SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The trigonometric ratio cot A is defined as the ratio of the Base to the Perpendicular in a right-angled triangle. Given cot A = ¹²⁄₅, we identify the Base as 12 units and the Perpendicular as 5 units. In any right-angled triangle, the relationship between the sides is governed by the Pythagorean theorem H² = B² + P². Substituting the given values, the Hypotenuse is calculated as H = √(12² + 5²) = √(144 + 25) = √(169) = 13. The value of sin A is defined as the ratio of the Perpendicular to the Hypotenuse. Therefore, sin A = (P)/(H) = ⁵⁄₁₃.

Explanation

The expression is tan 6° × tan 45° × tan 84°. We use the standard value tan 45° = 1 and the complementary angle identity tan(90° - θ) = cot θ. Since 84° = 90° - 6°, we have tan 84° = cot 6°. Substituting these into the expression gives tan 6° × 1 × cot 6°. Using the reciprocal identity tan θ × cot θ = 1, the calculation simplifies to 1 × 1 = 1.

Explanation

Given x = √5 - √4{√5 + √4} and y = √5 + √4{√5 - √4}, we observe that x and y are reciprocals, meaning xy = 1.

To find the sum x + y, we apply the algebraic identity (a-b)/(a+b) + (a+b)/(a-b) = (2(a² + b²))/(a² - b²). Let a = √5 and b = √4 = 2: x + y = 2((√5)² + 2²){(√5)² - 2²} = (2(5 + 4))/(5 - 4) = ²⁽⁹⁾⁄₁ = 18

The expression to evaluate is (x² - xy + y²)/(x² + xy + y²). We can rewrite the numerator and denominator in terms of (x + y) and xy: Numerator: x² + y² - xy = (x + y)² - 2xy - xy = (x + y)² - 3xy Denominator: x² + y² + xy = (x + y)² - 2xy + xy = (x + y)² - xy

Substituting the calculated values x + y = 18 and xy = 1: Result = (18² - 3(1))/(18² - 1) Result = (324 - 3)/(324 - 1) = ³²¹⁄₃₂₃

Explanation

For any angle θ, the identity relating the square of the sum and the square of the difference of sine and cosine is derived from the algebraic identity (a + b)² + (a - b)² = 2(a² + b²). Substituting a = sin θ and b = cos θ, we get:

(sin θ + cos θ)² + (sin θ - cos θ)² = 2(sin² θ + cos² θ)

Using the fundamental trigonometric identity sin² θ + cos² θ = 1, the equation becomes:

(sin θ + cos θ)² + (sin θ - cos θ)² = 2

Given that sin θ - cos θ = ⅘, substitute this into the equation:

(sin θ + cos θ)² + (⅘)² = 2

(sin θ + cos θ)² + ¹⁶⁄₂₅ = 2

(sin θ + cos θ)² = 2 - ¹⁶⁄₂₅

(sin θ + cos θ)² = (50 - 16)/(25) = ³⁴⁄₂₅

Taking the square root of both sides:

sin θ + cos θ = √(³⁴⁄₂₅) = √(34){5}

Explanation

The volume of a cylinder is given by the formula V = π r² h. When the iron rod is drawn into a wire, the total volume of the material remains constant. For the initial rod, the diameter is 4 cm, so the radius r = 2 cm, and the length h = 12 cm. The volume is calculated as: V = π × 2² × 12 = 48π cm³

The new wire has a length L = 12 m, which must be converted to centimeters to maintain consistent units: L = 1200 cm. Let R be the radius of the wire. The volume of the wire is: V = π R² × 1200

Equating the two volumes: 48π = π R² × 1200 R² = ⁴⁸⁄₁₂₀₀ = ⁴⁄₁₀₀ R = √(⁴⁄₁₀₀) = ²⁄₁₀ = 0.2 cm

The thickness of the wire refers to its diameter, which is 2R. Therefore, the thickness is 2 × 0.2 = 0.4 cm.

