SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

We can find the value of x using basic trigonometric identities. First, we use the property that sine is an odd function, which means sin(-θ) = -sin(θ). This gives:

sin(-330°) = -sin(330°)

Next, we can express the angle 330° as 360° - 30°. Since sin(360° - θ) = -sin(θ) in the fourth quadrant, we have:

-sin(330°) = -sin(360° - 30°) = -(-sin(30°)) = sin(30°)

The value of sin(30°) is ½. Therefore, the value of x is ½.

Explanation

Using the algebraic identity for the difference of squares: x² - y² = (x - y)(x + y)

Substituting the given values into the formula: x - y = -7 x + y = 13

x² - y² = (-7) × 13 = -91

Explanation

Average speed when equal distances are covered at two different speeds, x and y, is given by the formula:

Average speed = 2xy / (x + y)

Substituting the given values, x = 34 km/h and y = 66 km/h, into the formula:

Average speed = (2 × 34 × 66) / (34 + 66) Average speed = 4488 / 100 Average speed = 44.88 km/h

Explanation

The effective percentage for successive discounts of a% and b% can be calculated using the successive discount formula:

Effective discount = a + b - ab/100

Substituting the given values of a = 30 and b = 10 into the formula:

Effective discount = 30 + 10 - (30 × 10)/100 Effective discount = 40 - 300/100 Effective discount = 40 - 3 Effective discount = 37%

Explanation

The volume of sand remains constant when poured from the cylindrical vessel into a conical shape. Therefore, the volume of the cylinder is equal to the volume of the cone.

The formula for the volume of a cylinder is V = πr₁²h₁, where r₁ is the radius and h₁ is the height of the cylinder.

The formula for the volume of a cone is V = ⅓πr₂²h₂, where r₂ is the radius and h₂ is the height of the cone.

Given that r₁ = 5 cm, h₁ = 7 cm, and r₂ = 8 cm:

Volume of the cylinder = π × 5² × 7 = 175π cm³

Volume of the cone = ⅓ × π × 8² × h₂ = 64/3 × π × h₂ cm³

Equating the two volumes:

175π = 64/3 × π × h₂

175 = 64/3 × h₂

h₂ = 175 × 3 / 64

h₂ = 525 / 64 ≈ 8.2 cm

Explanation

By using standard trigonometric identities, we know that cosec²A - 1 = cot²A.

Substituting this into the given expression: cosecA / √(cosec²A - 1) = cosecA / √(cot²A) = cosecA / cotA

Since cosecA = 1/sinA and cotA = cosA/sinA, the expression simplifies as follows: cosecA / cotA = (1/sinA) / (cosA/sinA) = 1/cosA = secA

Explanation

In a right-angled triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuse into two segments such that the square of the altitude is equal to the product of these two segments. This relation is given by the geometric mean theorem:

BD² = AD × DC

Substituting the given values into the formula:

BD² = 8 × 50 BD² = 400 BD = √400 = 20 cm

Explanation

We are given: x = secA - tanA

Using the trigonometric identities secA = 1/cosA and tanA = sinA/cosA, we can write: x = (1 - sinA)/cosA

Squaring both sides: x² = (1 - sinA)²/cos²A

Using the identity cos²A = 1 - sin²A: x² = (1 - sinA)²/(1 - sin²A)

The denominator can be factored as (1 - sinA)(1 + sinA): x² = (1 - sinA)²/[(1 - sinA)(1 + sinA)]

Canceling the common term (1 - sinA) from the numerator and denominator: x² = (1 - sinA)/(1 + sinA)

Taking the square root on both sides: x = √[(1 - sinA)/(1 + sinA)]

Explanation

A diagonal of a rectangle divides it into two right-angled triangles. For any rectangle ABCD with diagonal AC, we can compare triangles ADC and CBA. The opposite sides of a rectangle are equal, which means AD = CB and DC = BA. The diagonal AC serves as a common side for both triangles. By the Side-Side-Side (SSS) congruence criterion, triangle ADC is congruent to triangle CBA (ΔADC ≅ ΔCBA). Thus, the diagonal of a rectangle always divides it into two congruent triangles.

