SSC CHSL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

Given that x + y + z = 0, we can write z = -x - y.

Substituting this value of z into the denominators of the three terms, we get:

x² - yz = x² - y(-x - y) = x² + xy + y²

y² - zx = y² - (-x - y)x = y² + x² + xy = x² + xy + y²

z² - xy = (-x - y)² - xy = x² + 2xy + y² - xy = x² + xy + y²

Since all three denominators are equal, we can combine the terms over the common denominator:

Expression = (x² + y² + z²) / (x² + xy + y²)

By substituting z² = (x + y)² = x² + 2xy + y² into the numerator, we obtain:

x² + y² + z² = x² + y² + (x² + 2xy + y²) = 2x² + 2xy + 2y² = 2(x² + xy + y²)

Therefore, the expression becomes:

2(x² + xy + y²) / (x² + xy + y²) = 2

Explanation

Using the trigonometric identity for the sum of two angles:

sin(A + B) = sin A cos B + cos A sin B

By substituting A = 35° and B = 55° into this formula, the expression can be simplified as:

sin 35° cos 55° + cos 35° sin 55° = sin(35° + 55°)

Simplifying the angle:

35° + 55° = 90°

Since the sine of 90° is equal to 1:

sin 90° = 1

Explanation

Using the algebraic identity for the sum of cubes, the numerator can be factored as:

r³ + 1/r³ = (r + 1/r)(r² - 1 + 1/r²)

Substitute this factored form into the given expression:

(r³ + 1/r³) / (r² + 1/r² - 1) = (r + 1/r)(r² + 1/r² - 1) / (r² + 1/r² - 1)

The common terms in the numerator and the denominator cancel out, simplifying the expression to:

r + 1/r

Given that r + 1/r = 11, the value of the expression is directly 11.

Explanation

The total volume of the three smaller cubes is the sum of their individual volumes: Volume = 3³ + 4³ + 5³ = 27 + 64 + 125 = 216 cm³

Since the three cubes are melted to form a single larger cube, the volume of the larger cube is also 216 cm³. Let the side of the larger cube be A. A³ = 216 A = ∛216 = 6 cm

The sum of the total surface areas of the three smaller cubes is: Total Surface Area (small) = 6 × (3² + 4² + 5²) Total Surface Area (small) = 6 × (9 + 16 + 25) = 6 × 50 = 300 cm²

The total surface area of the larger cube is: Total Surface Area (large) = 6 × A² = 6 × 6² = 6 × 36 = 216 cm²

The ratio of the smaller cubes' total surface area to the larger cube's total surface area is: Ratio = 300 : 216

Dividing both terms by their greatest common divisor, which is 12: Ratio = 25 : 18

Explanation

Total cost price (CP) of the machine includes both the purchase price and the transportation cost: CP = ₹5,000 + ₹250 = ₹5,250

The selling price (SP) is ₹5,075. Since the cost price is greater than the selling price, there is a loss.

Loss = CP - SP Loss = ₹5,250 - ₹5,075 = ₹175

Loss percentage = (Loss / CP) × 100 Loss percentage = (175 / 5250) × 100 Loss percentage = (1 / 30) × 100 = 3.33%

Explanation

The effective annual rate of interest, E, is calculated using the formula:

E = (1 + r/n)ⁿ - 1

where r is the nominal annual rate of interest and n is the compounding frequency per year.

Given that the nominal rate is 6% per annum (r = 0.06) and the compounding is half-yearly (n = 2), the values are substituted as follows:

E = (1 + 0.06/2)² - 1 E = (1.03)² - 1 E = 1.0609 - 1 E = 0.0609

This calculation yields an effective annual rate of interest of 0.0609.

Explanation

The length of an arc s that subtends an angle θ at the center of a circle is related to the total circumference C of the circle by the formula:

s = (θ / 360) × C

By substituting the given values of arc length s = 18 cm and angle θ = 28.8° into the formula:

18 = (28.8 / 360) × C

Rearranging the equation to solve for C:

C = (18 × 360) / 28.8

C = 6480 / 28.8

C = 225 cm

Explanation

An analysis of the census data across all four specified years shows that Andhra Pradesh consistently recorded the lowest literacy rate among the four states in every single census.

In the 1981 census, Andhra Pradesh had a literacy rate of 35.7%, which was lower than the rates of Gujarat (44.9%), Karnataka (46.2%), and Kerala (78.9%).

