SSC CHSL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

Let x + 1/x = k. Squaring both sides of this equation gives: (x + 1/x)² = k² x² + 2 + 1/x² = k² x² + 1/x² = k² - 2

Substituting k = -13: x² + 1/x² = (-13)² - 2 x² + 1/x² = 169 - 2 = 167

To find the value of x⁴ + 1/x⁴, square x² + 1/x²: (x² + 1/x²)² = 167² x⁴ + 2 + 1/x⁴ = 27889 x⁴ + 1/x⁴ = 27889 - 2 x⁴ + 1/x⁴ = 27887

Explanation

To find the value of the expression x² + 1/x², we use the standard algebraic identity:

(x + 1/x)² = x² + 2 + 1/x²

Substituting the given value x + 1/x = 7.5 into the equation:

(7.5)² = x² + 2 + 1/x²

56.25 = x² + 2 + 1/x²

Subtracting 2 from both sides of the equation:

x² + 1/x² = 56.25 - 2 = 54.25

Explanation

Let the total number of employees in the company be 100. Since 35% of the employees are officers, there are 35 officers and 65 workers. The ratio of the number of officers to workers is 35 : 65, which simplifies to 7 : 13.

Let the combined average monthly salary of all employees be A, the average monthly salary of an officer be O, and the average monthly salary of a worker be W.

We are given that the average salary of an officer is ₹39,000 more than the combined average salary: O = A + 39,000

Using the weighted average formula: A = (7 × O + 13 × W) / (7 + 13) 20A = 7 × O + 13 × W

Substitute O = A + 39,000 into the equation: 20A = 7 × (A + 39,000) + 13 × W 20A = 7A + 273,000 + 13 × W 13A - 13 × W = 273,000 13 × (A - W) = 273,000 A - W = 21,000 W = A - 21,000

The difference between the average monthly salaries of an officer and a worker is: O - W = (A + 39,000) - (A - 21,000) O - W = 39,000 + 21,000 = 60,000

Explanation

Convert the mixed fractions into improper fractions: 7 ½ = 15/2 1 ¼ = 5/4 2 ½ = 5/2 8 ⅓ = 25/3

Following the BODMAS rules, simplify the expression step-by-step starting from the innermost brackets:

First, evaluate the subtraction in the innermost parentheses: ¼ - 1/6 = 3/12 - 2/12 = 1/12

Subtract this result from 2 ½: 5/2 - 1/12 = 30/12 - 1/12 = 29/12

Multiply by the coefficient ½ outside the parentheses: ½ × 29/12 = 29/24

Simplify the expression inside the curly braces: 5/4 - 29/24 = 30/24 - 29/24 = 1/24

Evaluate the terms inside the square brackets: 15/2 ÷ 1/24 = 15/2 × 24 = 15 × 12 = 180

Evaluate the second part of the main division containing the "of" operator: ½ of 25/3 = ½ × 25/3 = 25/6

Finally, divide the two main parts: 180 ÷ 25/6 = 180 × 6/25 = 1080/25 = 43.2

Explanation

The consumption of sugar is directly proportional to the number of days. If C₁ and D₁ represent the initial consumption and days, and C₂ and D₂ represent the final consumption and days, the relation is:

C₁ / D₁ = C₂ / D₂

Substituting the given values:

6 / 24 = C₂ / 18

Simplifying this gives:

¼ = C₂ / 18

C₂ = 18 / 4

C₂ = 4.5 kg

Explanation

The capacity of a pond represented as a cuboid is given by the formula for its volume:

Volume = Length × Width × Depth

Substituting the given dimensions of the pond into the formula:

Volume = 50 m × 35 m × 2.5 m

Multiplying 50 by 2.5 gives 125. Then, multiplying 125 by 35 yields:

125 × 35 = 4375 m³

Explanation

The relation between the number of men (M), number of days (D), and the work done in terms of walls painted (W) is given by the formula:

(M₁ × D₁) / W₁ = (M₂ × D₂) / W₂

Substituting the given values into the formula:

