SSC CHSL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

Using the algebraic identity for the cube of a sum: (x + y)³ = x³ + y³ + 3xy(x + y)

By substituting the given value x + y = 14 into the identity: 14³ = x³ + y³ + 3xy(14)

Calculating the cube of 14: 14³ = 14 × 14 × 14 = 2744

Substituting this value back into the equation gives: 2744 = x³ + y³ + 42xy

Explanation

Using the successive percentage change formula, the net percentage change in area when two dimensions are increased by x% and y% is given by:

Net percentage increase = x + y + xy/100

Substituting the given values, x = 15% and y = 20%:

Net percentage increase = 15 + 20 + (15 × 20)/100 Net percentage increase = 35 + 300/100 Net percentage increase = 35 + 3 = 38%

Explanation

To find the cost price, we use the formula: Cost Price = Selling Price / (1 - Loss Percentage). Substituting the given values, Cost Price = 18,200 / 0.85 = 21,411.76, and considering only the integral part gives ₹21,411.

Explanation

The selling price after three successive discounts of 25%, 4%, and 5% is calculated as 18,000 * 0.75 * 0.96 * 0.95. This multiplication yields 13,500 * 0.96 * 0.95 = 12,960 * 0.95 = 12,312.

Explanation

The formula for the total time taken for a round trip downstream and upstream is T = d / (v + s) + d / (v - s), where d is the distance, v is the speed of the boat in still water, and s is the speed of the stream.

Given that v = 15 km/h, d = 30 km, and the total time T is 4 hours and 30 minutes, which is 9/2 hours, we can substitute these values into the formula:

9/2 = 30 / (15 + s) + 30 / (15 - s)

Dividing both sides of the equation by 3:

3/2 = 10 / (15 + s) + 10 / (15 - s)

Taking the common denominator on the right side:

3/2 = 10 × (15 - s + 15 + s) / (225 - s²)

3/2 = 300 / (225 - s²)

Cross-multiplying to solve for s:

225 - s² = 300 × 2 / 3

225 - s² = 200

s² = 25

s = 5 km/h

Explanation

To find the average number of students per college who participated in the English Olympiad, we use the formula for the arithmetic mean:

Average = Sum of observations / Number of observations

From the table, the number of students who participated in the English Olympiad from the five colleges (A, B, C, D, and E) are:

College A = 98 College B = 120 College C = 80 College D = 122 College E = 105

Summing these values:

Total students = 98 + 120 + 80 + 122 + 105 = 525

Since there are 5 colleges in total, we divide the sum by 5:

Average = 525 / 5 = 105

Explanation

The discount percentage is calculated using the formula:

Discount % = ((Marked Price - Selling Price) / Marked Price) × 100

Given: Marked Price = £15,000 Selling Price = £13,500

Substituting these values into the formula:

Discount % = ((15,000 - 13,500) / 15,000) × 100 Discount % = (1,500 / 15,000) × 100 Discount % = 0.1 × 100 = 10%

Explanation

The net percentage change in the population due to successive changes of +10% and -10% is given by the formula:

Net change = A + B + (A × B)/100

Substituting A = 10 and B = -10:

Net change = 10 - 10 + (10 × -10)/100 = -1%

This represents a net decrease of 1% over the two-year period.

Present population = Initial population × (1 - 1/100) Present population = 1000 × 0.99 = 990

Explanation

The weighted average of a combined group is given by the formula: Combined Average = (n₁w₁ + n₂w₂) / (n₁ + n₂)

Here, the first group has n₁ = 18 boys with an average weight of w₁ = 45 kg, and the second group has n₂ = 8 boys with an average weight of w₂ = 48.75 kg.

Substituting these values into the formula: Combined Average = (18 × 45 + 8 × 48.75) / (18 + 8) Combined Average = (810 + 390) / 26 Combined Average = 1200 / 26 Combined Average = 600 / 13 ≈ 46.15 kg

Explanation

The five consecutive natural numbers starting with m are m, m + 1, m + 2, m + 3, and m + 4.

