 children.

Dividing the total number of children by the total number of houses gives: 57 / 36 = 1.5833... which rounds to 1.58.

Explanation

By the reciprocal relation of trigonometric ratios: cosec θ = 1 / sin θ

Substituting θ = 135°: cosec 135° = 1 / sin 135°

Using the identity sin(180° - A) = sin A: sin 135° = sin(180° - 45°) = sin 45°

Substituting sin 45° = 1/2: cosec 135° = 1 / (1/√2) = √2

Explanation

To find the value of x, we solve the given compound inequality by splitting it into two separate inequalities.

First inequality: 4(x + 1) - 3 < 3 - x 4x + 4 - 3 < 3 - x 4x + 1 < 3 - x 5x < 2 x < 2/5

Second inequality: 3 - x < 2x + 5 -x - 2x < 5 - 3 -3x < 2 x > -2/3

Combining both inequalities gives the range: -2/3 < x < 2/5

Converting these fractions to decimals: -0.67 < x < 0.40

The value 0 lies within this interval, satisfying the given compound inequality.

Explanation

The volume of the remaining solid is obtained by subtracting the volume of the hollowed-out cone from the volume of the solid wooden cube.

Volume of the remaining cube = Volume of the cube - Volume of the cone

The volume of a cube with side a is given by: Volume of the cube = a³

Given that the side of the cube a = 7 cm, its volume is: Volume of the cube = 7³ = 343 cm³

The volume of a cone with radius r and height h is given by: Volume of the cone = (1/3) × π × r² × h

The diameter of the cone is equal to the side of the cube, which is 7 cm, making the radius r = 7/2 = 3.5 cm. The height h of the cone is also equal to the side of the cube, which is 7 cm. Using π ≈ 22/7: Volume of the cone = (1/3) × (22/7) × (7/2)² × 7 Volume of the cone = (1/3) × (22/7) × (49/4) × 7 Volume of the cone = 269.5 / 3 ≈ 89.83 cm³

Subtracting the volume of the cone from the volume of the cube yields: Volume of the remaining cube = 343 - 89.83 = 253.17 cm³

Explanation

Let the first number be x and the sum of the other two numbers be S.

The average of the three numbers is 72, which means their total sum is: x + S = 3 × 72 = 216

We are given that the first number is 2/7 times the sum of the other two: x = 2/7 × S S = 7/2 × x

Substituting the value of S into the sum equation: x + 7/2 × x = 216 9/2 × x = 216 x = 216 × 2/9 x = 24 × 2 x = 48

Explanation

Using the fundamental trigonometric identity:

1 + cot²A = cosec²A

Taking the square root on both sides:

√(1 + cot²A) = cosecA

Since the given expression is x = √(1 + cot²A), the value of x is cosecA.

Explanation

We can simplify the given trigonometric expression tan⁴A + tan²A by factoring out tan²A:

tan⁴A + tan²A = tan²A (tan²A + 1)

Using the fundamental trigonometric identity sec²A = 1 + tan²A, we can substitute sec²A for (tan²A + 1):

= tan²A · sec²A

Next, substituting tan²A = sec²A - 1 into the expression:

= (sec²A - 1) sec²A = sec⁴A - sec²A

Explanation

When a point R(a, b) is reflected in the origin, its coordinates change sign, giving R1(-a, -b).

When a point is reflected in the x-axis, its y-coordinate changes sign while its x-coordinate remains unchanged. Applying this reflection to R1(-a, -b) gives the coordinates (-a, b).

Equating (-a, b) to the given final coordinates (-3, 4) yields: -a = -3, which gives a = 3 b = 4

Therefore, the coordinates of the point R(a, b) are (3, 4).

Explanation

Using the algebraic expansion formulas (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b², we expand each binomial term:

(x + 5)² = x² + 10x + 25 (x - 3)² = x² - 6x + 9

Adding these expanded terms together: (x² + 10x + 25) + (x² - 6x + 9) = 2x² + 4x + 34

Factoring out the common factor of 2 from the simplified polynomial: 2(x² + 2x + 17)

Explanation

To find the simplest form of the fraction 3565/1495, we determine the greatest common divisor (GCD) of the numerator and the denominator using the Euclidean algorithm.

