SSC CGL Tier 1 2014 Quantitative Aptitude Questions with Solutions

Explanation

The actual length of the cloth is determined by multiplying the true length of the measuring instrument by the number of times it was applied. A standard metre stick is 100 cm long, but because this specific stick is one cm short, its true length is 100 cm - 1 cm = 99 cm. The recorded measurement of 100 metres indicates that the stick was used exactly 100 times, as 100 metres / 1 metre per stick = 100 applications. To find the total actual length in centimetres, we use the following relation:

Actual Length = Number of applications × True length per application

Substituting the values, we get:

Actual Length = 100 × 99 cm = 9,900 cm

This calculation corrects the measurement error by accounting for the 1 cm deficiency in the tool for every metre of cloth recorded.

Explanation

A diagonal of a parallelogram divides it into two triangles of equal area. Given the sides of the parallelogram are a = 60 m and b = 40 m, and one diagonal is d = 80 m, these three lengths form the sides of one such triangle. To find the area of this triangle, we use Heron's formula.

First, calculate the semi-perimeter s of the triangle: s = (a + b + d)/(2) = (60 + 40 + 80)/(2) = ¹⁸⁰⁄₂ = 90 m

Now, calculate the area of the triangle Δ using Heron's formula: Δ = √(s(s - a)(s - b)(s - d)) Δ = √(90(90 - 60)(90 - 40)(90 - 80)) Δ = √(90 × 30 × 50 × 10) Δ = √(1350000) Δ = √(90000 × 15) Δ = 300√(15) sq.m

Since the parallelogram consists of two such congruent triangles, the total area is: Area = 2 × Δ = 2 × 300√(15) = 600√(15) sq.m

Explanation

The relationship between the marked price (MP) and the cost price (CP) when both a profit percentage and a discount percentage are given is defined by the formula MP = CP × (100 + Profit%)/(100 - Discount%).

Substituting the given values where CP = 3,200, Profit = 25%, and Discount = 20%:

MP = 3200 × (100 + 25)/(100 - 20)

MP = 3200 × ¹²⁵⁄₈₀

Dividing 3,200 by 80 gives 40. Therefore:

MP = 40 × 125 = 5,000

Alternatively, the selling price required for a 25% profit on 3,200 is 3200 + 800 = 4,000. Since a 20% discount is offered on the marked price, the selling price represents 80% of the marked price. Solving 0.8 × MP = 4000 yields MP = 5,000.

Explanation

The relationship between the Marked Price (MP) and Cost Price (CP) is given by the formula: (MP)/(CP) = (100 + Profit%)/(100 - Discount%)

Substituting the given values CP = 210, Profit% = 20, and Discount% = 12.5 into the equation: (MP)/(210) = (100 + 20)/(100 - 12.5) (MP)/(210) = (120)/(87.5)

To find the Marked Price: MP = 210 × (120)/(87.5) MP = 210 × ¹²⁰⁰⁄₈₇₅

Simplifying the fraction by dividing both 1200 and 875 by 25: MP = 210 × ⁴⁸⁄₃₅ MP = 6 × 48 = 288

Alternatively, the Selling Price (SP) can be found by adding the 20% profit to the cost price: SP = 210 + (20% of 210) = 210 + 42 = 252. Since a discount of 12.5% (which is equal to the fraction 1/8) is allowed on the marked price, the selling price is 1 - 1/8 = 7/8 of the marked price: ⅞ × MP = 252 MP = (252 × 8)/(7) MP = 36 × 8 = 288

Explanation

The relationship between Marked Price (MP) and Cost Price (CP) when a discount percentage (d) and a profit percentage (p) are given is determined by the formula:

(MP)/(CP) = (100 + p)/(100 - d)

Given that the businessman wants a profit of 17% (p = 17) and allows a discount of 10% (d = 10), substitute these values into the ratio:

(MP)/(CP) = (100 + 17)/(100 - 10) = ¹¹⁷⁄₉₀

Simplifying the fraction by dividing both terms by 9:

(MP)/(CP) = ¹³⁄₁₀

The markup percentage represents how much the MP is above the CP. It is calculated as:

Markup% = (MP - CP)/(CP) × 100 = ((MP)/(CP) - 1) × 100

Markup% = (¹³⁄₁₀ - 1) × 100 = ³⁄₁₀ × 100 = 30%

Alternatively, the direct markup formula can be used:

Markup% = (Profit% + Discount%)/(100 - Discount%) × 100 = (17 + 10)/(100 - 10) × 100 = ²⁷⁄₉₀ × 100 = 30%

Explanation

The total number of bricks in a pile without gaps is the ratio of the total volume of the pile to the volume of an individual brick.