The average of these five numbers is calculated by dividing their sum by 5:

Average = (m + (m + 1) + (m + 2) + (m + 3) + (m + 4)) / 5 Average = (5m + 10) / 5 Average = m + 2

Alternatively, for any arithmetic progression with an odd number of terms, the average is always equal to the middle term. Since there are five terms, the middle (third) term is m + 2.

Explanation

The relationship between the number of men (M), days (D), and work done (W) is given by the work formula:

(M₁ × D₁) / W₁ = (M₂ × D₂) / W₂

Substituting the given values into the formula:

M₁ = 5 persons D₁ = 5 days W₁ = 5 tables M₂ = 25 persons W₂ = 25 tables

(5 × 5) / 5 = (25 × D₂) / 25

5 = D₂

Therefore, the number of days required is 5 days.

Explanation

Using the standard trigonometric identity for the triple-angle of sine: sin 3x = 3sin x - 4sin³x

We can substitute this identity directly into the given expression: 4sin³x - 3sin x + sin 3x = 4sin³x - 3sin x + (3sin x - 4sin³x)

Combining the like terms simplifies the expression: (4sin³x - 4sin³x) + (-3sin x + 3sin x) = 0 + 0 = 0

Explanation

Total weight of the 4 family members is: 4 × 45 = 180 kg

Total weight of the 6 family members after the grandparents arrive is: 6 × 48 = 288 kg

The combined weight of the grandfather and the grandmother is the difference between these two totals: 288 - 180 = 108 kg

The average weight of the grandparents is: 108 ÷ 2 = 54 kg

Explanation

A chord of a circle is defined as a straight line segment whose endpoints both lie on the circle. The longest possible chord in any circle is one that passes through the center, which by definition is the diameter of that circle. Since the problem specifies that a chord of length 10 cm passes through the center of the circle, this chord is the diameter.

Explanation

Using the algebraic identity (a + b + c)(a² + b² + c² - ab - bc - ca) = a³ + b³ + c³ - 3abc where a = 2x, b = 3y, and c = 4z, the expression simplifies to (2x)³ + (3y)³ + (4z)³ - 3(2x)(3y)(4z). This yields 8x³ + 27y³ + 64z³ - 72xyz, which matches option B.

Explanation

The initial number of shares held by Dhanya is calculated by taking her percentage share of the total issued shares: Dhanya's initial shares = 35% of 10000 = 0.35 × 10000 = 3500

When Tanmay sells 1000 of his shares to Dhanya, these shares are added to Dhanya's existing shareholding: Dhanya's final shares = 3500 + 1000 = 4500

Explanation

The mean proportional between two numbers a and b is given by the formula √(a × b).

Substituting the given values a = 144 and b = 225:

Mean proportional = √(144 × 225)

Using the property of square roots, this can be written as:

√(144 × 225) = √144 × √225

Since √144 = 12 and √225 = 15:

12 × 15 = 180

Explanation

For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. This relation is expressed as:

Area(ΔABC) / Area(ΔPQR) = (BC / QR)²

Substituting the given values into the formula:

64 / 121 = (BC / 15.4)²

Taking the square root on both sides:

8 / 11 = BC / 15.4

Solving for BC:

BC = (8 × 15.4) / 11 BC = 8 × 1.4 BC = 11.2 cm

Explanation

The weighted average is calculated using the formula:

Weighted Average = ∑ (Value × Weight)

First, convert the scores and the percentage weights into decimal form:

Safety: Value = 8/10 = 0.8, Weight = 40% = 0.4 Comfort: Value = 6/10 = 0.6, Weight = 20% = 0.2 Fuel Mileage: Value = 5/10 = 0.5, Weight = 30% = 0.3 Exterior looks: Value = 8/10 = 0.8, Weight = 10% = 0.1

Now, substitute these values into the formula:

Weighted Average = (0.8 × 0.4) + (0.6 × 0.2) + (0.5 × 0.3) + (0.8 × 0.1) Weighted Average = 0.32 + 0.12 + 0.15 + 0.08 Weighted Average = 0.67

Explanation

Let the time taken by Anamika to complete the work be T days and the time taken by Bani be 3T days, as their time ratio is 1:3.

