SSC CHSL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

We use the standard algebraic identity: (x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

Substituting the given values x + y + z = 10 and x² + y² + z² = 30: 10² = 30 + 2(xy + yz + zx) 100 = 30 + 2(xy + yz + zx) 70 = 2(xy + yz + zx) xy + yz + zx = 35

Now, we use the identity for the required expression: x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - (xy + yz + zx))

Substituting the known values into this equation: x³ + y³ + z³ - 3xyz = 10 × (30 - 35) = 10 × (-5) = -50

Explanation

The successive percentage change formula for two consecutive profit percentages a% and b% is: Total Profit Percentage = a + b + ab/100

Given that the first dealer makes a profit of 20% (a = 20) and the final cumulative profit is 44%, let the profit percentage made by the second dealer be b%.

44 = 20 + b + 20 × b / 100 24 = b + b/5 24 = 6b/5 b = 24 × 5 / 6 b = 20%

Explanation

To find the weighted average price, we can simplify the ratio of the quantities 20000 : 15000 : 5000 to 4 : 3 : 1.

The formula for the weighted average is: Weighted Average = (w₁x₁ + w₂x₂ + w₃x₃) / (w₁ + w₂ + w₃)

Substituting the simplified ratios and prices: Weighted Average = (4 × 1 + 3 × 1.15 + 1 × 2) / (4 + 3 + 1) Weighted Average = (4 + 3.45 + 2) / 8 Weighted Average = 9.45 / 8 = 1.18125

Rounding to two decimal places, the weighted average price of one unit is ₹1.18.

Explanation

The total number of cakes sold by bakery D on Monday, Thursday, and Sunday is calculated by adding the values corresponding to these days for bakery D from the given table.

Cakes sold on Monday = 221 Cakes sold on Thursday = 185 Cakes sold on Sunday = 280

Total cakes sold = 221 + 185 + 280 = 686

Explanation

To find the percentage of production of type-A cars in 1995 compared to its total production over all the years, we first calculate the total production of type-A cars from 1992 to 1996:

Total production of type-A cars = 160 + 210 + 130 + 190 + 260 = 950 thousand

The production of type-A cars in 1995 is given as 190 thousand.

Percentage = (Production of type-A in 1995 / Total production of type-A) × 100

Percentage = (190 / 950) × 100 = 1/5 × 100 = 20%

Explanation

The volume V of a sphere is given by the formula: V = (4/3)πr³

where r is the radius of the sphere.

Substituting the given volume of 972π cm³ into the formula: (4/3)πr³ = 972π

Dividing both sides by π: (4/3)r³ = 972

Multiplying both sides by 3/4 to solve for r³: r³ = 972 × (3/4) r³ = 243 × 3 r³ = 729

Taking the cube root of both sides: r = ∛729 r = 9 cm

Explanation

Using the weighted average formula: Combined Average = (Number of boys × Average of boys + Number of girls × Average of girls) / (Number of boys + Number of girls)

Let g be the number of girls. 30 = (15 × 26 + g × 36) / (15 + g)

Simplifying the equation: 30 × (15 + g) = 390 + 36g 450 + 30g = 390 + 36g 450 - 390 = 36g - 30g 60 = 6g g = 10

By the rule of alligation, the ratio of the number of boys to girls is: (Average of girls - Combined average) / (Combined average - Average of boys) = (36 - 30) / (30 - 26) = 6 / 4 = 3 / 2

Since the ratio of boys to girls is 3 to 2, and there are 15 boys: Number of girls = 15 × 2 / 3 = 10

Explanation

The total volume of the metallic solid cuboid is conserved when it is melted and recast into smaller cubes.

The volume of a cuboid is given by the formula: Volume of cuboid = length × width × height

Substituting the given dimensions: Volume of cuboid = 36 × 18 × 12

The volume of a single cube with side a is given by the formula: Volume of cube = a³

For a cube of side 6 cm: Volume of cube = 6 × 6 × 6

The number of cubes formed is the ratio of the volume of the cuboid to the volume of one cube: Number of cubes = (36 × 18 × 12) / (6 × 6 × 6)

By dividing the terms directly: Number of cubes = (36/6) × (18/6) × (12/6) Number of cubes = 6 × 3 × 2 Number of cubes = 36

Explanation

The volume V of a cylindrical pillar is calculated using the formula V = πr²h, where r is the radius of the cylinder and h is its height.

