SSC CHSL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

According to the SAS (Side-Angle-Side) congruence criterion, two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle.

In triangle ABC, the given sides are AC = 2.5 cm and BC = 5 cm, and the included angle between these two sides is ∠C = 75°.

In triangle DEF, the given sides are DE = 2.5 cm and DF = 5 cm. The included angle between these two sides is ∠D.

For the triangles to be congruent, their corresponding sides and the included angles must be equal. Under the correspondence where AC matches DE and BC matches DF, the included angle ∠C must match the included angle ∠D. Therefore, ∠D must equal ∠C, which gives ∠D = 75°.

Explanation

To find the value of x³ + 1/x³, we begin with the given quadratic equation:

8x² + 9x + 8 = 0

Dividing the entire equation by 8x, we get:

x + 9/8 + 1/x = 0

x + 1/x = -9/8

Now, using the algebraic identity for the sum of cubes:

x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x)

Substitute the value of x + 1/x into this identity:

x³ + 1/x³ = (-9/8)³ - 3(-9/8)

x³ + 1/x³ = -729/512 + 27/8

To simplify, make the denominators equal by multiplying 27/8 by 64/64:

x³ + 1/x³ = -729/512 + 1728/512

x³ + 1/x³ = 999/512

Explanation

Let CP be the cost price and SP be the selling price of the article.

The initial profit is given by: Profit = SP - CP

If the selling price is doubled, the new selling price becomes 2SP. The new profit is: New Profit = 2SP - CP

According to the problem, the profit triples when the selling price is doubled: 2SP - CP = 3 × (SP - CP) 2SP - CP = 3SP - 3CP 3CP - CP = 3SP - 2SP 2CP = SP

Using this relation, the original profit is: Profit = SP - CP = 2CP - CP = CP

The profit percentage is calculated as: Profit Percentage = (Profit / CP) × 100% Profit Percentage = (CP / CP) × 100% = 100%

Explanation

The area of a triangle is calculated using the formula:

Area = ½ × base × height

Here, the length of the given side is the base (b = 24 cm), and the length of the perpendicular drawn to this side from the opposite vertex is the height (h).

Substituting the given values into the formula:

96 = ½ × 24 × h

96 = 12 × h

h = 96 / 12

h = 8 cm

Explanation

Let the given expression be represented as: E = ((9³ + 9⁴ + 9⁵ + 9⁶ + 9⁷) / (9¹ + 9² + 9³ + 9⁴ + 9⁵))¹/²

By factoring out 9² from each term in the numerator, we obtain: 9³ + 9⁴ + 9⁵ + 9⁶ + 9⁷ = 9² × (9¹ + 9² + 9³ + 9⁴ + 9⁵)

Substituting this back into the expression: E = ((9² × (9¹ + 9² + 9³ + 9⁴ + 9⁵)) / (9¹ + 9² + 9³ + 9⁴ + 9⁵))¹/²

Canceling the common term (9¹ + 9² + 9³ + 9⁴ + 9⁵) from the numerator and the denominator simplifies the expression: E = (9²)¹/²

E = 9

Explanation

From the total row of the given table, the total number of scooters produced by the company in each month is: January: 100 February: 105 March: 108 April: 110 May: 110 June: 90

The total number of scooters produced of all types taken together is equal in April and May, with 110 scooters produced in each of these two months.

Explanation

Applying the BODMAS rule, we first solve the expression inside the parentheses: 5 + 7 = 12

The expression becomes: 42 ÷ 7 × 12 ÷ 3

Next, division and multiplication are performed sequentially from left to right: 42 ÷ 7 = 6 6 × 12 = 72 72 ÷ 3 = 24

Explanation

Let the total monthly salary of Bhanu be S.