The difference in time taken by them is 40 days: 3T - T = 40 2T = 40 T = 20 days

So, Anamika takes 20 days and Bani takes 3 × 20 = 60 days to complete the work individually.

When working together, the total time taken is: (Time by Anamika × Time by Bani) / (Time by Anamika + Time by Bani) = (20 × 60) / (20 + 60) = 1200 / 80 = 15 days

Explanation

Using the negative exponent rule, rewrite the term: a⁻¹ + b⁻¹ = 1/a + 1/b = (a + b) / ab

Substituting this back into the given expression: 3ab × (a + b)⁻¹ × (a⁻¹ + b⁻¹) = 3ab × (1 / (a + b)) × ((a + b) / ab)

Multiplying these terms together, the (a + b) and ab terms in the numerator and denominator cancel out: 3ab × (1 / (a + b)) × ((a + b) / ab) = 3

Explanation

Given the relation: x + 1/x = 5

We need to evaluate the expression: 3x / (2x² + 2 - 5x)

To simplify the expression, we divide both the numerator and the denominator by x: Numerator: 3x / x = 3 Denominator: (2x² + 2 - 5x) / x = 2x + 2/x - 5 = 2(x + 1/x) - 5

Substituting the given value of x + 1/x = 5 into the simplified denominator: Denominator = 2(5) - 5 = 10 - 5 = 5

Thus, the value of the expression is 3/5.

Explanation

Under compound interest, if a sum becomes n times in t years, then it will become nᵏ times in k × t years.

Here, the sum becomes 3 times in 4 years, so n = 3 and t = 4.

We want to find the time required for the sum to become 81 times. Writing 81 as a power of 3: 81 = 3⁴

This gives k = 4.

Applying the formula, the required time is: k × t = 4 × 4 = 16 years.

Explanation

First, convert the mixed fractions into improper fractions: 2 2/3 = 8/3 1 4/3 = 7/3 1 6/7 = 13/7 5 1/2 = 11/2

Substitute these improper fractions into the expression: 8/3 + 7/3 × 13/7 ÷ 11/2

Following the BODMAS order of operations, perform division first: 13/7 ÷ 11/2 = 13/7 × 2/11 = 26/77

Next, perform multiplication: 7/3 × 26/77 = 26/33

Finally, perform addition by finding a common denominator: 8/3 + 26/33 = 88/33 + 26/33 = 114/33

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 3: 114/33 = 38/11

Convert the improper fraction 38/11 back into a mixed fraction: 38/11 = 3 5/11

Explanation

Marked price of 12 pairs of shoes = ₹7,200 Discount percentage = 20%

Selling price of 12 pairs of shoes = 7200 × (100 - 20)/100 = 7200 × 0.80 = ₹5,760

Selling price of 1 pair of shoes = 5760 / 12 = ₹480

Number of pairs of shoes that can be bought with ₹1,440 = 1440 / 480 = 3

Explanation

The curved surface area (CSA) of a right circular cylinder with radius r and height h is 2πrh.

The total surface area (TSA) of the cylinder is 2πr(h + r).

The ratio of the curved surface area to the total surface area is 3:7.

CSA / TSA = 3/7

Substitute the formulas into the ratio: 2πrh / [2πr(h + r)] = 3/7

Simplify the fraction by cancelling 2πr from the numerator and denominator: h / (h + r) = 3/7

Cross-multiply to solve for the relationship between h and r: 7h = 3(h + r) 7h = 3h + 3r 4h = 3r h/r = 3/4

This yields the ratio of the height to the radius as h:r = 3:4.

Explanation

Let P and Q represent the salaries of P and Q respectively.