Given: Radius r = 11 m Height h = 21 m π = 22/7

Substituting these values into the formula: V = (22/7) × 11² × 21 V = (22/7) × 121 × 21 V = 22 × 121 × 3 V = 66 × 121 V = 7986 m³

Explanation

A number is exactly divisible by 42 if it is simultaneously divisible by its prime factors: 2, 3, and 7.

To check the divisibility of 25242:

First, for divisibility by 2, the units digit must be an even number. The last digit of 25242 is 2, which is even, so it is divisible by 2.

Second, for divisibility by 3, the sum of the digits must be divisible by 3. The sum of the digits of 25242 is 2 + 5 + 2 + 4 + 2 = 15. Since 15 is divisible by 3, the number is divisible by 3.

Third, for divisibility by 7, we can express the number as 25200 + 42. Since 25200 is exactly divisible by 7 (7 × 3600 = 25200) and 42 is also exactly divisible by 7 (7 × 6 = 42), their sum, 25242, is divisible by 7.

Since 25242 is divisible by 2, 3, and 7, it is exactly divisible by 42.

Explanation

According to the order of operations in simplification (BODMAS), the innermost parts of the expression are solved first. The term 9 - 3 contains a vinculum (a bar over 9 - 3, acting as a grouping symbol), which is evaluated first:

9 - 3 = 6

Substitute this value back into the expression:

9 - [10 - {20 - (15 - 6)}]

Next, solve the term inside the parentheses:

15 - 6 = 9

Substitute this value back:

9 - [10 - {20 - 9}]

Next, solve the term inside the curly brackets:

20 - 9 = 11

Substitute this value back:

9 - [10 - 11]

Next, solve the term inside the square brackets:

10 - 11 = -1

Substitute this value back:

9 - [-1] = 9 + 1 = 10

Explanation

Let the selling price of the cell phone be SP.

The gain is given as 12.5% calculated on the selling price. Gain = 12.5% of SP = (12.5/100) × SP = (1/8) × SP

The relationship between Selling Price (SP), Cost Price (CP), and Gain is: SP - CP = Gain

Substituting the given value of CP = ₹24,500 and the expression for Gain: SP - 24,500 = SP / 8 SP - SP / 8 = 24,500 (7/8) × SP = 24,500 SP = (24,500 × 8) / 7 SP = 3,500 × 8 SP = 28,000

Explanation

The position of the ladder forms a right-angled triangle with the building and the ground. According to the Pythagorean theorem, x² + y² = L², where L is the length of the ladder, x is the distance of the foot of the ladder from the building, and y is the height of the top of the ladder from the ground.

Given that L = 25 m and the initial distance of the foot of the ladder x₁ = 7 m, the initial height of the top of the ladder y₁ is calculated as:

y₁ = √(L² - x₁²) y₁ = √(25² - 7²) y₁ = √(625 - 49) = √576 = 24 m

If the top of the ladder slips down by 4 m, the new height y₂ becomes:

y₂ = 24 - 4 = 20 m

The new distance of the foot of the ladder from the building x₂ is calculated as:

x₂ = √(L² - y₂²) x₂ = √(25² - 20²) x₂ = √(625 - 400) = √225 = 15 m

The distance by which the foot of the ladder slides is the difference between the final and initial distances:

Slide distance = x₂ - x₁ = 15 - 7 = 8 m

Explanation

The third proportion c of two numbers a and b is defined by the relation a/b = b/c, which gives the formula c = b²/a.

Substituting the given values a = 15 and b = 24 into the formula:

c = 24² / 15 c = 576 / 15

Dividing both the numerator and the denominator by 3:

c = 192 / 5 c = 38 2/5

Explanation

Using the algebraic identity for the difference of squares, (a + b)(a - b) = a² - b², we can group and simplify the expression in two steps.