Explanation

Let the two fractions be x and y. A quadratic equation with roots x and y is given by the formula: t² - (x + y)t + xy = 0

Substituting the given sum of -32/15 and product of -4: t² - (-32/15)t + (-4) = 0 t² + (32/15)t - 4 = 0

Multiplying the entire equation by 15 to clear the fraction: 15t² + 32t - 60 = 0

Splitting the middle term to factor the quadratic: 15t² + 50t - 18t - 60 = 0 5t(3t + 10) - 6(3t + 10) = 0 (5t - 6)(3t + 10) = 0

Solving for t yields the two roots: t = 6/5 and t = -10/3

Thus, the two fractions are 6/5 and -10/3.

Explanation

The value of the total stock is the sum of the products of the quantity of stock and its average cost for each item.

Value of Mobile Phones = 52 × 15000 = Rs 780000 Value of Cameras = 76 × 8000 = Rs 608000 Value of TVs = 31 × 51000 = Rs 1581000 Value of Refrigerators = 71 × 26000 = Rs 1846000 Value of ACs = 70 × 20000 = Rs 1400000

Total value of stock = 780000 + 608000 + 1581000 + 1846000 + 1400000 = Rs 6215000

To find the value in lakh rupees, divide the total value by 100000: Value in lakh rupees = 6215000 / 100000 = 62.15

Explanation

The ratio of Import to Export in 2012 is 0.8. Since the imports in 2012 was Rs. 1200 crores, the exports in 2012 can be calculated as: Export in 2012 = Import in 2012 / 0.8 = 1200 / 0.8 = 1500 crores.

Given that the total exports in 2012 and 2013 together was Rs. 3200 crores, the exports in 2013 is: Export in 2013 = 3200 - 1500 = 1700 crores.

The ratio of Import to Export in 2013 is 1.2. Using this ratio, the imports in 2013 can be calculated as: Import in 2013 = Export in 2013 × 1.2 = 1700 × 1.2 = 2040 crores.

Explanation

The height of the child on the 5th birthday is 105 cms, and the height of the child on the 16th birthday is 175 cms.

The increase in height from the 5th birthday to the 16th birthday is calculated as:

Increase in height = Height on 16th birthday - Height on 5th birthday Increase in height = 175 cms - 105 cms = 70 cms

Explanation

The total duration of sleep tracked by the fitness band from 10:00 PM to 6:00 AM is 8 hours.

The values in the data table represent percentages, as their sum is 25 + 10 + 10 + 15 + 40 = 100%.

The combined percentage of time the user spent either Dreaming or Awake is: 10% + 40% = 50%

The total time spent Dreaming or Awake is: 8 hours × 50% = 4 hours

Explanation

Let the original number be x.

The correct value is 7/6 x, and the wrong value obtained is 6/7 x.

The formula for percentage error is: Percentage Error = |Correct Value - Wrong Value| ÷ Correct Value × 100

Substituting the values: Percentage Error = (7/6 x - 6/7 x) ÷ (7/6 x) × 100 Percentage Error = (13/42 x) ÷ (7/6 x) × 100 Percentage Error = 13/42 × 6/7 × 100 Percentage Error = 13/49 × 100 Percentage Error = 1300/49 % ≈ 26.53%

Explanation

The given compound inequality can be solved by breaking it down into two separate inequalities.

For the first part: 3x + 4(1-x) > 5x - 2 3x + 4 - 4x > 5x - 2 -x + 4 > 5x - 2 6 > 6x x < 1

For the second part: 5x - 2 > 3x - 4 2x > -2 x > -1

Combining the two inequalities gives the range: -1 < x < 1

Substituting x = 0 into the original compound inequality yields: 3(0) + 4(1 - 0) > 5(0) - 2 > 3(0) - 4 4 > -2 > -4

Since 4 is greater than -2 and -2 is greater than -4, the inequality holds true.

Explanation

Using the trigonometric identity for angles in the second quadrant, cos(180° - θ) = -cos θ:

cos 135° = cos(180° - 45°) = -cos 45°

Since the standard value of cos 45° is 1/√2, this gives:

cos 135° = -1/√2

Explanation

The total surface area of the canvas needed to construct the tent is the sum of the curved surface area of the cylindrical part and the curved surface area of the conical part.

