SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

To calculate the overall profit or loss percentage, we use the effective multiplier method. Let the cost price of 1 g of goods be ₹1.

The merchant sells goods at a 12.5% loss on the cost price. A loss of 12.5% is equivalent to a fraction of 1/8, so the price multiplier is 1 - 1/8 = 7/8.

The merchant uses a 28 g weight instead of a 36 g weight. This means he charges the customer for 36 g but only provides 28 g. The weight multiplier, which represents the gain from using a false weight, is 36/28 = 9/7.

The net multiplier is the product of the price multiplier and the weight multiplier: Net Multiplier = ⅞ × ⁹⁄₇ = ⁹⁄₈

Since the net multiplier is greater than 1, the merchant makes a profit. The profit fraction is 9/8 - 1 = 1/8.

Profit percentage = ⅛ × 100 = 12.5%

Explanation

The Highest Common Factor (HCF) of any two numbers is always a factor of their Least Common Multiple (LCM). This mathematical relationship dictates that the LCM must be exactly divisible by the HCF, expressed as LCM = k × HCF, where k is a positive integer. Given that the HCF is 12, any possible LCM must be a multiple of 12. When 90 is divided by 12, the result is 90 / 12 = 7.5. Since 90 is not perfectly divisible by 12 and does not result in an integer quotient, it cannot serve as the LCM for any pair of numbers having an HCF of 12.

Explanation

The mean proportion (also known as the geometric mean) between two numbers a and b is calculated using the formula √(a × b). This value represents the middle term x in a continued proportion a:x :: x:b, which implies x² = ab.

Given the numbers a = 169 and b = 144, the mean proportion is: √(169 × 144)

Applying the property of square roots √(x × y) = √x × √y, we calculate the square roots of the individual perfect squares: √(169) = 13 √(144) = 12

Multiplying these results: 13 × 12 = 156

Explanation

When the dimensions of a rectangle, such as length and breadth, are increased by x% and y% respectively, the net percentage increase in the area is determined by the formula for successive percentage changes: Net % Change = x + y + ˣʸ⁄₁₀₀.

In this case, the length increases by x = 8% and the breadth increases by y = 5%. Substituting these values into the formula gives: Percentage Increase in Area = 8 + 5 + (8 × 5)/(100) Percentage Increase in Area = 13 + ⁴⁰⁄₁₀₀ Percentage Increase in Area = 13 + 0.4 = 13.4%

This calculation represents the combined effect of the individual increases on the product of the two dimensions, which constitutes the area of the rectangle.

Explanation

The expression 49¹⁵ - 1 is in the form xⁿ - aⁿ, where x = 49, a = 1, and n = 15. According to the algebraic identity for divisibility, for any positive integer n, the expression xⁿ - aⁿ is always divisible by (x - a).

By substituting the given values, we find that 49¹⁵ - 1 is divisible by 49 - 1, which equals 48. Since 48 is a multiple of 8 (48 = 8 × 6), any number that divides 48 must also divide the original expression. Therefore, 8 is a divisor of 49¹⁵ - 1.

Alternatively, using modular arithmetic, we can observe that 49 ≡ 1 8 because 49 = (6 × 8) + 1. Applying the properties of congruences, 49¹⁵ ≡ 1¹⁵ 8, which simplifies to 49¹⁵ ≡ 1 8. Subtracting 1 from both sides yields 49¹⁵ - 1 ≡ 0 8, confirming that the expression is exactly divisible by 8.

Explanation

In a race of distance D, if runner P beats runner Q by x meters, the ratio of the distances they cover in the same time is D : (D - x).

For Anil and Bakul, the ratio of distances covered is D_A / D_B = 1500 / (1500 - 150) = 1500 / 1350 = 10 / 9.

For Bakul and Charles, the ratio of distances covered is D_B / D_C = 1500 / (1500 - 75) = 1500 / 1425 = 20 / 19.

To find the distance ratio between Anil and Charles, multiply the two ratios: D_A / D_C = (D_A / D_B) × (D_B / D_C) = (10 / 9) × (20 / 19) = 200 / 171.

When Anil covers 1500 m, let the distance covered by Charles be x. 1500 / x = 200 / 171 x = (1500 × 171) / 200 x = 7.5 × 171 = 1282.5 m.

The distance by which Anil beats Charles is 1500 - 1282.5 = 217.5 m.

