SSC CHSL Tier 1 2025 Quantitative Aptitude Questions with Solutions

Explanation

The curved surface area of a hemisphere with radius r is given by the formula: CSA = 2πr²

We are given that the curved surface area is 154π cm²: 2πr² = 154π r² = 77 r = √77

The volume V of a hemisphere is given by the formula: V = (2/3)πr³

Substituting r = √77 into the volume formula: V = (2/3)π(√77)³ V = (2/3)π(77√77) V = (154/3)π√77

Explanation

To arrange the fractions in descending order, convert each fraction into its decimal equivalent:

3/4 = 0.75 2/3 ≈ 0.67 1/3 ≈ 0.33 2/7 ≈ 0.29

Comparing these decimal values from largest to smallest: 0.75 > 0.67 > 0.33 > 0.29

Thus, the correct descending order of the fractions is 3/4, 2/3, 1/3, 2/7.

Explanation

The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. Using the standard approximation of π = 22/7, we can set up the equation:

2 × (22/7) × r = 88

Solving for the radius:

44/7 × r = 88 r = 88 × 7/44 r = 14 cm

The area of a circle is calculated using the formula A = πr². Substituting the value of the radius:

A = (22/7) × 14² A = (22/7) × 196 A = 22 × 28 A = 616 sq. cm.

Explanation

Using the algebraic identity for the difference of cubes, we have: p³ - q³ = (p - q)³ + 3pq(p - q)

Substituting p - q = 6 and pq = 7: p³ - q³ = 6³ + 3 × 7 × 6 p³ - q³ = 216 + 126 = 342

Next, using the identity relating the sum and difference of two numbers: (p + q)² = (p - q)² + 4pq

Substituting the given values: (p + q)² = 6² + 4 × 7 (p + q)² = 36 + 28 = 64

Now, substituting these calculated values into the required expression: p³ - q³ + 6(p + q)² = 342 + 6 × 64 p³ - q³ + 6(p + q)² = 342 + 384 = 726

Explanation

Using the laws of exponents, first express all bases in terms of the prime base 3. Since 81 = 3⁴, we can substitute this into the given equation:

81² ÷ 3⁴ = 3ⁿ (3⁴)² ÷ 3⁴ = 3ⁿ

Applying the power of a power rule, (aᵐ)ⁿ = aᵐⁿ:

3⁸ ÷ 3⁴ = 3ⁿ

Applying the quotient rule of exponents, aᵐ ÷ aⁿ = aᵐ⁻ⁿ:

3⁸⁻⁴ = 3ⁿ 3⁴ = 3ⁿ

By equating the exponents of the same base, we get n = 4.

Explanation

The volume of a cone is given by the formula V = ⅓ × π × r² × h, where r is the radius and h is the height.

From the formula, the volume is directly proportional to r² × h.

When the radius r is increased by 10%, the new radius becomes 1.1r. When the height h is increased by 20%, the new height becomes 1.2h.

The new volume Vₙew is calculated as: Vₙew = ⅓ × π × (1.1r)² × (1.2h) Vₙew = 1.21 × 1.2 × (⅓ × π × r² × h) Vₙew = 1.452 × V

The fractional increase in volume is: 1.452 - 1 = 0.452

Expressing this as a percentage: 0.452 × 100% = 45.2%

Explanation

To simplify the given expression, expand the numerator using the algebraic identity (a - b)² = a² + b² - 2ab:

(sin x - cos x)² = sin² x + cos² x - 2 sin x cos x

Applying the fundamental trigonometric identity sin² x + cos² x = 1, the numerator simplifies to:

1 - 2 sin x cos x

Substituting this back into the original fraction yields:

(1 - 2 sin x cos x) / (1 - sin x cos x)

Explanation

Using the fundamental trigonometric identity sin²A + cos²A = 1, we can find sin A for the acute angle A.

sin A = √(1 - cos²A)

Substituting the given value of cos A = 12/13:

sin A = √(1 - (12/13)²) sin A = √(1 - 144/169) sin A = √(25/169) sin A = 5/13

Since A is an acute angle, sin A must be positive, which yields 5/13.

Explanation

We are given sin θ = ½. In a right-angled triangle, this corresponds to θ = 30°.

The value of cos θ for θ = 30° is cos 30° = √3/2.

Substituting these values into the expression 2 sin θ cos θ:

2 sin θ cos θ = 2 × ½ × √3/2 = √3/2.