The total percentage of the salary spent on rent, stationery, transport, and electricity is: 20% + 10% + 14% + 18% = 62%

The percentage representing the remaining balance is: 100% - 62% = 38%

Since this remaining balance of 38% corresponds to ₹1,520, we calculate the total salary as: 38% of S = 1,520 (38 / 100) × S = 1,520 S = (1,520 × 100) / 38 S = 40 × 100 S = 4,000

Explanation

The formula for compound interest is: A = P(1 + r/100)ⁿ

To double the principal sum P, the amount A becomes 2P: 2P = P(1 + 26/100)ⁿ 2 = (1.26)ⁿ

The quickest way to find the doubling time is the Rule of 72 shortcut: Doubling time ≈ 72 ÷ Rate of interest Doubling time ≈ 72 ÷ 26 ≈ 2.77 years

We can also calculate the powers of 1.26 directly: 1.26¹ = 1.26 1.26² = 1.5876 1.26³ = 2.000376 ≈ 2

This confirms that the invested sum doubles in approximately 3 years.

Explanation

Let the speed of the watercraft in still water be x km/h and the speed of the river be y km/h.

The downstream speed is the speed along the river: x + y = 32 km / 4 hours = 8 km/h

The upstream speed is the speed against the river: x - y = 10 km / 5 hours = 2 km/h

Adding these two equations gives: 2x = 10 x = 5 km/h

Subtracting the second equation from the first gives: 2y = 6 y = 3 km/h

Explanation

The average speed for a round trip with equal onward and return distances is given by the formula 2xy/(x + y), where x and y are the two speeds. Substituting x = 66 km/h and y = 92.4 km/h into the formula gives (2 * 66 * 92.4) / (66 + 92.4) = 12196.8 / 158.4 = 77 km/h.

Explanation

Let the current ages of the two friends be 7x and 9x. According to the problem, after 10 years, the ratio of their ages is (7x + 10) / (9x + 10) = 19 / 23, which solves to x = 4. Thus, their current ages are 28 and 36 years, giving a current average age of 32 years. Therefore, their average age 7 years from now will be 32 + 7 = 39 years.

Explanation

Using the standard algebraic identity: If x - 1/x = k, then squaring both sides gives x² + 1/x² = k² + 2.

Substituting k = 11: x² + 1/x² = 11² + 2 = 121 + 2 = 123

Squaring this equation again gives: (x² + 1/x²)² = 123² x⁴ + 1/x⁴ + 2 = 15129 x⁴ + 1/x⁴ = 15129 - 2 = 15127

Explanation

The selling price of an item after a discount is given by the formula:

Selling Price = Marked Price × (1 - Discount / 100)

Substituting the given values:

Marked Price = ₹2,900 Discount = 37%

Selling Price = 2900 × (1 - 37 / 100) Selling Price = 2900 × 0.63 Selling Price = 29 × 63 = 1827

Explanation

The volume of a sphere is given by the formula V = (4/3) × π × r³, where r is the radius of the sphere.

Given that the diameter of the sphere is 16.8 cm, the radius r is: r = 16.8 ÷ 2 = 8.4 cm

Substituting the values into the formula with π = 22/7: V = (4/3) × (22/7) × 8.4 × 8.4 × 8.4 V = 4 × 22 × (8.4 ÷ 3) × (8.4 ÷ 7) × 8.4 V = 88 × 2.8 × 1.2 × 8.4 V = 246.4 × 10.08 V = 2483.712 cm³

Explanation

The volume V of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height.

Let the radii of the two cylinders be r₁ and r₂, and their heights be h₁ and h₂. The ratio of their volumes V₁/V₂ is:

V₁/V₂ = (π × r₁² × h₁) / (π × r₂² × h₂) V₁/V₂ = (r₁/r₂)² × (h₁/h₂)

Substituting the given ratios r₁/r₂ = 2/3 and h₁/h₂ = 5/3 into the equation:

V₁/V₂ = (2/3)² × (5/3) V₁/V₂ = (4/9) × (5/3) V₁/V₂ = 20/27

Explanation

Average speed is determined by the formula:

Average Speed = Total Distance / Total Time

The total distance traveled during the entire trip is the sum of the two segments: Total Distance = 10 km + 12 km = 22 km

The time taken for the first segment is: t₁ = 10 km / 12 km/h = 5/6 hours

The time taken for the second segment is: t₂ = 12 km / 10 km/h = 6/5 hours

The total time for the trip is: Total Time = 5/6 + 6/5 = (25 + 36) / 30 = 61/30 hours

Substituting these values into the average speed formula: Average Speed = 22 / (61/30) = (22 × 30) / 61 = 660 / 61 ≈ 10.82 km/h

This gives an approximate average speed of 10.8 km/h.