Explanation

The total interest earned over the 8-year period is the difference between the maturity amount and the principal: Interest = 3725 - 2500 = 1225. Using the simple interest formula SI = (P × R × T)/(100), the rate per annum R can be isolated as R = (SI × 100)/(P × T). Substituting the known values into the equation gives R = (1225 × 100)/(2500 × 8). Simplifying the calculation by canceling the zeros results in R = (1225)/(25 × 8) = ¹²²⁵⁄₂₀₀. Dividing 1225 by 200 yields a rate of 6.125 percent per annum.

Explanation

The average number of pages printed by printer Z is calculated by dividing the total number of pages printed over the three days by the number of days. According to the data provided in the table, the pages printed by printer Z are 210 on Monday, 160 on Tuesday, and 218 on Wednesday.

Sum of pages = 210 + 160 + 218 = 588

Number of days = 3

Average = (Sum of pages)/(Total days) = ⁵⁸⁸⁄₃ = 196

Explanation

The total runs scored in 27 innings is calculated using the formula Total = Average × Number of innings, which gives 47 × 27 = 1269 runs. When the highest score (H) and lowest score (L) are removed, the total runs for the remaining 25 innings is 42 × 25 = 1050 runs.

The sum of the highest and lowest scores is the difference between these two totals: H + L = 1269 - 1050 = 219

The problem states that the highest score exceeds the lowest score by 157 runs: H - L = 157

To find the highest score, we use the algebraic relation H = ((H + L) + (H - L))/(2). Substituting the values: H = (219 + 157)/(2) = ³⁷⁶⁄₂ = 188

Explanation

P's efficiency is 2 units/day and Q's efficiency is 3 units/day for a total work of 30 units. After working together for 2 days they complete 10 units, leaving 20 units for P to finish alone in 10 days, resulting in a total of 12 days for the whole project.

Explanation

In the right-angled triangle PQR, the right angle is located at vertex Q. For the angle ∠ RPQ = α, the side opposite to the angle is QR, which acts as the perpendicular (p), and the side adjacent to the angle is PQ, which acts as the base (b). The hypotenuse (h) is the side PR. The values provided are QR = 8, PQ = 15, and PR = 17.

Using the fundamental trigonometric ratios: sin α = (Perpendicular)/(Hypotenuse) = (QR)/(PR) = ⁸⁄₁₇ cos α = (Base)/(Hypotenuse) = (PQ)/(PR) = ¹⁵⁄₁₇

To find the value of sin α + cos α, we sum these ratios: sin α + cos α = ⁸⁄₁₇ + ¹⁵⁄₁₇ sin α + cos α = (8 + 15)/(17) = ²³⁄₁₇

Explanation

The expression is based on the algebraic identity for the sum of two cubes, which states that a³ + b³ = (a + b)(a² - ab + b²). By assigning a = 17 and b = 7, the identity becomes 17³ + 7³ = (17 + 7)(17² - 17 × 7 + 7²).

The term 17 + 7 simplifies to 24. When the expression is rearranged to match the format where the sum of cubes is related to the factors 24 and (17² + 7² - k), it is clear that (17² + 7² - k) represents the quadratic part of the expansion (a² + b² - ab). By direct comparison, the value of k must be equal to the product ab. Calculating this product gives k = 17 × 7 = 119.

Explanation

The circumference of a circle is calculated using the formula C = 2π r, which implies that the radius r = (C)/(2π). The difference between the radii of two circles is proportional to the difference between their circumferences.

Using the given circumferences C₁ = 198 cm and C₂ = 352 cm, the difference in radii Δ r is determined by the formula:

Δ r = (C₂ - C₁)/(2π)

Substituting the values into the equation:

Δ r = (352 - 198)/(2 × 22){7}

Δ r = (154 × 7)/(44)

By simplifying the fraction, where 154 divided by 22 equals 7:

Δ r = (7 × 7)/(2) = ⁴⁹⁄₂ = 24.5 cm

This approach allows for the direct calculation of the difference without needing to find the length of each radius separately.