Curved surface area of the cylindrical part = 2πrh Curved surface area of the conical part = πrl

Thus, the total surface area of the tent is: Area = 2πrh + πrl = πr(2h + l)

Substituting the given values: r = 7 m, h = 8 m, and l = 12 m with π = 22/7: Area = (22/7) × 7 × (2 × 8 + 12) Area = 22 × (16 + 12) Area = 22 × 28 Area = 616 sq m

An additional 20% extra canvas is required for folding and stitching. Total canvas required = 616 × (1 + 20/100) Total canvas required = 616 × 1.2 Total canvas required = 739.2 sq m

Explanation

The effective discount can be directly calculated using the standard formula for buy-and-get-free offers:

Effective Discount % = (Number of free articles / Total number of articles) × 100

In this case, 2 shirts are offered free on purchasing 3 shirts. This means the customer receives a total of 5 shirts (3 paid + 2 free) while only paying for 3.

Substituting these values into the formula:

Effective Discount % = (2 / 5) × 100 = 40%

The pricing of ₹600 per shirt is redundant for calculating the percentage discount, as the discount percentage remains 40% regardless of the individual price.

Explanation

Total quantity of rice purchased is 12 quintals.

Since 15% of the rice is lost in transportation, the remaining quantity available for sale is: Remaining rice = 12 × (1 - 0.15) = 12 × 0.85 = 10.2 quintals

The total cost price of the rice is Rs 1,870. To achieve a profit of 20%, the total selling price is calculated as: Total selling price = Cost price × (1 + Profit percentage) Total selling price = 1870 × 1.20 = Rs 2,244

The selling rate per quintal is the total selling price divided by the remaining quantity of rice: Selling rate = 2244 / 10.2 = Rs 220 per quintal

Explanation

Total collection is the product of the ticket price and the number of visitors:

Total Collection = Ticket Price × Number of Visitors

Let the initial ticket price and the new ticket price be represented as 13x and 16x respectively. Similarly, let the initial number of visitors and the new number of visitors be represented as 11y and 9y.

The initial total collection is 13x × 11y = 143xy.

The new total collection is 16x × 9y = 144xy.

The ratio of the initial total collection to the new total collection is 143xy : 144xy, which simplifies directly to 143 : 144. Since the new value of 144 is larger than the initial value of 143, the collection has increased.

Explanation

To simplify the expression 486b³x²a⁴z³ / 27a³b²z, divide the numerical coefficients and apply the laws of exponents for the variables, which state that pᵐ / pⁿ = pᵐ⁻ⁿ.

Dividing the coefficients: 486 / 27 = 18

Dividing the variable a: a⁴ / a³ = a⁴⁻³ = a

Dividing the variable b: b³ / b² = b³⁻² = b

Dividing the variable x: x² / 1 = x²

Dividing the variable z: z³ / z = z³⁻¹ = z²

Combining all the simplified terms yields 18bx²az².

Explanation

When a point (x, y) is reflected in the y-axis, its x-coordinate changes sign while its y-coordinate remains unchanged. The transformation rule for reflection in the y-axis is:

(x, y) → (-x, y)

Substituting the coordinates of the given point (4, -5) into the transformation rule:

x = 4 becomes -4 y = -5 remains -5

This yields the reflected coordinates as (-4, -5).

Explanation

To find the sum of the two fractions 5/9 and 7/8, determine a common denominator. Since 9 and 8 are coprime, their least common multiple is 9 × 8 = 72.

Adjusting the numerators accordingly: (5 × 8) / (9 × 8) = 40/72 (7 × 9) / (8 × 9) = 63/72

Adding the fractions: 40/72 + 63/72 = (40 + 63) / 72 = 103/72

Explanation

The fundamental trigonometric identity states that sin²A + cos²A = 1. Rearranging this identity gives cos²A = 1 - sin²A. Taking the square root of both sides yields cosA = √(1 - sin²A). Since the given expression is x = √(1 - sin²A), substituting the identity results in x = cosA.

Explanation

The equation of a line in slope-intercept form is y = mx + c, where m is the slope of the line and c is the y-intercept.

The slope m is given by tan(θ), where θ is the angle made with the positive x-axis. With θ = 45°: m = tan(45°) = 1

Substituting the slope m = 1 and the y-intercept c = -3 into the equation: y = 1x - 3 y = x - 3

Rearranging the terms yields: x - y = 3

Explanation

Let the speed of the Express train be E km/hr and the speed of the Rajdhani train be R km/hr.

The time taken by the Express train to travel 672 km is 672/E hours, and the time taken by the Rajdhani train is 672/R hours.