Explanation

The fourth proportional d to three numbers a, b, and c is defined by the ratio a:b = c:d, which can be expressed as the formula d = (b × c)/(a). Given the values a = 16, b = 26, and c = 32, the calculation is performed by substituting these into the formula:

d = (26 × 32)/(16)

By simplifying the fraction, 32 divided by 16 equals 2. Multiplying the remaining terms gives:

d = 26 × 2 = 52

Explanation

Given that tan(α + β) = √3 and α + β is an acute angle, we have α + β = 60°. Similarly, since tan(α - β) = 1 and α - β is an acute angle, we have α - β = 45°.

To find the value of 2α, we add these two equations: (α + β) + (α - β) = 60° + 45° 2α = 105°

Multiplying by 3 gives: 6α = 3 × 105° = 315°

The value of tan(6α) is: tan(315°) = tan(360° - 45°) Using the trigonometric identity tan(360° - θ) = -tanθ: tan(315°) = -tan(45°) = -1

Explanation

The given condition is based on the quadratic equation p² - 25p + 1 = 0. To find the value of the expression p + ¹⁄ₚ, we divide the entire equation by p:

(p²)/(p) - ²⁵ᵖ⁄ₚ + ¹⁄ₚ = ⁰⁄ₚ

p - 25 + ¹⁄ₚ = 0

Rearranging the terms to isolate the variable expression on one side:

p + ¹⁄ₚ = 25

Since the problem stipulates p ≠ 0, the division by p is a valid operation. This linear transformation of the quadratic equation directly yields the required value.

Explanation

The dividend is determined by the fundamental division identity: Dividend = (Divisor × Quotient) + Remainder.

From the given information, the remainder is 45. The divisor is stated to be four times the remainder, so Divisor = 4 × 45 = 180. It is also given that the divisor is ten times the quotient, which leads to the equation 10 × Quotient = 180. Solving for the quotient gives Quotient = 18.

By substituting these values into the identity: Dividend = (180 × 18) + 45 Dividend = 3240 + 45 Dividend = 3285

Explanation

First, calculate the total number of players participating in game L from schools D and E. Summing the values from the table, we get 82 + 68 = 150. Next, determine the total number of players participating in game M from schools H and I. Adding these values gives 86 + 108 = 194.

To find the percentage by which 150 is less than 194, use the formula for percentage decrease:

Percentage Less = (Higher Value - Lower Value)/(Higher Value) × 100

Substituting the calculated totals into the formula:

Percentage Less = (194 - 150)/(194) × 100 = ⁴⁴⁄₁₉₄ × 100

Simplifying the fraction results in:

²²⁄₉₇ × 100 = ²²⁰⁰⁄₉₇ ≈ 22.6804%

Rounding to two decimal places, the result is 22.68 percent.

Explanation

The Highest Common Factor (HCF) of 222, 642, and 1062 can be determined by finding the prime factorization of each number. For 222, the sum of the digits is 2 + 2 + 2 = 6, indicating divisibility by 3. Since the number is even, it is also divisible by 2. Thus, 222 = 2 × 3 × 37. For 642, the sum of the digits is 6 + 4 + 2 = 12, indicating divisibility by 3. Being even, it is also divisible by 2. Thus, 642 = 2 × 3 × 107. For 1062, the sum of the digits is 1 + 0 + 6 + 2 = 9, which is divisible by 3. Since it is an even number, it is also divisible by 2. Dividing 1062 by 6 gives 177, which can be further factored into 3 × 59. Thus, 1062 = 2 × 3 × 3 × 59.

The common prime factors present in all three factorizations are 2 and 3, each with a minimum power of one. Therefore, the HCF is calculated as 2¹ × 3¹ = 6. Alternatively, applying the difference method, we find that 642 - 222 = 420 and 1062 - 642 = 420. The HCF of the original numbers must also be a factor of their differences. Since 6 divides 222, 642, and 1062 exactly, and no higher common factor exists among the remaining prime factors (37, 107, and 3 × 59), 6 is the greatest common divisor.

Explanation

The selling price (SP) of an article is calculated using the relation SP = CP × (1 + (Profit Percentage)/(100)).