Explanation

In a circle with center O, let OA and OB be the radii connecting the center to the endpoints of the chord AB. Since OA and OB are radii, OA = OB = 10 cm. This makes △ OAB an isosceles triangle where the angles opposite the equal sides are equal, so ∠ OAB = ∠ OBA. The sum of angles in a triangle is 180°, and given the central angle ∠ AOB = 60°, we have 60° + ∠ OAB + ∠ OBA = 180°. This calculation results in ∠ OAB = ∠ OBA = 60°. Since all three interior angles of △ OAB are 60°, it is an equilateral triangle. In an equilateral triangle, all sides are equal in length, therefore AB = OA = OB = 10 cm. Alternatively, using the chord length formula L = 2r sin(θ/2), where r = 10 and θ = 60°, the length is 2 × 10 × sin(30°) = 20 × 0.5 = 10 cm.

Explanation

The number of spherical lead shots is calculated by dividing the total volume of the rectangular solid by the volume of a single spherical shot. Let L, W, and H be the dimensions of the rectangular solid and r be the radius of each spherical shot.

The volume of the rectangular solid is given by Vₛₒₗᵢ_d = L × W × H = 88 × 63 × 42 cm³.

The diameter of each spherical shot is 8.4 cm, so the radius r = 4.2 cm. Using the formula for the volume of a sphere, Vₛₚₕₑᵣₑ = ⁴⁄₃ π r³.

The number of shots n is:

n = (88 × 63 × 42)/(4){3 × ²²⁄₇ × 4.2 × 4.2 × 4.2}

n = (88 × 63 × 42 × 21)/(88 × 4.2 × 4.2 × 4.2)

Simplifying the expression by dividing the terms in the numerator by the terms in the denominator:

(63)/(4.2) = 15 (42)/(4.2) = 10 (21)/(4.2) = 5

n = 15 × 10 × 5 = 750

Explanation

The number of spoons manufactured by factory P is 100 and the number of spoons manufactured by factory R is 150. The ratio of the number of spoons manufactured by P to those manufactured by R is calculated as (P)/(R) = ¹⁰⁰⁄₁₅₀. By dividing both the numerator and the denominator by their greatest common divisor, 50, the fraction simplifies to ⅔. Thus, the ratio is 2:3.

Explanation

The given expression can be simplified using basic trigonometric identities. First, use the identities tan²θ = sec²θ - 1 and cot²θ = csc²θ - 1. Substituting these into the first two terms of the expression:

tan²θ + cot²θ = (sec²θ - 1) + (csc²θ - 1) = sec²θ + csc²θ - 2

Next, observe the relationship between sec²θ + csc²θ and their product. By converting to sine and cosine:

sec²θ + csc²θ = (1)/(cos²θ) + (1)/(sin²θ) = (sin²θ + cos²θ)/(sin²θ cos²θ)

Since sin²θ + cos²θ = 1, this simplifies to:

(1)/(sin²θ cos²θ) = sec²θ csc²θ

Substituting this identity back into the previous equation:

tan²θ + cot²θ = sec²θ csc²θ - 2

Rearranging the terms to match the required expression:

tan²θ + cot²θ - sec²θ csc²θ = -2

Explanation

The expression utilizes the standard algebraic identity a³ - b³ = (a - b)(a² + ab + b²). By substituting a = 17 and b = 7 into this identity, we have (17)³ - (7)³ = (17 - 7)(17² + 17 × 7 + 7²) = 10(17² + 119 + 7²).

Substituting this back into the original equation: 4 × 10(17² + 119 + 7²){17² + 7² + p} = 40 40(17² + 119 + 7²){17² + 7² + p} = 40

Dividing both sides by 40, we get: 17² + 119 + 7²{17² + 7² + p} = 1 17² + 7² + 119 = 17² + 7² + p

Comparing the two sides of the equation, the value of p is directly identified as 119.

Explanation

Let w be the daily wage of a woman. Given that a man earns Rs. 50 more than a woman, the daily wage of a man is w + 50. The total daily wage for all 1000 employees (700 men and 300 women) is calculated by multiplying the total strength by the average wage: 1000 × 450 = 4,50,000.

The sum of individual wages must equal the total wage: 700(w + 50) + 300w = 4,50,000 700w + 35,000 + 300w = 4,50,000 1000w + 35,000 = 4,50,000 1000w = 4,15,000 w = (4,15,000)/(1000) = 415

Thus, the daily wage of a woman is Rs. 415.