First, convert the total volume of the pile into cubic centimeters. Since 1 m³ = 1,000,000 cm³, the total volume is: Vₚᵢₗₑ = 20 × 1,000,000 = 20,000,000 cm³

Next, calculate the volume of a single brick using the dimensions provided: V_bᵣᵢ_cₖ = 25 cm × 12.5 cm × 8 cm Since 12.5 × 8 = 100, the calculation becomes: V_bᵣᵢ_cₖ = 25 × 100 = 2,500 cm³

The number of bricks n is given by the formula: n = Vₚᵢₗₑ{V_bᵣᵢ_cₖ} n = (20,000,000)/(2,500) n = (200,000)/(25) n = 8,000

Explanation

Let the total number of workers in the institution be N. The number of workers excluding the officers is N - 12. The sum of the salaries of all workers must equal the sum of the salaries of the two sub-groups.

60 × N = (12 × 400) + (N - 12) × 56 60N = 4800 + 56N - 672 60N - 56N = 4128 4N = 4128 N = 1032

Using the method of alligation, the ratio of the number of officers to the rest of the workers is determined by the differences between their individual averages and the overall average: Ratio (Officers : Rest) = (60 - 56) : (400 - 60) Ratio = 4 : 340 = 1 : 85

Since the number of officers corresponding to 1 part is 12, the number of rest workers corresponding to 85 parts is 85 × 12 = 1020. The total number of workers in the institution is the sum of these two groups: 12 + 1020 = 1032.

Explanation

The production cost (CP) is calculated by reversing the 30% markup applied to the labeled price (MP). CP = (MP)/(1 + 30){100} = (286)/(1.3) = 220

The effective profit percentage can be calculated using the successive percentage formula x + y + ˣʸ⁄₁₀₀, where x = 30 (markup) and y = -10 (discount). Effective Profit % = 30 - 10 - (30 × 10)/(100) = 17%

The total gain is 17% of the production cost. Gain = ¹⁷⁄₁₀₀ × 220 = 37.40

Explanation

In an election with two candidates, if the winner receives 84% of the total votes, the remaining votes go to the opponent. The loser’s share is 100% - 84% = 16%. The majority or margin of victory is the difference between the winner's percentage and the loser's percentage.

Majority percentage = 84% - 16% = 68%

We are given that this 68% majority corresponds to 476 votes. Let the total number of votes polled be x.

68% of x = 476

⁶⁸⁄₁₀₀ × x = 476

x = (476 × 100)/(68)

x = 7 × 100

x = 700

Explanation

The length of shadows cast by objects at the same time and place are proportional to their respective heights because the angle of elevation of the sun is the same. This creates similar right-angled triangles where the ratio of height to shadow length remains constant:

(Height of Man)/(Shadow of Man) = (Height of Pole)/(Shadow of Pole)

Substituting the given values into the formula:

¹⁶⁹⁄₁₃₀ = (Height of Pole)/(420)

Simplifying the fraction for the man:

¹⁶⁹⁄₁₃₀ = ¹³⁄₁₀ = 1.3

To find the height of the pole, multiply the length of its shadow by this ratio:

Height of Pole = 1.3 × 420 = 546 cm

Explanation

The intended ratio for dividing the amount is 1/4 : 1/5 : 1/6. To simplify this, multiply each term by the least common multiple of the denominators 4, 5, and 6, which is 60. The ratio becomes 15 : 12 : 10. The total parts in this ratio are 15 + 12 + 10 = 37. C's intended share is (10 / 37) × 555. Since 555 / 37 = 15, C's intended share is 10 × 15 = 150.

The mistaken ratio used is 4 : 5 : 6. The total parts in this ratio are 4 + 5 + 6 = 15. C's actual share under the mistaken ratio is (6 / 15) × 555. Since 555 / 15 = 37, C's actual share is 6 × 37 = 222.

The excess amount received by C is the difference between the actual share and the intended share, calculated as 222 - 150 = 72.

