SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The centroid G(x, y) of a triangle ABC with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) is calculated using the formula:

x = (x₁ + x₂ + x₃)/3 y = (y₁ + y₂ + y₃)/3

Given that the centroid is G(-1, 2), and two of the vertices are A(3, 0) and B(-1, -1), let the coordinates of vertex C be (x₃, y₃).

Substituting the x-coordinates into the centroid formula:

-1 = (3 - 1 + x₃)/3 -3 = 2 + x₃ x₃ = -5

Substituting the y-coordinates into the centroid formula:

2 = (0 - 1 + y₃)/3 6 = -1 + y₃ y₃ = 7

The coordinates of vertex C are (-5, 7).

Explanation

Using the method of alligation, the ratio of the quantities sold at 14% profit and 24% profit is calculated by comparing their individual profit percentages to the overall mean profit percentage:

Ratio = (24 - 18) : (18 - 14) = 6 : 4 = 3 : 2

This ratio represents the proportion of wheat sold at 14% profit to that sold at 24% profit.

Since 1 quintal is equal to 100 kg, the total quantity of wheat is: 13 quintals = 13 × 100 = 1300 kg

The quantity of wheat sold at 24% profit corresponds to 2 parts out of a total of 5 parts (3 + 2): Quantity = (2 / 5) × 1300 = 520 kg

Explanation

In a general parallelogram, opposite angles are equal to each other. They are not supplementary, meaning their sum is not equal to 180°, unless the parallelogram is a rectangle or a square where all angles are 90°. In any parallelogram, it is the consecutive or adjacent angles that are supplementary, summing to 180° because they are consecutive interior angles between parallel lines. Since the opposite angles of a general parallelogram are equal rather than supplementary, it fits the description.

Explanation

To find which quadratic equation has 5 as a root, we substitute x = 5 into the equation and verify if it satisfies the equation.

For the equation x² - 7x + 10 = 0, substituting x = 5 gives:

5² - 7(5) + 10 = 25 - 35 + 10 = 0

Since the left-hand side equals the right-hand side, 5 is a root of this equation.

Explanation

To find the average bonus, use the formula:

Average Bonus = Total Bonus ÷ Total Number of Employees

First, find the total number of employees: Total Employees = 1 Manager + 8 Executives + 2 Trainees = 11

Next, calculate the bonus paid to each employee category in rupees (where 1 lakh = ₹100,000):

Manager Bonus: Bonus per Manager = 30% of ₹36 lakhs = 0.30 × ₹3,600,000 = ₹1,080,000 Total Manager Bonus = 1 × ₹1,080,000 = ₹1,080,000

Executive Bonus: Bonus per Executive = 30% of ₹12 lakhs = 0.30 × ₹1,200,000 = ₹360,000 Total Executive Bonus = 8 × ₹360,000 = ₹2,880,000

Trainee Bonus: Bonus per Trainee = 20% of ₹4 lakhs = 0.20 × ₹400,000 = ₹80,000 Total Trainee Bonus = 2 × ₹80,000 = ₹160,000

Calculate the overall total bonus paid: Total Bonus = ₹1,080,000 + ₹2,880,000 + ₹160,000 = ₹4,120,000

Now, calculate the average bonus per employee: Average Bonus = ₹4,120,000 ÷ 11 ≈ ₹374,545.45

Explanation

To find the years with equal total exports, we calculate the sum of exports of all three companies for each year.

For the year 2011, the total export is 5000 + 2000 + 2000 = 9000 lakh rupees.

For the year 2015, the total export is 2000 + 2000 + 5000 = 9000 lakh rupees.

Since the total exports for both 2011 and 2015 are equal to 9000 lakh rupees, this pair of years has equal total exports.

Explanation

To find the percentage share of cotton in total exports, first calculate the sum of all exports from the given table:

Total exports = 625 + 625 + 825 + 1000 + 825 = 3900 million USThe export value of cotton is 825 million US.

Now, use the percentage formula:

Percentage share of cotton = (Cotton export / Total exports) × 100

Substituting the values into the formula:

Percentage share of cotton = (825 / 3900) × 100 = 825 / 39 = 21.15% (approximately)

Explanation

Let the efficiency of Mehdi be 1 unit/hour.