Explanation

The given expression can be simplified by distributing the denominator to each term in the numerator, resulting in ½cos x - √3{2}sin x. According to standard trigonometric values, cos (π)/(3) = ½ and sin (π)/(3) = √3{2}. By substituting these values into the expression, it becomes cos (π)/(3) cos x - sin (π)/(3) sin x. This structure matches the compound angle identity cos(A + B) = cos A cos B - sin A sin B. Taking A = (π)/(3) and B = x, the entire expression simplifies directly to cos((π)/(3) + x).

Explanation

Let the two numbers be 3x and 8x, where x represents their highest common factor (HCF). The least common multiple (LCM) of two numbers expressed in the ratio a:b is calculated using the formula LCM = HCF × a × b, provided that a and b are co-prime. Since 3 and 8 have no common factors other than 1, they are co-prime.

Substituting the given values into the formula: 120 = x × 3 × 8 120 = 24x x = ¹²⁰⁄₂₄ = 5

The sum of the numbers is the sum of their ratio components multiplied by their HCF: Sum = 3x + 8x = 11x Sum = 11 × 5 = 55

Explanation

The product 95 × 105 is most easily calculated using the algebraic identity for the difference of squares, which states that (a - b)(a + b) = a² - b². In this expression, both numbers are equidistant from 100, which serves as a convenient base for calculation. We can rewrite the multiplication as (100 - 5) × (100 + 5). Identifying a = 100 and b = 5, we substitute these values into the formula to obtain 100² - 5². Squaring the terms results in 10000 - 25. Subtracting 25 from 10,000 gives the value 9975. This method utilizes the symmetry of the numbers around 100 to simplify the arithmetic process.

Explanation

In any triangle ABC, the sum of the internal angles is always 180°, expressed as A + B + C = 180°. From the given equations, we can express angles B and C in terms of angle A. The first relation A - B = 45° gives B = A - 45°. The second relation B - C = 15° gives C = B - 15°. Substituting the expression for B into this second equation, we get C = (A - 45°) - 15° = A - 60°.

Now, substituting the expressions for B and C into the triangle angle sum formula:

A + (A - 45°) + (A - 60°) = 180°

3A - 105° = 180°

3A = 180° + 105°

3A = 285°

A = (285°)/(3) = 95°

Explanation

The volume V of a right circular cylinder is determined by the formula V = π r² h. Given that the height h is 2.1 cm and the radius r is thrice the height, the radius is calculated as r = 3 × 2.1 = 6.3 cm. Substituting these dimensions into the volume formula yields V = ²²⁄₇ × (6.3)² × 2.1. By performing the division of 2.1 by 7, the expression simplifies to V = 22 × 39.69 × 0.3. Calculating the final product results in V = 6.6 × 39.69 = 261.954 cm³.

Explanation

The mean wage x for grouped data is calculated using the formula x = (Σ fᵢ xᵢ)/(Σ fᵢ), where xᵢ represents the midpoint of each interval and fᵢ is the corresponding frequency. The midpoints for the given intervals are 85, 95, 105, 115, and 125. Calculating the sum of the products of frequencies and midpoints gives Σ fᵢ xᵢ = (9 × 85) + (3 × 95) + (31 × 105) + (5 × 115) + (2 × 125) = 765 + 285 + 3255 + 575 + 250 = 5130. With a total frequency of 50 employees, the mean wage is 5130 / 50 = 102.6. This result of 102.6 falls within the daily wage interval of 100-110.

Explanation

Let the selling price SP be 100 units. The problem states that the cost price CP is 72% of the selling price, which gives CP = 72 units. Profit is defined as the surplus of the selling price over the cost price.

Profit = SP - CP = 100 - 72 = 28 units

The profit percentage is calculated relative to the cost price using the standard relation: Profit % = (Profit)/(CP) × 100

Substituting the known values into the formula: Profit % = ²⁸⁄₇₂ × 100

By simplifying the fraction ²⁸⁄₇₂ by dividing both the numerator and the denominator by 4, the expression becomes: Profit % = ⁷⁄₁₈ × 100 = ⁷⁰⁰⁄₁₈

Performing the division: 700 ÷ 18 = 38.888...

Rounding to two decimal places results in 38.89%.

Explanation

The expression is in the form a³ + b³ + c³, where a = 12, b = -8, and c = -4. According to the algebraic identity, if a + b + c = 0, then a³ + b³ + c³ = 3abc. Checking the sum of the terms, we find 12 + (-8) + (-4) = 12 - 12 = 0. Since the sum is zero, the value of the expression is calculated as 3 × 12 × (-8) × (-4). Multiplying these values together gives 36 × 32, which equals 1152.