Explanation

The sum of the original 50 numbers is calculated by multiplying the count by the average: 50 × 38 = 1900. When the two numbers 45 and 55 are discarded, their total value of 45 + 55 = 100 is removed from the sum. The new sum for the remaining 50 - 2 = 48 numbers is 1900 - 100 = 1800. To find the new average, divide the remaining sum by the remaining count:

New Average = ¹⁸⁰⁰⁄₄₈

Simplifying the fraction by dividing both the numerator and denominator by 12 gives ¹⁵⁰⁄₄, which equals 37.5. Alternatively, the discarded numbers have a total deviation of (45 - 38) + (55 - 38) = 7 + 17 = 24 above the original average. Removing this surplus reduces the average of the remaining 48 numbers by ²⁴⁄₄₈ = 0.5, resulting in 38 - 0.5 = 37.5.

Explanation

Simple interest is calculated using the formula SI = (P × R × T)/(100), where P is the principal, R is the annual interest rate, and T is the time in years. The problem states that the interest generated by 8,000 at 3% for an unknown time T is equal to the interest generated by 6,000 at 4% for 5 years.

Equating the two interests: (8000 × 3 × T)/(100) = (6000 × 4 × 5)/(100)

By eliminating the common denominator of 100 and simplifying the equation: 8000 × 3 × T = 6000 × 4 × 5 24000 × T = 120000

To find T, divide the total interest value by the product of the first principal and rate: T = ¹²⁰⁰⁰⁰⁄₂₄₀₀₀ T = 5

Explanation

To find the reciprocal of the expression x + ¹⁄ₓ, first simplify the given expression by combining the terms over a common denominator:

x + ¹⁄ₓ = (x² + 1)/(x)

The reciprocal of a mathematical expression is defined as its multiplicative inverse, or 1 divided by the expression. For any fraction (a)/(b), the reciprocal is ᵇ⁄ₐ. Applying this definition to the simplified expression (x² + 1)/(x), we get:

(1)/(x² + 1){x} = (x)/(x² + 1)

Explanation

The given expression follows the algebraic identity (a³ - b³)/(a² + ab + b²). Let a = 2.75 and b = 2.25. The numerator is in the form a³ - b³, which can be expanded using the formula a³ - b³ = (a - b)(a² + ab + b²).

Substituting this into the expression, we get:

((a - b)(a² + ab + b²))/(a² + ab + b²)

The term (a² + ab + b²) in the numerator and denominator cancels out, leaving only (a - b). Substituting the values of a and b:

2.75 - 2.25 = 0.50

Converting the decimal 0.50 to a fraction, we get ½.

Explanation

Simplify the expression by evaluating the nested fractions from the innermost part outward. The innermost denominator is 1 + (a)/(1 - a), which simplifies to (1 - a + a)/(1 - a) = 1/(1 - a).

Next, substitute this into the middle term: 1/(1/(1 - a)) = 1 - a.

Substituting this result into the main denominator gives 1 - (1 - a), which simplifies to 1 - 1 + a = a.

The full expression now becomes 1 - a/a. Since any non-zero value divided by itself is 1, the expression reduces to 1 - 1 = 0.

Explanation

To solve the expression, express all terms with a common base of 3. Given that 243 = 3⁵ and 9 = 3², substitute these into the expression:

((3⁵)^ᶠʳᵃᶜ^ⁿ^⁾^^^/^⁽^⁵^⁾ ^× ³^²^ⁿ^⁺^¹{(3²)ⁿ × 3ⁿ⁻¹}

Apply the power of a power rule (aᵐ)ⁿ = aᵐⁿ to simplify the first term in the numerator and the first term in the denominator:

3ⁿ × 3²ⁿ⁺¹{3²ⁿ × 3ⁿ⁻¹}

According to the laws of indices, when multiplying terms with the same base, add their exponents (aᵐ × aⁿ = aᵐ⁺ⁿ).

Numerator: 3ⁿ⁺²ⁿ⁺¹ = 3³ⁿ⁺¹ Denominator: 3²ⁿ⁺ⁿ⁻¹ = 3³ⁿ⁻¹

The expression is now:

3³ⁿ⁺¹{3³ⁿ⁻¹}

When dividing terms with the same base, subtract the exponent of the denominator from the exponent of the numerator ((aᵐ)/(aⁿ) = aᵐ⁻ⁿ):

3⁽³ⁿ⁺¹⁾⁻⁽³ⁿ⁻¹⁾ = 3³ⁿ⁺¹⁻³ⁿ⁺¹ = 3²

Calculating the final value:

3² = 9

Explanation

To convert speed from kilometers per hour (km/hr) to meters per second (m/s), the value is multiplied by the conversion factor 5/18. This factor is derived from the relationship that 1 kilometer = 1000 meters and 1 hour = 3600 seconds, such that 1 km/hr = 1000/3600 m/s = 5/18 m/s.

