SSC CHSL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

Using the difference of squares identity, a² - b² = (a - b)(a + b), we can express the calculation 99 × 99 - 1 as:

99² - 1² = (99 - 1) × (99 + 1)

Simplifying the terms inside the parentheses:

98 × 100 = 9800

Alternatively, we can calculate the square of 99 using the identity (a - b)² = a² - 2ab + b²:

99² = (100 - 1)² = 100² - 2 × 100 × 1 + 1² = 10000 - 200 + 1 = 9801

Subtracting 1 from this result gives:

9801 - 1 = 9800

Explanation

When a body travels equal distances at two different speeds, say a and b, the average speed for the entire journey is given by the formula:

Average Speed = 2ab / (a + b)

Given: Speed during the onward journey (a) = 40 km/h Speed during the return journey (b) = 60 km/h

Substituting these values into the formula:

Average Speed = 2 × 40 × 60 / (40 + 60) Average Speed = 4800 / 100 Average Speed = 48 km/h

Explanation

The relation between the population two years ago and the present population is given by the compound growth formula: Present Population = P × (1 + R/100)ⁿ

In this scenario, the present population is 54,450, the annual growth rate R is 10%, and the time period n is 2 years. Let P be the population of the village two years ago.

Substituting the given values into the formula: 54450 = P × (1 + 10/100)² 54450 = P × (11/10)² 54450 = P × 121/100 P = (54450 × 100) ÷ 121 P = 450 × 100 P = 45000

Explanation

Correct Answer: D

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

By expanding the expression, we get: p³ + 3p² + 3p

We can rewrite this by adding and subtracting 1 to complete the cube: p³ + 3p² + 3p + 1 - 1 = (p + 1)³ - 1

Substitute the given value p = 38 into the simplified expression: (38 + 1)³ - 1 = 39³ - 1

Calculating the cube of 39: 39³ = 39 × 39 × 39 = 1521 × 39 = 59319

Subtracting 1 from this value: 59319 - 1 = 59318

Explanation

In triangle ABC, the sides AB = 3.6 cm and CA = 4 cm share the vertex A, meaning the included angle between them is angle A. Since the sum of the angles in a triangle is 180°, the measure of angle A is calculated as 180° - 52.4° - 45.5° = 82.1°.

In triangle JKL, the given sides are JK = 3.6 cm and LJ = 4 cm, which share the vertex J. The included angle between these sides is given as angle J = 52.4°. Since this included angle J (52.4°) is not equal to the corresponding included angle A (82.1°) in triangle ABC, the two triangles do not satisfy the SAS (Side-Angle-Side) congruence criterion. Consequently, triangle JKL is not congruent to triangle ABC.

Explanation

For compound interest compounded semi-annually, the effective interest rate per period is half of the annual rate, and the number of compounding periods is doubled.

Given: Principal (P) = ₹8,000 Annual interest rate (R) = 10% per annum Rate per half-year (r) = 10% / 2 = 5% Time (t) = 1 year Number of compounding periods (n) = 2

The formula for the total amount (A) after compounding is: A = P × (1 + r/100)ⁿ

Substituting the given values: A = 8000 × (1 + 5/100)² A = 8000 × (1.05)² A = 8000 × 1.1025 A = 8820

The compound interest (CI) is the difference between the total amount and the principal: CI = A - P CI = 8820 - 8000 = 820

Explanation

The net multiplier for the transaction can be calculated by multiplying the profit factor from the price markup and the profit factor from the false weight:

Net multiplier = (1 + markup percentage / 100) × (nominal weight / actual weight)

Substituting the given values into the formula: Net multiplier = (1 + 8/100) × (750 / 720) Net multiplier = 1.08 × (25 / 24) Net multiplier = (108 / 100) × (25 / 24) Net multiplier = (27 / 25) × (25 / 24) Net multiplier = 27 / 24 = 9 / 8 = 1.125

Actual profit percentage = (1.125 - 1) × 100 = 12.5%

Explanation

To find the volume of a cuboid, we use the standard formula:

Volume = length × breadth × height

Before multiplying, all dimensions must be converted to the same unit of measurement. The given dimensions of the cuboid are a length of 12 m, a breadth of 6 m, and a height of 50 cm.