First, simplify the product of the last two terms: (6p + 7q)(6p - 7q) = (6p)² - (7q)² = 36p² - 49q²

Now, substitute this back into the original expression: (36p² + 49q²)(36p² - 49q²)

Apply the difference of squares identity once more: (36p² + 49q²)(36p² - 49q²) = (36p²)² - (49q²)² = 1296p⁴ - 2401q⁴

Explanation

The discount percentage in buy-get schemes is calculated using the formula:

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

In this scheme, the customer gets 2 shirts free upon buying 3 shirts. This means the customer takes home a total of 5 shirts (3 + 2 = 5) while paying for only 3 shirts. The discount is equivalent to the value of the 2 free shirts out of the total of 5 shirts.

Discount Percentage = (2 / 5) × 100 = 40%

Explanation

Using the formula for the combined work rate of two individuals, the time taken T for both A and B to complete the work together is given by:

T = (T₁ × T₂) / (T₁ + T₂)

where T₁ is the time taken by A (12 days) and T₂ is the time taken by B (36 days).

Substituting the given values:

T = (12 × 36) / (12 + 36) T = 432 / 48 T = 9 days

Alternatively, taking the total work as the least common multiple (LCM) of 12 and 36, which is 36 units: The efficiency of A is 36 ÷ 12 = 3 units per day. The efficiency of B is 36 ÷ 36 = 1 unit per day. Their combined efficiency is 3 + 1 = 4 units per day. The number of days required to complete the work together is 36 ÷ 4 = 9 days.

Explanation

Let the number be the least common multiple of the denominators 3 and 5, which is 15.

The correct calculation is 15 × 5/3 = 25.

The incorrect calculation made by the pupil is 15 × 3/5 = 9.

The absolute error in the calculation is 25 - 9 = 16.

The percentage error is calculated by dividing the absolute error by the correct value and multiplying by 100:

Percentage error = (16 / 25) × 100 = 16 × 4 = 64%

Explanation

To find the value of (x²/yz) + (y²/zx) + (z²/xy), we find a common denominator for the fractions, which is xyz. Combining the terms under this common denominator gives:

(x³ + y³ + z³) / xyz

Using the standard algebraic identity: x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

Since we are given that x + y + z = 0, substituting this into the identity yields: x³ + y³ + z³ - 3xyz = 0 x³ + y³ + z³ = 3xyz

Substituting this result back into our combined fraction expression: 3xyz / xyz = 3

Explanation

Under the Buy 3, Get 1 Free scheme, a customer gets 4 items for the price of 3. Ramesh made purchases for which the amount payable under this scheme was ₹18,000.

Since the marked price of each item is ₹300, the number of items Ramesh paid for is: 18,000 ÷ 300 = 60 items.

According to the scheme, for every 3 items paid for, 1 item is given free. Therefore, for 60 paid items, the number of free items received is: 60 ÷ 3 = 20 items.

This means Ramesh received a total of 80 items (60 paid + 20 free). The total marked price of these 80 items is: 80 × 300 = ₹24,000.

Since the initial payable amount of ₹18,000 is more than ₹10,000, Ramesh is eligible for an additional 10% discount on this payable amount. Additional discount = 10% of 18,000 = ₹1,800.

Thus, the final amount paid by Ramesh is: 18,000 - 1,800 = ₹16,200.

The total discount received by Ramesh is: 24,000 - 16,200 = ₹7,800.

The effective percentage discount is: (7,800 ÷ 24,000) × 100 = 32.5%

Explanation

From the given equation, we have: m xᵐ - n xⁿ = 0 m xᵐ = n xⁿ xᵐ = (n/m) xⁿ

The given expression is: 1/(xᵐ + xⁿ) + 1/(xᵐ - xⁿ)

Combining the fractions by taking a common denominator: [ (xᵐ - xⁿ) + (xᵐ + xⁿ) ] / [ (xᵐ + xⁿ)(xᵐ - xⁿ) ] = 2 xᵐ / (x²ᵐ - x²ⁿ)

Substituting xᵐ = (n/m) xⁿ into the expression: = 2 (n/m) xⁿ / [ ( (n/m) xⁿ )² - x²ⁿ ] = 2 (n/m) xⁿ / [ (n²/m²) x²ⁿ - x²ⁿ ] = 2 (n/m) xⁿ / [ x²ⁿ ( (n² - m²) / m² ) ] = [ 2 (n/m) xⁿ ] × [ m² / ( x²ⁿ (n² - m²) ) ] = 2 m n xⁿ / [ x²ⁿ (n² - m²) ] = 2 m n / [ xⁿ (n² - m²) ]

Explanation

Let the total number of students in the class be N.