Using this formula for the given speed: Speed in m/s = 45 × ⁵⁄₁₈

By simplifying the fraction: Speed in m/s = (45 × 5)/(18) = (5 × 5)/(2) = ²⁵⁄₂ = 12.5

Explanation

The time taken T for a journey is determined by the formula T = (Distance)/(Speed). Based on the parameters of the problem, assuming the distance between place X and place Y is 110 km, the initial time taken at a speed of 55 km/hr is: T₁ = (110 km)/(55 km/hr) = 2 hours = 120 minutes

If the speed of the train is increased by 11 km/hr, the new speed becomes: V₂ = 55 km/hr + 11 km/hr = 66 km/hr

The time taken at this increased speed is: T₂ = (110 km)/(66 km/hr) = ⁵⁄₃ hours = 100 minutes

The reduction in the time of the journey is the difference between the initial time and the new time: Reduction = 120 minutes - 100 minutes = 20 minutes

Explanation

The Arithmetic Mean - Harmonic Mean (AM-HM) inequality for positive real numbers a, b, c states that (a + b + c)/(3) ≥ (3)/(1){a + (1)/(b) + (1)/(c)}. This relationship dictates that the average of the numbers is always greater than or equal to their harmonic average.

Given the constraint a + b + c = 1, we can substitute this value directly into the AM-HM inequality:

⅓ ≥ (3)/(1){a + (1)/(b) + (1)/(c)}

To find the minimum value of the expression, we rearrange the inequality by multiplying both sides by 3 and by (¹⁄ₐ + (1)/(b) + (1)/(c)):

¹⁄ₐ + (1)/(b) + (1)/(c) ≥ 3 × 3

¹⁄ₐ + (1)/(b) + (1)/(c) ≥ 9

The equality holds, and thus the least value is achieved, when the variables are equal. For a = b = c = ⅓, the sum is (1)/(1/3) + (1)/(1/3) + (1)/(1/3) = 3 + 3 + 3 = 9.

Explanation

In any triangle ABC, the sum of the interior angles is always ∠ A + ∠ B + ∠ C = 180°. We are given two equations: ∠ A + ∠ B = 118° and ∠ A + ∠ C = 96°. By adding these two equations together, we get:

(∠ A + ∠ B) + (∠ A + ∠ C) = 118° + 96° 2∠ A + ∠ B + ∠ C = 214°

This expression can be rearranged as ∠ A + (∠ A + ∠ B + ∠ C) = 214°. Substituting the property that the sum of angles in a triangle is 180°, the equation becomes:

∠ A + 180° = 214° ∠ A = 214° - 180° ∠ A = 34°

Explanation

In Δ ABC, since AD ⊥ BC, the altitude AD creates two right-angled triangles, Δ ABD and Δ ACD, both sharing the vertex D as the right angle. Applying the Pythagoras Theorem to Δ ABD, the square of the hypotenuse AB is expressed as AB² = AD² + BD². Similarly, applying the Pythagoras Theorem to Δ ACD, the square of the hypotenuse AC is expressed as AC² = AD² + CD².

To determine the value of AB² + CD², we can substitute the expression for AB² from the first triangle into the sum: AB² + CD² = (AD² + BD²) + CD²

By rearranging the terms using the associative property of addition, we group the terms related to the second right triangle: AB² + CD² = BD² + (AD² + CD²)

Substituting the known relation AD² + CD² = AC² from Δ ACD into the equation, we arrive at the final result: AB² + CD² = BD² + AC²

Explanation

In equilateral triangle ABC, each interior angle is 60°, so ∠ ACB = 60°. Since CD is the internal bisector of ∠ C, we have ∠ ACD = (60°)/(2) = 30°. Because DC is produced to E, the points D, C, and E lie on a straight line, meaning ∠ ACD and ∠ ACE are supplementary. Therefore, ∠ ACE = 180° - ∠ ACD = 180° - 30° = 150°.

In triangle ACE, it is given that AC = CE, making it an isosceles triangle where ∠ CAE = ∠ CEA. By the angle sum property of a triangle, the sum of angles in △ ACE is 180°. Thus, ∠ ACE + ∠ CAE + ∠ CEA = 180°. Substituting the known values, we get 150° + 2∠ CAE = 180°. Solving for ∠ CAE gives 2∠ CAE = 30°, which results in ∠ CAE = 15°.

