SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

Distance (d) is related to speed (v) and time (t) by the formula t = d/v.

Let the distance between City A and City B be d km.

The time taken by the slower car is d/30 hours, and the time taken by the faster car is d/48 hours.

Given that the difference in time is 3 hours: d/30 - d/48 = 3

Taking the LCM of 30 and 48, which is 240: (8d - 5d) / 240 = 3 3d / 240 = 3 d / 80 = 3 d = 240 km

Explanation

First, compute the compound interest on Rs 6000 for 2 years at 10% per annum. Using the formula for compound interest: CI = P × ((1 + R/100)ᵀ - 1)

Substituting the given values: CI = 6000 × ((1 + 10/100)² - 1) CI = 6000 × (1.21 - 1) CI = 6000 × 0.21 = 1260

According to the question, the simple interest (SI) is half of this compound interest: SI = 1260 / 2 = 630

Using the simple interest formula: SI = (P × R × T) / 100

Substituting SI = 630, T = 3, and R = 10: 630 = (P × 10 × 3) / 100 630 = 30P / 100 630 = 3P / 10 P = (630 × 10) / 3 P = 2100

Explanation

Discount % = (Number of Free Items / Total Number of Items) × 100

Here, the number of free dupattas is 3, and the customer gets a total of 6 + 3 = 9 dupattas.

Discount % = (3 / 9) × 100 = 33.33%

Since the discount percentage is the same for the entire transaction as it is for each individual dupatta, the effective discount on each dupatta is 33.33%. The individual price of Rs 1200 per dupatta does not alter this percentage.

Explanation

The relationship between two numbers and their HCF and LCM is given by the formula:

First Number × Second Number = HCF × LCM

Substituting the given values into the formula:

275 × Second Number = 11 × 825

Second Number = (11 × 825) ÷ 275

Second Number = 9075 ÷ 275

Second Number = 33

Explanation

Using standard trigonometric values:

cos 60° = 1/2 sec 30° = 2/√3

Substituting these values into the given equation:

x = cos 60° - sec 30° x = 1/2 - 2/√3

Taking the common denominator 2√3:

x = (√3 - 4) / 2√3

Explanation

Using the standard double-angle identity for cosine: cos 2θ = 1 - 2 sin² θ

By setting θ = A/2, the identity becomes: cos A = 1 - 2 sin²(A/2)

Rearranging this formula to express 2 sin²(A/2) on one side gives: 2 sin²(A/2) = 1 - cos A

Dividing both sides by 2 yields: sin²(A/2) = (1 - cos A)/2

Given that x = (1 - cos A)/2, substituting this relation directly results in: x = sin²(A/2)

Explanation

Let Gaganjyot's salary be G, Hafiz's salary be H, and Sayed's salary be S.

The relation of salaries can be written as: G = (7/6)H S = (8/7)H

To find the ratio of Gaganjyot's salary to Sayed's salary (G:S), divide G by S: G/S = ((7/6)H) / ((8/7)H)

Simplifying the fraction by canceling H: G/S = (7/6) / (8/7) G/S = (7/6) × (7/8) G/S = 49/48

This yields a ratio of 49:48.

Explanation

Let the selling price of one Timex watch be S.

The selling price of 20 watches is 20S.

The profit earned is equal to the selling price of 4 watches, which is 4S.

Since Profit = Selling Price - Cost Price, we can express the cost price of 20 watches as: Cost Price = Selling Price - Profit Cost Price = 20S - 4S = 16S

The profit percentage is calculated using the formula: Profit Percentage = (Profit / Cost Price) × 100 Profit Percentage = (4S / 16S) × 100 = ¼ × 100 = 25%

Explanation

Using the basic trigonometric definitions, csc A = 1/sin A and cot A = cos A/sin A.

Substituting these into the given expression: x = csc A - cot A x = (1/sin A) - (cos A/sin A) x = (1 - cos A)/sin A

Multiplying both the numerator and the denominator by (1 + cos A): x = ((1 - cos A)(1 + cos A)) / (sin A(1 + cos A)) x = (1 - cos² A) / (sin A(1 + cos A))

Using the fundamental trigonometric identity sin² A + cos² A = 1, we can replace (1 - cos² A) with sin² A: x = sin² A / (sin A(1 + cos A)) x = sin A / (1 + cos A)

Explanation

When a solid cone is melted and recast into a cylinder, the volume of the recast shape remains equal to the volume of the original shape.