Explanation

Price after trade discount = Marked Price × (1 - Trade Discount Percentage)

Substituting the given values: Price after trade discount = 35600 × (1 - 0.10) = 35600 × 0.90 = ₹32,040

Cash discount = Price after trade discount - Final Selling Price Cash discount = 32040 - 30000 = ₹2,040

Cash discount percentage = (Cash discount / Price after trade discount) × 100 Cash discount percentage = (2040 / 32040) × 100 = (17 / 267) × 100 ≈ 6.367%

Rounding off to two decimal places gives 6.37%.

Explanation

Let the shares of A and B be represented as 4x and 5x respectively.

The difference in their shares is given as 10 marbles, which can be written as: 5x - 4x = 10 x = 10

A's share is 4x, which equals: 4 × 10 = 40

Explanation

The algebraic identity for the square of a trinomial is: (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

Comparing this identity with the given expression 16a² + 4b² + 9c² - 16ab + 12bc - 24ca: x² = 16a² which gives x = ±4a y² = 4b² which gives y = ±2b z² = 9c² which gives z = ±3c

The sign of each term can be determined from the sign of the cross-product terms: The term -16ab is negative, which means x and y have opposite signs. The term 12bc is positive, which means y and z have the same sign. The term -24ca is negative, which means z and x have opposite signs.

By choosing x = 4a, y must have the opposite sign, giving y = -2b. Since z has the same sign as y, z = -3c.

Substituting these values back into the identity: (4a - 2b - 3c)² = (4a)² + (-2b)² + (-3c)² + 2(4a)(-2b) + 2(-2b)(-3c) + 2(-3c)(4a) = 16a² + 4b² + 9c² - 16ab + 12bc - 24ca

Explanation

To find the positive difference between the yearly average expenditure on salaries and taxes, we first calculate the average expenditure for each of these categories over the five-year period from 2016 to 2020.

First, calculate the total expenditure on Salaries: Total Salaries = 332 + 368 + 444 + 480 + 516 = 2140 lakhs Average Salaries = 2140 / 5 = 428 lakhs

Next, calculate the total expenditure on Taxes: Total Taxes = 102 + 128 + 88 + 96 + 116 = 530 lakhs Average Taxes = 530 / 5 = 106 lakhs

Now, calculate the positive difference between the two averages: Difference = 428 - 106 = 322 lakhs

Explanation

The given expression is in the form of the standard algebraic identity:

(a³ + b³ + c³ - 3abc) / (a² + b² + c² - ab - bc - ca)

Using the algebraic identity:

a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Dividing the numerator by the denominator simplifies the expression directly to:

a + b + c

By substituting the given values a = 2.01, b = 39.75, and c = 28.24:

a + b + c = 2.01 + 39.75 + 28.24 = 70

Explanation

To find the correct similarity relation between the two triangles, we establish the one-to-one correspondence of their vertices by analyzing the given ratio of corresponding sides: PX/ER = ZX/RF = PZ/EF.

By comparing the ratios PX/ER = ZX/RF, we see that the common vertex in the numerators is X, and the common vertex in the denominators is R. This means vertex X corresponds to vertex R.

Comparing the ratios PX/ER = PZ/EF, the common vertex in the numerators is P, and the common vertex in the denominators is E. This means vertex P corresponds to vertex E.

Comparing the ratios ZX/RF = PZ/EF, the common vertex in the numerators is Z, and the common vertex in the denominators is F. This means vertex Z corresponds to vertex F.

Combining these vertex correspondences, we get P ↔ E, X ↔ R, and Z ↔ F. Therefore, triangle PXZ is similar to triangle ERF, which is written as ΔPXZ is similar to ΔERF.