Explanation

The given equation of the line is 3x + 4y - 12 = 0.

By rewriting this equation in the slope-intercept form, y = mx + c, the slope m₁ of the line can be determined: 4y = -3x + 12 y = (-3/4)x + 3

Thus, the slope of the given line is m₁ = -3/4.

For any two perpendicular lines, the product of their slopes is -1: m₁ × m₂ = -1

Substituting the value of m₁ into the formula: (-3/4) × m₂ = -1 m₂ = 4/3

Explanation

For simple interest, the interest earned is directly proportional to the time period when the principal and rate of interest are constant.

When a sum becomes n times itself, the interest earned is (n - 1) times the principal. Therefore, the time T required for a sum to become n times itself is directly proportional to (n - 1):

T₁ / T₂ = (n₁ - 1) / (n₂ - 1)

Given that the sum becomes 3 times itself (n₁ = 3) in 10 years (T₁ = 10), we want to find the time T₂ for it to become 6 times itself (n₂ = 6):

10 / T₂ = (3 - 1) / (6 - 1)

10 / T₂ = 2 / 5

T₂ = 10 × 5 / 2

T₂ = 25 years

Explanation

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

Substitute the given values x + y + z = 15 and xy + yz + zx = 54 into the equation: 15² = x² + y² + z² + 2(54) 225 = x² + y² + z² + 108

Subtracting 108 from both sides gives the value: x² + y² + z² = 225 - 108 x² + y² + z² = 117

Explanation

Let the amount invested in Scheme Y be P_Y and the rate of interest of Scheme X be R%.

Since the investment of 3,000 in Scheme X is 25% less than the investment in Scheme Y: 3,000 = P_Y × (1 - 0.25) P_Y = 3,000 / 0.75 = 4,000

The rate of interest for Scheme Y is (R + 4)%.

Using the Simple Interest formula, SI = (Principal × Rate × Time) / 100: Interest from Scheme X = (3,000 × R × 3) / 100 = 90R Interest from Scheme Y = (4,000 × (R + 4) × 6) / 100 = 240R + 960

The total simple interest earned is 2,520: 90R + 240R + 960 = 2,520 330R + 960 = 2,520 330R = 1,560 R = 1,560 / 330 = 52 / 11 ≈ 4.73%

Explanation

The number 146689 is a perfect square because its square root is an integer. Specifically, the square of 383 is:

383 × 383 = 146689

Since the square root of 146689 is exactly 383, it is a perfect square.

Explanation

The weighted average formula for the percentage of water in the mixture is:

P = (V₁P₁ + V₂P₂) / (V₁ + V₂)

Where: V₁ = 5 liters, P₁ = 20% V₂ = 7 liters, P₂ = 35%

Substituting these values into the formula:

P = (5 × 20 + 7 × 35) / (5 + 7) P = (100 + 245) / 12 P = 345 / 12 P = 28.75%

Explanation

To find the Highest Common Factor (HCF) of numbers expressed in their prime factorization, we take the product of the lowest power of each common prime factor present in the given numbers.

Comparing the prime factorizations: First number: 2³ × 3³ × 5² × 11¹ Second number: 2¹ × 3¹ × 5¹ × 7¹ × 11²

The common prime factors are 2, 3, 5, and 11. Taking the lowest power for each: For 2, the lowest power is 2¹ For 3, the lowest power is 3¹ For 5, the lowest power is 5¹ For 11, the lowest power is 11¹

Multiplying these lowest powers gives the HCF: HCF = 2 × 3 × 5 × 11

Explanation

The final population after a series of percentage changes can be calculated using the successive percentage formula:

Final Population = P × (1 - r₁/100) × (1 + r₂/100) × (1 - r₃/100)

Here, the initial population P is 50,000. The rate of change for the first year r₁ is a 15% decrease, for the second year r₂ is a 20% increase, and for the third year r₃ is a 10% decrease.

Substituting these values into the formula:

Final Population = 50,000 × (1 - 15/100) × (1 + 20/100) × (1 - 10/100)

Final Population = 50,000 × 0.85 × 1.20 × 0.90

Final Population = 50,000 × 17/20 × 6/5 × 9/10

Simplifying the calculation:

Final Population = 50,000 × 918/1,000

Final Population = 50 × 918

Final Population = 45,900

Explanation

By the time the motorbike starts at 7:00 AM, the cyclist has already been riding for 1 hour.