Explanation

Given the equation x(x - 3) = -1, we can expand it to x² - 3x = -1. Dividing the entire equation by x, we obtain x - 3 = -1/x, which can be rearranged to x + 1/x = 3.

To find the value of the expression x³(x³ - 18), we use the algebraic identity for the cube of a sum: (a + b)³ = a³ + b³ + 3ab(a + b). Applying this to x + 1/x = 3, we get: (x + 1/x)³ = x³ + 1/x³ + 3(x)(1/x)(x + 1/x) 3³ = x³ + 1/x³ + 3(3) 27 = x³ + 1/x³ + 9 x³ + 1/x³ = 18

Rearranging this result, we find that x³ - 18 = -1/x³. Multiplying both sides by x³, we arrive at the required value: x³(x³ - 18) = -1

Explanation

The value is calculated using the algebraic identity for the cube of a binomial sum: (a + b)³ = a³ + 3a²b + 3ab² + b³. By setting a = 1000 and b = 1, the expression (1001)³ is rewritten as (1000 + 1)³.

Substituting these values into the formula yields: (1000 + 1)³ = (1000)³ + 3(1000)²(1) + 3(1000)(1)² + (1)³

Calculating each term individually: (1000)³ = 1,000,000,000 3(1000)² = 3,000,000 3(1000) = 3,000 1³ = 1

Summing these parts results in: 1,000,000,000 + 3,000,000 + 3,000 + 1 = 1,003,003,001

This method provides a direct way to solve the problem by breaking the base into a power of ten and a small integer, significantly simplifying the arithmetic compared to long multiplication.

Explanation

The given equation is a(2 + √3) = b(2 - √3). We know that (2 + √3)(2 - √3) = 2² - (√3)² = 4 - 3 = 1. This implies that 2 + √3 and 2 - √3 are reciprocals of each other.

By setting the product a(2 + √3) = b(2 - √3) = 1, we find that a = (1)/(2 + √(3)) = 2 - √3 and b = (1)/(2 - √(3)) = 2 + √3. Consequently, their product is ab = (2 - √3)(2 + √3) = 1, which means b² = (1)/(a²).

Substituting b² = (1)/(a²) into the required expression: (1)/(a² + 1) + (1)/(b² + 1) = (1)/(a² + 1) + (1)/(1){a² + 1} = (1)/(a² + 1) + (a²)/(1 + a²) = (1 + a²)/(a² + 1) = 1

Explanation

The values of the trigonometric ratios involved are sin 60° = √3{2}, sec 60° = 2, tan 30° = (1)/(√(3)), sin 45° = (1)/(√(2)), and tan 60° = √3. Substituting these values into the given equation:

x (√3{2})² - ³⁄₂ (2) ((1)/(√(3)))² + ⅘ ((1)/(√(2)))² (√3)² = 0

Simplifying each term:

x (¾) - ³⁄₂ (2) (⅓) + ⅘ (½) (3) = 0

³ˣ⁄₄ - 1 + ⁶⁄₅ = 0

Combining the constant terms using a common denominator:

³ˣ⁄₄ + (-5 + 6)/(5) = 0

³ˣ⁄₄ + ⅕ = 0

³ˣ⁄₄ = -⅕

x = -⅕ · ⁴⁄₃ = -⁴⁄₁₅

Explanation

In the right-angled triangle ABC, the side AB represents the perpendicular opposite to ∠ C, and AC represents the hypotenuse. According to the trigonometric ratio for sine, sin C = (AB)/(AC). Given that ∠ C = 30° and AB = 6 units, substituting these values into the formula gives sin 30° = (6)/(AC). Using the standard trigonometric value sin 30° = ½, the equation becomes ½ = (6)/(AC). Solving for AC, we find AC = 6 × 2 = 12 units.

Explanation

In a semicircle, any angle subtended by the diameter at the circumference is a right angle. Therefore, ∠ ACB = 90°, which makes △ ABC a right-angled triangle with the diameter AB as its hypotenuse. Given the ratio AC:BC = 3:4, let AC = 3x and BC = 4x. Applying the Pythagorean theorem:

AC² + BC² = AB² (3x)² + (4x)² = 5² 9x² + 16x² = 25 25x² = 25 x = 1

The lengths of the two legs are AC = 3 cm and BC = 4 cm. The area of a right-angled triangle is given by the formula:

Area = ½ × base × height Area = ½ × 3 × 4 = 6 sq. cm.