Explanation

The length of the diagonal of a cube with side length a is given by the formula d = a√3. Equating this to the given diagonal:

a√3 = 8√3 a = 8 cm

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

S = 6 × 8² S = 6 × 64 S = 384 cm²

Explanation

By definition, the cosecant function is the reciprocal of the sine function, which is expressed as csc 18° = 1 / sin 18°.

Substituting the given value of sin 18° = (√5 - 1)/4 into the formula:

csc 18° = 4 / (√5 - 1)

To simplify this fraction, multiply both the numerator and the denominator by the conjugate of the denominator, which is √5 + 1:

csc 18° = (4 × (√5 + 1)) / ((√5 - 1) × (√5 + 1))

Applying the algebraic identity (a - b)(a + b) = a² - b² to the denominator:

csc 18° = 4(√5 + 1) / ((√5)² - 1²)

csc 18° = 4(√5 + 1) / (5 - 1)

csc 18° = 4(√5 + 1) / 4

csc 18° = √5 + 1

Explanation

Evaluating the expression using the BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction):

For the first bracket, (7 + 8 - 9 ÷ 3 × 2), perform the division first: 9 ÷ 3 = 3

Next, perform the multiplication: 3 × 2 = 6

Now evaluate the addition and subtraction: 7 + 8 - 6 = 9

For the second bracket, [52 - 4 of 5 + 4 × 7], perform the "of" operation first: 4 of 5 = 20

Next, perform the multiplication: 4 × 7 = 28

Now evaluate the addition and subtraction: 52 - 20 + 28 = 60

Summing the values of both brackets yields: 9 + 60 = 69

Explanation

Let the total work be the Least Common Multiple (LCM) of 12, 15, and 20, which is 60 units.

The daily work rates of the pairs are: Rate of (P + Q) = 60 / 12 = 5 units per day Rate of (Q + R) = 60 / 15 = 4 units per day Rate of (R + P) = 60 / 20 = 3 units per day

Adding these rates together: 2 × (P + Q + R) = 5 + 4 + 3 = 12 units per day P + Q + R = 6 units per day

Subtracting the rate of (Q + R) from the combined rate of (P + Q + R) gives the individual rate of P: Rate of P = (P + Q + R) - (Q + R) = 6 - 4 = 2 units per day

The total number of days taken by P alone to complete the work is: Time = Total work / Rate of P = 60 / 2 = 30 days

Explanation

Let the speed of the man in still water be u = 6 km/h and the speed of the stream be v km/h.

The speed of the man downstream is u + v, and the speed upstream is u - v.

Since the distance covered is the same, and the time taken to swim upstream is thrice the time taken downstream, the downstream speed must be thrice the upstream speed because speed is inversely proportional to time for a constant distance.

u + v = 3(u - v)

Substituting u = 6 into the equation:

6 + v = 3(6 - v) 6 + v = 18 - 3v 4v = 12 v = 3 km/h

The speed of the man when swimming downstream is:

Downstream speed = u + v = 6 + 3 = 9 km/h.

Explanation

The total work done is constant, which means the relationship between the number of men and the number of days is inversely proportional. This is expressed by the formula:

M₁ × D₁ = M₂ × D₂

where M₁ and M₂ are the number of men, and D₁ and D₂ are the number of days.

Substituting the given values into the formula:

30 × 7 = 21 × D₂

210 = 21 × D₂

D₂ = 210 / 21 = 10 days

Explanation

The successive discounts on the marked price of the bottle are 15% and 35%. The relation between the marked price (MP) and the selling price (SP) with successive discounts d₁ and d₂ is given by the formula:

SP = MP × (1 - d₁/100) × (1 - d₂/100)

Substituting the given values:

1547 = MP × (1 - 15/100) × (1 - 35/100)

1547 = MP × 85/100 × 65/100

Simplifying the fractions to their lowest terms:

1547 = MP × 17/20 × 13/20

1547 = MP × 221/400

Solving for MP:

MP = (1547 × 400) / 221

Dividing 1547 by 13 gives 119, and dividing 119 by 17 gives 7. Therefore:

MP = 7 × 400

MP = 2800

Explanation

Let the batsman's average score after 10 innings be x.

