SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

Let the number be x. The correct calculation is 6/5 of x, and the incorrect calculation is 5/6 of x.

The error is the difference between the correct value and the incorrect value: Error = 6x/5 - 5x/6 = (36x - 25x)/30 = 11x/30

The percentage error is calculated as: Percentage Error = (Error / Correct Value) × 100% Percentage Error = ((11x/30) / (6x/5)) × 100% Percentage Error = (11/30) × (5/6) × 100% Percentage Error = (11 / 36) × 100% Percentage Error = 1100 / 36% ≈ 30.56%

This directly gives 30.56% as the error in the calculation.

Explanation

For a period of 2 years, the difference between compound interest and simple interest on a principal sum P at an annual interest rate R is given by the formula:

CI - SI = P × (R/100)²

Given that the difference is ₹900 and the rate R is 12%, we can substitute these values into the formula:

900 = P × (12/100)²

900 = P × (3/25)²

900 = P × (9/625)

P = (900 × 625) / 9

P = 100 × 625

P = 62500

The principal sum of money is ₹62500.

Explanation

To find the values x can take, we split the given compound inequality into two separate inequalities.

The first inequality is: 6x + 2(6 - x) > 2x - 2

Simplifying this: 6x + 12 - 2x > 2x - 2 4x + 12 > 2x - 2 2x > -14 x > -7

The second inequality is: 2x - 2 < 5x/2 - 3x/4

To clear the fractions, we multiply all terms by 4: 8x - 8 < 10x - 3x 8x - 8 < 7x x < 8

Combining both inequalities gives the range for x: -7 < x < 8

The value 5 satisfies this inequality.

Explanation

A kite is defined as a quadrilateral with two distinct pairs of adjacent sides of equal length. If we consider a kite ABCD where AB = AD and CB = CD, we can draw a diagonal AC. This diagonal splits the kite into two triangles, ABC and ADC. In these triangles, AB = AD and CB = CD, while AC is a shared side. By the Side-Side-Side (SSS) congruence theorem, triangle ABC is congruent to triangle ADC. As a result of this congruence, the corresponding angles ABC and ADC must be equal to each other. Since these two angles are opposite to each other, a kite always has at least one pair of congruent opposite angles.

Explanation

Total cost price of 25 quintals of rice is Rs 1825. To earn a profit of 20%, the total desired selling price is calculated as:

Total Selling Price = 1825 × 1.20 = Rs 2190

Since 27% of the rice is lost in transportation, the quantity of rice available for sale is:

Remaining quantity = 25 × (1 - 0.27) = 18.25 quintals

The selling price per quintal is obtained by dividing the total selling price by the remaining quantity of rice:

Selling rate per quintal = 2190 / 18.25 = Rs 120 per quintal

Explanation

The expansion of the product of two binomials of the form (x + a)(x + b) is given by the formula x² + (a + b)x + ab.

By comparing (x + 3)(x - 1) to this formula, we have a = 3 and b = -1.

Substituting these values into the formula yields:

x² + (3 - 1)x + (3 × (-1))

Simplifying the terms gives:

x² + 2x - 3

Explanation

Let the number be x. According to the given condition:

(4/5) × (6/7) × x = 216

Multiplying the fractions on the left side:

(24/35) × x = 216

Solving for x:

x = (216 × 35) / 24 x = 9 × 35 x = 315

Now, we calculate 8/9th of this number:

(8/9) × 315 = 8 × 35 = 280

Explanation

Let the number of selected and unselected candidates initially be 9x and 2x respectively. The total number of candidates who applied initially is 9x + 2x = 11x.

According to the new condition: The number of applied candidates becomes 11x - 80. The number of selected candidates becomes 9x - 20. The number of unselected candidates becomes (11x - 80) - (9x - 20) = 2x - 60.

Since the new ratio of selected to unselected is 5:1, we can write: (9x - 20) / (2x - 60) = 5 / 1

Cross-multiplying the equation: 9x - 20 = 5(2x - 60) 9x - 20 = 10x - 300 10x - 9x = 300 - 20 x = 280

The number of candidates who had applied initially is: 11x = 11 × 280 = 3080.