The initial distance between them is: Distance = Speed of cyclist × Time = 12 km/h × 1 h = 12 km

The relative speed of the motorbike chasing the cyclist is: Relative Speed = Speed of motorbike - Speed of cyclist = 24 km/h - 12 km/h = 12 km/h

The time required for the motorbike to cover this distance and catch the cyclist is: Time = Distance / Relative Speed = 12 km / 12 km/h = 1 hour

Adding this 1 hour to the motorbike's starting time of 7:00 AM gives 8:00 AM.

Explanation

The marked price (MP) of the article is the sum of the selling price (SP) and the discount offered: MP = SP + Discount MP = 1800 + 200 = 2000

Since the article is marked up by 25% above its cost price (CP): MP = CP × (1 + 25/100) 2000 = CP × 1.25 CP = 2000 / 1.25 = 1600

Explanation

The total cost price of the sugar mixture is calculated by summing the individual costs of both qualities: Total Cost Price = (40 × 45) + (60 × 55) = 1800 + 3300 = Rs. 5100

The total weight of the mixture is 40 + 60 = 100 kg.

The total selling price is calculated by multiplying the total weight by the selling rate per kg: Total Selling Price = 100 × 52 = Rs. 5200

Since the total selling price is greater than the total cost price, the transaction results in a profit: Profit = Total Selling Price - Total Cost Price = 5200 - 5100 = Rs. 100

Explanation

Using the LCM method, let the total work be the Least Common Multiple of 20 and 40, which is 40 units.

The efficiency of X is 40 / 20 = 2 units per day. The efficiency of Y is 40 / 40 = 1 unit per day.

Their combined efficiency is 2 + 1 = 3 units per day.

In 4 days, the work completed by X and Y together is 3 × 4 = 12 units.

The remaining work to be done is 40 - 12 = 28 units.

Since X leaves, the time taken by Y to finish the remaining work is 28 / 1 = 28 days.

Explanation

The net change in the sum of the 10 numbers is calculated by finding the total increase and subtracting the total decrease.

The total increase from the first 5 numbers is: 5 × 4 = 20

The total decrease from the remaining 5 numbers is: 5 × 2 = 10

The net change in the sum of the numbers is: 20 - 10 = 10

To find the change in the average, this net change in the sum is divided by the total number of values, which is 10: 10 / 10 = 1

Adding this change to the original average gives the new average: 38 + 1 = 39

Explanation

Let the certain number be 100.

The first number is 25% less than this number, which gives: 100 - 25 = 75

The second number is 40% more than this number, which gives: 100 + 40 = 140

The ratio of the first number to the second number is 75 : 140. Dividing both terms by their greatest common divisor, which is 5, simplifies the ratio to 15 : 28.

Explanation

The given expression is sin P cos Q + cos P sin Q.

First, find cos P and sin Q using the fundamental trigonometric identity sin²θ + cos²θ = 1:

cos P = √(1 - sin²P) = √(1 - (5/13)²) = √(1 - 25/169) = √(144/169) = 12/13

sin Q = √(1 - cos²Q) = √(1 - (12/13)²) = √(1 - 144/169) = √(25/169) = 5/13

Substituting these values into the expression:

sin P cos Q + cos P sin Q = (5/13) × (12/13) + (12/13) × (5/13) = 60/169 + 60/169 = 120/169

Explanation

The profit sharing ratio in a partnership is determined by the product of the investment amount and the time period of the investment.

X's equivalent investment for 12 months is: 72,000 × 12 = 8,64,000

Y's initial investment for the first 4 months is 54,000. He then increases his investment by 20%, making his new investment: 54,000 × 1.20 = 64,800 Y's equivalent investment for 12 months is: (54,000 × 4) + (64,800 × 8) = 2,16,000 + 5,18,400 = 7,34,400

Z's initial investment for the first 4 months is 36,000. He then reduces his investment by 15%, making his new investment: 36,000 × 0.85 = 30,600 Z's equivalent investment for 12 months is: (36,000 × 4) + (30,600 × 8) = 1,44,000 + 2,44,800 = 3,88,800

The ratio of their profit shares is X : Y : Z: 8,64,000 : 7,34,400 : 3,88,800 Dividing each by 43,200 simplifies this ratio to: 20 : 17 : 9

The sum of the ratio parts is 20 + 17 + 9 = 46.

