SSC CHSL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

Let the present ages of the father and the son be F and S, respectively.

According to the first condition, ten years later, the father's age will be double the son's age: F + 10 = 2(S + 10) F - 2S = 10

According to the second condition, ten years ago, the father's age was six times the son's age: F - 10 = 6(S - 10) F - 6S = -50

Subtracting the second equation from the first equation gives: 4S = 60 S = 15

Substituting S = 15 into the first equation: F - 2(15) = 10 F = 40

So, the present age of the father is 40 years and that of the son is 15 years.

Let t be the number of years from now when the ratio of their ages will be 3 : 2: (40 + t) / (15 + t) = 3 / 2 2(40 + t) = 3(15 + t) 80 + 2t = 45 + 3t t = 35

Explanation

To find the percentage reduction required to restore the original price after an increase of r%, the formula is:

Percentage reduction = (r / (100 + r)) × 100%

Given that the price of pulses has increased by r = 45%:

Percentage reduction = (45 / (100 + 45)) × 100% Percentage reduction = (45 / 145) × 100% Percentage reduction = (9 / 29) × 100% Percentage reduction = 900 / 29 % Percentage reduction ≈ 31.03%

Rounding to the nearest integer, the required percentage reduction is 31%.

Explanation

The total cost price of the batch of goods is ₹5,000. Since 5% of the goods are damaged, the remaining undamaged goods constitute 95% of the total batch.

The cost price of these remaining undamaged goods is calculated as: CPᵣemaining = 95% of ₹5,000 = 0.95 × 5000 = ₹4,750

To make a profit of 20% on these remaining goods, the retailer must sell them at 120% of their cost price: Selling Price = 120% of ₹4,750 = 1.20 × 4750 = ₹5,700

Explanation

Radius of the circle R = 26/2 = 13 cm.

The perpendicular drawn from the center of a circle to a chord bisects the chord. For a chord of length 10 cm, the length of the half-chord is 5 cm.

Using the Pythagorean theorem in the right-angled triangle formed by the radius, the perpendicular distance from the center to the chord d, and the half-chord: d² + 5² = 13² d² + 25 = 169 d² = 144 d = 12 cm.

Since the two equal chords are situated on opposite sides of the center of the circle, the distance h between them is the sum of their individual distances from the center: h = d + d = 12 + 12 = 24 cm.

Explanation

Let the present age of the younger boy be x years. Since the older boy is thrice as old as the younger boy, his present age is 3x years.

Nine years ago, their respective ages were x - 9 and 3x - 9.

According to the given condition, the older boy's age at that time was five times that of the younger boy: 3x - 9 = 5(x - 9)

Expanding and solving the linear equation: 3x - 9 = 5x - 45 2x = 36 x = 18

Thus, the present age of the younger boy is 18 years, and the present age of the older boy is 3 × 18 = 54 years.

Explanation

The curved surface area of a cylinder is 2πrh and its volume is πr²h, where r is the radius and h is the height.

The ratio of the curved surface area to the volume is: 2πrh / πr²h = 2/r

Given that this ratio is 2:3: 2/r = 2/3 r = 3

The ratio between the radius and height is given as 3:7: r/h = 3/7 3/h = 3/7 h = 7

The total surface area of a cylinder is calculated using the formula: Total Surface Area = 2πr(r + h)

Substituting the values of r and h: Total Surface Area = 2 × π × 3 × (3 + 7) = 6π × 10 = 60π cm²

Explanation

Average speed is defined as the total distance traveled divided by the total time taken.