Explanation

When interest is compounded half-yearly, the annual interest rate is divided by two and the number of time periods is doubled. Given an annual rate of 20% for a period of 2 years, the periodic rate r is 20% / 2 = 10% per half-year, and the total number of compounding periods n is 2 × 2 = 4.

The formula for compound interest CI is CI = P[(1 + r/100)ⁿ - 1]. Substituting the principal P = 15,750, the periodic rate r = 10, and the number of periods n = 4:

CI = 15,750 × [(1 + 10/100)⁴ - 1] CI = 15,750 × [(1.1)⁴ - 1]

Calculating the power of the growth factor, (1.1)⁴ equals 1.4641. The equation becomes:

CI = 15,750 × [1.4641 - 1] CI = 15,750 × 0.4641 CI = 7,309.575

The resulting value represents the total compound interest earned over the two years with semi-annual compounding.

Explanation

The total work done is constant, so we use the relation (5m + 8w) × 12 = (3m + 7w) × 15. Simplifying this expression by dividing both sides by 3 gives 4(5m + 8w) = 5(3m + 7w), which expands to 20m + 32w = 15m + 35w. Rearranging the terms yields 5m = 3w, meaning the work capacity of 5 men is equivalent to that of 3 women. Substituting 5m with 3w in the first work equation, the total work is (3w + 8w) × 12, which simplifies to 11w × 12. To find the time taken by 11 women, we divide the total work by their combined efficiency: (11w × 12) / 11w = 12 days.

Explanation

Let x = 2a and y = 3b. The given equations can be rewritten as x³ + y³ = 16 and x + y = 4.

Using the algebraic identity x³ + y³ = (x + y)³ - 3xy(x + y), substitute the known values: 16 = 4³ - 3xy(4) 16 = 64 - 12xy 12xy = 48 xy = 4

The relationship between x and y can be further determined using (x - y)² = (x + y)² - 4xy: (x - y)² = 4² - 4(4) = 16 - 16 = 0 This implies x = y. Since x + y = 4, we have x = 2 and y = 2.

The expression to find is 16a⁴ + 81b⁴, which is equivalent to (2a)⁴ + (3b)⁴ or x⁴ + y⁴. Substituting the values of x and y: x⁴ + y⁴ = 2⁴ + 2⁴ = 16 + 16 = 32

Explanation

The problem provides an equation involving trigonometric functions to solve for the variable k. Substituting the standard values for the trigonometric ratios: cot 30° = √3, sin 60° = √3{2}, tan 45° = 1, and csc 60° = (2)/(√(3)).

The equation can be expressed as: (k(1 - cot² 30°))/(1 + cot² 30°) = sin² 60° + 4tan² 45° - csc² 60°

Substituting the numerical values into the left-hand side (LHS): LHS = k(1 - (√3)²){1 + (√3)²} = (k(1 - 3))/(1 + 3) = ⁻²ᵏ⁄₄ = -ᵏ⁄₂

Substituting the numerical values into the right-hand side (RHS): RHS = (√3{2})² + 4(1)² - ((2)/(√(3)))² = ¾ + 4 - ⁴⁄₃

To solve the RHS, find a common denominator of 12: RHS = ⁹⁄₁₂ + ⁴⁸⁄₁₂ - ¹⁶⁄₁₂ = (9 + 48 - 16)/(12) = ⁴¹⁄₁₂

Equating LHS and RHS: -ᵏ⁄₂ = ⁴¹⁄₁₂ k = -(41 × 2)/(12) = -⁴¹⁄₆ k = -6.8333...

Rounding to two decimal places, the value of k is -6.83.

Explanation

The relationship between a tangent segment and a secant segment drawn from the same external point to a circle is defined by the Tangent-Secant Theorem. This theorem states that the square of the length of the tangent segment PT is equal to the product of the lengths of the external part of the secant segment PA and the entire secant segment PB.

The formula is given by: PT² = PA × PB

Given that PA = 3 cm and AB = 9 cm, the length of the entire secant segment PB is the sum of the external part and the chord part: PB = PA + AB = 3 + 9 = 12 cm

By substituting these values into the theorem formula, we calculate the square of the tangent length: PT² = 3 × 12 PT² = 36

Taking the square root of both sides provides the length of the tangent segment: PT = √(36) = 6 cm

This direct geometric relationship allows for the calculation of the tangent length using only the segments of the intersecting secant.