Explanation

Since OA, OB, and OC are radii of the same circle, we have OA = OB = OC. This equality makes △ OAB and △ OAC isosceles triangles. In △ OAB, the angles opposite to the equal sides are equal, so ∠ OAB = ∠ OBA. Using the angle sum property of a triangle: ∠ OAB = (180° - ∠ AOB)/(2) = (180° - 90°)/(2) = 45°

Similarly, in △ OAC, since OA = OC, we have ∠ OAC = ∠ OCA: ∠ OAC = (180° - ∠ AOC)/(2) = (180° - 110°)/(2) = 35°

The problem specifies that the centre O lies in the interior of ∠ BAC. This geometric configuration implies that the ray AO lies between the rays AB and AC. Therefore, the total angle ∠ BAC is the sum of the angles ∠ OAB and ∠ OAC: ∠ BAC = ∠ OAB + ∠ OAC = 45° + 35° = 80°

Explanation

A pie chart is a circular statistical graphic used to illustrate numerical proportions by dividing the circle into sectors. In this representation, the entire circle represents the total sum of the data, while each individual sector represents a specific category or component. The central angle of each sector is calculated using the formula θ = (Component Value)/(Total Value) × 360°, ensuring that the area and arc length of the sector are directly proportional to the quantity it represents relative to the whole. This format is effective for visualizing the composition of a dataset and comparing the relative sizes of different parts.

Explanation

To find the value of x, we substitute the standard trigonometric ratios into the given equation: csc 30° = 2, sin 60° = √3/2, and tan 30° = 1/√3.

Substituting these values into the equation 2 csc² 30° + x sin² 60° - ¾ tan² 30° = 10:

2(2)² + x(√3{2})² - ¾((1)/(√(3)))² = 10

2(4) + x(¾) - ¾(⅓) = 10

8 + ³ˣ⁄₄ - ¼ = 10

Subtracting 8 from both sides:

(3x - 1)/(4) = 2

Multiplying by 4:

3x - 1 = 8

3x = 9

x = 3

Explanation

The value of the expression tan² Θ - sec² Θ is derived directly from the fundamental trigonometric identity 1 + tan² Θ = sec² Θ. By rearranging the terms of this identity, we subtract sec² Θ from both sides and move the constant 1 to the right side, which gives tan² Θ - sec² Θ = -1.

This identity holds true for all values of Θ for which the functions are defined, making the specific condition 2 sin Θ + cos Θ = 7/3 irrelevant to the final calculation. The value of the expression is a constant and does not depend on the specific value of Θ provided in the equation.

Explanation

The given equation is 29 tan Θ = 31, which implies tan Θ = ³¹⁄₂₉. The expression to be evaluated is:

(1 + 2 sin Θ cos Θ)/(1 - 2 sin Θ cos Θ)

Using the identity sin² Θ + cos² Θ = 1, the expression can be rewritten as:

sin² Θ + cos² Θ + 2 sin Θ cos Θ{sin² Θ + cos² Θ - 2 sin Θ cos Θ} = (sin Θ + cos Θ)²{(sin Θ - cos Θ)²}

To simplify this in terms of tan Θ, divide both the numerator and the denominator by cos² Θ:

((sin Θ)/(cos Θ) + (cos Θ)/(cos Θ))²{((sin Θ)/(cos Θ) - (cos Θ)/(cos Θ))²} = ( (tan Θ + 1)/(tan Θ - 1) )²

Substituting tan Θ = ³¹⁄₂₉:

( (31)/(29) + 1{³¹⁄₂₉ - 1} )² = ( (31 + 29)/(29){(31 - 29)/(29)} )² = ( ⁶⁰⁄₂ )² = 30² = 900

Explanation

In descriptive statistics, a bar chart is used to represent the distribution of qualitative or categorical variables. The categories are placed on the horizontal axis, and the vertical axis measures the magnitude of each category. The height of each bar can represent the class frequency, which is the raw count of observations within that category. It can also represent the class relative frequency, defined by the formula Relative Frequency = f/n, where f is the frequency of the class and n is the total number of observations. Furthermore, the height may represent the class percentage, which is simply the relative frequency expressed as a percentage (Relative Frequency × 100). Since all three measures provide a valid proportional representation of the data categories, they are all commonly used to determine bar height.