For the first segment of the journey: Distance = 110 miles Speed = 55 miles per hour Time = 110 / 55 = 2 hours

For the second segment of the journey: Distance = 120 miles Speed = 60 miles per hour Time = 120 / 60 = 2 hours

For the third segment of the journey: Distance = 60 miles Time = 1 hour

Total distance = 110 + 120 + 60 = 290 miles Total time = 2 + 2 + 1 = 5 hours

Average speed = Total distance / Total time = 290 / 5 = 58 miles per hour

Explanation

Using the percentage decrease formula: New Price = Original Price × (1 - d/100)

Substituting the given values: New Price = 1,25,000 × (1 - 30/100) New Price = 1,25,000 × 0.70 New Price = 87,500

Explanation

The relationship between Marked Price (MP) and Cost Price (CP) when profit percentage (P) and discount percentage (D) are given is expressed by the formula:

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

In the first case, a profit of 35% is made with a discount of 35%:

MP / CP = (100 + 35) / (100 - 35) = 135 / 65 = 27 / 13

In the second case, the discount is 45%. Let the new profit percentage be P. Using the MP/CP ratio:

27 / 13 = (100 + P) / (100 - 45)

27 / 13 = (100 + P) / 55

100 + P = (27 × 55) / 13

100 + P = 1485 / 13

P = 1485 / 13 - 100

P = (1485 - 1300) / 13

P = 185 / 13 %

Converting 185 / 13 into a mixed fraction:

185 ÷ 13 = 14 with a remainder of 3, which gives 14 3/13 %

Explanation

In triangle ABC, we are given AB = AC. Since BC is extended to D such that CD = AB, it follows that CD = AC. This makes triangle ACD an isosceles triangle.

In the isosceles triangle ACD with AC = CD, the angles opposite to the equal sides are equal: angle CAD = angle ADC = 30°

The sum of angles in triangle ACD is 180°: angle ACD + angle CAD + angle ADC = 180° angle ACD + 30° + 30° = 180° angle ACD = 120°

Since B, C, and D lie on a straight line, the angles ACB and ACD form a linear pair: angle ACB + angle ACD = 180° angle ACB + 120° = 180° angle ACB = 60°

In triangle ABC, since AB = AC, the base angles are equal: angle ABC = angle ACB = 60°

The remaining angle BAC is: angle BAC = 180° - (angle ABC + angle ACB) = 180° - (60° + 60°) = 60°

Explanation

Let the initial quantities of alcohol and water in the mixture be 12x and 5x respectively.

When 14 liters of water is added to the mixture, the quantity of water becomes 5x + 14, while the quantity of alcohol remains unchanged at 12x.

According to the given condition, the new ratio of alcohol to water is 4 : 3: 12x / (5x + 14) = 4 / 3

Cross-multiplying the terms: 36x = 20x + 56 16x = 56 x = 3.5

The quantity of alcohol in the mixture is 12x = 12 × 3.5 = 42 liters.

Explanation

The average cost of a book is calculated by dividing the total cost of all books by the total number of books.

Total cost of the books = (4 × 50) + (5 × 60) + (6 × 70) = 200 + 300 + 420 = 920

Total number of books = 4 + 5 + 6 = 15

Average cost = 920 ÷ 15 = 61.3333...

Rounding to two decimal places, the average cost of a book is 61.33.

Explanation

In a circle, the angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circumference. Since angle AOQ is the angle subtended by arc AQ at the center O, and angle ABQ is the angle subtended by arc AQ at the circumference point B: angle ABQ = ½ × angle AOQ = ½ × 94° = 47°

Since AT is a tangent to the circle at the diameter endpoint A, the angle TAB is 90°.

Because the line TB intersects the circle at Q, the points T, Q, and B are collinear, which means angle ATQ is the same as angle ATB. In the right-angled triangle TAB, the sum of acute angles is 90°: angle ATB = 90° - angle ABT angle ATB = 90° - 47° = 43°

Thus, angle ATQ = 43°.

Explanation

According to the divisibility rule for 11, the difference between the sum of digits at odd positions and the sum of digits at even positions must be 0 or a multiple of 11.