The number of girls in the class is 70% of the total students: Number of girls = 0.7N

The number of girls who chose soccer as their favourite sport is 70% of the girls: Number of soccer-playing girls = 0.7 × 0.7N = 0.49N

Given that 98 girls chose soccer, we set up the equation: 0.49N = 98

Solving for N: N = 98 ÷ 0.49 = 200

Explanation

In city P, the total population is 46800 and the percentage of children is 20%.

Number of children in city P = 20% of 46800 = 46800 × 0.20 = 9360

The ratio of boys to girls among the children in city P is 5 : 4.

Number of girls among the children in city P = 9360 × 4/9 = 4160

In city R, the total population is 42000 and the percentage of children is 10%.

Number of children in city R = 10% of 42000 = 42000 × 0.10 = 4200

The ratio of boys to girls among the children in city R is 2 : 3.

Number of girls among the children in city R = 4200 × 3/5 = 2520

Total number of girls among the children in city P and city R together = 4160 + 2520 = 6680

Explanation

The principal loan amount P is 8,925. Since the interest rate is 8% per annum compounded half-yearly, the rate of interest per half-year period is r = 8% ÷ 2 = 4%.

The relation between the principal and the two equal half-yearly installments of value x is given by the formula: P = x / (1 + r/100) + x / (1 + r/100)²

Calculating the term: 1 + r/100 = 1 + 4/100 = 26/25

Substituting the values into the formula: 8925 = x / (26/25) + x / (26/25)² 8925 = 25x / 26 + 625x / 676 8925 = x × (25 × 26 + 625) / 676 8925 = x × (650 + 625) / 676 8925 = x × 1275 / 676

Solving for x: x = 8925 × 676 / 1275 x = 7 × 676 x = 4732

Explanation

The distance D between two points when traveled at two different speeds S₁ and S₂ with a difference in time Δt is given by the formula:

D = (S₁ × S₂ / |S₁ - S₂|) × Δt

Here, the first speed S₁ is 40 km/h. On the second day, the speed is increased by 10 km/h, making the second speed S₂ equal to 50 km/h. The difference in speed is 10 km/h.

On the first day, he is late by 25 minutes, and on the second day, he is late by 10 minutes. The difference in time taken is:

Δt = 25 minutes - 10 minutes = 15 minutes = 15/60 hours = ¼ hour

Substituting these values into the formula:

D = (40 × 50 / 10) × ¼ D = 200 × ¼ D = 50 km

Explanation

A tangent to a circle is perpendicular to the radius at the point of contact. Therefore, the radius, the tangent, and the line segment joining the center of the circle to the external point form a right-angled triangle, where the line segment from the center is the hypotenuse.

Using the Pythagoras theorem: d² = r² + t²

where d is the distance of the point from the center (18 cm), r is the radius of the circle (12 cm), and t is the length of the tangent.

Substituting the values: 18² = 12² + t² 324 = 144 + t² t² = 324 - 144 t² = 180 t = √180 t = √(36 × 5) t = 6√5 cm

Explanation

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

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

Substituting the values a = 20 and b = 10:

Equivalent Discount = 20 + 10 - (20 × 10)/100 = 30 - 2 = 28%

Explanation

The sum of a set of numbers is equal to the product of their average and the number of terms.

Total sum of 25 numbers = 25 × 13 = 325

Sum of first 13 numbers = 13 × 11.5 = 149.5

Sum of last 13 numbers = 13 × 14.5 = 188.5

Since there are 25 numbers in total, the first 13 numbers (from 1st to 13th) and the last 13 numbers (from 13th to 25th) overlap at the 13th number, which is the middle number.