Z's share of the total profit of Rs. 1,08,000 is: (9 / 46) × 1,08,000 = (9 × 54,000) / 23 = 4,86,000 / 23

Explanation

The formula for the compound interest amount after n years is Aₙ = P(1 + R/100)ⁿ, where P is the principal, R is the annual rate of interest, and n is the time in years.

The ratio of the amounts of two consecutive years gives the factor (1 + R/100):

A₄ / A₃ = (P(1 + R/100)⁴) / (P(1 + R/100)³) = 1 + R/100

Substituting the given values:

1 + R/100 = 14,641 / 13,310 = 1.1

Substituting this factor back into the formula for the amount at the end of the 3rd year:

A₃ = P(1 + R/100)³ 13,310 = P(1.1)³ 13,310 = P × 1.331 P = 13,310 / 1.331 P = 10,000

Explanation

Let the total work be the Least Common Multiple of the individual times taken by A and B. The Least Common Multiple of 24 and 16 is 48 units.

Efficiency of A = 48 / 24 = 2 units per day. Efficiency of B = 48 / 16 = 3 units per day.

Combined efficiency of A and B = 2 + 3 = 5 units per day.

In the first 4 days, A and B work together and complete: 5 units per day × 4 days = 20 units.

Remaining work = 48 - 20 = 28 units.

After B leaves, A works alone to complete the remaining work. Number of additional days required by A = Remaining work / Efficiency of A = 28 / 2 = 14 days.

Explanation

The reduction in the price of the pen is 20% of Rs. 160.

Saved amount = 20% of 160 = (20 / 100) × 160 = Rs. 32

Since this saved amount of Rs. 32 allows the man to buy 4 more pens, the new price of each pen is the saved amount divided by the number of extra pens.

New price of each pen = 32 / 4 = Rs. 8

Explanation

The sum of a set of numbers is given by the formula: Sum = Average × Number of terms.

The sum of all 9 numbers is: 9 × 40 = 360

The sum of the first 5 numbers is: 5 × 38 = 190

The sum of the last 5 numbers is: 5 × 44 = 220

Adding the sum of the first 5 numbers and the last 5 numbers counts the middle (fifth) number twice. Therefore, the middle number is the difference between this combined sum and the sum of all 9 numbers: Middle number = (190 + 220) - 360 = 410 - 360 = 50

Explanation

To find the combined ratio a:b:c, the common term b must be made equal in both ratios.

The given ratios are: a:b = 3:4 b:c = 5:6

The least common multiple of the b terms, 4 and 5, is 20.

Multiplying the terms of the first ratio by 5: a:b = (3 × 5) : (4 × 5) = 15:20

Multiplying the terms of the second ratio by 4: b:c = (5 × 4) : (6 × 4) = 20:24

Combining the two ratios yields: a:b:c = 15:20:24

Explanation

The distance d between two points when the speeds for going and returning are v₁ and v₂, and the total time taken is T, is given by the formula:

d = (v₁ × v₂ × T) / (v₁ + v₂)

Substituting the given values: v₁ = 6 km/h v₂ = 3 km/h T = 2 hours

d = (6 × 3 × 2) / (6 + 3) d = 36 / 9 d = 4 km

Explanation

The fundamental trigonometric identity states that sin²θ + cos²θ = 1.

Substituting the given value of cosθ = 7/25 into the identity yields: sin²θ + (7/25)² = 1 sin²θ + 49/625 = 1 sin²θ = 1 - 49/625 sin²θ = 576/625 sinθ = ±24/25

In Quadrant IV, the cosine function is positive and the sine function is negative. Since θ lies in Quadrant IV, sinθ must take the negative value, which gives sinθ = -24/25.

Explanation

For any pure recurring decimal where a block of digits repeats indefinitely, the decimal can be directly converted into a fraction by placing the repeating block in the numerator and an equal number of nines in the denominator.

In the recurring decimal 0.363636..., the repeating block is 36, which has a length of 2 digits.

Applying this rule: 0.363636... = 36/99

To reduce this fraction to its lowest terms, divide both the numerator and the denominator by their greatest common divisor, which is 9: 36 ÷ 9 = 4 99 ÷ 9 = 11

This gives the simplified fraction 4/11.