Using the volume formulas: Volume of cylinder = Volume of cone π × r² × h_cylinder = ⅓ × π × r² × h_cone

Since both shapes have the same radius r, we can divide both sides by π × r²: h_cylinder = ⅓ × h_cone

Given that the height of the cone is 12 cm, substituting this value yields: h_cylinder = ⅓ × 12 = 4 cm

Explanation

The number of boys in the class is 65 − 39 = 26.

The ratio of the number of girls to the number of boys is 39 : 26, which simplifies to 3 : 2.

Using the weighted average formula with this ratio:

Weighted Average = (Ratio of Girls × Average Weight of Girls + Ratio of Boys × Average Weight of Boys) / Total Ratio

Let W be the average weight of the boys:

64 = (3 × 60 + 2 × W) / (3 + 2)

64 = (180 + 2W) / 5

320 = 180 + 2W

140 = 2W

W = 70 kg

Explanation

The area of a square in terms of its diagonal length d is given by the formula:

Area = d² / 2

Given that the diagonal d is 14 cm, substituting this value into the formula yields:

Area = 14² / 2 Area = 196 / 2 Area = 98 sq cm

Explanation

Using the section formula, the coordinates of point P(x, y) that divides the segment joining A(0,4) and B(-5,9) in the ratio 2:3 are calculated as:

x = (2 × (-5) + 3 × 0) / (2 + 3) = -10/5 = -2 y = (2 × 9 + 3 × 4) / (2 + 3) = 30/5 = 6

So, the coordinates of P are (-2, 6).

The slope of the segment AB is: m = (9 - 4) / (-5 - 0) = 5/-5 = -1

Since the required line is perpendicular to AB, its slope (m') satisfies the relation m × m' = -1. Therefore: m' = -1 / -1 = 1

Using the point-slope form for the line passing through P(-2, 6) with slope 1: y - 6 = 1 × (x - (-2)) y - 6 = x + 2 x - y = -8

Explanation

The percentage error formula is given by:

Percentage Error = (|Correct Value - Incorrect Value| / Correct Value) × 100

Let the original number be x.

The correct value is (7/4)x and the incorrect value is (4/7)x.

Substituting these values into the formula:

Percentage Error = |(7/4)x - (4/7)x| / ((7/4)x) × 100

Percentage Error = (33x / 28) / (7x / 4) × 100

Percentage Error = (33 / 28) × (4 / 7) × 100

Percentage Error = (33 / 49) × 100

Percentage Error ≈ 0.673469 × 100 ≈ 67.35%

Explanation

Let the total work to be done be 120 units, which is the least common multiple of 40, 60, and 20.

The daily work rates of the painters are: Rate of Painter A = 120 / 40 = 3 units per day Rate of Painter B = 120 / 60 = 2 units per day Combined rate of Painter A, B, and C = 120 / 20 = 6 units per day

The rate of Painter C alone is: Rate of C = Combined rate - (Rate of A + Rate of B) Rate of C = 6 - (3 + 2) = 1 unit per day

Therefore, the number of days Painter C alone takes to complete the entire job is: Time taken by C = Total work / Rate of C = 120 / 1 = 120 days

Explanation

The sum of the measures of the exterior angles of any convex polygon is always 360°, regardless of the number of sides it has. Since a decagon is a polygon with 10 sides, the sum of all its exterior angles is exactly 360°.

Explanation

The diagonal of a square with side length a is given by the formula d = a√2. Given that the diagonal is 10√2 cm, we can set up the equation a√2 = 10√2, which simplifies directly to a = 10 cm.

The perimeter of a square is calculated using the formula P = 4a. Substituting the side length of 10 cm into this formula gives P = 4 × 10 = 40 cm.

Explanation

Expanding the algebraic expression (6x + y)(x - 6y) using the distributive property gives:

(6x + y)(x - 6y) = 6x(x - 6y) + y(x - 6y)

Expanding each term:

= 6x² - 36xy + xy - 6y²

Combining the like terms:

= 6x² - 35xy - 6y²

Explanation

Using the algebraic identity for the square of a difference:

(a - b)² = a² + b² - 2ab

Substituting the given values (a - b) = -5 and a² + b² = 73 into the identity:

(-5)² = 73 - 2ab

25 = 73 - 2ab

Rearranging the terms to solve for ab:

2ab = 73 - 25

2ab = 48

ab = 24

Explanation

Let the required non-zero number be x.