Since Mehdi takes 25 hours to complete the work, the total work is: Total Work = Efficiency × Time = 1 × 25 = 25 units

Jahnavi is 50% more efficient than Mehdi, which means Jahnavi's efficiency is: 1 + 50% of 1 = 1.5 units/hour

The combined efficiency of Mehdi and Jahnavi working together is: 1 + 1.5 = 2.5 units/hour

The time taken by both of them to complete the work together is: Time = Total Work / Combined Efficiency = 25 / 2.5 = 10 hours

Explanation

Solving the first inequality: x + 3 ≤ 4x + 4 3 - 4 ≤ 4x - x -1 ≤ 3x x ≥ -1/3

Solving the second inequality: 3(4 - x) - 4 ≥ 2x - 2 12 - 3x - 4 ≥ 2x - 2 8 - 3x ≥ 2x - 2 8 + 2 ≥ 2x + 3x 10 ≥ 5x x ≤ 2

Combining both inequalities yields the range: -1/3 ≤ x ≤ 2

Since 1 lies within this interval, it is a valid value for x.

Explanation

By the Basic Proportionality Theorem, in a triangle, if a line is drawn parallel to one side intersecting the other two sides, then it divides the two sides in the same ratio. Since GH is parallel to EF in triangle DEF, we have:

DG/GE = DH/HF

The point G divides DE in the ratio 3:2, which means:

DG/GE = 3/2

We are given that HF = 8 cm. Substituting these values into the ratio:

3/2 = DH/8

Solving for DH:

DH = (3 × 8) / 2 = 12 cm

The total length of DF is the sum of DH and HF:

DF = DH + HF = 12 + 8 = 20 cm

Explanation

To find the greatest common factor of the monomials 12a⁴b⁶, 18a⁶c², and 36a²b², we compute the greatest common divisor of their numerical coefficients and the lowest power of each variable present in all three terms.

The greatest common divisor of the numerical coefficients 12, 18, and 36 is 6.

For the variable a, the minimum exponent across the terms is min(4, 6, 2) = 2, which gives a².

Since the variables b and c do not appear in all three terms, their minimum exponent is 0, which gives b⁰ = 1 and c⁰ = 1.

Multiplying these components together gives the greatest common factor:

6 × a² × 1 × 1 = 6a²

Explanation

The total surface area (TSA) of a hemisphere is given by the formula 3πr², where r is the radius. The curved surface area (CSA) of a hemisphere is given by the formula 2πr².

Given that the total surface area is 462 sq cm: 3πr² = 462

Dividing both sides by 3 gives: πr² = 154

Using this value, we calculate the curved surface area: CSA = 2πr² = 2 × 154 = 308 sq cm

Explanation

Use the standard relation for overlapping terms in a sequence. Let S_total be the total revenue for 11 years, S_first6 be the sum of the first 6 years, and S_last6 be the sum of the last 6 years.

S_total = 11 × 69 = 759 lakhs S_first6 = 6 × 64 = 384 lakhs S_last6 = 6 × 76 = 456 lakhs

The revenue of the 6th year is counted twice, once in the first 6 years and once in the last 6 years. The revenue of the 6th year is given by:

Revenue of 6th year = S_first6 + S_last6 - S_total Revenue of 6th year = 384 + 456 - 759 Revenue of 6th year = 840 - 759 = 81 lakhs

Explanation

A rhombus with an internal angle of 60° is divided by its shorter diagonal into two congruent equilateral triangles. Since the length of the shorter diagonal is 8 cm, the side length of the rhombus is also 8 cm.

The area of an equilateral triangle with side length a is given by: Area = (√3/4) × a²

Since the rhombus consists of two such equilateral triangles, its total area is: Area = 2 × (√3/4) × a² = (√3/2) × a²

Substituting the side length a = 8 cm: Area = (√3/2) × 8² = (√3/2) × 64 = 32√3 sq cm

Explanation

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

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

In the first case, the discount is 20% and the profit is 34%. Substituting these values into the formula:

MP / CP = (100 + 34) / (100 - 20) = 134 / 80 = 1.675

In the second case, the discount is 15%. Let the new profit percentage be P. Using the relation:

MP / CP = (100 + P) / (100 - 15)

1.675 = (100 + P) / 85

100 + P = 1.675 × 85

100 + P = 142.375

P = 142.375 - 100 = 42.375%

Explanation

Let the original price of the article be 100.

After a reduction of 7%, the new price of the article becomes: 100 - 7 = 93

To restore the price to its original value of 100, the increase needed on the new price is 7.