For article P, the selling price is 450 × 1.9 = 855. For article Q, the selling price is 250 × 2 = 500. The combined selling price of articles P and Q is 855 + 500 = 1355.

For article R, the selling price is 600 × 1.6 = 960. For article S, the selling price is 730 × 1.7 = 1241. For article T, the selling price is 400 × 1.8 = 720. The combined selling price of articles R, S, and T is 960 + 1241 + 720 = 2921.

The required percentage is found by dividing the total selling price of P and Q by the total selling price of R, S, and T: Percentage = ¹³⁵⁵⁄₂₉₂₁ × 100 ≈ 46.388%

Rounding to two decimal places, the result is 46.38 percent.

Explanation

In a partnership business, profits are shared in the ratio of investments when the time period is constant. Given the investment ratio A:B = 9:5, the profit shared between them is also in the ratio 9:5. Let the total profit be P. Since 10% goes to charity, the remaining 90% of the profit is distributed between A and B.

A's share is calculated as: A's share = ⁹⁄₉₊₅ × 90% of P

Substituting the given value of A's share: 29840 = ⁹⁄₁₄ × 0.9P 29840 = (8.1P)/(14)

Rearranging to solve for P: P = (29840 × 14)/(8.1) P = (417760)/(8.1) P = ⁴¹⁷⁷⁶⁰⁰⁄₈₁ P ≈ 51575.308

Rounding to the nearest integer, the total profit is Rs. 51,575.

Explanation

The algebraic identity for y³ + (1)/(y³) states that if y + (1)/(y) = k, then y³ + (1)/(y³) = k³ - 3k. Given y + (1)/(y) = 3, we substitute k = 3 into the formula to get y³ + (1)/(y³) = 3³ - 3(3) = 27 - 9 = 18. The expression to be evaluated is y³ + (1)/(y³) + 2, which equals 18 + 2 = 20.

Explanation

The trigonometric identity for the sine of a double angle is given by the formula sin 2θ = (2 tan θ)/(1 + tan² θ). Substituting θ = 60° into this identity, we obtain:

(2 tan 60°)/(1 + tan² 60°) = sin(2 × 60°) = sin 120°

According to the supplementary angle identity sin(180° - A) = sin A, we can simplify sin 120° as sin(180° - 60°) = sin 60°.

The calculation can also be verified using the numerical value tan 60° = √3. Substituting this into the expression:

2(√3){1 + (√3)²} = 2√3{1 + 3} = 2√3{4} = √3{2}

The value √3{2} corresponds to sin 60° in the standard trigonometric table.

Explanation

Statement I is incorrect because the income of C1 in year P is 750 and the income of C2 in year Q is 250. The percentage of C1's income in P relative to C2's income in Q is calculated as ⁷⁵⁰⁄₂₅₀ × 100 = 300%, which does not equal 33.33%. Statement II is correct as the average income of C1 and C2 in year T is (530 + 270)/(2) = ⁸⁰⁰⁄₂ = 400. Since only the first statement is false, the selection for the incorrect statement is restricted to I.

Explanation

The single discount equivalent to two successive discounts of x% and y% is given by the formula x + y - ˣʸ⁄₁₀₀. Substituting the given values where x = 15 and y = 12:

Equivalent Discount = 15 + 12 - (15 × 12)/(100) Equivalent Discount = 27 - ¹⁸⁰⁄₁₀₀ Equivalent Discount = 27 - 1.8 = 25.2%

Alternatively, if the initial price is 100, a 15% discount reduces it to 85. Applying a 12% discount on 85 results in a further reduction of 0.12 × 85 = 10.2. The final price becomes 85 - 10.2 = 74.8. The total discount is 100 - 74.8 = 25.2%.

Explanation

When a solid object is reshaped, its volume remains constant. The volume of the copper sphere is equated to the volume of the cylindrical wire.