Explanation

To find the profit share of each partner, we compute the ratio of their equivalent investments over a 1-year period using the formula: Equivalent Investment = Investment × Time

For A, the investment is Rs. 80,000 for the first 6 months, and then Rs. 1,00,000 for the next 6 months: Equivalent Investment of A = (80,000 × 6) + (1,00,000 × 6) = 4,80,000 + 6,00,000 = 10,80,000

For B, the investment is Rs. 50,000 for the first 6 months, and then Rs. 40,000 for the next 6 months: Equivalent Investment of B = (50,000 × 6) + (40,000 × 6) = 3,00,000 + 2,40,000 = 5,40,000

For C, the investment is Rs. 70,000 for the entire 12 months: Equivalent Investment of C = 70,000 × 12 = 8,40,000

The ratio of profit sharing is: A : B : C = 10,80,000 : 5,40,000 : 8,40,000 A : B : C = 18 : 9 : 14

The sum of the ratio terms is 18 + 9 + 14 = 41.

B's share from the total profit of Rs. 2,40,000 is: B's share = (9 / 41) × 2,40,000 = 21,60,000 / 41 ≈ 52,683

Explanation

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

Difference = P × (R / 100)²

Where P is the principal sum and R is the rate of interest per annum.

Substituting the given values into the formula:

72 = P × (12 / 100)²

72 = P × (3 / 25)²

72 = P × (9 / 625)

P = (72 × 625) / 9

P = 8 × 625

P = 5000

The principal sum is Rs. 5,000.

Explanation

Parallel lines have the same slope. The slope-intercept form of a linear equation is y = mx + c, where m represents the slope. The given line y = -2x + 3 has a slope of m = -2. Therefore, any line parallel to it must also have a slope of -2.

Using the point-slope formula, the equation of a line with slope m passing through a point (x₁, y₁) is:

y - y₁ = m(x - x₁)

Substituting the slope m = -2 and the coordinates of the point (2, -1) into this formula:

y - (-1) = -2(x - 2)

y + 1 = -2x + 4

y = -2x + 3

Explanation

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

Area₁ / Area₂ = (Side₁ / Side₂)²

Substituting the given area ratio of 16:49 into the formula:

16 / 49 = (Side₁ / Side₂)²

Taking the square root of both sides to find the ratio of the corresponding sides:

Side₁ / Side₂ = √(16 / 49) = 4 / 7

This gives the ratio of the corresponding sides as 4:7.

Explanation

The vertices of the triangle are A(0,0), B(6,0), and C(0,8). The side AB lies along the x-axis, and the side AC lies along the y-axis. Because the x-axis and the y-axis are perpendicular, the angle at vertex A(0,0) is exactly 90°.

In any right-angled triangle, the altitudes from the two acute vertices lie along the perpendicular sides and intersect at the vertex containing the right angle. Therefore, the orthocenter, which is the intersection of the three altitudes, always coincides with the vertex of the right angle. This places the orthocenter at the origin, A(0,0).

Explanation

To find the required number, we first determine the Least Common Multiple (LCM) of the divisors 5, 8, and 12.

LCM(5, 8, 12) = 120

Any number that leaves a remainder of 2 when divided by 5, 8, and 12 must be of the form 120k + 2, where k is an integer.

The largest four-digit number is 9999. Dividing 9999 by 120 gives a quotient of 83 with a remainder of 39.

The largest multiple of 120 that is a four-digit number is: 120 × 83 = 9960

Adding the required remainder of 2 to this multiple: 9960 + 2 = 9962

Explanation

By the law of exponents, (aᵐ)ⁿ = aᵐⁿ. Applying this to the left side of the equation gives:

(2³)ˣ = 2³ˣ

We are given the equation:

2³ˣ = 2¹²

Since the bases on both sides are equal, we can equate their exponents:

3x = 12

Dividing both sides by 3:

x = 4

Explanation

The quantity of milk in the mixture remains constant because only water is being added.

Initially, water is 20% of the 60-litre mixture, which means milk constitutes 80% of the mixture. Volume of milk = 60 × 0.80 = 48 litres

In the final mixture, water becomes 40%, meaning milk now constitutes 60% of the total final volume. Let the final volume of the mixture be V.

0.60 × V = 48 V = 48 / 0.60 V = 80 litres

The quantity of water to be added is the difference between the final and initial volumes of the mixture: Water added = 80 - 60 = 20 litres

Explanation

Given that tanθ - cotθ = 0, we have tanθ = cotθ, which implies sinθ/cosθ = cosθ/sinθ, or sin²θ = cos²θ. Assuming θ is an acute angle, this gives sinθ = cosθ, and therefore sinθ - cosθ = 0.