For the number 19487161: Sum of digits at odd positions from right to left is 1 + 1 + 8 + 9 = 19. Sum of digits at even positions from right to left is 6 + 7 + 4 + 1 = 18. The difference is 19 - 18 = 1.

This difference of 1 indicates that 19487161 leaves a remainder of 1 when divided by 11, or 19487161 ≡ 1 mod 11.

To make the sum divisible by 11, the smallest integer x to add must satisfy: 1 + x = 11 x = 10

Explanation

Using the ratio of distances covered by the runners in the same duration of time:

When A covers 100 m, B covers 100 - 20 = 80 m. The ratio of their distances is D_A / D_B = 100 / 80 = 5/4.

When B covers 100 m, C covers 100 - 5 = 95 m. The ratio of their distances is D_B / D_C = 100 / 95 = 20/19.

To find the distance covered by C when A covers 100 m, we multiply the ratios: D_C / D_A = (D_C / D_B) × (D_B / D_A) D_C / 100 = (19 / 20) × (80 / 100) D_C / 100 = 76 / 100 D_C = 76 m

When A completes the 100 m race, C has covered 76 m. Thus, the distance by which A beats C is: 100 - 76 = 24 m.

Explanation

Using the method of alligation, the overall annual interest rate is calculated first. Since the total interest earned over 2 years is ₹720, the annual interest earned is ₹720 ÷ 2 = ₹360.

The overall annual interest rate is (360 ÷ 8000) × 100% = 4.5%.

By the rule of alligation, the ratio of the money invested in Scheme 1 (at 5%) to Scheme 2 (at 4%) is: (4.5 - 4) : (5 - 4.5) = 0.5 : 0.5 = 1 : 1

Since the investment is split in a 1 : 1 ratio, the amount invested in Scheme 1 is half of the total investment: 8000 × ½ = ₹4,000.

Explanation

The volume of a sphere is given by the formula V = 4/3 πr³, where r is the radius.

The diameters of the three spheres are 6 cm, 8 cm, and 10 cm, which gives their respective radii as r₁ = 3 cm, r₂ = 4 cm, and r₃ = 5 cm.

When the three spheres are melted and recast into a single solid sphere of radius R, the volume of the new sphere is equal to the sum of the volumes of the three smaller spheres: 4/3 πR³ = 4/3 πr₁³ + 4/3 πr₂³ + 4/3 πr₃³

Dividing both sides of the equation by 4/3 π simplifies the relation to: R³ = r₁³ + r₂³ + r₃³

Substituting the values of the radii into the equation: R³ = 3³ + 4³ + 5³ R³ = 27 + 64 + 125 R³ = 216

Taking the cube root on both sides: R = ∛216 = 6 cm

Explanation

The average production of a food grain is calculated by dividing the sum of its production over the five years by the total number of years.

To find the difference between the average production of wheat and maize, the total production for each food grain is calculated first.

Total production of Wheat over the five years: 650 + 800 + 680 + 700 + 640 = 3470 ten lakh tonnes

Average production of Wheat: 3470 / 5 = 694 ten lakh tonnes

Total production of Maize over the five years: 200 + 240 + 220 + 260 + 300 = 1220 ten lakh tonnes

Average production of Maize: 1220 / 5 = 244 ten lakh tonnes

The difference between the average production of wheat and maize is: 694 - 244 = 450 ten lakh tonnes

Explanation

To find the required selling rate per kg, use the relation:

Total Desired Revenue = Total Cost Price × (1 + Profit Percentage) Total Desired Revenue = 32400 × (1 + 0.20) = 32400 × 1.2 = ₹38,880

Next, calculate the remaining quantity of wheat after 20% is spoiled:

Remaining Wheat = Total Wheat × (1 - Spoiled Percentage) Remaining Wheat = 1800 × (1 - 0.20) = 1800 × 0.8 = 1440 kg

The selling price per kg for the remaining wheat is:

Rate = Total Desired Revenue ÷ Remaining Wheat Rate = 38880 ÷ 1440 = 27

Thus, the shopkeeper should sell the remaining wheat at ₹27 per kg.