Explanation

Using the standard values of trigonometric functions: sin 30° = ½ cos 45° = 1/√2

We can rewrite 1/√2 by rationalizing the denominator, which gives √2/2.

Substituting these values into the equation for X: X = sin 30° + cos 45° X = ½ + √2/2 X = (1 + √2)/2

Explanation

Let the two-digit number be represented as 10x + y, where x is the tens digit and y is the units digit.

According to the given conditions, the product of the digits is: x × y = 18

Adding 63 to the number reverses its digits: 10x + y + 63 = 10y + x 9y - 9x = 63 y - x = 7

Substitute y = x + 7 into the product equation: x(x + 7) = 18 x² + 7x - 18 = 0 (x + 9)(x - 2) = 0

Since x represents a digit of a two-digit number, it must be a positive single-digit integer, which gives x = 2. Using y = x + 7, we get y = 9.

Thus, the tens digit is 2 and the units digit is 9, making the number 29.

Explanation

Let the weights of Somdev, Gurdeep, and Ritu be represented by S, G, and R respectively.

Using the relationship Sum = Average × Number of observations:

The sum of weights of Somdev, Gurdeep, and Ritu is: S + G + R = 3 × 85 = 255 kg

The sum of weights of Somdev and Gurdeep is: S + G = 2 × 79 = 158 kg

The sum of weights of Gurdeep and Ritu is: G + R = 2 × 73 = 146 kg

Adding these two equations gives: S + 2G + R = 158 + 146 = 304 kg

Subtracting the sum of all three (S + G + R) from this combined equation isolates Gurdeep's weight: G = (S + 2G + R) - (S + G + R) G = 304 - 255 = 49 kg

Explanation

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

The goods train passes the station 4 hours before the second train. The second train, traveling at 72 km/hr, overtakes the goods train 8 hours after passing the station.

This means that by the time the second train overtakes the goods train, the goods train has been traveling for a total of 4 + 8 = 12 hours. The second train has traveled for 8 hours.

Since both trains start their measurements from the same station and meet at the overtaking point, they cover the same distance:

Distance covered by goods train = Distance covered by second train v × 12 = 72 × 8 v = (72 × 8) / 12 v = 6 × 8 v = 48 km/hr

Explanation

The coordinates of the midpoint M(x, y) of a line segment joining two points (x₁, y₁) and (x₂, y₂) are calculated using the midpoint formula:

x = (x₁ + x₂)/2 y = (y₁ + y₂)/2

Substituting the coordinates of C(3, -5) and D(-7, 3) into the formula:

x = (3 + (-7))/2 = -4/2 = -2 y = (-5 + 3)/2 = -2/2 = -1

The coordinates of the midpoint are (-2, -1).

Explanation

To find the point of intersection M, we first determine the equation of the line passing through the origin that is perpendicular to the given line 2x + 3y = 6.

The slope of the line L₁: 2x + 3y = 6 is m₁ = -2/3.

Since the second line L₂ is perpendicular to L₁ and passes through the origin (0, 0), its slope m₂ is: m₂ = -1 / m₁ = 3/2

The equation of L₂ is: y = (3/2)x

To find the point of intersection M, we substitute y = (3/2)x into the equation of L₁: 2x + 3((3/2)x) = 6 2x + (9/2)x = 6

Simplifying this algebraic expression yields: 2x + 9x = 6 11x = 6 x = 6/11

Substituting x = 6/11 back into the equation for L₂ gives: y = (3/2) × (6/11) = 9/11

Thus, the coordinates of the point M are (6/11, 9/11).

Explanation

Let the total work of the project be the least common multiple of 6 and 12, which is 12 units.

The daily work efficiency of P is 12 ÷ 6 = 2 units per day. The daily work efficiency of Q is 12 ÷ 12 = 1 unit per day. Their combined daily work efficiency when working together is 2 + 1 = 3 units per day.

To find the time needed to complete 25% of the project, we first calculate 25% of the total work: 25% of 12 units = 0.25 × 12 = 3 units.

The number of days they take to complete this portion of the project is: Work to be done ÷ Combined efficiency = 3 units ÷ 3 units per day = 1 day.