Converting the height from centimeters to meters: height = 50 cm = 50 ÷ 100 m = 0.5 m

Substituting these values into the formula: Volume = 12 m × 6 m × 0.5 m Volume = 36 m³

Explanation

In the initial alloy, the ratio of copper to bronze is 3:4, and in the final alloy, the ratio becomes 4:3 after adding copper. Since the quantity of bronze remains constant, we can make the bronze parts equal in both ratios by multiplying the initial ratio of 3:4 by 3 and the final ratio of 4:3 by 4.

This gives: Initial ratio of copper to bronze = 9:12 Final ratio of copper to bronze = 16:12

The initial total mixture is represented by 9 + 12 = 21 units. Given that the initial mixture weight is 350 gm, we have: 21 units = 350 gm 1 unit = 350 / 21 gm = 50 / 3 gm

The increase in copper quantity is 16 - 9 = 7 units. Therefore, the quantity of copper to be added is: 7 units = 7 × (50 / 3) gm = 350 / 3 gm = 116.67 gm

Explanation

Substitute the standard trigonometric values for A = 30°: sin 30° = ½ cosec 30° = 2 cot 30° = √3 cot² 30° = 3

First, evaluate the bracketed term in the numerator: 6 sin A + 9 cosec A - cot² A = 6(½) + 9(2) - 3 = 3 + 18 - 3 = 18

Next, evaluate the term in the denominator: 12 sin A = 12(½) = 6

The given expression represents the division of the bracketed term by the remaining term: [6 sin A + 9 cosec A - cot² A] / (12 sin A) = 18 / 6 = 3

Explanation

The volume of a cuboid is calculated using the standard formula:

Volume = length × width × height

Substituting the given dimensions of 15 m, 22 m, and 5 m into the formula:

Volume = 15 × 22 × 5

To simplify the calculation, first multiply 15 by 5:

Volume = 75 × 22

Calculating the final product:

Volume = 1650 m³

Explanation

Let T be the maximum aggregate marks of the examination.

The minimum marks required to pass are 340. Since the student scores 244 marks and fails by 12% of the total marks, the difference between the passing marks and the student's score must be equal to 12% of the total marks.

By establishing this relation: 340 - 244 = 12% of T 96 = 0.12 × T T = 96 / 0.12 T = 800

The maximum aggregate marks a student can get are 800.

Explanation

The area of a right-angled triangle is given by the formula:

Area = ½ × base × height

In a right-angled triangle, the two sides containing the right angle serve as the base and the height. Substituting the given area of 42 cm² and one side of 10 cm into the formula:

42 = ½ × 10 × height

42 = 5 × height

height = 42 / 5 = 8.4 cm

Explanation

The net percentage change when there is a successive increase and decrease is given by the formula:

Net change % = a + b + (ab / 100)

Here, the markup represents an increase of 30% (a = +30) and the discount represents a decrease of 15% (b = -15).

Substituting these values into the formula:

Net change % = 30 - 15 + ((30 × -15) / 100)

Net change % = 15 - 4.5 = 10.5%

Explanation

The mean proportional between two numbers a and b is defined as the square root of their product: Mean Proportional = √(a × b)

Given that the mean proportional between an unknown number x and 20 is 50, we set up the relation: √(x × 20) = 50

Squaring both sides of the equation to eliminate the square root gives: 20x = 50² 20x = 2500

Solving for x: x = 2500 / 20 x = 125

Explanation

The total runs of the initial 8 players can be calculated as: 8 × 6.5 = 52

When 2 new players are included, the total number of players becomes 10, and their new average run increases by 1 to become 7.5. The total runs of all 10 players is: 10 × 7.5 = 75