The sum of their salaries is given by the equation: P + Q = 1,20,000

Since P spends 95% of his salary, his savings are: 100% - 95% = 5% of P

Since Q spends 85% of his salary, his savings are: 100% - 85% = 15% of Q

Their savings are equal, which gives: 5% of P = 15% of Q 0.05 × P = 0.15 × Q P = 3 × Q

Substituting P = 3 × Q into the first equation: 3 × Q + Q = 1,20,000 4 × Q = 1,20,000 Q = 30,000

Using the value of Q to find P: P = 3 × 30,000 = 90,000

Explanation

Initial savings are calculated as the difference between initial income and initial expenditure: Initial Savings = 1,20,000 - 55,000 = 65,000

The increase in income is 22% of 1,20,000: Increase in Income = 0.22 × 1,20,000 = 26,400

The increase in expenditure is 10% of 55,000: Increase in Expenditure = 0.10 × 55,000 = 5,500

The net increase in savings is the difference between the increase in income and the increase in expenditure: Increase in Savings = 26,400 - 5,500 = 20,900

The percentage increase in savings is the increase in savings divided by the initial savings, multiplied by 100: Percentage Increase = (20,900 / 65,000) × 100 = 2,090 / 65 = 32.15%

Explanation

The relationship between Input, Output, and Ratio is given by the formula: Ratio = Input / Output

This can be rearranged to find the Output: Output = Input / Ratio

For the year 1998: Input in 1998 = 1,200 crores Ratio in 1998 = 0.9 Output in 1998 = 1200 / 0.9 = 4000/3 crores

The total output for the years 1998 and 2000 combined is 2,500 crores. Therefore: Output in 2000 = 2500 - 4000/3 = (7500 - 4000) / 3 = 3500/3 crores

For the year 2000, the ratio of Input to Output is 1.8. To find the input in 2000: Input in 2000 = Ratio in 2000 × Output in 2000 Input in 2000 = 1.8 × (3500/3) = 0.6 × 3500 = 2,100 crores

Explanation

Let the required number be x. The product of this number when multiplied by 11 is a six-digit number containing only 6s, which is 666666.

To find x, set up the equation: x × 11 = 666666

Now, divide 666666 by 11: x = 666666 ÷ 11 x = 60606

Explanation

To find the college with the maximum aggregate number of participants, we calculate the total sum of participants across all five subjects for each college.

The aggregate participant count for College A is 110 + 98 + 130 + 100 + 182 = 620. The aggregate participant count for College B is 100 + 120 + 110 + 100 + 200 = 630. The aggregate participant count for College C is 125 + 80 + 250 + 150 + 120 = 725. The aggregate participant count for College D is 103 + 122 + 160 + 200 + 130 = 715. The aggregate participant count for College E is 112 + 105 + 180 + 80 + 183 = 660.

Comparing these aggregates, College C has the highest total with 725 participants.

Explanation

Let the marked price of the item be 100.

Tanvir pays 75% of the marked price in cash and the remaining 25% through a credit card.

Cash portion of the marked price = 75 Credit card portion of the marked price = 25

A discount of 32% is offered on the cash portion, so the actual payment made in cash is: 75 × (1 - 0.32) = 75 × 0.68 = 51

An extra 1.2% is added to the credit card portion, so the actual payment made through the credit card is: 25 × (1 + 0.012) = 25 × 1.012 = 25.3

His total final payment is the sum of both payments: 51 + 25.3 = 76.3

Since the initial marked price was 100, the total final payment of 76.3 represents 76.3% of the marked price.

Explanation

The length of the longest bamboo pole that can be placed in a room is equal to the length of the diagonal of the cuboid-shaped room.

The formula for the diagonal d of a cuboid with length l, breadth b, and height h is: d = √(l² + b² + h²)

Substituting the given values, l = 16 m, b = 12 m, and h = 10 m, into the formula: d = √(16² + 12² + 10²) d = √(256 + 144 + 100) d = √500 d = 10√5 m

Explanation

Using the direct formula for replacement: Weight of new person = Weight of removed person + (Total number of persons × Increase in average weight)

Given: Weight of removed person = 45 kg Total number of persons = 8 Increase in average weight = 2.5 kg

Substituting the values into the formula: Weight of new person = 45 + 8 × 2.5 Weight of new person = 45 + 20 = 65 kg

Explanation

The mean proportional between two numbers a and b is given by the formula √(a × b).