The percentage increase required is calculated as: Percentage Increase = (Increase / New Price) × 100 Percentage Increase = (7 / 93) × 100 = 700/93 ≈ 7.53%

Explanation

Let the number of 1 rupee, 50 paise, and 10 paise coins be 3x, 2x, and 3x respectively. Since 50 paise is Rs 0.50 and 10 paise is Rs 0.10, the total value equation is:

3x × 1.00 + 2x × 0.50 + 3x × 0.10 = 12.90

Simplifying this expression yields:

3x + 1x + 0.3x = 12.90 4.3x = 12.90 x = 3

The number of 50 paise coins in the bag is 2x, which equals 2 × 3 = 6.

Explanation

For a standard quadratic equation of the form ax² + bx + c = 0, the sum of the roots is given by the formula: Sum of roots = -b/a

For the equation x² - 11x + 18 = 0, we identify the coefficients as: a = 1 b = -11 c = 18

Substituting these values into the formula gives: Sum of roots = -(-11)/1 = 11

Explanation

The fundamental trigonometric identity states: sec²A - tan²A = 1

Using the algebraic difference of squares formula, a² - b² = (a - b)(a + b), we can factorize this identity as: (secA - tanA)(secA + tanA) = 1

Dividing both sides by (secA + tanA) yields: 1/(secA + tanA) = secA - tanA

Explanation

Using the half-angle trigonometric identity, tan(A/2) can be expressed as:

tan(A/2) = sin(A/2) / cos(A/2)

Multiplying both the numerator and the denominator by 2 sin(A/2) yields:

tan(A/2) = (2 sin²(A/2)) / (2 sin(A/2) cos(A/2))

Using standard trigonometric identities, we substitute 2 sin²(A/2) = 1 - cosA and 2 sin(A/2) cos(A/2) = sinA:

tan(A/2) = (1 - cosA) / sinA

Splitting the fraction gives:

tan(A/2) = 1/sinA - cosA/sinA

Since the reciprocal of sinA is cosecA and the ratio cosA/sinA is cotA, we get:

tan(A/2) = cosecA - cotA

Explanation

Using the difference of squares identity, a² - b² = (a - b)(a + b), the expression on the right side of the equation can be simplified directly:

19² - 11² = (19 - 11)(19 + 11) = 8 × 30 = 240

Substituting this value back into the original equation gives:

12x = 240

Dividing both sides by 12 yields:

x = 240 / 12 = 20

Explanation

The secant function is an even function, which satisfies the identity sec(-θ) = sec(θ). Applying this to the given expression gives:

sec(-2π/3) = sec(2π/3)

The secant function is the reciprocal of the cosine function:

sec(2π/3) = 1 / cos(2π/3)

Using the trigonometric identity cos(π - θ) = -cos(θ), we can find the value of cos(2π/3):

cos(2π/3) = cos(π - π/3) = -cos(π/3)

Since the standard trigonometric value of cos(π/3) is ½, we get:

cos(2π/3) = -½

Substituting this back into the reciprocal relation:

sec(-2π/3) = 1 / (-½) = -2

Explanation

The given equation can be written as: (1/4 × x) - (4/5 × 6/7) = -9/7

First, calculate the product of the fractions in the second term: 4/5 × 6/7 = 24/35

Substitute this value back into the equation: x/4 - 24/35 = -9/7

Add 24/35 to both sides of the equation to isolate the term with x: x/4 = 24/35 - 9/7

Convert the fraction -9/7 to a fraction with a denominator of 35 by multiplying both its numerator and denominator by 5: -9/7 = -45/35

Now, combine the terms on the right side: x/4 = 24/35 - 45/35 x/4 = -21/35

Simplify the fraction -21/35 by dividing both the numerator and the denominator by their common factor of 7: x/4 = -3/5

Multiply both sides by 4 to find the value of x: x = -12/5 x = -2.4

Explanation

Average speed is calculated as total distance divided by total time.

Average speed in km/h = 333 km / 2 hours = 166.5 km/h

To convert the speed from kilometers per hour (km/h) to meters per second (m/s), we multiply the speed in km/h by the conversion factor of 5/18.

Average speed in m/s = 166.5 × 5/18 = 832.5 / 18 = 46.25 m/s

Explanation

Let the marked price of the item be ₹100. A discount of 10% reduces the price of the item to 100 - 10 = ₹90.