The sum of the runs of the 2 new players is the difference between the two totals: 75 - 52 = 23

The average run of these 2 new players is: 23 ÷ 2 = 11.5

Explanation

Let the initial population of the city be P. The final population A after successive percentage increases of r₁, r₂, and r₃ over three years is given by the formula:

A = P × (1 + r₁/100) × (1 + r₂/100) × (1 + r₃/100)

Substituting the given values r₁ = 5, r₂ = 2, r₃ = 6, and A = 56763 into the formula:

56763 = P × (1 + 5/100) × (1 + 2/100) × (1 + 6/100)

56763 = P × (105/100) × (102/100) × (106/100)

Simplifying the fractions:

56763 = P × (21/20) × (51/50) × (53/50)

56763 = P × (21 × 51 × 53) / (20 × 50 × 50)

Since 21 × 51 × 53 = 56763, the expression simplifies directly to:

56763 = P × 56763 / 50000

P = 50000

Explanation

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

Substituting the given values, we get: (√7)² + (√5)² = 2(a² + b²) 7 + 5 = 2(a² + b²) 12 = 2(a² + b²) a² + b² = 6

Using another identity for the product ab: (a + b)² - (a - b)² = 4ab 7 - 5 = 4ab 2 = 4ab ab = ½

Now, substitute these values into the required expression: 8ab(a² + b²) - (a - b)² = 8 × ½ × 6 - (√5)² = 4 × 6 - 5 = 24 - 5 = 19

Explanation

Using the fundamental trigonometric identity: cosec²θ - cot²θ = 1

We can express cosec²θ in terms of cot²θ: cosec²θ = 1 + cot²θ

Substitute this relation into the given equation: cosec²θ + cot²θ = 5/3 (1 + cot²θ) + cot²θ = 5/3 1 + 2 cot²θ = 5/3

Subtracting 1 from both sides: 2 cot²θ = 5/3 - 1 2 cot²θ = 2/3

Dividing by 2 yields: cot²θ = 1/3

Explanation

Following the BODMAS rule, we first evaluate the terms in parentheses and exponents.

The term in the parentheses is evaluated as 1 + 2 + 3 + 4 = 10. The exponent term is evaluated as 5² = 25.

Substituting these values, the expression simplifies to: 96 ÷ 48 × 29/2 + 25 - 10

Next, we perform division and multiplication from left to right. 96 ÷ 48 = 2 Multiplying this result by 29/2 gives: 2 × 29/2 = 29

Finally, we perform addition and subtraction: 29 + 25 - 10 = 54 - 10 = 44

Explanation

For any arithmetic progression, the average of the terms is equal to the average of the first and last terms.

The odd numbers from 21 to 81 form an arithmetic progression: 21, 23, 25, ..., 81.

Here, the first term is 21 and the last term is 81.

Average = (First term + Last term) / 2 Average = (21 + 81) / 2 Average = 102 / 2 Average = 51

Explanation

The total surface area (TSA) of a solid cylinder is given by the formula:

TSA = 2πr(r + h)

where r is the radius and h is the height.

Given that height h = r/2, substituting this value into the formula:

TSA = 2πr(r + r/2)

TSA = 2πr(3r/2)

TSA = 3πr²

Explanation

For two successive discount percentages of d₁ and d₂, the net discount percentage is calculated using the formula:

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

Substituting the values d₁ = 10 and d₂ = 5:

Net Discount = 10 + 5 - (10 × 5)/100 Net Discount = 15 - 50/100 Net Discount = 15 - 0.5 Net Discount = 14.5%

Explanation

Using the rule of alligation, the ratio in which the two types of pulses must be mixed is calculated as:

Ratio of cheaper pulse to dearer pulse = (Price of dearer pulse - Mean price) / (Mean price - Price of cheaper pulse)

Substituting the given values:

Ratio = (95 - 85) / (85 - 80)

Ratio = 10 / 5 = 2 / 1

This gives the required ratio as 2:1.