Let the unknown number be x. According to the formula: √(6 × x) = 30

Squaring both sides of the equation yields: 6 × x = 30² 6 × x = 900

Dividing both sides by 6 gives: x = 150

Explanation

The relative speed of the policeman with respect to the thief is the difference in their speeds because they are running in the same direction: Relative Speed = 23 km/h - 19 km/h = 4 km/h

Converting this relative speed into meters per second: Relative Speed = 4 × 5/18 = 10/9 m/s

The time taken by the policeman to catch the thief is the initial separation distance divided by their relative speed: Time = 480 ÷ (10/9) = 480 × 9/10 = 432 seconds

The distance run by the thief during this time is: Distance = Speed of thief × Time

Converting the thief's speed of 19 km/h into meters per second: Speed of thief = 19 × 5/18 = 95/18 m/s

Distance = 95/18 × 432 = 95 × 24 = 2280 m

Explanation

The mixture ratio can be determined using the rule of alligation:

Ratio of cheaper wheat to dearer wheat = (Cost of dearer wheat - Mean price) / (Mean price - Cost of cheaper wheat)

Given: Cost of cheaper wheat = ₹20.40 per kg Cost of dearer wheat = ₹25.50 per kg Mean price of the mixture = ₹23.80 per kg

Substituting these values into the formula:

Difference 1 (Dearer - Mean) = 25.50 - 23.80 = 1.70 Difference 2 (Mean - Cheaper) = 23.80 - 20.40 = 3.40

Required ratio = 1.70 : 3.40 = 1 : 2

Explanation

By simplifying the expression and correcting a typographical square in the numerator, we get cos²(90 - theta) - [cos(90 - theta) * cos(theta)] / cot(theta). This simplifies to sin²(theta) - [sin(theta) * cos(theta)] / (cos(theta) / sin(theta)) = sin²(theta) - sin²(theta) = 0.

Explanation

The volume of a cube is given by the formula V = a³, where a is the side length of the cube. Given that the volume is 1331 cm³, the side length is found by taking the cube root:

a = ∛1331 = 11 cm

The length of the diagonal of a cube is given by the formula d = a√3. Substituting the side length into this formula:

d = 11√3 cm

Explanation

Let k = x + 1/x.

Using the algebraic identity: x² + 1/x² = (x + 1/x)² - 2 = k² - 2

Squaring this relation gives: x⁴ + 1/x⁴ = (x² + 1/x²)² - 2 = (k² - 2)² - 2

Substitute the given value x⁴ + 1/x⁴ = 14159: (k² - 2)² - 2 = 14159 (k² - 2)² = 14161

Taking the square root: k² - 2 = 119 k² = 121 k = ±11

Explanation

When a person travels equal distances at two different speeds, say v₁ and v₂, the overall average speed for the entire journey is given by the harmonic mean formula:

Average Speed = 2v₁v₂ / (v₁ + v₂)

Given that the speed from A to B is v₁ = 60 km/h, the average speed is 48 km/h, and letting the return speed from B to A be v₂:

48 = 2 × 60 × v₂ / (60 + v₂)

48 = 120v₂ / (60 + v₂)

48 × (60 + v₂) = 120v₂

2880 + 48v₂ = 120v₂

2880 = 72v₂

v₂ = 2880 / 72 = 40

Explanation

Distance is calculated using the formula:

Distance = Speed × Time

First, convert the speed of the bus from kilometers per hour (km/h) to meters per second (m/s) by multiplying by 5/18:

Speed = 108 × 5/18 = 30 m/s

Next, substitute the speed and the given time of 40 seconds into the formula to find the distance:

Distance = 30 m/s × 40 s = 1200 m

Since 1000 meters equals 1 kilometer, 1200 meters is equivalent to 1 km 200 m.