Middle Number = (Sum of first 13 numbers + Sum of last 13 numbers) - Total sum of 25 numbers

Middle Number = (149.5 + 188.5) - 325 = 338 - 325 = 13

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 64 cm³, the side length is:

a³ = 64 a = 4 cm

The total surface area of a cube is given by the formula A = 6a². Substituting the side length:

A = 6 × 4² A = 6 × 16 A = 96 cm²

Explanation

The volume of a cube is given by the formula V = a³, where a is the side length of the cube. Given V = 27 m³, we can find the side length as:

a³ = 27 a = 3 m

The total surface area of a cube is calculated using the formula A = 6a². Substituting the value of a:

A = 6 × 3² A = 6 × 9 = 54 m²

Explanation

Net profit percentage when an item is marked up by M% and then sold at a discount of D% is given by the successive percentage formula: Net Profit % = M - D - (M × D)/100

Substituting the given values M = 45 and D = 21: Net Profit % = 45 - 21 - (45 × 21)/100 Net Profit % = 24 - 945/100 Net Profit % = 24 - 9.45 Net Profit % = 14.55%

Rounding 14.55% to the nearest integer yields 15%.

Explanation

To check if 3661 is divisible by 7, we can apply the divisibility rule for 7. This rule states that we double the last digit of the number and subtract it from the remaining part of the number. If the resulting number is divisible by 7, then the original number is also divisible by 7.

Applying this rule to 3661: The last digit is 1. Doubling it gives 2. Subtracting 2 from the remaining part (366) gives: 366 − 2 = 364

Repeating the process for 364: The last digit is 4. Doubling it gives 8. Subtracting 8 from the remaining part (36) gives: 36 − 8 = 28

Since 28 is divisible by 7 (7 × 4 = 28), the original number 3661 is also divisible by 7. Alternatively, direct division shows that 3661 ÷ 7 = 523, which leaves a remainder of 0.

Explanation

Let the cost price of the mobile be ₹100.

The marked price is 25% above the cost price: Marked Price = 100 × 1.25 = ₹125

To gain a profit of 20%, the selling price must be: Selling Price = 100 × 1.20 = ₹120

The required discount is the difference between the marked price and the selling price: Discount = 125 - 120 = ₹5

The discount percentage is calculated on the marked price: Discount Percentage = (Discount / Marked Price) × 100% Discount Percentage = (5 / 125) × 100% = 4%

Explanation

To find the values of a and b, we solve the system of equations given by a + b = 12 and ab = 35.

By inspection, the two numbers that add up to 12 and multiply to 35 are 7 and 5. Let a = 7 and b = 5.

Now, we calculate the value of the required expression: a⁴ + b⁴ - 10³ = 7⁴ + 5⁴ - 10³

Evaluating the powers of 7 and 5: 7⁴ = 2401 5⁴ = 625

Substituting these values: a⁴ + b⁴ - 10³ = 2401 + 625 - 1000 a⁴ + b⁴ - 10³ = 3026 - 1000 a⁴ + b⁴ - 10³ = 2026

Alternatively, using algebraic identities: a² + b² = (a + b)² - 2ab a² + b² = 12² - 2(35) = 144 - 70 = 74

Now, squaring this result: a⁴ + b⁴ = (a² + b²)² - 2(ab)² a⁴ + b⁴ = 74² - 2(35)² a⁴ + b⁴ = 5476 - 2(1225) a⁴ + b⁴ = 5476 - 2450 = 3026

Subtracting 10³ from this result yields: 3026 - 1000 = 2026

Explanation

To find the required ratio, first calculate the sum of the seats won by the Con party in 2014, Kanta Dal in 2019, and the Shiv Party in 2019:

Seats won by Con party in 2014 = 43 Seats won by Kanta Dal in 2019 = 16 Seats won by Shiv Party in 2019 = 18

Sum of these seats = 43 + 16 + 18 = 77

Next, identify the number of seats won by AGP in 2014:

Seats won by AGP in 2014 = 282

The ratio of the sum of the seats to the seats won by AGP in 2014 is 77:282.