For the sphere, the radius R is half the diameter, so R = ¹⁸⁄₂ = 9 cm. The volume of the sphere Vₛ is given by: Vₛ = ⁴⁄₃ π R³ = ⁴⁄₃ π (9)³ = ⁴⁄₃ π × 729 = 4 π × 243 = 972 π cm³

For the wire, which is cylindrical, the diameter is 6 mm, so the radius r is 3 mm. Converting this to centimeters to match the sphere's units, r = 0.3 cm. Let L be the length of the wire in centimeters. The volume of the wire V_w is: V_w = π r² L = π (0.3)² L = 0.09 π L cm³

Equating the two volumes: 0.09 π L = 972 π L = (972)/(0.09) = ⁹⁷²⁰⁰⁄₉ = 10800 cm

To convert the length from centimeters to meters, divide by 100: L = ¹⁰⁸⁰⁰⁄₁₀₀ = 108 m

Explanation

The final population after successive percentage changes can be calculated using the formula P_fᵢₙₐₗ = Pᵢₙᵢₜᵢₐₗ × (1 + (r₁)/(100)) × (1 + (r₂)/(100)) × (1 + (r₃)/(100)), where r is the rate of increase (positive) or decrease (negative).

Given the final population is 9,47,625 and the successive changes are -5%, -5%, and +5%, we have:

9,47,625 = Pᵢₙᵢₜᵢₐₗ × (1 - ⁵⁄₁₀₀) × (1 - ⁵⁄₁₀₀) × (1 + ⁵⁄₁₀₀)

9,47,625 = Pᵢₙᵢₜᵢₐₗ × ⁹⁵⁄₁₀₀ × ⁹⁵⁄₁₀₀ × ¹⁰⁵⁄₁₀₀

To simplify the calculation, note that ⁹⁵⁄₁₀₀ × ¹⁰⁵⁄₁₀₀ = ((100 - 5)(100 + 5))/(10000) = (100² - 5²)/(10000) = ⁹⁹⁷⁵⁄₁₀₀₀₀ = 0.9975.

Substituting this back into the equation:

9,47,625 = Pᵢₙᵢₜᵢₐₗ × 0.95 × 0.9975

9,47,625 = Pᵢₙᵢₜᵢₐₗ × 0.947625

Pᵢₙᵢₜᵢₐₗ = (9,47,625)/(0.947625) = 10,00,000

Thus, the population at the beginning of the first year was 10,00,000.

Explanation

If the cost price is 100, a 20% markup makes the marked price 120, and a 10% discount on that price results in a selling price of 108. The net gain is 8% (108 - 100).

Explanation

The volume of a cube is calculated using the formula V = a³, where a represents the side length of the cube. Given that the volume is 343 m³, the side length is determined by taking the cube root of the volume: a = ³√(343) = 7 m.

The total surface area of a cube is calculated using the formula TSA = 6a². By substituting the side length a = 7 m into this formula, the calculation is as follows: TSA = 6 × (7)² = 6 × 49 = 294 m².

Explanation

The fourth proportional x to three numbers a, b, and c is defined by the ratio a:b = c:x, which leads to the formula x = (b × c)/(a). Substituting the given values a = 17, b = 272, and c = 19 into the formula:

x = (272 × 19)/(17)

First, divide 272 by 17: 272 ÷ 17 = 16

Now, multiply the result by 19: 16 × 19 = 16 × (20 - 1) = 320 - 16 = 304

Explanation

The total increase in marks is 20 (65 - 45). Given that the average marks increased by 1/3, the number of students is calculated by dividing the total increase by the average increase: 20 / (1/3) = 60. Although the options include percentage signs, the numerical value 60 matches option A.

Explanation

When a solid metallic sphere is melted and recast into a cone, the volume remains constant. The volume of a sphere is given by Vₛ = ⁴⁄₃ π R³ and the volume of a cone is V_c = ⅓ π r² h.