According to the given condition: x + 10/x = 7

Multiplying the equation by x: x² + 10 = 7x

Rearranging the terms: x² - 7x + 10 = 0

Factoring the quadratic equation: (x - 5)(x - 2) = 0

Solving for x gives: x = 5 or x = 2

Substituting x = 5 into the original expression: 5 + 10/5 = 5 + 2 = 7

This verifies that 5 is the correct number.

Explanation

We can break the compound inequality into two separate inequalities.

First, consider the left inequality: 2x + 2(4 + 3x) < 2 + 3x 2x + 8 + 6x < 2 + 3x 8x + 8 < 2 + 3x 5x < -6 x < -1.2

Next, consider the right inequality: 2 + 3x > 2x + x/2 2 + 3x > 2.5x 0.5x > -2 x > -4

Combining both conditions yields the interval: -4 < x < -1.2

Of the given values, only -3 lies within this range.

Explanation

The ratio of Import to Export for any year is given as: Import / Export = Ratio

For the year 2012, the ratio is 0.7 and the imports were ₹1100 crores. Using this relation: 1100 / E₂₀₁₂ = 0.7 E₂₀₁₂ = 1100 / 0.7 ≈ 1571.43 crores

The total exports for the years 2012 and 2013 together was ₹5200 crores: E₂₀₁₂ + E₂₀₁₃ = 5200 1571.43 + E₂₀₁₃ = 5200 E₂₀₁₃ = 5200 - 1571.43 = 3628.57 crores

For the year 2013, the ratio of Import to Export is 0.8: I₂₀₁₃ / E₂₀₁₃ = 0.8 I₂₀₁₃ = 0.8 × 3628.57 ≈ 2902.86 crores

Rounding this value to the nearest integer gives 2903 crores.

Explanation

To find the increase in the height of the child from the 8th Birthday to the 15th Birthday, we subtract the height on the 8th Birthday from the height on the 15th Birthday.

From the given table: Height of the child on the 15th Birthday = 180 cm Height of the child on the 8th Birthday = 135 cm

Increase in height = 180 cm - 135 cm = 45 cm

Explanation

The table values represent percentages of the total tracked time, as their sum is 100 (25 + 5 + 10 + 5 + 55 = 100).

The total time duration recorded by the fitness band between 10 PM and 6 AM is 8 hours.

The combined percentage of time the user spent either Dreaming or in Extremely light sleep is: 5% + 5% = 10%

To find the duration in hours, apply the percentage formula: Duration = (Combined percentage / 100) × Total hours Duration = (10 / 100) × 8 = 0.8 hours

Explanation

In triangle ABC, since DE is parallel to BC, triangle ADE is similar to triangle ABC by Angle-Angle similarity.

The ratio of their corresponding sides must be equal: AD/AB = DE/BC

First, find the total length of side AB: AB = AD + DB = 10 + 5 = 15 cm

Now, substitute the known values into the ratio: 10/15 = 6/BC

Simplify the fraction on the left side: 2/3 = 6/BC

Cross-multiplying to solve for BC: 2 × BC = 18 BC = 9 cm

Explanation

We solve the two given inequalities to find the range of values that x can take.

The first inequality is: 2x - 1 < 5x + 2 -3 < 3x x > -1

The second inequality is: 2x + 5 < 6 - 3x 5x < 1 x < 1/5

Combining both inequalities, we get the solution interval: -1 < x < 1/5

The value 0 lies within this interval. Substituting x = 0 into the given inequalities confirms this: 2(0) - 1 < 5(0) + 2 simplifies to -1 < 2, which is true. 2(0) + 5 < 6 - 3(0) simplifies to 5 < 6, which is also true.

Explanation

The total work required to build one cupboard is completed in 60 hours.

The fraction of the cupboard built in 15 hours is: 15 / 60 = 1/4 = 0.25

The fraction of the cupboard that is yet to be built is: 1 - 0.25 = 0.75

Explanation

The simple interest formula is SI = P × R × T / 100. Since the amount increases by 75% in 5 years, the simple interest SI is 75% of the principal P.

0.75 × P = P × R × 5 / 100 R = 75 / 5 = 15% per annum

For compound interest over 2 years, the effective rate of interest is: Effective Rate = R + R + (R × R) / 100 Effective Rate = 15 + 15 + (15 × 15) / 100 = 32.25%

The compound interest on Rs 40000 for 2 years at this effective rate is: CI = 40000 × 32.25% = 40000 × 0.3225 = Rs 12900

Explanation

For a quadratic equation of the form ax² + bx + c = 0, the sum of its roots is given by the formula -b/a.