Explanation

The formula for the length of an arc L of a circle with radius r and corresponding central angle θ is:

L = (θ / 360°) × 2πr

Given that the arc length L = 33 cm and the central angle θ = 90°, we can substitute these values using π ≈ 22/7 to find the radius:

33 = (90 / 360) × 2 × (22 / 7) × r

33 = (1 / 4) × (44 / 7) × r

33 = (11 / 7) × r

Rearranging the equation to solve for r:

r = (33 × 7) / 11

r = 3 × 7

r = 21 cm

Explanation

When a sphere is completely immersed in a cylindrical vessel, the volume of water displaced is equal to the volume of the sphere.

Let the rise in the water level be h.

Volume of the sphere = 4/3 × π × (radius of sphere)³ Volume of the displaced water in the cylinder = π × (radius of cylinder)² × h

Equating the two volumes: π × 6² × h = 4/3 × π × 5³

Simplifying the equation: 36 × h = 500/3 h = 500 / (3 × 36) h = 500 / 108 h = 125 / 27 h ≈ 4.63 cm

Explanation

The number of students who scored marks 20 or more but less than 40 is calculated by subtracting the number of students scoring 40 and above from the number of students scoring 20 and above.

From the given data table, the number of students scoring 20 and above is 55, and the number of students scoring 40 and above is 26.

Applying the subtraction: 55 - 26 = 29

Explanation

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

The GDP of the country at the end of 2013 is given as 1 trillion. The GDP growth rate for the year 2014 is 6%.

The GDP at the end of 2014 can be calculated using the formula:

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

Substituting the given values into the formula:

GDP at the end of 2014 = 1 × (1 + 6/100) = 1 × 1.06 = 1.06 trillion USD

Since the GDP at the end of 2014 is the same as the GDP at the beginning of 2015, the GDP at the beginning of 2015 was1.06 trillion.

Explanation

The formula for the average marks is: Average = Sum of marks / Number of subjects

First, calculate the sum of the marks scored across all five subjects: Sum of marks = 80 + 40 + 30 + 80 + 30 = 260

Since there are 5 subjects, the student's average marks before deduction is: Average = 260 ÷ 5 = 52

The net average marks after deducting 5 points for poor attendance is: Net average = 52 - 5 = 47

This calculation yields 47 as the final net average marks.

Explanation

Let the original number be x.

Increasing the number by 24 makes it 110% of itself, which represents an increase of 10% (since 110% - 100% = 10%).

Therefore, 10% of the number is equal to 24.

10% of x = 24 0.10 × x = 24 x = 24 ÷ 0.10 x = 240

Explanation

To reduce the fraction 3596 / 4292 to its lowest terms, find the Greatest Common Divisor (GCD) of the numerator and the denominator using the Euclidean algorithm:

4292 = 3596 × 1 + 696 3596 = 696 × 5 + 116 696 = 116 × 6 + 0

The GCD of 3596 and 4292 is 116. Dividing both the numerator and the denominator by 116 gives:

Numerator: 3596 ÷ 116 = 31 Denominator: 4292 ÷ 116 = 37

This simplifies the fraction to 31/37.

Explanation

A completes 60% (which is 3/5) of the work in 15 days.

The rate of work of A per day is (3/5) ÷ 15 = 1/25 of the total work.

The remaining work to be completed is 100% - 60% = 40% (which is 2/5) of the work.

A and B together complete this remaining 2/5 of the work in 5 days.

The combined rate of work of A and B per day is (2/5) ÷ 5 = 2/25 of the total work.

The rate of work of B per day is obtained by subtracting A's rate from the combined rate:

Rate of B = 2/25 - 1/25 = 1/25 of the work per day.

Since B does 1/25 of the work in one day, B alone will take 25 days to complete the entire work.