Explanation

In triangle ABC, since the line PQ is parallel to BC, the triangles APQ and ABC are similar by Angle-Angle similarity.

For similar triangles, the ratio of corresponding sides is equal, which gives: PQ / BC = AP / AB

We are given the relation AB = 6PB. Let PB be represented as x, which means AB is equal to 6x.

The length of AP can be expressed as: AP = AB - PB AP = 6x - x = 5x

Substituting these values into the ratio of the sides: PQ / BC = AP / AB = 5x / 6x = 5/6

This gives the ratio PQ : BC as 5:6.

Explanation

The given table represents a cumulative frequency distribution. The total number of students in the class is 48.

According to the table, the number of students with a height up to 160 cm is 18.

Therefore, the number of students whose height is more than 160 cm is calculated by subtracting the cumulative frequency up to 160 cm from the total number of students: 48 - 18 = 30

To find the percentage of these students: Percentage = (30 / 48) × 100

Simplifying the fraction: 30 / 48 = 5 / 8

Percentage = (5 / 8) × 100 = 62.5%

Explanation

The given expression contains vinculums (bar brackets) acting as grouping symbols over p + q and q - p. The expression is written as: 3p - [3p - (p + q) - {3p - (p - (q - p))}]

Applying the order of operations, first simplify the innermost parenthesis containing the bar bracket: p - (q - p) = p - q + p = 2p - q

Next, simplify the expression within the curly brackets: 3p - (2p - q) = 3p - 2p + q = p + q

Then, simplify the expression within the square brackets: 3p - (p + q) - (p + q) = 3p - p - q - p - q = p - 2q

Finally, simplify the outer expression: 3p - (p - 2q) = 3p - p + 2q = 2p + 2q

Explanation

The ratio of girls to boys is 40% to 60%, which simplifies to 2:3.

Let the average score of the girls be x. Since the average score of the boys is 20% less than that of the girls, the average score of the boys is 0.8x.

Using the weighted average formula: Weighted Average = (Ratio of girls × Average of girls + Ratio of boys × Average of boys) / (Sum of ratios)

Substituting the values into the formula: 66 = (2 × x + 3 × 0.8x) / (2 + 3) 66 = (2x + 2.4x) / 5 66 × 5 = 4.4x 330 = 4.4x x = 330 / 4.4 x = 75

Explanation

The curved surface area (CSA) of a cylinder is given by the formula:

CSA = 2 × π × r × h

where r is the base radius and h is the height of the cylinder.

Given that r = 4 cm, h = 5 cm, and π = 3.14, we can substitute these values directly into the formula:

CSA = 2 × 3.14 × 4 × 5

CSA = 40 × 3.14

CSA = 125.6 cm²

Explanation

By the properties of trigonometric functions, the sine function is classified as an odd function. For any odd function f(x), the mathematical relationship f(-x) = -f(x) holds true for all values in its domain. Applying this identity directly to the sine function gives:

sin(-A) = -sin A

This behavior can also be understood geometrically using the unit circle. An angle of -A is measured in the clockwise direction from the positive x-axis, placing it in the fourth quadrant if A is in the first quadrant. Since the sine of an angle corresponds to the y-coordinate of the point on the unit circle, rotating in the clockwise direction negates the y-coordinate while keeping its absolute value the same, which results in the negative value of the sine of the positive angle.

Explanation

The net annual population growth rate is the difference between the annual birth rate and the annual death rate.

Net growth rate = 12.5% - 2.5% = 10% per annum

The population after n years can be calculated using the compound growth formula:

A = P × (1 + R/100)ⁿ

Substituting the given values where initial population P = 9000, net growth rate R = 10, and time n = 3 years:

A = 9000 × (1 + 10/100)³

A = 9000 × (1.1)³

A = 9000 × 1.331

A = 11979

Explanation

The percentage of valid votes is: 100% - 25% = 75%

The total number of valid votes is: 75% of 6000 = 0.75 × 6000 = 4500

Since Ravi gets 65% of the total valid votes, the other candidate Sushil gets the remaining valid votes: 100% - 65% = 35%

The number of valid votes that Sushil gets is: 35% of 4500 = 0.35 × 4500 = 1575

Explanation

The average of a given dataset is calculated by dividing the sum of all the data values by the total number of values.