The customer then receives a 20% cashback on this discounted price. The cashback amount is 20% of 90, which equals ₹18.

The final effective price paid by the customer is 90 - 18 = ₹72.

The total effective discount is the difference between the original marked price and the final price paid, which is 100 - 72 = 28%.

Alternatively, using the successive discount formula: Effective Discount = d₁ + d₂ - (d₁ × d₂)/100 Effective Discount = 10 + 20 - (10 × 20)/100 = 30 - 2 = 28%

Explanation

For any two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding side lengths.

Perimeter(ΔABC) ÷ Perimeter(ΔDEF) = AB ÷ DE

Substituting the given values:

Ratio = 10 ÷ 6 = 5 ÷ 3

This gives the ratio of the perimeters as 5:3.

Explanation

The compound interest amount formula is A = P(1 + r)ⁿ, where P is the principal, r is the rate of interest per period, and n is the number of periods.

Here, the interest rate is r = 20% = 0.20 per half-year.

For the first deposit of ₹7600 made on 1st January, the interest is compounded for two half-years (n = 2) up to 31st December: A₁ = 7600 × (1 + 0.20)² = 7600 × 1.44 = ₹10944

For the second deposit of ₹7600 made on 1st July, the interest is compounded for one half-year (n = 1) up to 31st December: A₂ = 7600 × (1 + 0.20)¹ = 7600 × 1.20 = ₹9120

The total accumulated amount at the end of the year is: Total Amount = A₁ + A₂ = 10944 + 9120 = ₹20064

The total principal deposited during the year is: Total Principal = 7600 + 7600 = ₹15200

The total interest gained by the customer is the difference between the total accumulated amount and the total principal deposited: Interest Gained = 20064 - 15200 = ₹4864

Explanation

To find the point where a line intercepts the Y-axis, set the x-coordinate to 0.

Substituting x = 0 into the given equation 3x + 2y = -12:

3(0) + 2y = -12

2y = -12

y = -6

This gives the coordinates of the Y-intercept as (0, -6).

Explanation

The total value of the stock is calculated by summing the products of the quantity and the average cost for each item.

Mobile Phones value = 29 × 18000 = 5,22,000 Cameras value = 22 × 6000 = 1,32,000 TVs value = 63 × 51000 = 32,13,000 Refrigerators value = 45 × 49000 = 22,05,000 ACs value = 27 × 25000 = 6,75,000

Total stock value = 5,22,000 + 1,32,000 + 32,13,000 + 22,05,000 + 6,75,000 = 67,47,000

To convert the total value from rupees to lakh rupees, divide by 1,00,000: 67,47,000 ÷ 1,00,000 = 67.47 lakhs

Explanation

For the year 2012, the ratio of imports to exports is 1.4. Since the imports in 2012 was ₹600 crores, the exports in 2012 is: Exports in 2012 = 600 ÷ 1.4 = 3000/7 crores

The combined exports in 2012 and 2013 is ₹2400 crores. This gives the exports in 2013 as: Exports in 2013 = 2400 - 3000/7 = 13800/7 crores

For the year 2013, the ratio of imports to exports is 0.7. The imports in 2013 is: Imports in 2013 = 0.7 × (13800/7) = (7/10) × (13800/7) = 1380 crores

Explanation

To find the increase in height from the 7th birthday to the 11th birthday, identify the heights recorded at both ages from the data table. The height of the child at age 7 is 125 cms, and the height at age 11 is 160 cms.

The increase in height is the difference between these two values:

160 cms - 125 cms = 35 cms

Subtracting the height at age 7 from the height at age 11 yields an increase of 35 cms.

Explanation

The total time duration between 10 pm and 6 am is 8 hours.

From the given table, the percentage of time spent in Light sleep is 10% and in Extremely light sleep is 5%.

The combined percentage of time spent in either Light sleep or Extremely light sleep is 10% + 5% = 15%.

To find the actual duration, calculate 15% of the total 8 hours:

Duration = 8 × 0.15 = 1.2 hours

Explanation

Using the algebraic identity for the difference of squares, a² - b² = (a - b)(a + b), the right side of the equation can be simplified directly.