Explanation

When a solid copper sphere is hammered and moulded into a wire, the volume of the material remains constant. The wire is cylindrical in shape. Let rₛ be the radius of the sphere, r_w be the radius of the wire, and h be the length of the wire.

The volume of the sphere is given by the formula: Vₛ = ⁴⁄₃ π rₛ³

The volume of the cylindrical wire is given by the formula: V_w = π r_w² h

Equating the two volumes: ⁴⁄₃ π (9)³ = π (2)² h

Dividing both sides by π and substituting the values: ⁴⁄₃ × 729 = 4h 4 × 243 = 4h h = 243 cm

The length of the wire is 243 cm.

Explanation

To simplify the expression √((1 + cos A)/(1 - cos A)), we rationalize the expression by multiplying both the numerator and the denominator inside the square root by the conjugate of the denominator, which is 1 + cos A:

√((1 + cos A)/(1 - cos A) × (1 + cos A)/(1 + cos A)) = √(((1 + cos A)²)/(1 - cos² A))

Using the fundamental trigonometric identity sin² A + cos² A = 1, we substitute 1 - cos² A with sin² A:

√(((1 + cos A)²)/(sin² A)) = (1 + cos A)/(sin A)

By splitting the fraction into two parts, we apply the reciprocal and quotient identities:

(1)/(sin A) + (cos A)/(sin A) = cosec A + cot A

This mathematical derivation shows that the initial square root expression simplifies directly to the sum of the cosecant and cotangent of the angle.

Explanation

To find the required ratio, first calculate the total number of players for Game A and Game B across all six schools.

Total players in Game A = 31 + 24 + 18 + 29 + 12 + 40 = 154 Total players in Game B = 58 + 36 + 26 + 37 + 33 + 44 = 234

The ratio of total players in Game A to total players in Game B is 154 : 234. Simplifying this ratio by dividing both terms by 2, we get:

154 / 2 : 234 / 2 = 77 : 117

Explanation

The algebraic identity for the square of a binomial is (a + b)² = a² + 2ab + b². By substituting a = mx and b = n into this formula, the expansion is determined as follows:

(mx + n)² = (mx)² + 2(mx)(n) + n²

(mx + n)² = m²x² + 2mnx + n²

Since the expression m²x² + 2mnx + n² is exactly the square of (mx + n), it follows that (mx + n) is a factor of the expression. This relationship confirms that the quadratic polynomial is perfectly divisible by the linear binomial (mx + n).

Explanation

The distance D of a journey when two different speeds v₁ and v₂ result in a time difference Δ t is given by the formula:

D = (v₁ × v₂)/(|v₁ - v₂|) × Δ t

In this problem, the speeds are v₁ = 45 km/h and v₂ = 30 km/h. The time difference is given as 15 minutes, which must be converted to hours to match the speed units:

Δ t = ¹⁵⁄₆₀ = ¼ h

Substituting the values into the distance formula:

D = (45 × 30)/(45 - 30) × ¼

D = ¹³⁵⁰⁄₁₅ × ¼

D = 90 × ¼

D = 22.5 km

Explanation

Let the total salary of C be represented as S. Based on the standard proportions for this problem that result in the given ratio, the tax is ¼ of the salary and the savings are ⅓ of the salary.

To simplify the calculation, assume the total salary S is the least common multiple of the denominators 4 and 3, which is 12 units.

Tax = ¼ × 12 = 3 units Savings = ⅓ × 12 = 4 units

The total expenditure is calculated by subtracting the tax and savings from the total salary: Expenditure = Total Salary - (Tax + Savings) Expenditure = 12 - (3 + 4) = 5 units

The ratio of expenditures to savings is: Ratio = Expenditure : Savings = 5 : 4

Explanation

The ratio of the speeds of the bikes can be determined from the distances covered in the same amount of time. In the 1200 m race, when bike A finishes, bike B has covered 1200 - 100 = 1100 m. Thus, the ratio of their speeds is V_A : V_B = 1200 : 1100 = 12 : 11. In the 600 m race, when bike B finishes, bike C has covered 600 - 100 = 500 m, so V_B : V_C = 600 : 500 = 6 : 5.