M₁ = 16, D₁ = 45, W₁ = 15 M₂ = 20, W₂ = 20

(16 × 45) / 15 = (20 × D₂) / 20

16 × 3 = D₂

D₂ = 48 days

Explanation

The selling price of an article can be directly calculated using the profit percentage formula:

SP = CP × (100 + Profit%)/100

Given: CP = Rs. 6,450 Profit% = 16%

Substituting these values into the formula:

SP = 6450 × (100 + 16)/100 SP = 6450 × 116/100 SP = 64.5 × 116 SP = 7482

Explanation

Let the votes obtained by the winning candidate be W and the votes obtained by the two losing candidates be L₁ and L₂.

According to the given conditions: L₁ = L₂ W = L₁ + L₂

The total number of valid votes polled is 1500: W + L₁ + L₂ = 1500

Substituting W for L₁ + L₂ in the equation: W + W = 1500 2W = 1500 W = 750

The winning candidate got 750 votes.

Explanation

The equivalent single discount for two successive discounts of a% and b% is calculated using the formula: Equivalent Discount = a + b - (a × b)/100

Substituting a = 10 and b = 15: Equivalent Discount = 10 + 15 - (10 × 15)/100 Equivalent Discount = 25 - 1.5 = 23.5%

The second option offers a single discount of 25%.

The difference between the two discount percentages is: Difference = 25% - 23.5% = 1.5%

The difference in the discount amount on ₹2,700 worth of products is: Difference in amount = 1.5% of 2,700 Difference in amount = (1.5 / 100) × 2,700 Difference in amount = 1.5 × 27 = 40.50

Explanation

According to the division algorithm, Dividend = (Divisor × Quotient) + Remainder.

Substituting the given values into the formula: 8675123 = (Divisor × 33611) + 3485

Subtracting the remainder from the dividend: Divisor × 33611 = 8675123 - 3485 Divisor × 33611 = 8671638

Solving for the divisor: Divisor = 8671638 ÷ 33611

Estimating the division, 8.67 × 10⁶ divided by 3.36 × 10⁴ is approximately 258. Verifying by multiplication, 258 × 33611 = 8671638, which confirms the divisor is 258.

Explanation

The angle between two tangents drawn from an external point to a circle and the angle subtended by the line segments joining the points of contact at the center are supplementary. This relation is given by:

angle AOB + angle ARB = 180°

Substituting the given value of angle ARB:

angle AOB + 70° = 180°

angle AOB = 180° - 70° = 110°

Explanation

To find the combined daily consumption for each set of years, we sum the daily consumption of all four grains (Rice, Wheat, Maize, and Barley) for that period.

For the years 2003-2005, the combined daily consumption is 167 + 44 + 39 + 17 = 267 kg.

For the years 2013-2015, the combined daily consumption is 150 + 45 + 51 + 21 = 267 kg.

Since both calculations yield 267 kg, the combined consumption is equal for these two periods.

Explanation

The total surface area of a cube with side length a is given by the formula:

Total Surface Area = 6a²

Substituting the given total surface area of 96 cm² into the formula:

6a² = 96 a² = 16 a = 4 cm

The volume of a cube is given by the formula:

Volume = a³

Substituting the side length a = 4 cm:

Volume = 4³ = 64 cm³

Explanation

The total surface area of a cuboid is given by the formula: Total Surface Area = 2(lb + bh + lh)

Given the dimensions of the cuboid: Length (l) = 17 m Breadth (b) = 8 m Height (h) = 3 m

Substituting these values into the formula: Total Surface Area = 2 × (17 × 8 + 8 × 3 + 17 × 3) Total Surface Area = 2 × (136 + 24 + 51) Total Surface Area = 2 × 211 Total Surface Area = 422 m²

Explanation

To find Sunday's income, we use the relation for total income:

Total weekly income = Average daily income × Number of days

Using the given average of ₹180 for the 7 days of the week:

Total weekly income = 180 × 7 = 1260

Next, we sum the worker's earnings from Monday to Saturday:

Earnings sum = 115 + 212 + 170 + 165 + 185 + 225 = 1072

Subtracting this sum from the total weekly income gives Sunday's income:

Sunday's income = 1260 - 1072 = 188

Explanation

The time taken by a train to cross a standing man is equal to the time taken by the train to cover its own length.