4x² = 15² - 9² 4x² = (15 - 9)(15 + 9) 4x² = 6 × 24 4x² = 144

Dividing both sides of the equation by 4:

x² = 36

Taking the square root of both sides:

x = 6

Explanation

Subtracting the first equation, 2x - 5y = 5, from the second equation, 2x - y = 9, eliminates x:

(2x - y) - (2x - 5y) = 9 - 5 4y = 4 y = 1

Substituting y = 1 back into the second equation:

2x - 1 = 9 2x = 10 x = 5

Using these values, the expression x - y is:

5 - 1 = 4

Explanation

Using the trigonometric property of negative angles, cosec(-θ) = -cosec(θ). This gives:

cosec(-7π/6) = -cosec(7π/6)

We can rewrite the angle 7π/6 as π + π/6. Using the identity cosec(π + θ) = -cosec(θ) because the cosecant function is negative in the third quadrant, we have:

cosec(7π/6) = cosec(π + π/6) = -cosec(π/6)

Substituting this value back into the original expression:

cosec(-7π/6) = -[-cosec(π/6)] = cosec(π/6)

Since sin(π/6) = ½, its reciprocal cosec(π/6) is equal to 2.

Explanation

In a general trapezium, only one pair of opposite sides is parallel, and the non-parallel sides are of unequal lengths. Because of this lack of symmetry, drawing either diagonal does not divide the quadrilateral into two congruent triangles. When both diagonals are drawn, they intersect to divide the trapezium into four smaller triangles. The two triangles adjacent to the parallel bases are similar but not congruent because their corresponding side lengths are different. The other two triangles adjacent to the non-parallel sides have equal areas but are not congruent. Consequently, the diagonals of a trapezium do not form at least two congruent triangles.

Explanation

Since segment PQ is parallel to side BC, triangle APQ is similar to triangle ABC.

The ratio of the corresponding sides is: AP/AB = AP / (AP + PB)

Using the ratio AP:PB = 1:4, we get: AP/AB = 1 / (1 + 4) = 1/5

The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides: Area(Δ APQ) / Area(Δ ABC) = (AP/AB)² = (1/5)² = 1/25

Given that the area of Δ APQ is 4 sq cm, we can find the area of Δ ABC: Area(Δ ABC) = 25 × Area(Δ APQ) = 25 × 4 = 100 sq cm

The area of trapezium PQCB is obtained by subtracting the area of Δ APQ from the total area of Δ ABC: Area(trapezium PQCB) = Area(Δ ABC) - Area(Δ APQ) = 100 - 4 = 96 sq cm

Explanation

To find the range of x, we solve the system of inequalities.

From the first inequality: 2x - 4 ≤ 2 - x/3 Multiplying the entire equation by 3 to clear the fraction: 6x - 12 ≤ 6 - x 7x ≤ 18 x ≤ 18/7

From the second inequality: 2(2x + 5) > 3x - 5 4x + 10 > 3x - 5 x > -15

Combining these two results, we obtain the interval: -15 < x ≤ 18/7

The value -14 satisfies this combined inequality as it is strictly greater than -15 and less than or equal to 18/7.

Explanation

The fraction of the fence yet to be painted is given by the formula: Remaining fraction = 1 - (Time worked / Total time required)

Substituting the given values: Remaining fraction = 1 - 6/24 = 1 - ¼ = ¾ = 0.75

Explanation

The given mathematical statement can be written as the algebraic equation:

x/6 - (7/2 × 3/7) = -7/4

First, simplify the multiplication inside the parenthesis:

7/2 × 3/7 = 3/2

Now, substitute this value back into the equation:

x/6 - 3/2 = -7/4

Add 3/2 to both sides of the equation to isolate the term with x:

x/6 = 3/2 - 7/4

Find a common denominator to subtract the fractions on the right side:

x/6 = 6/4 - 7/4

x/6 = -1/4

Multiply both sides by 6 to solve for x:

x = -6/4

x = -1.5

Explanation

The mean proportion of two numbers x and y is given by the formula √(xy). Given: √(xy) = 9 xy = 81 x = 81/y

The third proportion of x and y is given by the formula y²/x. Given: y²/x = 243 y² = 243x

Substituting the value of x into the third proportion equation: y² = 243 × (81/y) y³ = 243 × 81 y³ = 3⁵ × 3⁴ y³ = 3⁹ y = 3³ = 27

Using the value of y to find x: x = 81 / 27 = 3

The values of x and y are 3 and 27.

Explanation

If the discount on the electricity bill is 21%, the percentage of the bill still to be paid is: 100% - 21% = 79%

Let the original bill amount be x. The relation is: 79% of x = 1817 0.79 × x = 1817 x = 1817 / 0.79 x = 2300

Explanation

Let the revenues of the seven years be represented from R₁ to R₇.