Explanation

The perpendicular distance from the center of a circle of radius R to a chord of length L is given by the Pythagorean theorem as d = √(R² - (L/2)²).

For the first chord of length 5 units: d₁ = √(7² - 2.5²) = √(49 - 6.25) = √42.75 ≈ 6.538 units

For the second chord of length 8 units: d₂ = √(7² - 4²) = √(49 - 16) = √33 ≈ 5.745 units

Since the two parallel chords lie on opposite sides of the center, the total distance between them is the sum of their perpendicular distances from the center: Distance = d₁ + d₂ = 6.538 + 5.745 = 12.283 units

Rounding to two decimal places gives 12.28 units.

Explanation

Let Ramu's initial income be 100.

Since he spends 72% of his income: Initial expenditure = 72 Initial savings y = 100 - 72 = 28

His income increases by 12%: New income = 100 × 1.12 = 112

His expenditure increases by 5%: New expenditure = 72 × 1.05 = 75.6

His new savings x is the difference between the new income and new expenditure: x = 112 - 75.6 = 36.4

The percentage change in savings is given by: ((x - y) / y) × 100% = ((36.4 - 28) / 28) × 100% = (8.4 / 28) × 100% = 0.3 × 100% = 30%

Explanation

An angle subtended by a diameter at any point on the circumference of a circle is a right angle. Since PQ is the diameter of the circle and R is a point on the circle, the angle ∠PRQ = 90°.

In the right-angled triangle PRQ, the sum of the angles is 180°: ∠PRQ + ∠RPQ + ∠RQP = 180°

Substituting the given values: 90° + 27° + ∠RQP = 180° 117° + ∠RQP = 180° ∠RQP = 180° - 117° = 63°

Explanation

The ratio of the lateral surface area (LSA) to the total surface area (TSA) of a cylinder is given by the formula:

Ratio = LSA / TSA = (2πrh) / (2πr(h + r)) = h / (h + r)

Given that the base diameter of the cylinder is 60 cm, the radius r is half of the diameter, which is 30 cm. The height h is given as 10 cm.

Substituting these values into the ratio formula:

Ratio = 10 / (10 + 30) = 10 / 40 = 1/4

This simplifies to the ratio 1:4.

Explanation

For a pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have infinitely many solutions, the ratio of their coefficients must satisfy the condition: a₁/a₂ = b₁/b₂ = c₁/c₂

Comparing the given equations kx + 3y - (k - 3) = 0 and 12x + ky - k = 0 with the standard form gives: a₁ = k, b₁ = 3, c₁ = -(k - 3) a₂ = 12, b₂ = k, c₂ = -k

Substituting these values into the condition: k/12 = 3/k = -(k - 3)/(-k)

Equating the first two ratios: k/12 = 3/k k² = 36 k = ±6

Since we are looking for positive values of k, we take k = 6. Substituting k = 6 into the third ratio yields (6 - 3)/6 = 3/6 = ½, which is equal to the ratio 3/k = 3/6 = ½. Thus, the positive value of k that gives infinitely many solutions is 6.

Explanation

The length of the arc of a sector of a circle is calculated using the formula L = (θ / 360°) × 2πr, where θ is the central angle and r is the radius of the circle.

Substituting the given values, L = 5.5 cm, θ = 30°, and π = 22/7, into the formula:

5.5 = (30 / 360) × 2 × (22 / 7) × r

Simplifying the fraction:

5.5 = (1 / 12) × (44 / 7) × r

5.5 = (11 / 21) × r

Solving for r:

r = (5.5 × 21) / 11

r = 0.5 × 21

r = 10.5 cm

Explanation

To find the percentage by which Monica's income is less than Neha's income, use the standard percentage relationship formula:

Percentage less = [r / (100 + r)] × 100 %

Here, r represents the percentage by which Neha's income is more than Monica's income, which is 5%.