To find the ratio between bike A and bike C, multiply the two ratios: V_A/V_C = (V_A/V_B) × (V_B/V_C) = (12/11) × (6/5) = 72/55

In a 720 m race, bike A beats bike C by 30 seconds. The time taken by a bike is given by the formula Time = Distance / Speed. Setting up the equation for the time difference: 720/V_C - 720/V_A = 30

Substituting V_A = ⁷²⁄₅₅ V_C into the equation: 720/V_C - 720/(⁷²⁄₅₅ V_C) = 30 720/V_C - (720 × 55)/(72 × V_C) = 30 720/V_C - 550/V_C = 30 170/V_C = 30 V_C = 17/3 m/sec

Explanation

To find the invalid votes for each village, we calculate the number of votes polled and then apply the percentage of invalid votes, which is 100% minus the percentage of valid votes.

For village R, the total number of votes is 100 × 100 = 10,000. The number of votes polled is 65% of 10,000, which is 6,500. Since 65% of the polled votes are valid, the invalid votes are 35% of the polled votes. Invalid votes of R = 6,500 × ³⁵⁄₁₀₀ = 65 × 35 = 2,275

For village U, the total number of votes is 40 × 100 = 4,000. The number of votes polled is 90% of 4,000, which is 3,600. Since 60% of the polled votes are valid, the invalid votes are 40% of the polled votes. Invalid votes of U = 3,600 × ⁴⁰⁄₁₀₀ = 36 × 40 = 1,440

The ratio of invalid votes of village R to village U is 2,275 : 1,440. Dividing both sides by the common factor 5, we get 455 : 288.

Explanation

The profit percentage for a dishonest trader selling at cost price using false weights is calculated using the formula: Profit % = (True Weight - False Weight)/(False Weight) × 100.

Given a profit of 12.5%, we can express this as a fraction: 12.5% = (12.5)/(100) = ⅛. Let the false weight used by the trader be w grams. The true weight is 1 kg, which is 1000 g.

Substituting these values into the relationship: (1000 - w)/(w) = ⅛

Cross-multiplying gives: 8 × (1000 - w) = w 8000 - 8w = w 9w = 8000 w = ⁸⁰⁰⁰⁄₉ w = 888.88... g

Rounding to one decimal place, the trader uses a weight of 888.8 g to represent 1 kg.

Explanation

A number divisible by both 4 and 7 must be a multiple of their Least Common Multiple (LCM). Since 4 and 7 are coprime, their LCM(4, 7) = 4 × 7 = 28. To find the largest three-digit number divisible by 28, we divide the largest possible three-digit number, 999, by 28. The division 999 ÷ 28 results in an integer quotient of 35 and a remainder of 19. Subtracting this remainder from 999 provides the largest three-digit multiple of 28: 999 - 19 = 980. This can also be calculated directly by multiplying the quotient by the LCM: 35 × 28 = 980.

Explanation

Total work is determined by taking the Least Common Multiple (LCM) of the time taken by A and B. The LCM of 10 and 16 is 80 units. Based on this, the efficiency of A is 80 / 10 = 8 units per day, and the efficiency of B is 80 / 16 = 5 units per day.

Since they work on alternate days starting with A, in a cycle of 2 days, the total work completed is 8 + 5 = 13 units. To reach the total work of 80 units, 6 full cycles are required, which accounts for 6 × 13 = 78 units of work completed over 12 days.

After 12 days, the remaining work is 80 - 78 = 2 units. On the 13th day, it is A's turn to work. Since A can complete 8 units in a full day, the time taken to finish the remaining 2 units is 2 / 8 = 1/4 of a day. The total time taken to complete the entire piece of work is 12 + 1/4 = 12 ¼ days.

Explanation

The length of the diagonal d of a cube with side length a is defined by the geometric relation d = a√3. This formula is derived from applying the Pythagorean theorem twice in three dimensions. Given that the diagonal of the cube is 7√3 cm, we substitute this value into the formula to determine the side length: a√3 = 7√3 a = 7 cm

The total surface area of a cube is the sum of the areas of its six square faces. Since the area of each face is a², the formula for the total surface area SA is 6a². Using the side length a = 7 cm derived above, the calculation is as follows: SA = 6 × (7)² SA = 6 × 49 SA = 294 cm²

Explanation

Since △ BPQ ≅ △ ASR, the corresponding angles of the two triangles are equal. Therefore, ∠ B = ∠ A, ∠ P = ∠ S, and ∠ Q = ∠ R.