Explanation

Using standard trigonometric values: cot 60° = 1/√3 cosec 60° = 2/√3

Substituting these values into the given equation: x = cot 60° + cosec 60° x = 1/√3 + 2/√3 x = (1 + 2)/√3 x = 3/√3 x = √3

Explanation

When the distance D is constant, the relationship between two speeds S₁, S₂ and the difference in travel times Δt is given by the formula:

D = (S₁ × S₂ × Δt) / (S₂ - S₁)

Substituting the given values, S₁ = 42 km/hr, S₂ = 60 km/hr, and Δt = 2 hours:

D = (42 × 60 × 2) / (60 - 42)

D = 5040 / 18

D = 280 km

Explanation

Using the half-angle trigonometric identity for sine: sin²(θ) = (1 - cos(2θ))/2

Substituting θ = A/2 into the identity: sin²(A/2) = (1 - cos A)/2

Taking the square root on both sides: sin(A/2) = √( (1 - cos A)/2 )

Given that x = √( (1 - cos A)/2 ), we directly get: x = sin(A/2)

Explanation

We can simplify the expression in option C to show that it is equal to 1 + sec A.

Starting with the expression: sin A tan A / (1 - cos A)

Using the identity tan A = sin A / cos A, we get: sin A × (sin A / cos A) / (1 - cos A) = sin² A / [cos A (1 - cos A)]

Using the Pythagorean identity sin² A = 1 - cos² A, we can rewrite the numerator: (1 - cos² A) / [cos A (1 - cos A)]

Applying the algebraic identity a² - b² = (a - b)(a + b) to the numerator: (1 - cos A)(1 + cos A) / [cos A (1 - cos A)]

Canceling the common term (1 - cos A) in the numerator and denominator, we are left with: (1 + cos A) / cos A

Dividing term by term: 1 / cos A + cos A / cos A = sec A + 1

This is equivalent to 1 + sec A, which is the given value of x.

Explanation

The volume of water in the cone is equal to the volume of water in the cylindrical vessel.

The volume of a cone is: V = (1/3) × π × r₁² × h₁

The volume of a cylinder is: V = π × r₂² × h₂

Equating the two volumes: (1/3) × π × r₁² × h₁ = π × r₂² × h₂

Cancelling π on both sides: (1/3) × r₁² × h₁ = r₂² × h₂

Substituting the given values, where the cone has radius r₁ = 7 cm and height h₁ = 12 cm, and the cylinder has radius r₂ = 3.5 cm (which can be written as 7/2 cm): (1/3) × 7² × 12 = (7/2)² × h₂

4 × 49 = (49/4) × h₂

16 × 49 = 49 × h₂

h₂ = 16 cm

Explanation

The area of an equilateral triangle is given by the formula: Area = (√3/4)a² where a is the length of the side of the triangle.

Substituting the given area of 64√3 sq cm into the formula: 64√3 = (√3/4)a²

Dividing both sides by √3: 64 = a²/4

Multiplying both sides by 4: a² = 256

Taking the square root of both sides: a = √256 a = 16 cm

Explanation

The cost price of 4 pens is Rs 5, and the selling price of the same 4 pens is Rs 3.

Since the selling price is less than the cost price, there is a loss in this transaction.

Loss = Cost Price - Selling Price = 5 - 3 = Rs 2

Loss percentage = (Loss / Cost Price) × 100

Substituting the values:

Loss percentage = (2 / 5) × 100 = 40%

Explanation

Effective rate of compound interest for 2 years at 10% per annum is calculated using the successive percentage formula: Rate = 10 + 10 + (10 × 10)/100 = 21%

Compound interest on Rs. 9000 for 2 years = 21% of 9000 CI = (21/100) × 9000 = Rs. 1890

According to the question, the simple interest is half of this compound interest: SI = 1890 / 2 = Rs. 945

Using the simple interest formula, SI = (P × R × T)/100, where P is the principal sum, R is the rate of 18%, and T is the time of 3 years: 945 = (P × 18 × 3) / 100 945 = 54P / 100 P = (945 × 100) / 54 P = 1750

Explanation

For two lines a₁x + b₁y = c₁ and a₂x + b₂y = c₂ to be perpendicular, the product of their slopes must be -1, which yields the standard relation a₁a₂ + b₁b₂ = 0.