Substituting r = 5 into the formula:

Percentage less = [5 / (100 + 5)] × 100 % = [5 / 105] × 100 % = [1 / 21] × 100 % = 100 / 21 % = 4 16/21 %

Explanation

Using the definitions of trigonometric ratios in a right-angled triangle, cot θ = base/perpendicular = 3/4.

Since θ is an acute angle, the base is 3, the perpendicular is 4, and the hypotenuse is computed using the Pythagorean theorem: Hypotenuse = √(3² + 4²) = 5

From this, the remaining trigonometric ratios are: sin θ = perpendicular/hypotenuse = 4/5 cos θ = base/hypotenuse = 3/5 tan θ = 1/cot θ = 4/3

Substituting these values into the expression: sin θ + cos θ - tan θ = 4/5 + 3/5 - 4/3 = 7/5 - 4/3 = (21 - 20) / 15 = 1/15

Explanation

When two people can complete a task individually in A and B days, the time taken for them to complete the work together is given by the formula (A × B) / (A + B).

Substituting the given values where Akhil's time A = 24 days and Shyam's time B = 12 days:

Time taken together = (24 × 12) / (24 + 12) = 288 / 36 = 8 days.

Explanation

Simple Interest (SI) is calculated using the formula: SI = (P × R × T) / 100

Here, the number of years (T) is equal to the rate of interest (R), so T = R. Substituting this into the formula gives: SI = (P × R²) / 100

Given: Principal (P) = ₹12,00,000 Simple Interest (SI) = ₹9,72,000

Substituting these values into the formula: 9,72,000 = (12,00,000 × R²) / 100 9,72,000 = 12,000 × R² R² = 9,72,000 / 12,000 R² = 81 R = √81 R = 9

Explanation

When two circles of radii r₁ and r₂ touch each other externally, the length of their common tangent AB is given by the formula:

AB = 2 × √(r₁ × r₂)

Given the radii of the two circles are r₁ = 9 cm and r₂ = 4 cm, we substitute these values into the formula:

AB = 2 × √(9 × 4) AB = 2 × √36 AB = 2 × 6 AB = 12 cm

Explanation

The length of the transverse common tangent (T) between two circles with radii r₁ and r₂ and distance between their centres d is given by the formula:

T = √(d² - (r₁ + r₂)²)

Substituting the given values: d = 15 cm r₁ = 7 cm r₂ = 5 cm

T = √(15² - (7 + 5)²) T = √(225 - 12²) T = √(225 - 144) T = √81 T = 9 cm

Explanation

Let the length, breadth, and height of the cuboid be 3x, 4x, and 5x, respectively.

The volume of a cuboid is calculated using the formula: Volume = length × breadth × height

Substitute the given volume and ratio terms into the formula: 1620 = 3x × 4x × 5x 1620 = 60x³ x³ = 27 x = 3

Using this value of x, the dimensions of the cuboid are: Length = 3 × 3 = 9 m Breadth = 4 × 3 = 12 m Height = 5 × 3 = 15 m

The total surface area of a cuboid is given by the formula: Total Surface Area = 2(length × breadth + breadth × height + height × length)

Substitute the dimensions into this formula: Total Surface Area = 2(9 × 12 + 12 × 15 + 15 × 9) Total Surface Area = 2(108 + 180 + 135) Total Surface Area = 2 × 423 Total Surface Area = 846 m²

Explanation

To find the combined ratio a:b:c:d, we first equalize the common terms across the given ratios.

Given: a:b = 3:4 b:c = 2:3

To combine these, we make the term for b equal in both ratios. Since the value of b is 4 in the first ratio and 2 in the second, we multiply the second ratio by 2: b:c = 4:6

This gives: a:b:c = 3:4:6

Next, we combine this with the ratio c:d = 5:6. We make the term for c equal in both ratios by finding the least common multiple of 6 and 5, which is 30.