Sum of the data values = 1 + 2 + 3 + 5 + 17 + 21 + 28 = 77

Total number of data values = 7

Average = Sum of the values / Total number of values = 77 / 7 = 11

Explanation

The rate of work of A and B together is 1/12 of the work per day, while the rate of work of A alone is 1/18 of the work per day.

The rate of work of B alone is the difference between their combined rate and A's individual rate:

Rate of B = 1/12 - 1/18

To subtract these fractions, find the least common multiple of 12 and 18, which is 36:

Rate of B = 3/36 - 2/36 = 1/36

Since B completes 1/36 of the work in one day, B alone will take 36 days to complete the entire work.

Explanation

The speed of rowing in still water is u = 5 km/h, and the speed of the current is v = 2.5 km/h.

The downstream speed is given by the formula: d = u + v = 5 + 2.5 = 7.5 km/h

The upstream speed is given by the formula: s = u - v = 5 - 2.5 = 2.5 km/h

Let the distance be D km. The difference in time between upstream and downstream travel is 4 hours, which gives the equation: D/s - D/d = 4

Substituting the values of upstream and downstream speeds: D/2.5 - D/7.5 = 4

Taking a common denominator of 7.5: (3D - D) / 7.5 = 4 2D / 7.5 = 4 2D = 30 D = 15 km

Explanation

The final value of the land after the periods of depreciation and appreciation can be calculated directly using the compound formula.

Final Value = Initial Value × (1 - d/100)² × (1 + a/100)²

Here, the initial value of the land is ₹4,32,000. It depreciates at a rate of d = 25% per annum for the first 2 years, and then appreciates at a rate of a = 50% per annum for the next 2 years.

Substituting the values into the formula:

Final Value = 4,32,000 × (1 - 25/100)² × (1 + 50/100)²

Simplifying the percentage terms into fractions:

1 - 25/100 = 3/4 1 + 50/100 = 3/2

Substitute these fractions back into the equation:

Final Value = 4,32,000 × (3/4)² × (3/2)² Final Value = 4,32,000 × (9/16) × (9/4) Final Value = 4,32,000 × 81/64 Final Value = 6,750 × 81 Final Value = 5,46,750

Explanation

Let the cost price of 1 g of goods be ₹1.

The shopkeeper marks up the price of 1000 g of goods by 30%, which makes the marked price: 1000 × (1 + 30/100) = ₹1300

He then offers a 10% discount on this marked price, so the selling price becomes: 1300 × (1 - 10/100) = ₹1170

Due to the faulty balance, the shopkeeper delivers only 870 g of goods while charging the customer for 1000 g. The actual cost price of 870 g of goods to the shopkeeper is: 870 × ₹1 = ₹870

The net profit earned on this transaction is: Net Profit = Selling Price - Cost Price = 1170 - 870 = ₹300

The net profit percentage is: (Net Profit / Cost Price) × 100 = (300 / 870) × 100 = 34.48%

Explanation

To find the distance by which Radha beats Shyamal, we determine how far Shyamal has run in the time it takes Radha to complete the race.

Radha completes the 1 km (1000 m) race in 8 min.

Shyamal's speed is calculated using the formula: Speed = Distance / Time Shyamal's speed = 1000 m / 10 min = 100 m/min

In the 8 min that Radha takes to finish the race, Shyamal covers: Distance = Speed × Time Distance = 100 m/min × 8 min = 800 m

The distance by which Radha beats Shyamal is the remaining distance Shyamal has yet to cover when Radha finishes: 1000 m - 800 m = 200 m

Explanation

The sum of the first n natural numbers is calculated using the formula S = n(n + 1)/2.

Substituting n = 78 into the formula:

S = 78 × (78 + 1) / 2

S = 78 × 79 / 2

S = 39 × 79

Since the sum can be written as the product of 39 and 79, it is directly divisible by 79.