The total runs scored in the first 10 innings is 10x. After scoring 88 runs in his 11th inning, his total runs become 10x + 88, and his new average increases by 4, becoming x + 4.

We can set up the equation for the new average as: (10x + 88) / 11 = x + 4

Multiplying both sides by 11: 10x + 88 = 11(x + 4) 10x + 88 = 11x + 44

Solving for x: 11x - 10x = 88 - 44 x = 44

The average after 10 innings was 44. The new average after the 11th inning is x + 4 = 44 + 4 = 48.

Alternatively, using a shortcut: An increase of 4 runs per inning over 11 innings requires an extra 11 × 4 = 44 runs. Subtracting these extra runs from his 11th inning score gives his previous average: 88 - 44 = 44

Adding the average increase of 4 runs gives the new average after the 11th inning: 44 + 4 = 48

Explanation

First, calculate the total number of valid votes. Since 30% of the votes were invalid, the valid votes comprise 70% of the total votes.

Total valid votes = 70% of 7500 = 0.70 × 7500 = 5250

One candidate received 40% of these valid votes. This means the other candidate received the remaining 60% of the valid votes.

Valid votes received by the other candidate = 60% of 5250 = 0.60 × 5250 = 3150

This directly gives the required number of valid votes obtained by the other candidate.

Explanation

To find the required percentage, we first calculate the total number of cakes sold by Bakery C on Saturday and Sunday. According to the table, Bakery C sold 195 cakes on Saturday and 235 cakes on Sunday.

Total cakes sold by Bakery C on Saturday and Sunday = 195 + 235 = 430

Next, we find the number of cakes sold by Bakery E on Tuesday. From the table, Bakery E sold 325 cakes on Tuesday.

Now, we calculate the percentage using the formula:

Percentage = (Cakes sold by Bakery C on Saturday and Sunday / Cakes sold by Bakery E on Tuesday) × 100

Percentage = (430 / 325) × 100

Percentage = (86 / 65) × 100

Percentage ≈ 1.323 × 100 ≈ 132.3%

This is approximately equal to 132%.

Explanation

The area of a rhombus is given by the formula: Area = ½ × d₁ × d₂

Substituting the given area of 48 cm² and diagonal d₁ = 8 cm: 48 = ½ × 8 × d₂ 48 = 4 × d₂ d₂ = 12 cm

The diagonals of a rhombus bisect each other at right angles, dividing the rhombus into four congruent right-angled triangles. The legs of each right-angled triangle are half of the diagonals: d₁/2 = 8/2 = 4 cm d₂/2 = 12/2 = 6 cm

Using the Pythagorean theorem, the side of the rhombus (s) is the hypotenuse of these right-angled triangles: s² = (d₁/2)² + (d₂/2)² s² = 4² + 6² s² = 16 + 36 s² = 52 s = √52 = 2√13 cm

Explanation

The dealer sells sugar at ₹40 per kg, claiming to incur a loss of ₹5 per kg. This means the actual cost price (CP) of 1 kg (1000 g) of sugar is: CP of 1000 g = ₹40 + ₹5 = ₹45

However, the dealer uses a faulty weight and gives only 800 g instead of 1000 g. The actual cost price of the 800 g delivered to the customer is: CP of 800 g = (₹45 / 1000) × 800 = ₹36

The dealer charges the customer ₹40 for this quantity, meaning his actual selling price (SP) for the 800 g is ₹40.

Since the actual selling price is higher than the actual cost price, he makes a profit. Actual Gain = SP - CP = ₹40 - ₹36 = ₹4

Gain percent = (Actual Gain / CP of delivered quantity) × 100 Gain percent = (4 / 36) × 100 = (1 / 9) × 100 = 11 1/9%

Explanation

For a number to be divisible by 6, it must be divisible by both 2 and 3.

Divisibility by 2 is satisfied because the units digit of 5132832 is 2, which is even.

Divisibility by 3 is satisfied because the sum of the digits of the number is 5 + 1 + 3 + 2 + 8 + 3 + 2 = 24, which is a multiple of 3.