Given the radius of the sphere R = 13 cm and the diameter of the cone base as 13 cm, the radius of the cone is r = ¹³⁄₂ cm. Equating the two volumes:

⁴⁄₃ π R³ = ⅓ π r² h

Dividing both sides by ⅓ π results in:

4 R³ = r² h

Substitute the known values into the equation:

4 × (13)³ = (¹³⁄₂)² × h

4 × 13 × 13 × 13 = (13 × 13)/(4) × h

Cancel 13² from both sides:

4 × 13 = ʰ⁄₄

h = 4 × 4 × 13

h = 16 × 13 = 208 cm

Explanation

The average sale of bikes for companies P, Q, and R is calculated by summing their individual bike sales and dividing by 3. Summing the values from the table, we get 200 + 100 + 500 = 800, thus J₁ = ⁸⁰⁰⁄₃. Similarly, the average sale of cars for companies R, S, and T is calculated by summing their car sales and dividing by 3. Summing these values gives 700 + 1000 + 800 = 2500, thus J₂ = ²⁵⁰⁰⁄₃. The required ratio J₁ : J₂ is (⁸⁰⁰⁄₃) : (²⁵⁰⁰⁄₃). By canceling the common denominator and dividing both terms by 100, the ratio simplifies directly to 8 : 25.

Explanation

To find the Highest Common Factor (HCF) of 4.08 and 6.63, first convert the decimal numbers into integers by multiplying each by 100, which gives 408 and 663. Applying the division method for HCF:

663 = 408 × 1 + 255 408 = 255 × 1 + 153 255 = 153 × 1 + 102 153 = 102 × 1 + 51 102 = 51 × 2 + 0

The HCF of 408 and 663 is 51. Since the original numbers were multiplied by 100, divide this result by 100 to find the HCF of the decimals: 51 / 100 = 0.51. Alternatively, prime factorization shows 408 = 2³ × 3 × 17 and 663 = 3 × 13 × 17. The common prime factors are 3 and 17, and their product 3 × 17 = 51 leads to the same result.

Explanation

The given equation a² = b² + (ab)² can be rearranged by factoring b² on the right-hand side to obtain a² = b²(1 + a²). Solving for b², we get b² = a²{1 + a²}. Substituting a = cot x into the right-hand side and applying the trigonometric identity 1 + cot² x = csc² x: b² = cot² x{1 + cot² x} = cot² x{csc² x} By expanding the terms into sines and cosines: b² = cos² x / sin² x{1 / sin² x} = cos² x Taking the square root of both sides gives b = cos x. This confirms that the values a = cot x and b = cos x satisfy the identity.

Explanation

Let the original number be x. Following the steps provided in the question, we can represent the operations as an algebraic equation. First, 7 is added to x, resulting in (x + 7). This sum is then multiplied by 5 to get 5(x + 7). Dividing this product by 3 yields the quotient (5(x + 7))/(3). Finally, 4 is subtracted from this quotient to arrive at the final result of 16.

The equation is: (5(x + 7))/(3) - 4 = 16

To find the value of x, we solve the equation: (5(x + 7))/(3) = 16 + 4 (5(x + 7))/(3) = 20 5(x + 7) = 20 × 3 5(x + 7) = 60 x + 7 = ⁶⁰⁄₅ x + 7 = 12 x = 12 - 7 x = 5

By substituting 5 back into the original sequence: (5 + 7) = 12; 12 × 5 = 60; 60 ÷ 3 = 20; and 20 - 4 = 16. This confirms that 5 is the original number.

Explanation

The product-to-sum trigonometric identity states that 2 cos θ sin γ = sin(θ + γ) - sin(θ - γ). Given the values θ = 135° and γ = 15°, the expression can be evaluated as follows:

2 cos(135°) sin(15°) = sin(135° + 15°) - sin(135° - 15°) 2 cos(135°) sin(15°) = sin(150°) - sin(120°)

Using the identity sin(180° - A) = sin A, the specific sine values are determined: sin(150°) = sin(180° - 30°) = sin(30°) = 1/2 sin(120°) = sin(180° - 60°) = sin(60°) = √3/2

Substituting these values into the simplified expression: 1/2 - √3/2 = 1 - √3{2}

Explanation

The expression is a sum of three algebraic fractions of the form (1)/(1 + a + b⁻¹⁾ + (1)/(1 + b + 5c⁻¹⁾ + (1)/(1 + c){5 + a⁻¹}. Given abc = 5, we can simplify each term to a common denominator to find the total value.