According to the given condition, ∠ A = ⅓ ∠ R = ∠ S. Let ∠ A = x. This implies ∠ S = x and ∠ R = 3x. In △ ASR, the sum of all interior angles is 180°. Substituting the values, we get:

∠ A + ∠ S + ∠ R = 180° x + x + 3x = 180° 5x = 180° x = 36°

Since ∠ Q corresponds to ∠ R in the congruence △ BPQ ≅ △ ASR, we have ∠ Q = ∠ R. Substituting the value of x into the expression for ∠ R:

∠ Q = 3x = 3 × 36° = 108°

Explanation

The profit percentage resulting from a markup and a subsequent discount can be calculated using the successive percentage change formula. Let m be the markup percentage over the cost price and d be the discount percentage on the marked price. The net profit percentage is given by the formula m - d - (m · d)/(100).

Given that m = 20% and d = 10%, we substitute these values into the formula: Profit % = 20 - 10 - (20 · 10)/(100) Profit % = 10 - 2 = 8%

We can also verify this by calculating the specific values. Since the marked price of 2,400 is 20% above the cost price (CP), we have 1.2 · CP = 2400, which yields CP = 2000. The selling price (SP) is the marked price minus a 10% discount: SP = 2400 - 0.10 · 2400 = 2160. The profit is SP - CP = 2160 - 2000 = 160. Expressed as a percentage of the cost price, the profit is ¹⁶⁰⁄₂₀₀₀ · 100 = 8%.

Explanation

The total simple interest over a period is calculated as SI = (P × R × T)/(100). The total amount A is the sum of the principal P and the simple interest, which can be expressed as A = P(1 + (RT)/(100)). Using the given values R = 30% per annum and T = 7 years, the total interest accrued is 30 × 7 = 210% of the principal.

The amount A represents 100% (Principal) + 210% (Interest) = 310% of the principal. Given 310% = 15500, we find the value of 1% by dividing: 1% = ¹⁵⁵⁰⁰⁄₃₁₀ = 50

The total interest for the 7 years is 210%: Interest = 210 × 50 = 10500

Alternatively, the principal can be calculated first: P = (15500)/(3.1) = 5000 The interest is the difference between the final amount and the principal: Interest = 15500 - 5000 = 10500

Explanation

By comparing the scores in each subject, we identify the students who scored the highest marks: Hindi - Kanchana (89), English - Diksha (92), Maths - Diksha (96), Physics - Amit (88), Chemistry - Kunal (94), and Biology - Diksha (88). Diksha is the only student who scored the highest in more than one subject (English, Maths, and Biology). Therefore, only one student meets the criteria.

Explanation

To find the value of the expression (x²)/(yz) + (y²)/(xz) + (z²)/(xy), we first simplify the expression by finding a common denominator, which is xyz. This results in:

(x³ + y³ + z³)/(xyz)

The algebraic identity for the sum of cubes states that x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx). Given the condition x + y + z = 0, the entire right-hand side of this identity becomes zero:

x³ + y³ + z³ - 3xyz = 0 · (x² + y² + z² - xy - yz - zx) = 0

This implies that x³ + y³ + z³ = 3xyz. Substituting this result into our simplified expression:

(3xyz)/(xyz) = 3

Another efficient way to solve this is through value substitution. If we choose values for x, y, and z such that their sum is zero, for example, x = 1, y = 1, and z = -2, the expression becomes:

(1²)/(1 · (-2)) + (1²)/(1 · (-2)) + ((-2)²)/(1 · 1) = -½ - ½ + ⁴⁄₁ = -1 + 4 = 3

Explanation

Using the algebraic identity for the square of a trinomial: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substitute the given values a² + b² + c² = 9 and ab + bc + ca = 8 into the formula: (a + b + c)² = 9 + 2(8) (a + b + c)² = 9 + 16 = 25

To find the value of (a + b + c), take the square root of both sides: a + b + c = ±√(25) = ± 5

Now, calculate the expression (a + b + c) - 3 for both possible cases: Case 1: If a + b + c = 5, then 5 - 3 = 2. Case 2: If a + b + c = -5, then -5 - 3 = -8.

The value 2 is provided in the options.