First, divide 3565 by 1495: 3565 = 1495 × 2 + 575

Next, divide 1495 by the remainder 575: 1495 = 575 × 2 + 345

Next, divide 575 by the remainder 345: 575 = 345 × 1 + 230

Next, divide 345 by the remainder 230: 345 = 230 × 1 + 115

Finally, divide 230 by the remainder 115: 230 = 115 × 2 + 0

Since the remainder is 0, the GCD is 115.

Dividing both the numerator and the denominator by their GCD of 115 gives: 3565 ÷ 115 = 31 1495 ÷ 115 = 13

Thus, the simplest form of the fraction is 31/13.

Explanation

Let the number be x. According to the given condition, increasing the number by 42 makes it 121% of itself. This relationship can be expressed as:

x + 42 = 1.21x

Subtracting x from both sides of the equation gives:

0.21x = 42

Solving for x:

x = 42 / 0.21 x = 4200 / 21 x = 200

Explanation

The company has issued a total of six lakh shares, which is equal to 6,00,000 shares.

Ejaz initially holds 45% of the total shares.

Initial number of shares held by Ejaz = 45% of 6,00,000 = (45 ÷ 100) × 6,00,000 = 2,70,000 shares

When Basu sells 15,000 shares to Ejaz, these shares are added directly to Ejaz's initial holdings.

Total shares held by Ejaz after purchasing from Basu = 2,70,000 + 15,000 = 2,85,000 shares

Explanation

When two objects move towards each other, their relative speed is the sum of their individual speeds.

Relative speed = 15 km/hr + 20 km/hr = 35 km/hr

The meeting time is given as 42 minutes. To convert this into hours, divide by 60:

Time = 42/60 hours = 0.7 hours

The initial distance between them is the product of their relative speed and the time they took to meet:

Distance = Relative speed × Time Distance = 35 km/hr × 0.7 hours = 24.5 km

Explanation

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

Substituting the given values, a = 7 and b = 21:

x = 21² / 7 x = 441 / 7 x = 63

Explanation

The effective discount of two successive discounts (or a discount followed by cashback) can be calculated using the successive percentage formula:

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

Here, the initial discount d₁ is 30% and the cashback discount d₂ is 5%.

Substituting these values into the formula:

Effective Discount = 30 + 5 - (30 × 5) / 100 Effective Discount = 35 - 150 / 100 Effective Discount = 35 - 1.5 Effective Discount = 33.5%

Explanation

The simple interest is calculated using the formula:

SI = (P × R × T) / 100

where SI is the simple interest, P is the principal sum, R is the rate of interest per year, and T is the time in years.

Substituting the given values into the formula:

7056 = (P × 8 × 7) / 100

7056 = (P × 56) / 100

P = (7056 × 100) / 56

P = 126 × 100

P = 12600

The sum is Rs. 12600.

Explanation

The total time required by the mason to build the wall is 18 hours. This means the mason completes 1/18 of the wall in 1 hour.

In 9 hours, the fraction of the wall completed is: 9 × 1/18 = 9/18 = 0.5

The fraction of the wall that is yet to be built is: 1 - 0.5 = 0.5

Explanation

The median AD of triangle ABC connects the vertex A to the midpoint D of the opposite side BC.

First, determine the coordinates of the midpoint D of side BC using the midpoint formula with the coordinates of B(5, -4) and C(3, 2): D = ((5 + 3)/2, (-4 + 2)/2) = (4, -1)

Next, find the equation of the line passing through A(-1, -3) and D(4, -1). The slope m of this line is calculated as: m = (-1 - (-3)) / (4 - (-1)) = 2/5

Using the point-slope form of the linear equation with the point D(4, -1): y - (-1) = (2/5)(x - 4) y + 1 = (2/5)(x - 4) 5y + 5 = 2x - 8 2x - 5y = 13

Explanation

Let the selling price of one iphone be S. The selling price of 8 iphones is 8S.

The profit earned is equal to the selling price of 3 iphones, which is 3S.