In the quadratic equation x² - 5x + 6 = 0, the coefficients are: a = 1 b = -5 c = 6

Using the formula: Sum of roots = -b/a = -(-5)/1 = 5

Factoring the equation also confirms this: x² - 5x + 6 = 0 x² - 2x - 3x + 6 = 0 x(x - 2) - 3(x - 2) = 0 (x - 2)(x - 3) = 0

This gives the roots as 2 and 3. The sum of these roots is 2 + 3 = 5.

Explanation

In an isosceles trapezium, let the shorter parallel side be DC = 10 cm and the equal non-parallel sides be AD = BC = 10 cm. The base angles are 45°.

Draw perpendiculars DP and CQ from the upper vertices D and C to the lower base AB. This divides the base AB into three segments: AP, PQ, and QB, where PQ = DC = 10 cm.

In the right-angled triangle APD, the angle DAP is 45° and the hypotenuse AD is 10 cm. Using trigonometric ratios: Height h = DP = AD × sin(45°) = 10 × (1/√2) = 5√2 cm Base segment x = AP = AD × cos(45°) = 10 × (1/√2) = 5√2 cm

Since the trapezium is symmetric, QB = AP = 5√2 cm. The length of the longer parallel side AB is: AB = AP + PQ + QB = 5√2 + 10 + 5√2 = 10√2 + 10 cm

The area of a trapezium is calculated using the formula: Area = ½ × (sum of parallel sides) × height Area = ½ × (AB + DC) × h Area = ½ × (10√2 + 10 + 10) × 5√2 Area = ½ × (10√2 + 20) × 5√2 Area = (5√2 + 10) × 5√2 Area = 50 + 50√2 sq cm

Explanation

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

When the machine is sold for Rs 50 lakhs, the loss incurred is: Loss = CP - 50

When the machine is sold for Rs 60 lakhs, the gain made is: Gain = 60 - CP

As given, the gain is 7 times the earlier loss: Gain = 7 × Loss

Substituting the values into the relation: 60 - CP = 7 × (CP - 50) 60 - CP = 7 × CP - 350 8 × CP = 410 CP = 410 / 8 = 51.25 lakhs

Explanation

The area of a circle is directly proportional to the square of its radius, given by the formula A = πr². When a quantity depends on the product of two variables (in this case, r × r), the net percentage change can be calculated using the successive percentage change formula:

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

Here, the radius increases by 27%, so we substitute x = 27 and y = 27 into the formula:

Net percentage increase in area = 27 + 27 + (27 × 27 / 100) = 54 + 729 / 100 = 54 + 7.29 = 61.29%

Explanation

Let Hariram's salary be H.

Girilal's salary G is given as 11/7 times of Hariram's salary: G = (11/7) × H

Shekhar's salary S is given as 3/4 times of Hariram's salary: S = (3/4) × H

The ratio of Girilal's salary to Shekhar's salary is calculated as: G / S = ((11/7) × H) / ((3/4) × H) G / S = (11/7) / (3/4) G / S = (11/7) × (4/3) G / S = 44/21

Explanation

The algebraic identity for the sum of the squares of a sum and a difference is: (a + b)² + (a - b)² = 2(a² + b²)

By substituting a = cos α and b = sin α into this identity: (cos α + sin α)² + (cos α - sin α)² = 2(cos² α + sin² α)

Using the fundamental trigonometric identity cos² α + sin² α = 1: 2(cos² α + sin² α) = 2 × 1 = 2

Explanation

Let CP be the Cost Price and L be the List Price of the goods.

In the first scenario, a discount of 20% on the list price results in a selling price of 80% of the List Price (0.8L). This transaction yields a loss of 16%, meaning this same selling price is equal to 84% of the Cost Price (0.84CP).

Equating these two expressions for the selling price: 0.8L = 0.84CP L = 1.05CP

In the second scenario, a discount of 10% is offered on the list price. The new selling price is 90% of the List Price: New Selling Price = 0.9L

Substituting the relationship L = 1.05CP into the formula: New Selling Price = 0.9 × 1.05CP = 0.945CP

Since the new selling price is less than the Cost Price, the merchant incurs a loss. Loss percentage = (1 - 0.945) × 100 = 5.5%

Explanation

The prime numbers between 30 and 42 are 31, 37, and 41. These are integers greater than 1 that have no positive divisors other than 1 and themselves.