Explanation

For a right circular cylinder of radius r and height h inscribed inside a sphere of diameter D, the diagonal of the longitudinal cross-section of the cylinder represents the diameter of the sphere. This cross-section is a rectangle with a width equal to the cylinder's diameter (2r) and a height equal to the cylinder's height (h).

Applying the Pythagorean theorem to this right-angled triangle formed by the cylinder's diameter, height, and the sphere's diameter gives:

D² = (2r)² + h²

D² = 4r² + h²

Rearranging the terms to express h in terms of D and r:

h² = D² - 4r²

h = √(D² - 4r²)

Explanation

The Selling Price (SP) of 10 books is Rs. 500. Therefore, the SP of one book is: SP of one book = 500 / 10 = Rs. 50

The trader suffers a loss equal to the selling price of 2 books: Loss = 2 × SP of one book = 2 × 50 = Rs. 100

Using the fundamental relationship for loss: Cost Price (CP) = Selling Price (SP) + Loss

Substitute the values for 10 books: CP of 10 books = 500 + 100 = Rs. 600

Therefore, the cost price of one book is: CP of one book = CP of 10 books / 10 = 600 / 10 = Rs. 60

Explanation

Using the simple interest formula, Simple Interest (SI) = P × R × T / 100 and Amount (A) = P + SI.

For the second case, the principal is Rs. 10,000 and it amounts to Rs. 12,000. SI = 12,000 - 10,000 = Rs. 2,000 Using the formula for interest: 2,000 = 10,000 × 10 × T / 100 T = 2 years

For the first case, the time T is 2 years, the rate of interest is 8% per annum, and the amount is Rs. 9,280. Amount A = x × (1 + R × T / 100) 9,280 = x × (1 + 8 × 2 / 100) 9,280 = x × (1 + 16 / 100) 9,280 = x × (116 / 100) x = 9,280 × 100 / 116 x = 8,000

Explanation

The ratio of the dimensions of the model to the building is 1:500.

To find the height of the model, we use the scale ratio formula: Height of model = Height of building ÷ Scale factor

First, convert the height of the building from meters to centimeters: 75 m = 75 × 100 cm = 7500 cm

Using the scale factor of 500: Height of model = 7500 cm ÷ 500 = 15 cm

Explanation

Using the algebraic identity where if a + b + c = 0, then a³ + b³ + c³ = 3abc.

Let a = x² - y², b = y² - z², and c = z² - x².

Adding these three terms: a + b + c = (x² - y²) + (y² - z²) + (z² - x²) = 0

Since the sum is zero, the sum of their cubes is: (x² - y²)³ + (y² - z²)³ + (z² - x²)³ = 3(x² - y²)(y² - z²)(z² - x²)

Applying the difference of squares formula, a² - b² = (a - b)(a + b), to each factor: x² - y² = (x - y)(x + y) y² - z² = (y - z)(y + z) z² - x² = (z - x)(z + x)

Substituting these expansions into the expression yields: 3(x - y)(x + y)(y - z)(y + z)(z - x)(z + x)

Rearranging the terms gives: 3(x - y)(y - z)(z - x)(x + y)(y + z)(z + x)

Explanation

The volume of metal in a hemispherical bowl is determined by subtracting the volume of the inner hemisphere from the volume of the outer hemisphere. The formula is V = (2/3)π(R³ - r³), where R is the outer radius and r is the inner radius.

Given that the inner radius r is 7 cm and the thickness of the metal is 3 cm, the outer radius R is calculated as: R = 7 + 3 = 10 cm

Substituting these values into the volume formula: V = (2/3)π(10³ - 7³) V = (2/3)π(1000 - 343) V = (2/3)π × 657 V = 2π × 219 V = 438π cm³

Explanation

Since the square root of 81 is exactly 9 and the square root of 80 is approximately 8.94, the sum of these values is approximately 17.94. Therefore, the integer part of the expression is 17.

Explanation

The simple interest is calculated using the formula:

Simple Interest = (P × R × T) / 100

Where: P represents the principal amount, which is Rs. 15,000. R represents the rate of interest per annum, which is 10%. T represents the time period, which is 4 years.

Substituting the given values into the formula:

Simple Interest = (15000 × 10 × 4) / 100 Simple Interest = 15000 × 0.40 Simple Interest = 6000

The total interest to be paid at the end of 4 years is Rs. 6,000.