Explanation

Final Amount = Price × (1 - d₁/100) × (1 - d₂/100)

Here, the original price of the laptop is ₹35,000, the online payment discount d₁ is 20%, and the credit card discount d₂ is 8%.

Substituting these values into the formula:

Final Amount = 35000 × (1 - 20/100) × (1 - 8/100) Final Amount = 35000 × 0.80 × 0.92 Final Amount = 28000 × 0.92 Final Amount = 25760

Explanation

For two similar triangles, the ratio of their corresponding sides is equal. Since triangle ABC is similar to triangle DEF, the ratio of side AB to side DE is equal to the ratio of side BC to side EF:

AB / DE = BC / EF

Substituting the given values into the formula:

6 / 9 = 8 / EF

Simplifying the fraction 6 / 9 gives 2 / 3:

2 / 3 = 8 / EF

Cross-multiplying to solve for EF:

2 × EF = 8 × 3

2 × EF = 24

EF = 12 cm

Explanation

Number of puffs sold by bakeries A and B together in February = 109 + 87 = 196

Number of puffs sold by bakeries C and D together in February = 220 + 132 = 352

The ratio of puffs sold by A and B together to those sold by C and D together in February is 196 : 352.

Dividing both terms by their common factor of 4 yields the simplified ratio of 49:88.

Explanation

The relationship between two selling prices at different gain or loss percentages for the same cost price is given by the formula:

SP₂ = SP₁ × (100 + Gain%) / (100 - Loss%)

Given: First selling price (SP₁) = ₹14,400 Loss% = 25% Target Gain% = 20%

Substituting these values into the formula:

SP₂ = 14400 × (100 + 20) / (100 - 25) SP₂ = 14400 × 120 / 75 SP₂ = 14400 × 8/5 SP₂ = 2880 × 8 SP₂ = ₹23,040

Explanation

A number is divisible by 150 if it is simultaneously divisible by both 3 and 50. A number is divisible by 50 if its last two digits are either 00 or 50. A number is divisible by 3 if the sum of its individual digits is divisible by 3.

For the number 463750, the last two digits are 50, which satisfies the divisibility condition for 50. However, the sum of its digits is 4 + 6 + 3 + 7 + 5 + 0 = 25. Since 25 is not divisible by 3, the number 463750 is not divisible by 3, and hence it is not divisible by 150.

Explanation

Let the radius of the spherical balloon be r cm. The surface area of a sphere is given by the formula 4πr².

By equating the given surface area to the formula, we have: 4πr² = 400π r² = 100 r = 10 cm

The volume of a sphere, representing the volume of air inside the balloon, is given by the formula 4/3 πr³.

Substituting the radius r = 10 cm into this formula gives: Volume = 4/3 × π × 10³ = 4000/3 π cm³

Explanation

Let a = x - 8 and b = y - 5.

Sum of these terms: a + b = (x - 8) + (y - 5) = x + y - 13

Since x + y = 13, substituting this value gives: a + b = 13 - 13 = 0

This means: b = -a

Now, substituting b = -a into the expression: (x - 8)³ + (y - 5)³ = a³ + b³ = a³ + (-a)³ = a³ - a³ = 0

Explanation

To find the three consecutive odd numbers, we can factorize the given product, 693.

By dividing 693 by its prime factors: 693 = 7 × 99 = 7 × 9 × 11

The factors 7, 9, and 11 are indeed three consecutive odd numbers whose product is 693.

Comparing these three numbers, 7 is the smallest of them.

Explanation

Let the five numbers in order be a, b, c, d, and e.