The cost price of these 8 iphones is the difference between their selling price and the profit earned: Cost Price = Selling Price - Profit Cost Price = 8S - 3S = 5S

Profit percentage is given by the formula: Profit Percentage = (Profit / Cost Price) × 100 Profit Percentage = (3S / 5S) × 100 = 60%

Explanation

The algebraic identity for the sum of cubes of two numbers is: a³ + b³ = (a + b)³ - 3ab(a + b)

Substituting the given values of a + b = 10 and ab = 24 into the formula: a³ + b³ = 10³ - 3 × 24 × 10 a³ + b³ = 1000 - 720 a³ + b³ = 280

Explanation

Let the measure of the angle be x.

Since the sum of two complementary angles is 90°, the complementary angle of x is 90° - x.

According to the given condition, the angle is greater than its complementary angle by 40°: x = (90° - x) + 40° 2x = 130° x = 65°

Explanation

A rhombus is a quadrilateral with all four sides equal in length. When a diagonal is drawn in a rhombus, it divides the figure into two triangles. For instance, in a rhombus ABCD with diagonal AC, we have two triangles, ABC and ADC. In these triangles, AB = AD and BC = DC because all sides of a rhombus are equal. The diagonal AC is a common side to both triangles. By the Side-Side-Side (SSS) congruence criterion, triangle ABC is congruent to triangle ADC. Similarly, diagonal BD divides the rhombus into two congruent triangles, ABD and CBD. Therefore, each diagonal of a rhombus divides it into two congruent triangles.

Explanation

Let the number be x. According to the problem, the sum of thrice the number and twice its reciprocal is 97/4. This can be represented by the equation:

3x + 2/x = 97/4

Multiplying the entire equation by 4x to clear the denominators:

12x² + 8 = 97x

Rearranging this into a standard quadratic equation form:

12x² - 97x + 8 = 0

Solving this by factoring:

12x² - 96x - x + 8 = 0 12x(x - 8) - 1(x - 8) = 0 (12x - 1)(x - 8) = 0

This yields x = 8 or x = 1/12. Since 8 is the integer solution, we can verify it by substitution:

3(8) + 2/8 = 24 + ¼ = 97/4

This perfectly satisfies the given condition.

Explanation

To find the average bonus in rupees, we first calculate the total number of employees and the total bonus paid.

The total number of employees is 2 + 12 + 1 = 15.

The total bonus paid to 2 Managers is 2 × (40% of ₹3,000,000) = 2 × ₹1,200,000 = ₹2,400,000.

The total bonus paid to 12 Executives is 12 × (20% of ₹800,000) = 12 × ₹160,000 = ₹1,920,000.

The total bonus paid to 1 Trainee is 1 × (10% of ₹400,000) = ₹40,000.

Summing these amounts, the total bonus paid to all employees is ₹2,400,000 + ₹1,920,000 + ₹40,000 = ₹4,360,000.

The average bonus per employee is calculated by dividing the total bonus by the total number of employees: Average bonus = 4,360,000 / 15 ≈ ₹290,667.

Explanation

To find the pair of years for which the total exports from the three companies together are equal, we calculate the sum of exports of Company A, Company B, and Company C for each year:

For 2011: 1000 + 5000 + 1000 = 7000 lakh rupees For 2012: 1000 + 1000 + 3000 = 5000 lakh rupees For 2013: 1000 + 3000 + 2000 = 6000 lakh rupees For 2014: 3000 + 2000 + 4000 = 9000 lakh rupees For 2015: 1000 + 4000 + 4000 = 9000 lakh rupees

The total exports for the years 2014 and 2015 are equal, both amounting to 9000 lakh rupees.

Explanation

The total profit or loss of the company over the last five years is obtained by finding the algebraic sum of the profits and losses for each year from 2011 to 2015.

Total = 10 + (-25) + 20 + (-20) + 25

Combining the values: Total = 10 - 25 + 20 - 20 + 25

By cancelling out the additive inverses, where +20 and -20 cancel each other and -25 and +25 cancel each other: Total = 10

A positive result of 10 indicates a net profit of Rs 10 crores.

Explanation

The percentage of jewellery exports out of the total exports is calculated using the formula:

Percentage = (Value of Jewellery exports ÷ Total value of exports) × 100

First, calculate the total value of India's exports by summing the values of all the listed sectors:

Total exports = 625 + 550 + 850 + 850 + 525 = 3400 million USThe value of jewellery exports is 625 million US.