Based on the first condition: 672/E - 672/R = 14

Based on the second condition, when the speed of the Express train is doubled: 672/R - 672/2E = 8

By representing 672/E as x and 672/R as y, the system of equations is: x - y = 14 y - x/2 = 8

From the first equation, x = y + 14. Substituting this value into the second equation: y - (y + 14)/2 = 8 y - y/2 - 7 = 8 y/2 = 15 y = 30

Substituting the value of y back to find the speed of the Rajdhani train: 672/R = 30 R = 672/30 = 22.4 km/hr

Explanation

Let the first, second, and third numbers be F, S, and T respectively.

According to the problem: F = 2S F = 3T

To simplify calculations, we can express all three numbers in terms of a single variable x. Since F is a multiple of both 2 and 3, let F = 6x. This gives: S = 3x T = 2x

The average of the three numbers is given as 517: (F + S + T) / 3 = 517 (6x + 3x + 2x) / 3 = 517 11x / 3 = 517 11x = 1551 x = 141

The difference between the first and the third number is: F - T = 6x - 2x = 4x 4 × 141 = 564

Explanation

Under compound interest, the amount at the end of any year acts as the principal for the succeeding year. The relation between the amounts at the end of two consecutive years is:

A₃ = A₂ × (1 + R/100)

Here, A₂ is the amount at the end of the 2nd year (Rs 29160), A₃ is the amount at the end of the 3rd year (Rs 31492.8), and R is the rate of interest.

Substituting the given values:

31492.8 = 29160 × (1 + R/100)

1 + R/100 = 31492.8 / 29160

1 + R/100 = 1.08

R/100 = 0.08

R = 8 percent

Explanation

Using the trigonometric identity tan A = sin A / cos A, the given expression can be written as:

sin A tan A / (1 - cos A) = sin² A / [cos A (1 - cos A)]

Substituting sin² A = 1 - cos² A:

= (1 - cos² A) / [cos A (1 - cos A)]

Since 1 - cos² A can be factored as (1 - cos A)(1 + cos A) using the algebraic identity a² - b² = (a - b)(a + b), we get:

= (1 - cos A)(1 + cos A) / [cos A (1 - cos A)]

Canceling the common term (1 - cos A) in the numerator and denominator:

= (1 + cos A) / cos A

Dividing each term in the numerator by the denominator:

= 1/cos A + cos A/cos A

= sec A + 1

Explanation

Using the algebraic identity for the square of a sum:

(x + y)² = x² + y² + 2xy

Substitute the given values x + y = 12 and x² + y² = 94 into the formula:

12² = 94 + 2xy

144 = 94 + 2xy

Subtract 94 from both sides to isolate the term with xy:

2xy = 144 - 94

2xy = 50

Divide both sides by 2:

xy = 25

Explanation

A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. According to the inscribed angle theorem, the angle subtended by an arc at the center of a circle is twice the angle subtended by it at any point on the remaining part of the circle. For any two opposite angles of a cyclic quadrilateral, their corresponding intercepted arcs together form the complete circumference of the circle, which subtends a total angle of 360° at the center. Therefore, the sum of these two opposite inscribed angles is half of 360°, which equals 180°, making them supplementary.

Explanation

Let the efficiency of Sahay be 1 unit/day. Since Mahanath works 2 times as fast as Sahay, Mahanath's efficiency is 2 units/day.

The total work is the product of Sahay's efficiency and the time he takes to complete the job: Total Work = 1 unit/day × 21 days = 21 units

The combined efficiency of Mahanath and Sahay working together is: Combined Efficiency = 2 units/day + 1 unit/day = 3 units/day

The number of days they take together to complete the job is: Time = Total Work ÷ Combined Efficiency = 21 units ÷ 3 units/day = 7 days

Explanation

Let the tens digit of the two-digit number be x and the units digit be y. The number can be represented as 10x + y.

According to the given conditions, the sum of the digits is 11: x + y = 11

When 45 is added to the number, the digits are interchanged: 10x + y + 45 = 10y + x 9y - 9x = 45 y - x = 5

We now have a system of two linear equations: x + y = 11 y - x = 5

Adding these two equations: 2y = 16 y = 8

Substituting y = 8 into the first equation: x + 8 = 11 x = 3

Thus, the original number is 38.

Explanation

To find the percentage of students who chose Engineering who are girls, use the percentage formula:

Percentage = (Number of girls in Engineering / Total students in Engineering) × 100

From the table: Number of girls who chose Engineering = 30 Number of boys who chose Engineering = 40

Total number of students in Engineering = 40 + 30 = 70

Substitute the values into the formula:

Percentage = (30 / 70) × 100 = 300 / 7 = 42.86%

Explanation

The cumulative production of cars up to the end of March is 1650, and the cumulative production up to the end of May is 2540.