The first term is (1)/(1 + a + b⁻¹⁾ = (1)/(1 + a + 1){b} = (b)/(b + ab + 1).

For the second term, we use the given condition abc = 5, which implies 5c⁻¹ = (5)/(c) = (abc)/(c) = ab. Substituting this into the denominator gives (1)/(1 + b + 5c⁻¹⁾ = (1)/(1 + b + ab).

For the third term, we multiply the numerator and denominator by ab: (ab)/(ab(1 + c){5 + a⁻¹)} = (ab)/(ab + abc){5 + b}. Substituting abc = 5 into the middle term of the denominator gives (ab)/(ab + 5){5 + b} = (ab)/(ab + 1 + b).

Now, adding the three simplified terms together: (b)/(b + ab + 1) + (1)/(1 + b + ab) + (ab)/(ab + 1 + b) = (b + 1 + ab)/(1 + b + ab) = 1

Explanation

In a circle, the tangent at any point is perpendicular to the radius through the point of contact. Therefore, the radius, the tangent, and the line segment joining the center of the circle to the external point form a right-angled triangle where the distance from the center is the hypotenuse.

Using the Pythagoras theorem: t² + r² = d²

where t is the length of the tangent, r is the radius, and d is the distance from the center. Substituting the given values r = 5 cm and d = 10 cm: t² + 5² = 10² t² + 25 = 100 t² = 75 t = √(75) = √(25 × 3) = 5√3 cm

Explanation

The volume V of a hollow sphere is the difference between the volume of the outer sphere and the volume of the inner sphere. This is expressed by the formula V = ⁴⁄₃π(R³ - r³), where R is the outer radius and r is the inner radius. Given R = 6 cm and r = 3 cm, the calculation is as follows:

V = ⁴⁄₃π(6³ - 3³) V = ⁴⁄₃π(216 - 27) V = ⁴⁄₃π(189)

Multiplying ⁴⁄₃ by 189 results in 4 × 63, which equals 252. Thus, the volume of the hollow sphere is 252π cm³.

Explanation

In simple interest, the time required for a principal to become n times its value is directly proportional to (n - 1), as the interest earned is SI = P(n - 1). This relationship can be expressed using the formula T₁{n₁ - 1} = T₂{n₂ - 1}, where T represents time and n represents the multiple of the principal.

In the first scenario, the amount becomes 6 times in 20 years, so n₁ = 6 and T₁ = 20. In the second scenario, the amount becomes 5 times, so n₂ = 5. Substituting these values into the ratio:

(20)/(6 - 1) = T₂{5 - 1}

²⁰⁄₅ = T₂{4}

4 = T₂{4}

T₂ = 4 × 4 = 16 years.

Explanation

To find the difference between the total earnings of A and B, we first calculate the total earnings for each person across the five days.

Total earnings of A = 85 + 76 + 45 + 95 + 110 = 411 Total earnings of B = 130 + 90 + 102 + 86 + 48 = 456

The difference between their total earnings is 456 - 411 = 45.

Alternatively, this can be solved by summing the individual daily differences (B - A) for each day: Tuesday: 130 - 85 = 45 Wednesday: 90 - 76 = 14 Thursday: 102 - 45 = 57 Friday: 86 - 95 = -9 Saturday: 48 - 110 = -62

Sum of differences = 45 + 14 + 57 - 9 - 62 = 116 - 71 = 45.

Explanation

The sum of the interior angles of a triangle is always 180°. Given the three angles as (x - 6)°, (x + 26)°, and 8x°, we can establish the following equation:

(x - 6) + (x + 26) + 8x = 180

Combining the terms involving x and the constant terms separately:

10x + 20 = 180

Subtracting 20 from both sides of the equation:

10x = 160

Dividing by 10, we find the value of x:

x = 16

The question requires the value of 2x. Multiplying the result for x by 2:

2x = 2 × 16 = 32

Explanation

To find the sum of the coefficients A + B + C for the expression Az² + Bz + C, evaluate the polynomial at z = 1. Substituting z = 1 into the identity 4(z + 7)(2z - 1) = Az² + Bz + C:

A(1)² + B(1) + C = 4(1 + 7)(2(1) - 1)

A + B + C = 4(8)(1)

A + B + C = 32

This substitution method directly yields the sum of coefficients without requiring the full expansion of the algebraic expression.