The sum of the revenues of all 7 years is: 7 × 79 = 553 lakhs

The sum of the revenues of the first 4 years is: 4 × 74 = 296 lakhs

The sum of the revenues of the last 4 years is: 4 × 86 = 344 lakhs

Adding the sum of the first 4 years and the last 4 years yields the total sum of all 7 years with the 4th year counted twice. Subtracting the sum of all 7 years from this total gives the revenue of the 4th year: (296 + 344) - 553 = 640 - 553 = 87 lakhs

Explanation

The Cost Price of the dried apricots is calculated using the relation: Cost Price = Selling Price × 100 / (100 - Loss%)

Substituting the given values: Cost Price = 1210 × 100 / (100 - 12) = 1210 × 100 / 88 = 1375

So, the Cost Price is ₹1375 per kg.

If the dried apricots are sold at the new Selling Price of ₹1331 per kg, the loss incurred is: Loss = Cost Price - New Selling Price = 1375 - 1331 = 44

The loss percentage is: Loss% = (Loss / Cost Price) × 100 = (44 / 1375) × 100 = 3.2%

Explanation

The formula for the discount percentage is: Discount % = (Marked Price - Selling Price) / Marked Price × 100

Substituting the given values into the formula: Marked Price = Rs 2000 Selling Price = Rs 1600

Discount % = (2000 - 1600) / 2000 × 100 Discount % = 400 / 2000 × 100 Discount % = 1/5 × 100 Discount % = 20%

Explanation

Using the standard trigonometric identity: cosec²A - cot²A = 1

Applying the algebraic difference of squares formula, a² - b² = (a - b)(a + b), the expression becomes: (cosecA - cotA)(cosecA + cotA) = 1

Substituting the given relation cosecA + cotA = x into this identity: (cosecA - cotA) × x = 1

Rearranging the equation to solve for x: x = 1/(cosecA - cotA)

Explanation

Using the formula for Simple Interest, SI = P × R × T / 100, where P is the principal, R is the rate of interest per annum, and T is the time period in years:

Total Interest = SI received from Jaabir + SI received from Kabir

Substitute the given values into the equation: 1422 = (10800 × R × 3 / 100) + (7500 × R × 2 / 100) 1422 = 324R + 150R 1422 = 474R R = 1422 / 474 R = 3

The rate of interest per annum is 3%.

Explanation

Using the standard trigonometric definitions: tan A = sin A / cos A sec A = 1 / cos A

Substituting these expressions into option A: tan A / (1 + sec A) = (sin A / cos A) / (1 + 1 / cos A) = (sin A / cos A) / ((cos A + 1) / cos A) = sin A / (1 + cos A)

Applying the half-angle trigonometric formulas: sin A = 2 sin(A/2) cos(A/2) 1 + cos A = 2 cos²(A/2)

Substituting these half-angle formulas: sin A / (1 + cos A) = 2 sin(A/2) cos(A/2) / 2 cos²(A/2) = sin(A/2) / cos(A/2) = tan(A/2)

Explanation

By Thales's Theorem (Basic Proportionality Theorem), since ST is parallel to QR in triangle PQR, the line ST divides the sides PQ and PR in the same ratio:

PS / SQ = PT / TR

The total length of PR is PT + TR = 12.5 cm, which means PT = 12.5 - TR.

Substituting the given values into the ratio:

6 / 9 = (12.5 - TR) / TR

Simplifying 6 / 9 to 2 / 3:

2 / 3 = (12.5 - TR) / TR

Cross-multiplying to solve for TR:

2 × TR = 3 × (12.5 - TR) 2 × TR = 37.5 - 3 × TR 5 × TR = 37.5 TR = 37.5 / 5 TR = 7.5 cm

Explanation

The number of small spheres that can be cast from a larger sphere is obtained by dividing the volume of the larger sphere by the volume of a single small sphere.

Volume of a sphere = 4/3 π r³

For the larger sphere: Radius (R) = Diameter / 2 = 16 / 2 = 8 cm

For the smaller sphere: Radius (r) = 2 cm

Number of smaller balls (n) = Volume of the larger ball / Volume of a smaller ball n = (4/3 π R³) / (4/3 π r³) n = R³ / r³ n = (R / r)³ n = (8 / 2)³ n = 4³ n = 64