Calculating the sum of these prime numbers: 31 + 37 + 41 = 109

Explanation

The relation between the volume (V) and the curved surface area (CSA) of a hemisphere of radius r is given by the formula: V / CSA = ((2/3)πr³) / (2πr²) = r/3

Rearranging this relation to solve for the radius directly: r = 3 × V / CSA

Substituting the given values of V = 19404 and CSA = 2772 into the equation: r = 3 × 19404 / 2772 r = 3 × 7 r = 21 cm

Alternatively, this can also be verified using the formula for the curved surface area: 2πr² = 2772 2 × (22/7) × r² = 2772 r² = (2772 × 7) / 44 r² = 441 r = √441 = 21 cm

Explanation

Using the standard sum-to-product trigonometric identity, the difference between two cosines is given by:

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

By applying the negative angle identity for the sine function, sin(-θ) = -sin θ, the negative sign can be incorporated into the second sine term:

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

Substituting this relationship back into the original identity gives:

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

Explanation

To find the value of x for which the two expressions 12 - 6x and 4x + 2 are equal, we set up the linear equation:

12 - 6x = 4x + 2

We rearrange the equation to group the terms with x on one side and the constant terms on the other side. Adding 6x to both sides gives:

12 = 10x + 2

Subtracting 2 from both sides gives:

10 = 10x

Dividing both sides by 10 yields:

x = 1

Explanation

For any two similar triangles, the ratio of their areas is equal to the ratio of the squares of their corresponding sides.

Area(Δ DEF) / Area(Δ GHI) = (DE / GH)²

Substituting the given values, DE = 4 cm and GH = 9 cm:

Area(Δ DEF) / Area(Δ GHI) = (4 / 9)² = 16 / 81

Thus, the ratio of the areas of Δ DEF and Δ GHI is 16:81.

Explanation

Let the time taken by the Rajdhani train to travel 816 km be T hours.

According to the first condition, the Express train takes 9 hours more than Rajdhani, so the time taken by the Express train is T + 9 hours.

When the speed of a train is doubled, the time taken to cover the same distance is halved. Therefore, the new time taken by the Express train is (T + 9) / 2 hours.

According to the second condition, this new time is 4 hours less than Rajdhani's time: (T + 9) / 2 = T - 4

Solving for T: T + 9 = 2(T - 4) T + 9 = 2T - 8 T = 17 hours

Thus, the Rajdhani train takes 17 hours to cover 816 km.

The speed of Rajdhani = Distance / Time = 816 / 17 = 48 km/hr.

Explanation

The equation of the given line is y = 2x + 3. Comparing this with the slope-intercept form y = mx + c, the slope of this line m₁ is 2.

Since the required line is perpendicular to the given line, the product of their slopes must be -1. Therefore, the slope of the required line m₂ is given by the relation: m₁ × m₂ = -1 2 × m₂ = -1 m₂ = -½

The line intercepts the x-axis at -5, which means it passes through the point (-5, 0).

Using the point-slope form of a linear equation, y - y₁ = m(x - x₁), we substitute m = -½ and the point (-5, 0): y - 0 = -½(x - (-5)) y = -½(x + 5) 2y = -x - 5 x + 2y = -5

Explanation

Using the standard algebraic identity (a + b)² - (a - b)² = 4ab, let a = x and b = 2y.

Substituting these values into the right side of the equation: (x + 2y)² - (x - 2y)² = 4(x)(2y) = 8xy

Equating this to the left-hand side: 4pxy = 8xy

Dividing both sides by 4xy gives: p = 8/4 p = 2

Explanation

Runs scored in the first 26 overs = 26 × 5.4 = 140.4

Remaining runs required to reach the target of 294 runs = 294 - 140.4 = 153.6

Required run rate in the remaining 24 overs = Remaining runs ÷ Remaining overs = 153.6 ÷ 24 = 6.4 runs/over

Explanation

By the periodic property of the tangent function, tan(θ + π) = tan(θ) for any angle θ.

Substituting θ = -5π/6:

tan(-5π/6) = tan(-5π/6 + π)

Simplifying the term inside the tangent function:

-5π/6 + π = π/6

This gives:

tan(-5π/6) = tan(π/6)

Using the standard trigonometric value, tan(π/6) = 1/√3. Therefore, the value of tan(-5π/6) is 1/√3.

Explanation

To find the pair of years with equal total exports, we calculate the sum of exports of all three companies for the respective years.

For the year 2013, the total exports of Company A, Company B, and Company C are 3000 + 1000 + 4000 = 8000 lakh rupees.

For the year 2014, the total exports of Company A, Company B, and Company C are 1000 + 5000 + 2000 = 8000 lakh rupees.

Since both 2013 and 2014 have equal total exports of 8000 lakh rupees, this pair of years meets the given condition.