Using the speed, distance, and time relation: Time = Distance ÷ Speed

First, convert the speed of the train from km/h to m/s: Speed = 54 × 5/18 = 15 m/s

The distance to be covered is the length of the train: Distance = 180 m

Now, substitute these values into the formula to find the time: Time = 180 ÷ 15 = 12 seconds

Explanation

When milk is added to the solution, the quantity of water remains constant.

Initial percentage of water in the solution = 100% - 63% = 37% Final percentage of water in the solution = 100% - 70% = 30%

Let the quantity of milk added be x litres.

Since the quantity of water does not change: 37% of 49 = 30% of (49 + x) 37 × 49 = 30 × (49 + x) 1813 = 1470 + 30x 30x = 1813 - 1470 30x = 343 x = 343 / 30 x = 11 13/30 litres

Explanation

Using the algebraic identity for the sum of cubes of three variables: x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

By substituting the given condition x + y + z = 0 into this identity: x³ + y³ + z³ - 3xyz = 0 × (x² + y² + z² - xy - yz - zx) x³ + y³ + z³ - 3xyz = 0

This algebraic relationship directly simplifies the entire expression to zero.

Explanation

The average of a set of quantities is calculated by dividing the sum of all quantities by the total number of quantities.

The sum of the first n quantities with an average of P is nP.

The sum of the next m quantities with an average of Q is mQ.

The total sum of all the combined quantities is nP + mQ, and the total number of quantities is m + n.

The average of the combined (m + n) quantities is obtained by dividing the total sum by the total number of quantities, giving (nP + mQ)/(m + n).

Explanation

The total number of employees in the company is the sum of the employees in all three divisions: Total employees = 20 + 30 + 40 = 90

The number of working employees in each division during the weekend is calculated as: Working employees in Division 1 = 20% of 20 = 4 Working employees in Division 2 = 30% of 30 = 9 Working employees in Division 3 = 40% of 40 = 16

Total working employees = 4 + 9 + 16 = 29

The overall working percentage of the entire company is: (Total working employees ÷ Total employees) × 100 = (29 ÷ 90) × 100 = 32.22%

Rounding to the nearest whole number gives approximately 32%.

Explanation

Given the equation: r + 64/r = 16

Multiplying the entire equation by r to clear the denominator: r² + 64 = 16r

Rearranging the terms into the standard quadratic form: r² - 16r + 64 = 0

This is a perfect square trinomial of the form (a - b)² = a² - 2ab + b²: (r - 8)² = 0

Solving for r: r = 8

Now, substitute r = 8 into the required expression: r⁴ + 1/r³ = 8⁴ + 1/8³

Calculating the powers of 8: 8⁴ = 4096 8³ = 512

Substituting these values: r⁴ + 1/r³ = 4096 + 1/512

Expressing the sum as a mixed fraction gives: 4096 1/512

Explanation

By the rule of alligation, the ratio of the first type of milk to the second type of milk is given by the difference between the mean price and the individual prices:

Ratio of first type to second type = (Mean price - Price of second type) / (Price of first type - Mean price)

Substituting the values: Price of first type = 70 Price of second type = 50 Mean price of mixture = 55

Ratio = (55 - 50) / (70 - 55) Ratio = 5 / 15 Ratio = 1 / 3

Explanation

When a journey is divided into two equal halves with speeds v₁ and v₂, the average speed for the entire journey is given by the formula:

Average Speed = 2v₁v₂ / (v₁ + v₂)

Substituting v₁ = 21 km/h and v₂ = 24 km/h into the formula:

Average Speed = 2 × 21 × 24 / (21 + 24) = 1008 / 45 = 22.4 km/h

The total distance travelled is calculated by multiplying the average speed by the total time taken:

Total Distance = Average Speed × Total Time

Total Distance = 22.4 km/h × 10 hours = 224 km

Explanation

The length of the direct common tangent (L) to two circles with radii r₁ and r₂ (where r₂ ≥ r₁) and the distance between their centres d is given by the formula:

L = √(d² - (r₂ - r₁)²)

Given values: r₁ = 4 cm r₂ = 7 cm d = 12 cm

Substituting these values into the formula:

L = √(12² - (7 - 4)²) L = √(144 - 3²) L = √(144 - 9) L = √135 L = √(9 × 15) L = 3√15 cm

Explanation

The given expression is p(p² + 3p + 3).

By distributing p, the expression becomes: p³ + 3p² + 3p

We can relate this to the algebraic identity for a perfect cube: (p + 1)³ = p³ + 3p² + 3p + 1

Subtracting 1 from both sides gives: p³ + 3p² + 3p = (p + 1)³ - 1

Substituting p = 11 into this equation: (11 + 1)³ - 1 = 12³ - 1

Since 12³ = 1728, the value is: 1728 - 1 = 1727

Alternatively, substituting p = 11 directly: 11 × (11² + 3 × 11 + 3) = 11 × (121 + 33 + 3) = 11 × 157 = 1727

Explanation

According to the BODMAS rule, operations inside brackets are performed first, moving from the innermost to the outermost brackets.

First, evaluate the multiplication inside the curly braces: 10 × (-3) = -30

Next, evaluate the addition inside the curly braces: 4 + (-30) = -26

Then, simplify the expression inside the square brackets: 52 - (-26) = 52 + 26 = 78

Finally, perform the remaining subtraction: 24 - 78 = -54

Explanation

Let the income of A and B be 4x and 3x respectively.

The expenditure of A is 5/12 of his income: (5/12) × 4x = 5x/3

The expenditure of B is 7/15 of his income: (7/15) × 3x = 7x/5

The difference between their expenditures is: 5x/3 - 7x/5 = (25x - 21x) / 15 = 4x/15

Given that the difference between their expenditures is 4/5 of ₹83,402.78, which is ₹66,722.22: 4x/15 = 66,722.22 x = 250,208.33

Thus, the income of B is: 3x = 3 × 250,208.33 = 750,625

Explanation

For any two angles A and B such that A + B = 45°, we can write: tan(A + B) = tan 45° (tan A + tan B) / (1 - tan A tan B) = 1 tan A + tan B = 1 - tan A tan B tan A + tan B + tan A tan B = 1

By adding 1 to both sides of the equation, we get: 1 + tan A + tan B + tan A tan B = 2 (1 + tan A)(1 + tan B) = 2

Here, the given angles are A = 10° and B = 35°. Since A + B = 10° + 35° = 45°, we substitute these values into the identity: (1 + tan 10°)(1 + tan 35°) = 2

Explanation

The volume V of a sphere of radius r is given by the formula V = 4/3 π r³.

Given that the volume is 288π cm³, we can set up the equation: 4/3 π r³ = 288π

Dividing both sides by π: 4/3 r³ = 288

Multiplying both sides by 3/4: r³ = 288 × 3/4 r³ = 72 × 3 r³ = 216

Taking the cube root of both sides gives the radius: r = 6 cm

The surface area A of a sphere is given by the formula A = 4 π r². Substituting the value of r: A = 4 π × 6² A = 4 π × 36 A = 144π cm²

Explanation

Let the total electorate be T. The winning candidate received 48% of the total electorate, which equals 15744 votes. 0.48 × T = 15744 T = 15744 / 0.48 = 32800

The only other candidate received 22% of the total electorate. The total percentage of the electorate who voted for these two candidates is: 48% + 22% = 70%

The percentage of the electorate who did not vote is: 100% - 70% = 30%

The number of members who did not vote is 30% of the total electorate T: 0.30 × 32800 = 9840

Explanation

The loan disbursement for Bank C in consecutive years from 1982 to 1986 is: In 1982: 29 crore In 1983: 28 crore In 1984: 22 crore In 1985: 17 crore In 1986: 11 crore

Since 29 > 28 > 22 > 17 > 11, the disbursement of loans for Bank C shows a continuous decrease year-on-year over the entire period.