In the 1991 census, the literacy rate of Andhra Pradesh rose to 44.1%, but this remained the lowest among the group, with Karnataka at 56.0%, Gujarat at 61.3%, and Kerala at 89.8%.

During the 2001 census, Andhra Pradesh recorded 60.5% literacy, which was again the lowest compared to Karnataka (66.6%), Gujarat (69.1%), and Kerala (90.9%).

Finally, in the 2011 census, Andhra Pradesh registered a literacy rate of 67.7%, continuing to be the lowest among the four states, while Karnataka stood at 75.6%, Gujarat at 79.3%, and Kerala at 93.9%.

Explanation

A cylinder circumscribing a sphere has a radius r equal to the radius of the sphere, and a height h equal to the diameter of the sphere.

Given the diameter of the sphere is 14 cm: r = 14 / 2 = 7 cm h = 14 cm

The volume V of the cylinder is given by the formula: V = π r² h

Substituting the given values: V = (22/7) × 7² × 14 V = 22 × 7 × 14 V = 154 × 14 V = 2156 cm³

Explanation

The compound interest formula when interest is compounded half-yearly is A = P(1 + r)ⁿ, where A is the maturity amount, P is the principal sum, r is the rate of interest per compounding period, and n is the total number of compounding periods.

For half-yearly compounding over 18 months: The half-yearly rate r = 10% / 2 = 5% = 0.05 The number of half-yearly periods n = 18 / 6 = 3

Substituting the given maturity amount A = ₹83,349: 83349 = P(1 + 0.05)³ 83349 = P(1.05)³ 83349 = P × 1.157625

Solving for P: P = 83349 / 1.157625 = 72000

Explanation

The relationship between the marked price, discount percentage, and the selling price is given by the formula:

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

Given that the selling price is ₹975 and the discount is 35%, we can substitute these values into the formula:

975 = Marked Price × (100 - 35) / 100

975 = Marked Price × 65 / 100

Marked Price = (975 × 100) / 65

Marked Price = 15 × 100

Marked Price = 1500

Explanation

The third proportional of two numbers a and b is defined as the number x such that a : b = b : x, which simplifies to the formula:

x = b² / a

In this problem, the two given numbers are a = 14 and b = 28. Substituting these values into the formula:

x = 28² / 14 x = (28 × 28) / 14 x = 2 × 28 x = 56

Explanation

To expand the polynomial f(x) = x² + 2x + 3 about x = -2, we can perform a simple change of variable. Let y = x + 2, which implies x = y - 2.

Substituting x = y - 2 into the original expression: f(x) = (y - 2)² + 2(y - 2) + 3

Expanding the terms gives: f(x) = (y² - 4y + 4) + (2y - 4) + 3 f(x) = y² - 2y + 3

Substituting y = x + 2 back into the expression yields: f(x) = (x + 2)² - 2(x + 2) + 3

Explanation

Let CP, MP, and SP be the cost price, marked price, and selling price of a single item, respectively.

According to the first condition: 42 × MP = 70 × CP MP/CP = 70/42 = 5/3 MP = (5/3) × CP

According to the second condition: 25 × SP = 21 × MP SP/MP = 21/25 SP = (21/25) × MP

By substituting the value of MP from the first equation into the second equation: SP = (21/25) × (5/3) × CP SP = (7/5) × CP SP = 1.4 × CP

The profit percentage is calculated using the formula: Profit % = ((SP - CP)/CP) × 100 Profit % = ((1.4 × CP - CP)/CP) × 100 Profit % = 0.4 × 100 = 40% profit

Explanation

When a journey covers equal distances at two different speeds, v₁ and v₂, the average speed is calculated using the formula:

Average Speed = 2 × v₁ × v₂ / (v₁ + v₂)

Given that the speed of the car while going is v₁ = 21 km/h and the speed while returning is v₂ = 17 km/h, substitute these values into the formula:

Average Speed = (2 × 21 × 17) / (21 + 17)

Average Speed = 714 / 38

Dividing both the numerator and the denominator by 2 simplifies the fraction to 357 / 19.

Converting 357 / 19 into a mixed fraction: 357 ÷ 19 = 18 with a remainder of 15, which is written as 18 15/19 km/h.

Explanation

Two angles are supplementary when their sum is 180 degrees. Let the two angles be x and y, with x being the larger angle.