The sum of all five numbers is: a + b + c + d + e = 5 × 32 = 160

The sum of the first three numbers is: a + b + c = 3 × 28 = 84

The sum of the last three numbers is: c + d + e = 3 × 34 = 102

Adding these two sums together: (a + b + c) + (c + d + e) = 84 + 102 = 186 (a + b + c + d + e) + c = 186

Substituting the sum of all five numbers into the equation: 160 + c = 186 c = 26

Subtracting the value of c from the sum of the first three numbers gives the sum of the first two numbers: a + b = 84 - 26 = 58

The average of the first two numbers is: (a + b) / 2 = 58 / 2 = 29

Explanation

To simplify the expression 11.5 - [5 + 0.5 × (9 - 3 × 2)], apply the BODMAS rule, which dictates the order of operations: Parentheses, Orders, Division and Multiplication, and Addition and Subtraction.

First, simplify the innermost parentheses by performing multiplication before subtraction: 9 - 3 × 2 = 9 - 6 = 3

Next, substitute this result back into the expression and simplify the square brackets, performing multiplication before addition: 5 + 0.5 × 3 = 5 + 1.5 = 6.5

Finally, subtract this value from 11.5: 11.5 - 6.5 = 5

Explanation

A number is divisible by 75 if it is divisible by both 3 and 25.

For divisibility by 25, the last two digits of the number must be 00, 25, 50, or 75. For divisibility by 3, the sum of the digits of the number must be divisible by 3.

The number 163750 ends in 50, which satisfies the divisibility rule for 25. However, the sum of its digits is 1 + 6 + 3 + 7 + 5 + 0 = 22. Since 22 is not divisible by 3, the number 163750 is not divisible by 3, making it not divisible by 75.

Explanation

For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides:

Area(XYZ) / Area(ABC) = (Side of XYZ / Side of ABC)²

Substituting the given values:

361 / 225 = (38 / s)²

Taking the square root on both sides:

19 / 15 = 38 / s

Solving for s:

s = (38 × 15) / 19 = 30 cm

Explanation

The difference between compound interest and simple interest for a period of 2 years is given by the formula:

D = P × (r/100)²

where D is the difference, P is the principal sum, and r is the annual rate of interest.

Substituting the given values, D = 80 and r = 20, into the formula:

80 = P × (20/100)²

80 = P × (1/5)²

80 = P × 1/25

P = 80 × 25

P = 2000

Explanation

When a quantity is successively changed by x% and y%, the net percentage change is given by the formula:

Net change% = x + y + xy/100

Here, the price is first increased by 8% (x = +8) and then decreased by 8% (y = -8). Substituting these values into the formula:

Net change% = 8 - 8 + (8 × (-8))/100 = -0.64%

The negative sign indicates a net decrease of 0.64%.

Explanation

The formula for calculating the discount given on a marked price is:

Discount = (Discount Percentage / 100) × Marked Price

Given that the marked price of the book is £600 and the discount percentage is 18%:

Discount = (18 / 100) × 600 Discount = 18 × 6 Discount = 108

Explanation

The net percentage change when a value is successively increased by a% and then decreased by b% can be calculated using the successive percentage formula:

Net change = a + b + ab/100

Here, the price is first increased by 15% (so a = 15) and then decreased by 20% (so b = -20). Substituting these values into the formula:

Net change = 15 - 20 + (15 × -20)/100

Net change = -5 - 300/100

Net change = -5 - 3 = -8%

The negative sign indicates a net decrease of 8% in the final price of the pen.

Explanation

The given expression is of the algebraic form a² - 2ab + b², which simplifies to (a - b)².

By setting a = 0.14 and b = 5.14, the expression can be rewritten as:

(0.14 - 5.14)²

Simplifying the term inside the parentheses:

-5² = 25

Explanation

Speed is calculated by dividing the distance traveled by the time taken.