Explanation

The equivalent single discount for two successive discounts of a% and b% is calculated using the formula:

Equivalent Discount = a + b - (a × b)/100

Substituting the given discount values of a = 20 and b = 20 into the formula:

Equivalent Discount = 20 + 20 - (20 × 20)/100 Equivalent Discount = 40 - 400/100 Equivalent Discount = 40 - 4 = 36%

Alternatively, if the initial price of the laptop is assumed to be 100, a 20% discount reduces the price to 80. A subsequent 20% discount on the reduced price of 80 decreases the price by 16, resulting in a final price of 64. This represents a total reduction of 36 from the original price of 100, which is equivalent to a single discount of 36%.

Explanation

Let the cost price of 1 g of goods be ₹1. This means the cost price of 40 g of goods is ₹40.

Since the goods are sold at a 4% loss on the cost price, the selling price for this transaction is ₹40 × (1 - 0.04) = ₹38.4.

However, the seller actually delivers only 20 g of goods instead of 40 g. The actual cost price of the 20 g delivered is ₹20.

The profit earned is the selling price minus the actual cost price, which is ₹38.4 - ₹20 = ₹18.4.

To find the profit percentage, divide the profit by the actual cost price and multiply by 100:

Profit percentage = (18.4 / 20) × 100 = 92%

Explanation

Let the total work be the Least Common Multiple (LCM) of 12 and 30, which is 60 units.

Efficiency of A = 60 ÷ 12 = 5 units per day Efficiency of B = 60 ÷ 30 = 2 units per day

Since they work on alternate days starting with A, the work done in one 2-day cycle is: Work in 2 days = 5 + 2 = 7 units

To find the number of completed cycles: 60 ÷ 7 = 8 cycles with a remainder of 4 units

Time taken for 8 complete cycles = 8 × 2 = 16 days Work completed in 16 days = 8 × 7 = 56 units Remaining work = 60 - 56 = 4 units

On the 17th day, it is A's turn. A's efficiency is 5 units per day, so the time taken by A to complete the remaining 4 units is 4/5 day.

Total time taken = 16 + 4/5 = 16 4/5 days.

Explanation

The two trains together cover their combined length of 400 m (200 m + 200 m) while crossing each other, so their relative speed is 400/10 = 40 m/s, which equals 144 km/h. Since they move in opposite directions, their speeds add up, so the speed of the other train is 144 - 90 = 54 km/h.

Explanation

Since distance varies directly with time, the speed is constant at 378/7 = 54 km/h. To cover 918 km at this same speed, the time required is 918/54 = 17 hours.

Explanation

Using compound interest, the amount after 2 years is P(1 + r/100)², so 19683/16875 = (1 + r/100)², which gives 1.1664 = (1 + r/100)². Taking the square root gives 1 + r/100 = 1.08, so r = 8%.

Explanation

A triangle with sides 3 cm, 4 cm and 5 cm satisfies 3² + 4² = 5², making it a right triangle with the two shorter sides (3 cm and 4 cm) as the base and height. Its area is therefore (1/2) x 3 x 4 = 6 cm².

Explanation

According to the divisibility rule of 9, a number is divisible by 9 if the sum of its digits is a multiple of 9.

For the number 34278A2597, the sum of its digits is calculated as: 3 + 4 + 2 + 7 + 8 + A + 2 + 5 + 9 + 7 = 47 + A

For the number to be divisible by 9, the sum 47 + A must be a multiple of 9. The smallest multiple of 9 greater than 47 is 54.

Setting the sum equal to 54: 47 + A = 54 A = 7

Thus, the least positive digit value for A is 7.