Multiply the terms of a:b:c by 5: a:b:c = 15:20:30

Multiply the terms of c:d by 6: c:d = 30:36

Combining these results in: a:b:c:d = 15:20:30:36

Explanation

For the number 3c2933k to be divisible by 5, its units digit k must be 0 or 5. Since k is a single-digit natural number, k must be 5.

Replacing k with 5, the number becomes 3c29335.

For this number to be divisible by 11, the difference between the sum of digits at odd positions and the sum of digits at even positions must be a multiple of 11.

Sum of digits at odd positions (from left to right): 3 + 2 + 3 + 5 = 13 Sum of digits at even positions (from left to right): c + 9 + 3 = c + 12

The difference is: 13 - (c + 12) = 1 - c

For this difference to be a multiple of 11, where c is a single-digit natural number from 1 to 9, the only possible value is when the difference is 0: 1 - c = 0 c = 1

Thus, c + k = 1 + 5 = 6.

Explanation

Following the order of operations (BODMAS), we simplify the expression step-by-step from the innermost brackets outward.

First, calculate the terms inside the parentheses: 13 × 13 = 169

Using this, simplify the first division: 169 ÷ 169 = 1

Next, simplify the innermost bracket: 3 - 17 + 10 = -14 + 10 = -4

Now substitute these results back into the larger square bracket: 1 - (-4) - (-4) = 1 + 4 + 4 = 9

Finally, subtract this result from the first term: 99 - 9 = 90

Explanation

The total surface area of a cylinder is the sum of its curved surface area and the area of its two circular bases.

Total Surface Area = Curved Surface Area + 2 × πr²

Given: Radius, r = 5 m Total Surface Area = 628 m²

Using π ≈ 22/7: Area of the two bases = 2 × (22/7) × 5² Area of the two bases = 2 × (22/7) × 25 = 1100/7 ≈ 157.14 m²

We can find the curved surface area by subtracting the area of the two bases from the total surface area: Curved Surface Area = Total Surface Area - Area of the two bases Curved Surface Area = 628 - 157.14 = 470.86 m²

Rounding to the nearest decimal value matches 470.8 m².

Explanation

Using the work-rate relation for two individuals working together: 1/T_together = 1/T_Sapna + 1/T_Riya

Here, T_Sapna is the time taken by Sapna alone, which is 12 hours, and T_together is the time taken by both working together, which is 8 hours. Let T_Riya be the time taken by Riya alone.

Substituting the given values into the formula: 1/8 = 1/12 + 1/T_Riya

Rearranging the terms to solve for T_Riya: 1/T_Riya = 1/8 - 1/12

Finding a common denominator of 24: 1/T_Riya = 3/24 - 2/24 1/T_Riya = 1/24

T_Riya = 24

This calculation shows that Riya takes 24 hours to complete the job working alone.

Explanation

From the given equations, express B in terms of A: cot B = a / tan A cosec B = b / sin A

Using the standard trigonometric identity: cosec²B - cot²B = 1

Substitute the values of cosec B and cot B into the identity: (b / sin A)² - (a / tan A)² = 1 b² / sin²A - a² cos²A / sin²A = 1 b² - a² cos²A = sin²A

Substitute sin²A = 1 - cos²A: b² - a² cos²A = 1 - cos²A b² - 1 = a² cos²A - cos²A b² - 1 = cos²A (a² - 1) cos²A = (b² - 1) / (a² - 1)

Explanation

The surface area of a spherical ball is calculated using the formula:

A = 4πr²

The diameter of the ball is 28 cm, which gives a radius r of 14 cm.