Since the number is divisible by both 2 and 3, it is divisible by 6.

Explanation

Let the length and width of the rectangle be L and W respectively.

The perimeter of the rectangle is given as 46 cm: 2(L + W) = 46 L + W = 23

The diagonal of the rectangle is given as 17 cm. By Pythagoras' theorem: L² + W² = 17² L² + W² = 289

We use the algebraic identity: (L + W)² = L² + W² + 2LW

Substitute the values of (L + W) and (L² + W²): 23² = 289 + 2LW 529 = 289 + 2LW 2LW = 529 - 289 2LW = 240 LW = 120

Since the area of the rectangle is given by L × W, the area is 120 cm².

Explanation

Using the algebraic identity: (a + b)² - (a - b)² = 4ab

Substituting the given values of (a + b) = 8 and (a - b) = 6: 8² - 6² = 4ab 64 - 36 = 4ab 28 = 4ab ab = 7

Explanation

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

Sum of the data points = 1.2 + 1.7 + 3 + 4.8 + 10.5 + 17.3 + 24.5 = 63

Total number of observations = 7

Average = 63 ÷ 7 = 9

Explanation

Let the marked price of the watch be MP.

The equivalent discount for two successive discounts of 20% and 20% is determined using the successive discount formula: Equivalent Discount = d₁ + d₂ - (d₁ × d₂)/100 Equivalent Discount = 20 + 20 - (20 × 20)/100 = 36%

Since the total discount is 36%, the selling price constitutes 64% of the marked price: 64% of MP = 640 (64/100) × MP = 640 MP = 640 × (100/64) MP = 1000

Explanation

The formula relating the present value and the future amount under compound interest is: A = PV × (1 + r/100)ⁿ

Here: A = Amount due = ₹ 484 r = Rate of interest = 10% per annum n = Time period = 2 years PV = Present value

Substituting these values into the formula: 484 = PV × (1 + 10/100)² 484 = PV × (1.1)² 484 = PV × 1.21 PV = 484 / 1.21 PV = 400

Explanation

The average speed of the entire journey is given by the formula: Average Speed = Total Distance / Total Time

For the first segment from A to B: Speed = 60 km/h Time = 3 hours Distance = Speed × Time = 60 × 3 = 180 km

For the second segment from B to C: Speed = 60 + (50% of 60) = 90 km/h Time = 2 hours Distance = Speed × Time = 90 × 2 = 180 km

Calculating the overall journey details: Total Distance = 180 + 180 = 360 km Total Time = 3 + 2 = 5 hours

Average Speed = 360 / 5 = 72 km/h

Explanation

The total number of votes polled is 1,89,000. Since the fourth candidate got 18,000 votes, the total votes polled by the three candidates from the national parties is:

1,89,000 - 18,000 = 1,71,000

The votes obtained by these three candidates are in the ratio of 2:3:4. Let their votes be represented as 2x, 3x, and 4x.

The sum of their votes is: 2x + 3x + 4x = 9x

Equating this to the combined votes: 9x = 1,71,000 x = 1,71,000 ÷ 9 = 19,000

The votes obtained by the three candidates are: First candidate: 2 × 19,000 = 38,000 Second candidate: 3 × 19,000 = 57,000 Third candidate: 4 × 19,000 = 76,000

Explanation

Under the Buy one, get one free scheme, a customer receives two items but pays the price of only one. The effective discount percentage is calculated by dividing the number of free items by the total number of items:

Discount Percentage = (1 / 2) × 100 = 50%

Since this scheme yields an effective discount of 50%, which is higher than the effective discounts of all other schemes, it is the most beneficial option for the customer.

Explanation

To find the time taken by the faster train to cross the slower train completely, we use the concepts of relative speed and total distance.

Total distance to be covered when one train crosses another completely is the sum of their lengths: Distance = 640 + 540 = 1180 metres

Since both trains are travelling in the same direction, their relative speed is the difference between their speeds: Relative speed = 80 - 72 = 8 km/h

Convert the relative speed from km/h to m/s by multiplying by 5/18: Relative speed = 8 × 5/18 = 20/9 m/s

Using the formula: Time = Distance / Relative speed

Substitute the values: Time = 1180 / (20/9) = (1180 × 9) / 20 = 59 × 9 = 531 seconds

Convert 531 seconds into minutes and seconds: 531 seconds = 8 minutes and 51 seconds

Explanation

To find the number of valid votes polled for the winning candidate, we first determine the total number of valid votes.