According to the given conditions: x + y = 180° x - y = 74°

Subtracting the second equation from the first equation gives: 2y = 180° - 74° 2y = 106° y = 53°

Substituting the value of y back into the first equation: x + 53° = 180° x = 127°

Thus, the two supplementary angles are 127° and 53°. Among the given choices, 53° represents one of these angles.

Explanation

The Cost Price (CP) of the article can be calculated using the formula: CP = Selling Price × 100 / (100 - Loss %)

Given that the Selling Price is £255 and the loss is 15%: CP = 255 × 100 / (100 - 15) CP = 255 × 100 / 85 CP = 3 × 100 = 300

When the article is sold for a new Selling Price of £357, the net result is a profit because the Selling Price is greater than the Cost Price.

Profit = New Selling Price - CP Profit = 357 - 300 = 57

The profit percentage is calculated as: Profit % = (Profit / CP) × 100% Profit % = (57 / 300) × 100% = 19%

Explanation

The difference between a single discount of (x + y)% and two successive discounts of x% and y% on a principal amount P is given by the formula:

Difference = (x × y / 100)% of P

In this case, the single discount of 20% is equal to the sum of the two successive discounts of 16% and 4% (since 16 + 4 = 20).

Here, x = 16, y = 4, and P = 50,000.

Substituting these values into the formula:

Difference = (16 × 4 / 100)% of 50,000 Difference = 0.64% of 50,000 Difference = (0.64 / 100) × 50,000 Difference = 0.64 × 500 Difference = 320

Explanation

The increase in the average weight of 8 men when one man is replaced is 2 kg. This means the overall sum of the weights of the 8 men increases by:

Total increase = Number of men × Increase in average weight Total increase = 8 × 2 = 16 kg

Since the weight of the replaced man is 90 kg, the weight of the new man must be 16 kg more than this to cause the 16 kg increase in the total weight:

Weight of the new man = Weight of the replaced man + Total increase Weight of the new man = 90 + 16 = 106 kg

Explanation

The relationship between the Cost Price (CP) and the Marked Price (MP) is given by the formula:

CP/MP = (100 - Discount%) / (100 + Profit%)

Substituting the given values: Marked Price (MP) = ₹2,500 Discount% = 10% Profit% = 20%

CP/2500 = (100 - 10) / (100 + 20) CP/2500 = 90/120 CP/2500 = 3/4 CP = 2500 × 3/4 = 1875

Explanation

The weight of the child aged 8 years is 27 kg, and the weight of the child aged 9 years is 25 kg. Adding these two values gives:

27 kg + 25 kg = 52 kg

Since 52 kg is greater than 50 kg, the sum of the weights of the children of these two ages exceeds 50 kg.

Explanation

Let the total work be the least common multiple of 6 and 12, which is 12 units.

The efficiency of X is 12 ÷ 6 = 2 units per day. The efficiency of Y is 12 ÷ 12 = 1 unit per day.

They work on alternate days, beginning with X: On Day 1, X completes 2 units of work. On Day 2, Y completes 1 unit of work.

Thus, in a 2-day cycle, the total work completed is 2 + 1 = 3 units.

To complete the entire 12 units of work, the number of cycles required is 12 ÷ 3 = 4 cycles.

Since each cycle consists of 2 days, the total time taken to finish the work is 4 × 2 = 8 days.

Explanation

To find the number that should be subtracted from 246837 to make it divisible by 13, we must find the remainder when 246837 is divided by 13.

Performing the division: 246837 ÷ 13 = 18987 with a remainder of 6

Subtracting this remainder of 6 from the original number yields 246831, which is perfectly divisible by 13 (13 × 18987 = 246831).

Explanation

If four numbers a, b, c, and d are in proportion, then the ratio of the first two is equal to the ratio of the last two, expressed as a/b = c/d, where d is the fourth proportional to a, b, and c.

Substituting the given values a = 14, b = 21, and c = 28 into the relation:

d = (b × c) / a

d = (21 × 28) / 14

d = 21 × 2

d = 42

Explanation

Using the successive percentage change formula, the remaining amount after consecutive expenditures is given by:

Remaining Amount = Salary × (1 - r₁) × (1 - r₂)

Here, r₁ = 20% = 0.20 and r₂ = 30% = 0.30.