Using π = 22/7, the surface area is:

A = 4 × (22/7) × 14 × 14 A = 4 × 22 × 2 × 14 A = 2464 cm²

The painter charges ₹2 per cm² to paint the ball. Therefore, the total cost of painting is:

Cost = Area × Rate Cost = 2464 × 2 = 4928

Explanation

In a trapezium ABCD with AB parallel to DC, the diagonals AC and BD intersect at O such that triangle AOB is similar to triangle COD. This similarity gives the relation:

AO / OC = OB / OD

Substituting the given values into the relation:

(3x - 15) / (x - 5) = (x + 9) / 5

Cross-multiplying the terms:

5(3x - 15) = (x - 5)(x + 9) 15x - 75 = x² + 4x - 45 x² - 11x + 30 = 0

Factoring the quadratic equation:

(x - 5)(x - 6) = 0

This yields the two values of x:

x₁ = 5 x₂ = 6

Calculating the sum of their squares:

x₁² + x₂² = 5² + 6² = 25 + 36 = 61

Explanation

To find the school with the maximum average number of candidates selected over the years, we calculate the sum of selected candidates for each school from 2012 to 2017 and divide by the total number of years, which is 6.

For School M: Total selected = 84 + 55 + 42 + 89 + 92 + 94 = 456 Average selected = 456 ÷ 6 = 76

For School N: Total selected = 34 + 56 + 66 + 72 + 86 + 92 = 406 Average selected = 406 ÷ 6 ≈ 67.67

For School O: Total selected = 27 + 36 + 48 + 57 + 68 + 89 = 325 Average selected = 325 ÷ 6 ≈ 54.17

For School P: Total selected = 23 + 56 + 67 + 87 + 89 + 93 = 415 Average selected = 415 ÷ 6 ≈ 69.17

Comparing the average values, School M has the highest average number of selected candidates with 76.

Explanation

Using the order of operations (BODMAS), we first evaluate the numerator: 7 + 8 × 7 - 8 + 7 - 7 + 7 = 7 + 56 - 8 + 7 - 7 + 7 = 63 - 8 + 7 - 7 + 7 = 55 + 7 = 62

Evaluating the expression with the denominator of 434: 62 ÷ 434 = 31/217

Explanation

Loss is calculated using the formula: Loss = Cost Price (CP) - Selling Price (SP)

Substituting the given values: CP = ₹4,960 SP = ₹4,712

Loss = 4,960 - 4,712 = ₹248

The loss percentage is given by the formula: Loss Percentage = (Loss / CP) × 100

Substituting the calculated loss and CP: Loss Percentage = (248 / 4,960) × 100 = (1 / 20) × 100 = 5%

Explanation

To find if and when Raju and Ramesh will have the same amount of money, we can set up an algebraic equation where w represents the number of weeks.

Raju's total money after w weeks is represented by the formula: 11000 + 5000w

Ramesh's total money after w weeks is represented by the formula: 60000 - 2000w

To find the week when they have the same amount, set these two expressions equal to each other: 11000 + 5000w = 60000 - 2000w

Add 2000w to both sides of the equation: 11000 + 7000w = 60000

Subtract 11000 from both sides of the equation: 7000w = 49000

Divide by 7000: w = 7

Since w is a positive integer, they will have the exact same amount of money after 7 weeks.

Explanation

The length of the bridge is the distance covered by the girl in the given time.

Using the formula: Distance = Speed × Time

First, convert the speed from kilometers per hour to meters per minute: Speed = 9 km/h = (9 × 1000) meters / 60 minutes = 150 meters/minute

Next, calculate the distance by multiplying the speed in meters per minute by the time in minutes: Distance = 150 meters/minute × 45 minutes = 6750 meters

Explanation

The net percentage change after two successive percentage changes of a% and b% is given by the formula:

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

Here, the population first decreases by 16%, so a = -16. The population then increases by 21%, so b = 21.

Substituting these values into the formula:

Net Change % = -16 + 21 + ((-16 × 21) / 100) = 5 - (336 / 100) = 5 - 3.36 = +1.64%

Since the net result is positive, there is a net increase of 1.64% in the population.