Substituting these values into the formula:

Percentage = (625 ÷ 3400) × 100 = 625 ÷ 34 ≈ 18.38%

Explanation

The given expression represents a system of two inequalities.

Solving the first inequality: 5(4 - x) - 4 < 3x - 2 20 - 5x - 4 < 3x - 2 16 - 5x < 3x - 2 18 < 8x x > 18/8 x > 2.25

Solving the second inequality: 3x - 2 > 4x - 6 4 > x x < 4

Combining these results gives the range of x as: 2.25 < x < 4

The only integer value in this range is 3.

Explanation

Mother's baking rate is 20 cakes / 4 hours = 5 cakes per hour.

The combined baking rate of Mother and Dadima is 60 cakes / 10 hours = 6 cakes per hour.

By subtracting Mother's rate from the combined rate, Dadima's baking rate is 6 - 5 = 1 cake per hour.

The number of cakes Dadima can bake in 30 hours is 1 cake per hour × 30 hours = 30 cakes.

Explanation

Let the three numbers be x, y, and z. Since their average is 70, their sum is: x + y + z = 3 × 70 = 210

The first number, x, is 1/9 times the sum of the other two numbers, which is y + z: x = (y + z) / 9 y + z = 9x

Substituting this relationship into the sum of the three numbers gives: x + 9x = 210 10x = 210 x = 21

Explanation

The midpoint formula states that if P(x, y) is the midpoint of the line segment joining A(x₁, y₁) and B(x₂, y₂), then:

x = (x₁ + x₂) / 2 y = (y₁ + y₂) / 2

Substituting the given coordinates P(-1, 1) and A(4, 3):

-1 = (4 + x₂) / 2 -2 = 4 + x₂ x₂ = -6

1 = (3 + y₂) / 2 2 = 3 + y₂ y₂ = -1

This gives the coordinates of B as (-6, -1).

Explanation

The amount at the end of consecutive years under compound interest grows by a factor of (1 + r/100) each year, where r is the annual rate of interest.

Thus, the relationship between the amount at the end of the second year (A₂) and the third year (A₃) is: A₃ = A₂ × (1 + r/100)

This can be rewritten to find the interest rate: r = ((A₃ - A₂) / A₂) × 100%

Substituting the given values A₂ = 33708 and A₃ = 35730.48: r = ((35730.48 - 33708) / 33708) × 100% r = (2022.48 / 33708) × 100% r = 202248 / 33708 % r = 6%

Explanation

In a rectangle ABCD, adjacent sides DA and AB are perpendicular to each other. The product of the slopes of two perpendicular lines is always equal to -1.

If the slope of the side DA is m₁ = 1/3 and the slope of the side AB is m₂, then:

m₁ × m₂ = -1

Substituting the given slope:

(1/3) × m₂ = -1

Solving for m₂ gives:

m₂ = -3

Explanation

Let the selling price of one shoe be SP and the cost price of one shoe be CP.

The profit from selling 23 shoes is: Profit = 23 × SP - 23 × CP

Given that the profit is equal to the selling price of 7 shoes: 7 × SP = 23 × SP - 23 × CP 23 × CP = 16 × SP

Profit percentage is calculated on the cost price: Profit Percentage = (Profit / Total Cost Price) × 100 Profit Percentage = (7 × SP / (23 × CP)) × 100

Substituting 23 × CP = 16 × SP into the equation: Profit Percentage = (7 × SP / (16 × SP)) × 100 Profit Percentage = 7/16 × 100 Profit Percentage = 43.75%

Explanation

Solving the linear equation for x: 3x - 7 = 5x - 1 2x = -6 x = -3

Substituting x = -3 into the given expression: (x - 4)² = (-3 - 4)² = (-7)² = 49

Explanation

Correct Answer: D

Let Hariprasad's salary be represented by H.

Giri's salary is 9/5 times of Hariprasad's, which is equal to 9/5 × H. Shaukat's salary is 9/2 times of Hariprasad's, which is equal to 9/2 × H.

The ratio of Giri's salary to Shaukat's salary is calculated by dividing Giri's salary by Shaukat's salary:

(9/5 × H) ÷ (9/2 × H) = (9/5) ÷ (9/2) = (9/5) × (2/9) = 2/5 = 2:5