Substituting the given values:

8400 = Salary × (1 - 0.20) × (1 - 0.30) 8400 = Salary × 0.80 × 0.70 8400 = Salary × 0.56 Salary = 8400 / 0.56 Salary = 840000 / 56 Salary = 15000

Explanation

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

Total Distance = 10 + 20 + 30 = 60 km

Total Time is the sum of the times taken for each part of the journey: Time 1 = 10 km ÷ 30 km/h = 1/3 h Time 2 = 20 km ÷ 40 km/h = 1/2 h Time 3 = 30 km ÷ 60 km/h = 1/2 h

Total Time = 1/3 + 1/2 + 1/2 = 1/3 + 1 = 4/3 h

Average Speed = Total Distance ÷ Total Time = 60 ÷ (4/3) = 60 × 3/4 = 45 km/h

Explanation

To find the value of 7x² - 50xy + 7y², we first calculate the sum and product of x and y.

Sum of x and y: x + y = (√10 + √11) + (√10 - √11) = 2√10

Product of x and y: xy = (√10 + √11)(√10 - √11) = (√10)² - (√11)² = 10 - 11 = -1

We can express x² + y² using the standard algebraic identity: x² + y² = (x + y)² - 2xy x² + y² = (2√10)² - 2(-1) x² + y² = 40 + 2 = 42

Now, substitute these values into the target expression: 7x² - 50xy + 7y² = 7(x² + y²) - 50xy = 7(42) - 50(-1) = 294 + 50 = 344

Explanation

When a journey covers equal distances at two different speeds, say v₁ and v₂, the average speed for the entire journey is given by the harmonic mean of the speeds, represented by the formula:

Average speed = 2 × v₁ × v₂ / (v₁ + v₂)

Given: v₁ = 84 km/h v₂ = 56 km/h

Substituting these values into the formula:

Average speed = 2 × 84 × 56 / (84 + 56) Average speed = 9408 / 140 Average speed = 67.2 km/h

The distance of 778 km between the stations does not affect the calculation of the average speed since the distance traveled at each speed is equal.

Explanation

The population of the city after n years is calculated using the compound growth formula:

P_future = P × (1 + R/100)ⁿ

Here, the present population P is 21000, the annual growth rate R is 10%, and the time period n is 3 years.

Substituting these values into the formula:

P_future = 21000 × (1 + 10/100)³ P_future = 21000 × (1.1)³ P_future = 21000 × 1.331 P_future = 21 × 1331 P_future = 27951

Explanation

When a journey covers two equal distances of 50 km each at different speeds, v₁ and v₂, the average speed of the entire journey is given by the harmonic mean of the two speeds:

Average Speed = 2 × v₁ × v₂ / (v₁ + v₂)

Given that the speed for the first 50 km (v₁) is 25 km/h, the average speed for the whole journey is 30 km/h, and the speed for the next 50 km is v₂:

30 = 2 × 25 × v₂ / (25 + v₂)

30 = 50 × v₂ / (25 + v₂)

3 × (25 + v₂) = 5 × v₂

75 + 3 × v₂ = 5 × v₂

2 × v₂ = 75

v₂ = 37.5 km/h

Explanation

A number is divisible by 15 if it is divisible by both 3 and 5.

For a number to be divisible by 5, its last digit must be either 0 or 5. The number 3,06,045 ends in 5, which satisfies this condition.

For a number to be divisible by 3, the sum of its digits must be a multiple of 3. Finding the sum of the digits for 3,06,045: 3 + 0 + 6 + 0 + 4 + 5 = 18

Since 18 is divisible by 3, the number 3,06,045 is also divisible by 3.

Since 3,06,045 is divisible by both 3 and 5, it is divisible by 15.

Explanation

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the remaining side. If we take the two shorter sides, their sum must exceed the length of the longest side.

For the side lengths 3 cm, 5 cm, and 1 cm, the two shorter sides are 3 cm and 1 cm. Their sum is 3 + 1 = 4 cm. Since this sum is less than the longest side of 5 cm, they violate the triangle inequality theorem and cannot form the sides of a triangle.