Explanation

In triangle XYZ, since XY = YZ, the triangle is isosceles. In an isosceles triangle, the angles opposite to the equal sides are equal. The angle opposite to side XY is m∠Z, and the angle opposite to side YZ is m∠X. Therefore, m∠Z = m∠X = 78°.

According to the angle sum property of a triangle, the sum of all interior angles is 180°: m∠X + m∠Y + m∠Z = 180°

Substituting the known values: 78° + m∠Y + 78° = 180° 156° + m∠Y = 180° m∠Y = 180° - 156° m∠Y = 24°

Explanation

The surface area of a sphere of radius r is given by the formula S = 4πr², and its volume is given by V = 4/3 πr³.

The ratio of the surface areas of the two spheres is: S₁/S₂ = r₁²/r₂² = 25/36

Taking the square root on both sides gives the ratio of their radii: r₁/r₂ = 5/6

The ratio of their volumes is the cube of the ratio of their radii: V₁/V₂ = (r₁/r₂)³ = (5/6)³ = 125/216

Explanation

To evaluate the given expression, apply the BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction) by first converting the mixed fractions to improper fractions: 4 ½ = 9/2 5 ⅓ = 16/3 2 ½ = 5/2 7 ⅞ = 63/8 5 ⅑ = 46/9 9 ⁹/₂₀ = 189/20

Substitute these improper fractions into the expression: (4/7) × (9/2) ÷ [(16/3) of (5/2)] - [(63/8) ÷ ((46/9) of (189/20))] + ½

Evaluate the "of" operations first: First "of" term: (16/3) of (5/2) = (16/3) × (5/2) = 40/3

Second "of" term: (46/9) of (189/20) = (46/9) × (189/20) = (23/1) × (21/10) = 483/10

Now substitute these values back into the expression: (4/7) × (9/2) ÷ (40/3) - [(63/8) ÷ (483/10)] + ½

Next, solve the division operations: For the first term: (9/2) ÷ (40/3) = (9/2) × (3/40) = 27/80 Multiplying by (4/7): (4/7) × (27/80) = 27/140

For the term inside the bracket: (63/8) ÷ (483/10) = (63/8) × (10/483) = (63/483) × (10/8) = (3/23) × (5/4) = 15/92

Now substitute these simplified values back: 27/140 - 15/92 + ½

To add and subtract these fractions, find the least common multiple of the denominators 140, 92, and 2: 140 = 4 × 5 × 7 92 = 4 × 23 LCM(140, 92, 2) = 4 × 5 × 7 × 23 = 3220

Convert each fraction with the common denominator 3220: 27/140 = (27 × 23) / 3220 = 621/3220 15/92 = (15 × 35) / 3220 = 525/3220 ½ = 1610/3220

Perform the final arithmetic operations: (621 - 525 + 1610) / 3220 = 1706 / 3220

Simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor, which is 2: 1706 ÷ 2 = 853 3220 ÷ 2 = 1610

This yields 853/1610.

Explanation

Let the dimensions of the cuboid be 3x, 4x, and 5x.

The total surface area of a cuboid is given by the formula 2(lb + bh + lh). Substituting the dimensions: Total surface area = 2((3x × 4x) + (4x × 5x) + (5x × 3x)) Total surface area = 2(12x² + 20x² + 15x²) = 94x²

The total surface area of a cube with side s is 6s². For a side of 6 units: Surface area of the cube = 6 × 6² = 216

Equating the surface areas of the cuboid and the cube: 94x² = 216 x² = 216 / 94 ≈ 2.298 x = √2.298 ≈ 1.516

The volume of the cuboid is given by l × b × h: Volume = 3x × 4x × 5x = 60x³

Substituting the value of x: Volume ≈ 60 × (1.516)³ ≈ 60 × 3.483 = 208.98

Rounding to the nearest integer gives approximately 210.