The number of cars manufactured in the months of April and May is calculated by subtracting the cumulative production at the end of March from the cumulative production at the end of May:

Production in April and May = Cumulative production in May - Cumulative production in March Production in April and May = 2540 - 1650 = 890

Alternatively, the individual production for each month can be computed and summed: Production in April = 2090 - 1650 = 440 Production in May = 2540 - 2090 = 450 Total production in April and May = 440 + 450 = 890

Explanation

The total distance jogged during the week is the sum of the daily distances: 3.5 + 0.5 + 2 + 1.5 + 1.5 + 1 + 2 = 12 km

Since 400 calories are burned by jogging 5 km, the rate of calorie burn is: 400 / 5 = 80 calories per km

Multiplying the total distance by this rate gives the total calories burned during the week: 12 × 80 = 960 calories

Explanation

To find the percentage of Rent and Taxes relative to the total yearly expenses, we use the percentage formula:

Percentage = (Expenses on Rent and Taxes ÷ Total Expenses) × 100

First, calculate the total yearly expenses by summing the expenses of all the given items: Total Expenses = 5 + 4 + 2 + 6 + 4 = 21 lakhs

Next, find the combined expenses of Rent and Taxes: Rent and Taxes = 2 + 4 = 6 lakhs

Now, substitute these values into the formula: Percentage = (6 ÷ 21) × 100 Percentage = (2 ÷ 7) × 100 Percentage ≈ 28.57%

Explanation

To solve the system of inequalities, it can be split into two parts.

First part: 2(5x - 4) - 2 < 2x - 1 10x - 8 - 2 < 2x - 1 10x - 10 < 2x - 1 8x < 9 x < 9/8

Second part: 2x - 1 < 4x - 2 1 < 2x x > 1/2

Combining these two inequalities gives the range of x: 1/2 < x < 9/8 0.5 < x < 1.125

The value x = 1 satisfies this relation. Substituting x = 1 into the original inequality gives: 2(5(1) - 4) - 2 < 2(1) - 1 < 4(1) - 2 0 < 1 < 2

Explanation

First, expand the squared term: (2x + 5)² = 4x² + 20x + 25

Next, multiply this result by (4x - 1): (4x² + 20x + 25)(4x - 1) = 4x²(4x - 1) + 20x(4x - 1) + 25(4x - 1) = 16x³ - 4x² + 80x² - 20x + 100x - 25 = 16x³ + 76x² + 80x - 25

Now, subtract the second polynomial from this result: (16x³ + 76x² + 80x - 25) - (3x³ - 16x² + 25x - 21) = (16 - 3)x³ + (76 - (-16))x² + (80 - 25)x + (-25 - (-21)) = 13x³ + (76 + 16)x² + 55x - 25 + 21 = 13x³ + 92x² + 55x - 4

Explanation

When the sphere is completely immersed in the water, the volume of water displaced in the cylindrical vessel is exactly equal to the volume of the sphere.

The volume of the sphere is given by the formula: V = 4/3 × π × r_s³ where r_s is the radius of the sphere, which is 7 cm.

The volume of the displaced water in the cylinder is given by: V = π × r_c² × h where r_c is the radius of the cylinder, which is 8 cm, and h is the rise in the water level.

Equating the two volumes: π × r_c² × h = 4/3 × π × r_s³

By cancelling π from both sides, the equation becomes: r_c² × h = 4/3 × r_s³

Substituting the given values into the equation: 8² × h = 4/3 × 7³ 64 × h = 4/3 × 343 64 × h = 1372/3 h = 1372 / (3 × 64) h = 1372 / 192 h ≈ 7.15 cm

Explanation

The algebraic identity for the difference of two cubes is: a³ - b³ = (a - b)³ + 3ab(a - b)

Substituting the given values, a - b = 6 and ab = 16, into the identity: a³ - b³ = 6³ + 3 × 16 × 6 a³ - b³ = 216 + 288 a³ - b³ = 504

Alternatively, by inspection, we find two numbers whose difference is 6 and product is 16. These numbers are a = 8 and b = 2, because 8 - 2 = 6 and 8 × 2 = 16. Substituting these values: a³ - b³ = 8³ - 2³ a³ - b³ = 512 - 8 a³ - b³ = 504

Explanation

Let Jaikishen's efficiency be 1 unit per hour. Since he completes the work in 15 hours, the total work is 15 units.