Explanation

To find the time taken by Arun and Shama to finish the work together, we use the formula for combined work:

T = (A × B) / (A + B)

Here, Arun's time A is 12 days and Shama's time B is 15 days. Substituting these values into the formula gives:

T = (12 × 15) / (12 + 15) T = 180 / 27 T = 20 / 3 days T = 6 2/3 days

Explanation

Convert the mixed fractions into improper fractions: 3 6/7 = (3 × 7 + 6)/7 = 27/7 5 1/7 = (5 × 7 + 1)/7 = 36/7 2 3/7 = (2 × 7 + 3)/7 = 17/7 3 5/7 = (3 × 7 + 5)/7 = 26/7 4 = 28/7

The expression can be written as: 27/7 - {36/7 - (17/7 - (26/7 - 28/7))}

Following BODMAS rules, first simplify the innermost part containing the vinculum (bar bracket): 26/7 - 28/7 = -2/7

Now evaluate the expression inside the round brackets: 17/7 - (-2/7) = 17/7 + 2/7 = 19/7

Next, evaluate the expression inside the curly brackets: 36/7 - 19/7 = 17/7

Finally, complete the subtraction: 27/7 - 17/7 = 10/7

Explanation

From the given equation: a + b = 2c

We can rearrange the terms to establish a relationship between the variables: a - c = c - b

Multiplying both sides by -1 gives: b - c = -(a - c)

Substitute this relationship directly into the expression to find its value: a / (a - c) + c / (b - c)

Replacing the term (b - c) with -(a - c) in the denominator: a / (a - c) + c / (-(a - c)) = a / (a - c) - c / (a - c) = (a - c) / (a - c) = 1

Explanation

The sum of runs in all 20 matches is: 20 × 30 = 600

The sum of runs in the first 12 matches is: 12 × 40 = 480

The sum of runs in the remaining 8 matches is: 600 - 480 = 120

The average of the last 8 matches is: 120 ÷ 8 = 15

Explanation

To find the colleges with the minimum number of participating students, the total number of participants across all four extra-curricular activities is calculated for each college.

The sum for College D is 50 + 50 + 125 + 275 = 500.

The sum for College G is 150 + 50 + 150 + 150 = 500.

The sum for College A is 150 + 50 + 100 + 250 = 550, while College B has a sum of 250 + 150 + 50 + 125 = 575. College C has a sum of 450 + 150 + 400 + 350 = 1350, College E has a sum of 350 + 50 + 100 + 250 = 750, and College F has a sum of 250 + 50 + 500 + 150 = 950.

College D and College G have the lowest overall student participation with 500 students each.

Explanation

The volume of a cuboid is calculated using the formula: V = l × b × h where l is the length, b is the breadth, and h is the height.

The breadth (b) is given as 8 cm.

Since the breadth is twice the height (b = 2h), the height is: h = 8 / 2 = 4 cm

The length (l) is twice more than its breadth, which means it is equal to the breadth plus twice the breadth: l = b + 2b = 3b = 3 × 8 = 24 cm

Now, substituting these values to find the volume: Volume = 24 × 8 × 4 = 768 cm³

Explanation

We are given x - 1/x = 11 and x > 0.

Using the algebraic identity: (x + 1/x)² = (x - 1/x)² + 4

Substitute the given value: (x + 1/x)² = 11² + 4 (x + 1/x)² = 121 + 4 = 125

Since x > 0, the sum x + 1/x must be positive: x + 1/x = √125 = 5√5

Now, using the difference of squares identity: x² - 1/x² = (x - 1/x)(x + 1/x)

Substitute the known values: x² - 1/x² = 11 × 5√5 = 55√5

Explanation

Let the cost price and the marked price of the article be 4x and 7x respectively.

Since a discount of 25% is offered on the marked price, the selling price is 75% of the marked price: Selling Price = 7x × 0.75 = 5.25x

The profit earned by the shopkeeper is the difference between the selling price and the cost price: Profit = 5.25x - 4x = 1.25x

Equating this to the given profit of ₹50: 1.25x = 50 x = 40

Substituting the value of x to find the cost price: Cost Price = 4 × 40 = 160

Explanation

The cost price of 1000 g of sugar is ₹30. Since the shopkeeper gives 750 g instead of 1000 g, the actual cost price incurred by him is only for 750 g.

Actual Cost Price of 750 g sugar = (30 ÷ 1000) × 750 = ₹22.50

The selling price for this quantity is ₹27.

Actual Selling Price = ₹27

Profit = Selling Price - Cost Price = 27 - 22.50 = ₹4.50

Profit percentage = (Profit ÷ Cost Price) × 100 = (4.50 ÷ 22.50) × 100 = 20%

Explanation

When two objects move towards each other, their relative speed is the sum of their individual speeds.

Relative Speed = 160 km/h + 200 km/h = 360 km/h

Converting the relative speed into m/s: 360 × 5/18 = 100 m/s

To completely cross each other, the total distance they must cover together is the sum of their lengths and the initial distance separating them: Total Distance = 230 m + 240 m + 130 m = 600 m

Using the formula Time = Distance / Speed: Time = 600 / 100 = 6 seconds

Explanation

Let the Cost Price of the article be CP and the Selling Price be SP.