Explanation

The centroid of a triangle is defined as the point of concurrency where its three medians intersect. A median is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. In △ ABC, the segments AD, BE, and CF are the medians because they connect the vertices A, B, and C to the midpoints D, E, and F of the opposite sides BC, CA, and AB, respectively. According to the properties of triangle geometry, the three medians always meet at a single common point, which is the centroid. This point acts as the geometric center of the triangle and divides each median in a ratio of 2:1, where the distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the side.

Explanation

Based on the ratio of expenditure to savings of 5:1, let the expenditure be 500 and the savings be 100. The total income is the sum of expenditure and savings, which is 500 + 100 = 600.

The new income after a 10% increase is 600 + (10% of 600) = 600 + 60 = 660. The new expenditure after a 20% increase is 500 + (20% of 500) = 500 + 100 = 600.

The new savings is calculated by subtracting the new expenditure from the new income: New Savings = 660 - 600 = 60

The change in savings is 60 - 100 = -40. To find the percentage change, divide the change by the initial savings and multiply by 100: Percentage change = ⁻⁴⁰⁄₁₀₀ × 100 = -40%

The negative sign indicates a decrease, resulting in a 40% change in her savings.

Explanation

The number of female students in a school opting for Biology is calculated by multiplying the total number of students by the percentage who opted for Biology, then multiplying by the female share of the male-to-female ratio.

For School B, the total number of students opting for Biology is 400 × 36% = 144. Given the male to female ratio is 5:7, the total ratio parts are 12. The number of female students is 144 × ⁷⁄₁₂ = 84.

For School D, the total number of students opting for Biology is 800 × 18% = 144. Given the male to female ratio is 3:9, the total ratio parts are also 12. The number of female students is 144 × ⁹⁄₁₂ = 108.

The ratio of the number of female students in School B to the number of female students in School D is 84 : 108. Dividing both sides by their greatest common divisor, which is 12, the ratio simplifies to 7 : 9.

Alternatively, because the total number of Biology students in both schools is identical (400 × 0.36 = 144 and 800 × 0.18 = 144), the ratio of female students is determined directly by their respective ratio proportions: ⁷⁄₁₂ : ⁹⁄₁₂, which is 7 : 9.

Explanation

According to the Angle Bisector Theorem, the internal bisector of an angle of a triangle divides the opposite side into segments that are proportional to the other two sides of the triangle. For triangle ABC where AD is the bisector of ∠ BAC, the relation is:

(AB)/(AC) = (BD)/(DC)

We are given BD = 6 and the total length of the side BC = 14. Since D lies on the line segment BC, we calculate the length of DC as:

DC = BC - BD = 14 - 6 = 8

Now, substitute the values of BD and DC into the ratio formula:

(AB)/(AC) = ⁶⁄₈

Simplifying the ratio by dividing both terms by their greatest common divisor, 2, results in:

(AB)/(AC) = ¾

The ratio AB : AC is therefore 3:4.

Explanation

Let the time taken by A to complete the work be x days. Based on the given conditions, the time taken by B is x/4 days and the time taken by C is x/5 days.

Efficiency is inversely proportional to the time taken. Therefore, the daily work rates are 1/x for A, 4/x for B, and 5/x for C. When working together, their combined daily work rate is the sum of their individual rates:

¹⁄ₓ + ⁴⁄ₓ + ⁵⁄ₓ = ¹⁰⁄ₓ

The problem states that they finish the work together in 4 days, meaning their combined daily rate is 1/4. By equating the two expressions for the combined rate:

¹⁰⁄ₓ = ¼

Solving for x, we get x = 40 days. Since B takes x/4 days to complete the work alone, the time taken by B is 40/4 = 10 days.