First, convert the distance from meters to kilometers: Distance = 3600 m = 3.6 km

Next, convert the time from minutes to hours: Time = 12 minutes = 12/60 hours = 0.2 hours

Now, apply the speed formula: Speed = Distance / Time = 3.6 / 0.2 = 18 km/h

Explanation

To find the smallest ratio, we can convert each given ratio into its decimal equivalent. The ratio 1:3 is equal to 1 ÷ 3, which is approximately 0.33. Comparing this to the other ratios, which have decimal values of 0.40 (for 2:5), 0.50 (for 1:2), and 0.75 (for 3:4), it is clear that 1:3 represents the smallest fractional value.

Explanation

The monthly aggregate production of all four types of scooters (X, Y, Z, and T) is given by the TOTAL row in the table.

Calculating the total production for each month: January = 25 + 25 + 25 + 25 = 100 February = 25 + 27 + 27 + 26 = 105 March = 18 + 50 + 15 + 25 = 108 April = 40 + 45 + 25 + 0 = 110 May = 20 + 30 + 30 + 30 = 110 June = 15 + 20 + 20 + 35 = 90

Comparing these totals, the lowest aggregate number of scooters produced is 90, which occurred in June.

Explanation

Using the weighted average formula:

Total Salary = (Number of technicians × Average salary of technicians) + (Number of remaining workers × Average salary of remaining workers)

Let N be the total number of workers. The number of remaining workers is N - 9.

8850 × N = 9 × 10000 + (N - 9) × 7780

8850 × N = 90000 + 7780 × N - 70020

8850 × N - 7780 × N = 19980

1070 × N = 19980

N = 19980 / 1070

N ≈ 18.67

Taking the integral part of N gives 18.

Explanation

The required percentage is calculated using the formula: Percentage = (Students with at least 40% in A ÷ Students with at least 20% in C) × 100

Since the maximum marks are 100, obtaining at least 40% marks in A corresponds to the category of 40 and above in subject A. According to the table, the number of students in this category is 81.

Similarly, obtaining at least 20% marks in C corresponds to the category of 20 and above in subject C. From the table, this number of students is 87.

Substituting these values into the formula yields: Percentage = (81 ÷ 87) × 100 ≈ 0.93103 × 100 ≈ 93.1%

Rounding to the nearest whole number gives approximately 93%.

Explanation

When a value is successively increased by two percentages, say a% and b%, the net percentage increase can be calculated directly using the successive percentage formula:

Net Percentage Increase = a + b + (a × b)/100

In this case, Asha secured 10% more marks than Sachin, so b = 10. Gopal secured 15% more marks than Asha, so a = 15.

Substituting these values into the formula:

Net Percentage Increase = 15 + 10 + (15 × 10)/100 Net Percentage Increase = 25 + 150/100 Net Percentage Increase = 25 + 1.5 = 26.5%

Explanation

Let the time taken by Pranjal to finish the work alone be x days.

Pranjal takes twice as much time as Vikram, which means Vikram takes x/2 days. Pranjal takes thrice as much time as Ashwin, which means Ashwin takes x/3 days.

The sum of their individual daily work rates equals their combined daily work rate. Since they complete the work together in 6 days, their combined rate is 1/6.

1/x + 2/x + 3/x = 1/6

6/x = 1/6

x = 36 days

The time taken by Vikram to complete the work alone is x/2 = 36/2 = 18 days.

Explanation

A month of 30 days starting with a Saturday will have Sundays on the 2nd, 9th, 16th, 23rd, and 30th days. This gives a total of 5 Sundays and 25 other days in the month.

The total number of visitors during the month is: (5 × 330) + (25 × 240) = 1650 + 6000 = 7650

The average number of visitors per day is: 7650 ÷ 30 = 255

Explanation

According to the BODMAS rule, brackets must be solved first, followed by of, division, multiplication, addition, and subtraction.

First, simplify the expression inside the parentheses: 226 + 506 = 732

Next, evaluate the of operation: 732 of 3/4 = 732 × 3/4 = 183 × 3 = 549

Now, perform the division: 3294 ÷ 549 = 6

Finally, add the remaining term: 6 + 500 = 506