Jhanvi is 50% more efficient than Jaikishen, which means her efficiency is 1.5 units per hour.

When working together, their combined efficiency is 1 + 1.5 = 2.5 units per hour.

The time taken by both of them together to complete the work is: Time = Total Work ÷ Combined Efficiency = 15 ÷ 2.5 = 6 hours.

Explanation

Let the number be x.

Increasing the number by 34 makes it 117% of itself. This means the increase of 34 represents the 17% increase over the original 100% value of the number.

We can set up the percentage relation: 17% of x = 34 (17 / 100) × x = 34 x = (34 × 100) / 17 x = 2 × 100 x = 200

Explanation

The coordinates of the midpoint of a line segment joining two points (x₁, y₁) and (x₂, y₂) are given by the midpoint formula:

( (x₁ + x₂)/2, (y₁ + y₂)/2 )

Substituting the coordinates of C(0, -3) and D(8, 5) into this formula:

x-coordinate = (0 + 8)/2 = 4

y-coordinate = (-3 + 5)/2 = 1

This yields the midpoint coordinates as (4, 1).

Explanation

The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula m = (y₂ - y₁) / (x₂ - x₁).

The slope m₁ of the line passing through (-1, x) and (3, 0) is: m₁ = (0 - x) / (3 - (-1)) = -x / 4

The slope m₂ of the line passing through (5, 2) and (4, 4) is: m₂ = (4 - 2) / (4 - 5) = 2 / -1 = -2

Two lines are perpendicular if the product of their slopes is -1, which is represented by the formula: m₁ × m₂ = -1

Substituting the slopes into this relation: (-x / 4) × (-2) = -1 2x / 4 = -1 x / 2 = -1 x = -2

Explanation

Let the measure of the angle be x.

Two angles are supplementary if their sum is 180°. Therefore, the supplementary angle of x is 180° - x.

According to the given condition, the angle is smaller than its supplementary angle by 70°:

(180° - x) - x = 70°

180° - 2x = 70°

2x = 180° - 70°

2x = 110°

x = 55°

Thus, the measure of the angle is 55°.

Explanation

The effective annual rate of interest (r_eff) is calculated using the formula:

r_eff = (1 + r/m)ᵐ - 1

where r is the nominal annual interest rate and m is the number of compounding periods per year.

Given that the nominal rate r is 7% per annum (0.07) and compounding is done half-yearly (m = 2):

r_eff = (1 + 0.07/2)² - 1 r_eff = (1 + 0.035)² - 1 r_eff = 1.035² - 1 r_eff = 1.071225 - 1 r_eff = 0.071225

Converting this value to a percentage yields:

r_eff = 0.071225 × 100 = 7.1225%

Rounding to two decimal places, the effective annual rate is 7.12%.

Explanation

Let the two fractions be x and y.

The sum of the fractions is x + y = -13/40, and their product is xy = -9/10.

For the fractions 4/5 and -9/8:

Sum = 4/5 + (-9/8) To add these fractions, we find the common denominator, which is 40: 4/5 = 32/40 -9/8 = -45/40 Sum = 32/40 - 45/40 = -13/40

Product = 4/5 × (-9/8) Product = (4 × -9) / (5 × 8) = -36/40 Simplifying the fraction by dividing the numerator and denominator by 4 gives: Product = -9/10

Both given conditions are satisfied.

Explanation

Let CP be the cost price and MP be the marked price of the shirt.

When a discount of 50% is given, the selling price is: SP₁ = MP × (1 - 0.50) = 0.5 × MP

Since this sale yields a profit of 25%, the selling price can also be expressed as: SP₁ = CP × (1 + 0.25) = 1.25 × CP

Equating the two expressions for SP₁: 0.5 × MP = 1.25 × CP MP = (1.25 / 0.5) × CP = 2.5 × CP

When the discount is changed to 10%, the new selling price is: SP₂ = MP × (1 - 0.10) = 0.9 × MP

Substituting MP = 2.5 × CP: SP₂ = 0.9 × 2.5 × CP = 2.25 × CP

The profit is the difference between this new selling price and the cost price: Profit = SP₂ - CP = 2.25 × CP - CP = 1.25 × CP

Expressing this profit as a percentage of the cost price: Profit % = (1.25 × CP / CP) × 100 = 125%