Comparing this with the given equations: 4x - y = 1, where a₁ = 4 and b₁ = -1 5x - ky = 2, where a₂ = 5 and b₂ = -k

Substituting these values into the perpendicularity condition: 4 × 5 + (-1) × (-k) = 0 20 + k = 0 k = -20

Explanation

The mean of the combined groups is calculated using the weighted mean formula:

Weighted Mean = (n₁x̄₁ + n₂x̄₂ + n₃x̄₃) / (n₁ + n₂ + n₃)

Here, the number of students in divisions A, B, and C are n₁ = 55, n₂ = 45, and n₃ = 75 respectively. The corresponding mean marks are x̄₁ = 58, x̄₂ = 54, and x̄₃ = 52.

To simplify the calculation, the ratio of the number of students can be reduced by dividing each by 5: Ratio of students = 11 : 9 : 15 Sum of the ratio terms = 11 + 9 + 15 = 35

Applying the weighted mean formula with these reduced weights: Mean = (11 × 58 + 9 × 54 + 15 × 52) / 35 Mean = (638 + 486 + 780) / 35 Mean = 1904 / 35 Mean = 54.4

Explanation

The formula for the length of an arc of a circle is:

Arc Length = (θ / 360°) × 2πr

Substituting the given central angle θ = 90°, radius r = 3.5 cm (or 7/2 cm), and using π ≈ 22/7:

Arc Length = (90 / 360) × 2 × (22/7) × (7/2)

Arc Length = ¼ × 22

Arc Length = 5.5 cm

Explanation

Let the present age of Ratnabali be 8x and that of Shaukat be 5x.

According to the given condition, after 22 years, the ratio of their ages becomes 10:9. This can be represented as:

(8x + 22) / (5x + 22) = 10 / 9

By cross-multiplying the terms:

9 × (8x + 22) = 10 × (5x + 22)

72x + 198 = 50x + 220

72x - 50x = 220 - 198

22x = 22

x = 1

Thus, the present age of Ratnabali is 8x = 8 × 1 = 8 years.

Explanation

The formula for the area of an equilateral triangle with side length a is Area = (√3/4)a².

Given that the area of the triangle is 9√3 sq cm, we can find the side length:

(√3/4)a² = 9√3

a²/4 = 9

a² = 36

a = 6 cm

The formula for the height (h) of an equilateral triangle is h = (√3/2)a.

Substituting the value of a:

h = (√3/2) × 6 = 3√3 cm

Explanation

To find the discount percentage, first calculate the total original cost of the tickets without any offer.

Original cost of 23 adult tickets = 23 × 2200 = Rs 50600 Original cost of 11 children tickets = 11 × 990 = Rs 10890 Total original cost = 50600 + 10890 = Rs 61490

According to the offer, one child goes free with two adults. With 23 adults, the group has 11 pairs of adults (with 1 adult leftover). Therefore, up to 11 children can go for free. Since the group has exactly 11 children, all of them receive free entry.

Total discount obtained = Cost of 11 children tickets = Rs 10890

Discount percentage = (Total discount ÷ Total original cost) × 100 Discount percentage = (10890 ÷ 61490) × 100 = 17.71%

Explanation

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

(8 - 10x) - (13x - 2) = -9

First, remove the parentheses and distribute the subtraction sign:

8 - 10x - 13x + 2 = -9

Combine the constant terms and the variable terms:

10 - 23x = -9

Subtract 10 from both sides of the equation:

-23x = -19

Divide both sides by -23:

x = 19/23

Explanation

Using the algebraic identity: (x + y)² = x² + y² + 2xy

Substitute the given values xy = -18 and x² + y² = 85 into the formula: (x + y)² = 85 + 2(-18) (x + y)² = 85 - 36 (x + y)² = 49

Taking the square root on both sides gives: x + y = ±7

Explanation

Let the number be x. According to the given condition, the difference between the number and twice its reciprocal is 31/4:

x - 2/x = 31/4

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

4x² - 8 = 31x

Rearranging this into the standard quadratic equation form:

4x² - 31x - 8 = 0

Splitting the middle term to factorize the equation:

4x² - 32x + x - 8 = 0

4x(x - 8) + 1(x - 8) = 0

(4x + 1)(x - 8) = 0

This yields the solutions x = 8 and x = -1/4.