According to the given condition: 40% of CP = 30% of SP

This equation can be written as: 0.40 × CP = 0.30 × SP

By simplifying the relation, we get: 4 × CP = 3 × SP SP / CP = 4/3

This ratio indicates that if the Cost Price is 3 units, the Selling Price is 4 units. Since the Selling Price is greater than the Cost Price, there is a profit.

Profit = SP - CP = 4 - 3 = 1 unit

The profit percentage is calculated on the Cost Price: Profit percentage = (Profit / CP) × 100 Profit percentage = (1/3) × 100 ≈ 33.33%

Explanation

The standard equation of a circle is x² + y² + 2gx + 2fy + c = 0, where the radius r is given by the formula r = √(g² + f² - c).

Comparing the given equation x² + y² - 4x + 6y - 12 = 0 with the standard equation, we get: 2g = -4, which gives g = -2 2f = 6, which gives f = 3 c = -12

Substituting these values into the radius formula: r = √((-2)² + 3² - (-12)) r = √(4 + 9 + 12) r = √25 r = 5

Explanation

The formula for the amount under half-yearly compound interest is:

A = P × (1 + R / 200)ⁿ

where A is the amount, P is the principal, R is the annual interest rate, and n is the number of half-years.

Substituting the given values into the formula:

4,56,976 = 3,90,625 × (1 + 8 / 200)ⁿ

4,56,976 / 3,90,625 = (1 + 1 / 25)ⁿ

4,56,976 / 3,90,625 = (26 / 25)ⁿ

Expressing the fraction on the left side with base 26/25:

25⁴ = 3,90,625 26⁴ = 4,56,976

So, (26 / 25)⁴ = (26 / 25)ⁿ

Comparing the exponents gives n = 4 half-years.

Since there are 2 half-years in a year, the time period is 4 / 2 = 2 years.

Explanation

When interest is compounded half-yearly, the rate of interest is halved and the number of compounding periods is doubled.

Here, Principal (P) is 1,60,000, the annual rate (R) is 10%, and the time (t) is 2 years.

The half-yearly interest rate (r) is 10% / 2 = 5%. The 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 given values into the formula: CI = 1,60,000 × ((1 + 5/100)⁴ - 1) CI = 1,60,000 × ((21/20)⁴ - 1) CI = 1,60,000 × (194481/160000 - 1) CI = 1,60,000 × (34481/160000) CI = 34,481

Explanation

To find the time taken when they work together, we first determine the total work by taking the Least Common Multiple (LCM) of the individual days, which are 5, 6, and 10.

LCM(5, 6, 10) = 30 units.

The efficiency (work done per day) of each person is: Efficiency of A = 30 ÷ 5 = 6 units/day Efficiency of B = 30 ÷ 6 = 5 units/day Efficiency of C = 30 ÷ 10 = 3 units/day

The combined efficiency of A, B, and C working together is: 6 + 5 + 3 = 14 units/day

The number of days they will take to finish the work together is: Total work ÷ Combined efficiency = 30 ÷ 14 = 15/7 = 2 1/7 days.

Explanation

The radius of the base of the cylinder is half of its diameter, so r = 12 ÷ 2 = 6 cm.

The height of the cylinder is 2.45 times its radius, which gives h = 2.45 × 6 = 14.7 cm.

The volume of a right-circular cylinder is calculated using the formula V = πr²h.

Substituting the given values into the formula with π = 22/7:

V = (22/7) × 6² × 14.7

V = (22/7) × 36 × 14.7

V = 22 × 36 × 2.1

V = 792 × 2.1

V = 1663.2 cm³

Explanation

Since y varies directly with x, their relationship is defined by the direct variation formula:

y₁ / x₁ = y₂ / x₂

Substituting the given values into the equation:

61 / 42 = y₂ / 54

Solving for y₂:

y₂ = (61 × 54) / 42

y₂ = (61 × 9) / 7

y₂ = 549 / 7

y₂ ≈ 78.43

Considering the nearest integral value, y is equal to 78.

Explanation

The area of an equilateral triangle with side length a is given by the formula:

Area = (√3 / 4) × a²

Substituting the given side length a = 12 cm into the formula:

Area = (√3 / 4) × 12 × 12 Area = √3 × 3 × 12 Area = 36√3 cm²