Explanation

The length of an arc of a circle is proportional to the angle it subtends at the center. The relation between the arc length, the central angle θ, and the total circumference C of the circle is given by the formula:

Arc Length = (θ / 360°) × C

Substituting the given arc length of 15 cm and the central angle of 24°:

15 = (24° / 360°) × C

Simplifying the ratio of the angles:

24 / 360 = 1 / 15

Substituting this back into the formula yields:

15 = (1 / 15) × C

Solving for C:

C = 15 × 15 = 225 cm

Explanation

The volume V of a right circular cylinder is given by the formula: V = πr²h

Let the radii of the two cylinders be r₁ and r₂, and their heights be h₁ and h₂. The ratio of their volumes V₁/V₂ is: V₁/V₂ = (π r₁² h₁) / (π r₂² h₂) = (r₁/r₂)² × (h₁/h₂)

Substituting the given ratios r₁/r₂ = 2/3 and h₁/h₂ = 5/3: V₁/V₂ = (2/3)² × (5/3) = 4/9 × 5/3 = 20/27

Explanation

Let the marked price of the article be MP. The final selling price after successive discounts of 10% and 40% is given by:

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

Substituting the given values of 10% and 40% discounts and the final cash payment of ₹2,700:

2,700 = MP × (1 - 0.10) × (1 - 0.40) 2,700 = MP × 0.90 × 0.60 2,700 = MP × 0.54 MP = 2,700 / 0.54 MP = 5,000

Explanation

The relationship between the marked price (MP), cost price (CP), profit percentage (P), and discount percentage (D) is given by the formula:

MP / CP = (100 + P) / (100 - D)

Given that the cost price CP is ₹2,500, the profit P is 15%, and the discount D is 15%, substitute these values into the formula:

MP / 2500 = (100 + 15) / (100 - 15) MP / 2500 = 115 / 85 MP / 2500 = 23 / 17

Solving for MP:

MP = (2500 × 23) / 17 MP = 57500 / 17 MP ≈ 3382.35

Rounding to the nearest integer, the marked price is ₹3,382.

Explanation

The area of a trapezium is given by the formula:

Area = ½ × (a + b) × h

where a and b are the lengths of the parallel sides, and h is the perpendicular distance between them.

Substituting the given values into the formula:

a = 20 cm b = 25 cm h = 14 cm

Area = ½ × (20 + 25) × 14 Area = ½ × 45 × 14 Area = 45 × 7 Area = 315 cm²

Explanation

The effective rate of compound interest for 2 years at 8% per annum can be calculated using the successive percentage formula:

Effective Rate = r + r + r² / 100 %

Substituting r = 8:

Effective Rate = 8 + 8 + 64 / 100 = 16.64%

The compound interest is:

Compound Interest = 16.64% of ₹10,625 Compound Interest = (16.64 / 100) × 10,625 = 0.1664 × 10,625 = ₹1,768

Explanation

To find the smallest four-digit number divisible by 4, 8, 12, and 16, we first find the Least Common Multiple (LCM) of these numbers.

The prime factorizations are: 4 = 2² 8 = 2³ 12 = 2² × 3 16 = 2⁴

The LCM is calculated by taking the highest power of each prime factor: LCM = 2⁴ × 3 = 16 × 3 = 48

The smallest four-digit number is 1000. Dividing 1000 by 48 yields: 1000 ÷ 48 = 20 with a remainder of 40

To find the first multiple of 48 that is a four-digit number, we multiply 48 by the next whole number, which is 21: 48 × 21 = 1008

Explanation

We can find the value of x² + 1/x² using the algebraic identity: (a - b)² = a² + b² - 2ab

By substituting a = x and b = 1/x, we get: (x - 1/x)² = x² + 1/x² - 2(x)(1/x) (x - 1/x)² = x² + 1/x² - 2

By rearranging this equation, we can express x² + 1/x² as: x² + 1/x² = (x - 1/x)² + 2

We are given that (x - 1/x) = 7.5. Substituting this value into the equation yields: x² + 1/x² = (7.5)² + 2

Calculating the square of 7.5: (7.5)² = 56.25

Adding 2 to this value: x² + 1/x² = 56.25 + 2 = 58.25