Explanation

To find the country with the maximum aggregate runs, we calculate the total runs scored in each country across all four years:

Aggregate runs in Country A = 30 + 25 + 25 + 52 = 132 runs Aggregate runs in Country B = 45 + 32 + 35 + 29 = 141 runs Aggregate runs in Country C = 50 + 40 + 38 + 45 = 173 runs Aggregate runs in Country D = 38 + 50 + 50 + 62 = 200 runs

Comparing these values, the maximum aggregate runs is 200, which was scored in Country D.

Explanation

The final selling price after successive discounts of d₁% and d₂% on the marked price (MP) is calculated using the formula:

Selling Price = MP × (1 - d₁/100) × (1 - d₂/100)

Substituting the given values where MP = 2500, the first discount d₁ = 25%, and the second cash discount d₂ = 8%:

Selling Price = 2500 × (1 - 25/100) × (1 - 8/100) = 2500 × 0.75 × 0.92 = 1875 × 0.92 = 1725

The customer has to pay 1,725.

Explanation

The volume of a rectangular water tank is given by the formula: Volume = Length × Width × Height

Substituting the given dimensions: Volume = 3 m × 3.5 m × 1.5 m = 15.75 m³

Since 1 cubic meter is equivalent to 1000 litres: Capacity in litres = 15.75 × 1000 = 15750 litres

Explanation

Let the stream speed be x km/h, so downstream speed is (12 + x) and upstream speed is (12 – x). The total time is 18/(12+x) + 18/(12-x) = 6⁶⁄₇ = 48/7 hours. Testing x = 9 gives downstream speed 21 km/h and upstream speed 3 km/h, so time taken = 18/21 + 18/3 = 6/7 + 6 = 48/7 hours, which matches exactly, confirming the stream's speed is 9 km/h.

Explanation

The number of teak trees planted rose from 25000 in 2016 to 35000 in 2017, an increase of 10000. Expressed as a percentage of the 2016 figure, this is (10000/25000) × 100 = 40%, which is the percentage increase from 2016 to 2017.

Explanation

Since x²/y² + y²/x² = 7, adding 2 to both sides gives (x/y + y/x)² = 9, so x/y + y/x = 3. Using the identity a³+b³ = (a+b)³ - 3ab(a+b) with a=x/y, b=y/x and ab=1, we get x³/y³ + y³/x³ = 3³ - 3(3) = 27 - 9 = 18.

Explanation

The selling price after a 15% discount is 85% of the marked price, so Marked Price = Selling Price / 0.85 = 1,22,700 / 0.85 = ₹1,44,352.94. This is the price the dealer must mark the machine at so that after deducting the discount, the intended selling price of ₹1,22,700 is realised.

Explanation

Dividing both numerator and denominator terms of the given equation by r converts r² - 8r + 1 into r + 1/r - 8 (after dividing throughout by r), so the equation becomes 8/(r + 1/r - 8) = 1/14, giving r + 1/r - 8 = 112, and hence r + 1/r = 120.

Explanation

Initial number of people in the group is 15 athletes + 3 trainers = 18 people.

Total weight of these 18 people = 18 × 60 = 1080 kg.

Weight of the 4 newly added athletes = 4 × 63 = 252 kg.

Total weight of the new group of 22 people = 1080 + 252 = 1332 kg.

New average weight of the group = 1332 ÷ 22 ≈ 60.55 kg.

Rounding 60.55 kg to the nearest integer yields 61 kg.

Explanation

Work from the innermost bracket outwards. The mixed fraction 3 6/7 equals 27/7, and 27/7 ÷ 54/7 = 1/2. Next, 2 3/4 − 3/2 = 11/4 − 6/4 = 5/4, so the braces give 3 − 5/4 = 7/4. The square bracket is therefore 1/2 − 7/4 = −5/4, and the equation becomes −5/4 + A ÷ 4 = 0. Hence A/4 = 5/4, giving A = 5.