Explanation

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

First, solve the left inequality: 2x - 3(x + 2) < 5 - 2x 2x - 3x - 6 < 5 - 2x -x - 6 < 5 - 2x x < 11

Next, solve the right inequality: 5 - 2x < -x + 2 5 - 2 < 2x - x 3 < x

Combining the two solutions gives the range for x: 3 < x < 11

The value 10 lies within this valid range.

Explanation

The weight to height ratio is calculated by dividing weight in kilograms by height in meters. For Ahilya, this ratio is 60 ÷ 1.8 = 33.33. Comparing this with the ratios of the other individuals, which are approximately 41.18, 37.5, and 38.07, Ahilya has the lowest value.

Explanation

The average number of children per house is calculated using the formula for the weighted mean:

Mean = ∑(f × x) / ∑f

where x represents the number of children in a house and f represents the frequency, which is the number of houses.

First, calculate the total number of children by summing the product of the number of children and the number of houses for each row: (0 × 3) + (1 × 10) + (2 × 12) + (3 × 3) = 0 + 10 + 24 + 9 = 43

Next, calculate the total number of houses by summing the frequencies: 3 + 10 + 12 + 3 = 28

Now, divide the total number of children by the total number of houses to find the average: Average = 43 / 28 ≈ 1.5357

Rounding this value to two decimal places gives 1.54.

Explanation

The total number of shares issued by the company is 6,00,000. Dhiraj initially holds a 10% share of the company.

Dhiraj's initial number of shares = 10% of 6,00,000 = 60,000 shares.

When Chinmay sells 15,000 of his shares to Dhiraj, Dhiraj's total shareholding increases by that amount.

Dhiraj's final number of shares = 60,000 + 15,000 = 75,000 shares.

Explanation

To find the value of x for which the two expressions are equal, set them equal to each other and solve the linear equation:

15 - 7x = 15x + 7

Rearranging the terms by adding 7x to both sides and subtracting 7 from both sides gives:

15 - 7 = 15x + 7x

8 = 22x

Solving for x:

x = 8/22

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

x = 4/11

Explanation

The largest 5-digit number is 99999. Dividing 99999 by 95 yields a quotient of 1052 and a remainder of 59.

99999 = (95 × 1052) + 59

Subtracting this remainder from the largest 5-digit number gives the largest 5-digit multiple of 95:

99999 - 59 = 99940

Explanation

Since the ratio a:b is given as 3:8, we can assume a = 3k and b = 8k, where k is a non-zero constant.

Substituting these values into the expression:

(5a - 3b) / (2a + b) = (5(3k) - 3(8k)) / (2(3k) + 8k) = (15k - 24k) / (6k + 8k) = -9k / 14k = -9/14

Explanation

Solving the first inequality: 5 - 3x < 4 - x 5 - 4 < 3x - x 1 < 2x x > ½

Solving the second inequality: 5(2 - x) > 2 - 2x 10 - 5x > 2 - 2x 10 - 2 > 5x - 2x 8 > 3x x < 8/3

Combining the two inequalities gives the range: ½ < x < 8/3

Since 8/3 is approximately 2.67, the interval is 0.5 < x < 2.67. The value 1 satisfies this condition.

Explanation

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

3.352 - (9.759 - x) - 19.64 = 7.052

Expand the expression by distributing the negative sign over the terms inside the parentheses:

3.352 - 9.759 + x - 19.64 = 7.052

Group and simplify the constant terms on the left side:

-6.407 + x - 19.64 = 7.052 x - 26.047 = 7.052

Solve for x by adding 26.047 to both sides:

x = 7.052 + 26.047 x = 33.099

Explanation

The sum of the revenues of all 9 years is obtained by multiplying the number of years by their average revenue: 9 × 76 = 684 lakhs

Similarly, the sum of the revenues for the first 5 years is: 5 × 71 = 355 lakhs

The sum of the revenues for the last 5 years is: 5 × 83 = 415 lakhs

Since the 5th year is common to both the first 5 years and the last 5 years, its revenue can be found using the relation: Revenue of 5th year = (Sum of first 5 years + Sum of last 5 years) - Sum of all 9 years

Substituting the values: Revenue of 5th year = (355 + 415) - 684 Revenue of 5th year = 770 - 684 = 86 lakhs