The total number of voters is 3,20,000. Since 5% of the total votes are declared invalid, the percentage of valid votes is: 100% - 5% = 95%

Thus, the total number of valid votes is: 95% of 3,20,000 = (95/100) × 3,20,000 = 3,04,000

The winning candidate received 56% of these valid votes. Therefore, the number of valid votes secured by the winner is: 56% of 3,04,000 = (56/100) × 3,04,000 = 56 × 3,040 = 1,70,240

Explanation

Let x = √(2 + √3) + √(2 - √3).

Squaring both sides of the equation using the identity (a + b)² = a² + b² + 2ab:

x² = (2 + √3) + (2 - √3) + 2 × √((2 + √3)(2 - √3))

Simplifying the terms using the difference of squares identity:

x² = 4 + 2 × √(2² - (√3)²) x² = 4 + 2 × √(4 - 3) x² = 4 + 2 × √1 x² = 6

Since x is the sum of two positive square roots, it must be positive. Taking the square root of both sides gives:

x = √6

Explanation

To simplify the expression, we apply the BODMAS rule, which dictates the order of operations as Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, we simplify the terms inside the parentheses: 4/3 + 5/3 of 6/25 Since of represents multiplication with a higher precedence than addition, we calculate: 5/3 of 6/25 = 5/3 × 6/25 = 2/5 Now, we perform the addition: 4/3 + 2/5 = (20 + 6)/15 = 26/15

Next, we simplify the expression inside the square brackets: 8 5/9 + 26/15 Converting the mixed fraction to an improper fraction: 8 5/9 = (8 × 9 + 5)/9 = 77/9 Using the least common multiple of 9 and 15, which is 45, we get: 77/9 + 26/15 = 385/45 + 78/45 = 463/45

Now, we evaluate the first part of the expression: 6/5 ÷ 6/5 of 6/5 × 5/6 Evaluating of first: 6/5 of 6/5 = 36/25 Evaluating division next: 6/5 ÷ 36/25 = 6/5 × 25/36 = 5/6 Evaluating multiplication: 5/6 × 5/6 = 25/36

Then, we evaluate the second part of the expression: 9/4 ÷ 4/5 of 15/2 Evaluating of first: 4/5 of 15/2 = (4 × 15)/(5 × 2) = 60/10 = 6 Evaluating division: 9/4 ÷ 6 = 9/4 × 1/6 = 3/8

Combining all the calculated parts together: 25/36 + 3/8 - 463/45 The least common multiple of 36, 8, and 45 is 360. We convert each fraction to a common denominator of 360: 25/36 = 250/360 3/8 = 135/360 463/45 = 3704/360

Now, we compute the final sum and difference: 250/360 + 135/360 - 3704/360 = (250 + 135 - 3704)/360 = (385 - 3704)/360 = -3319/360 = -9 79/360

Explanation

The prime numbers on a six-sided dice are 2, 3, and 5.

Using the frequencies given in the table for these prime outcomes: Frequency of obtaining 2 = 43 Frequency of obtaining 3 = 33 Frequency of obtaining 5 = 10

To find the total number of times a prime number appears, sum these individual frequencies: Total times = 43 + 33 + 10 = 86

Explanation

A number is divisible by 55 if it is divisible by both 5 and 11. Since 185625 ends with the digit 5, it is divisible by 5.

To check for divisibility by 11, we find the difference between the sum of the digits at odd positions and the sum of the digits at even positions. For the number 185625:

Sum of digits at odd positions = 1 + 5 + 2 = 8 Sum of digits at even positions = 8 + 6 + 5 = 19

The difference between these sums is 19 - 8 = 11. Since this difference is a multiple of 11, the number 185625 is divisible by 11. Consequently, it is divisible by 55.