Explanation

Let u be the speed of the swimmer in still water and v be the speed of the current, both in meters per minute. The distance covered while swimming upstream for 6 minutes from point P to some point R is PR = (u - v) × 6. The distance covered while swimming downstream for the next 6 minutes from point R to point Q is RQ = (u + v) × 6. Since the swimmer returns along the same path and ends at point Q, the distance PQ is the difference between the downstream distance and the upstream distance: PQ = RQ - PR.

Substituting the given distance PQ = 120 meters into the equation gives 120 = (u + v) × 6 - (u - v) × 6. Expanding the terms, 120 = 6u + 6v - 6u + 6v, which simplifies to 120 = 12v. Solving for v results in v = 10 meters per minute. To convert this speed to kilometers per hour, multiply by 60 to find the distance per hour and divide by 1000 to convert meters to kilometers: v = (10 × 60)/(1000) = 0.6 km/h.

Explanation

The expression is simplified using the complementary angle relations sin(90° - θ) = cos θ and cos(90° - θ) = sin θ. Since 63° + 27° = 90° and 17° + 73° = 90°, we can transform the terms to align the angles.

In the numerator, substitute sin 27° with cos 63°: sin² 63° + sin² 27° = sin² 63° + sin²(90° - 63°) = sin² 63° + cos² 63°

In the denominator, substitute cos 73° with sin 17°: cos² 17° + cos² 73° = cos² 17° + cos²(90° - 17°) = cos² 17° + sin² 17°

Applying the trigonometric identity sin² θ + cos² θ = 1, both the numerator and the denominator evaluate to 1. Thus, the total value is: ¹⁄₁ = 1

Explanation

To find the ratio a:c when given the intermediate ratios a:b and b:c, we multiply the two ratios. The relationship is expressed as:

(a)/(c) = (a)/(b) × (b)/(c)

Substituting the given values:

(a)/(c) = ⅘ × ³⁄₁₀

Multiplying the numerators and denominators results in:

(a)/(c) = (4 × 3)/(5 × 10) = ¹²⁄₅₀

To simplify the ratio ¹²⁄₅₀, we divide both the numerator and the denominator by their greatest common divisor, which is 2:

(12 ÷ 2)/(50 ÷ 2) = ⁶⁄₂₅

Thus, the ratio a:c is 6:25.

Explanation

The error in entering the marks resulted in an increase in the total sum of marks for the class. The difference between the wrongly entered marks and the actual marks is 98 - 89 = 9. This extra value of 9 marks was distributed across the entire class, causing the average to increase by 0.25.

The number of students (n) in the class can be determined using the formula n = (Total Change in Sum)/(Change in Average). Substituting the given values into the formula, we get n = (9)/(0.25). Since 0.25 is equivalent to ¼, the calculation becomes n = 9 × 4 = 36.

Explanation

The difference in the interest earned is directly proportional to the difference in the interest rate applied to the same principal over the same period. This relationship is expressed by the formula for the change in simple interest: Δ SI = (P × Δ R × T)/(100), where Δ SI is the additional interest fetched, P is the principal sum, Δ R is the increase in the annual interest rate, and T is the time in years.

Given that Δ SI = 600, Δ R = 5%, and T = 3 years, the equation becomes:

600 = (P × 5 × 3)/(100)

600 = (15P)/(100)

To find the principal sum P, rearrange the equation:

P = (600 × 100)/(15)

P = 40 × 100

P = 4000

Explanation

To find the required speed of the police jeep, we first determine the total distance the thief travels from the time of the theft until the scheduled time of arrest. The thief travels from 8:15 p.m. to 9:00 p.m., which is a duration of 45 minutes or ¾ hours. Given the thief travels at a speed of 60 km/h, the distance covered from the shop is:

Distance = Speed × Time = 60 × ¾ = 45 km

The police receive the information and begin the pursuit at 8:30 p.m. To arrest the thief at 9:00 p.m., they must cover the same 45 km distance in the 30-minute interval between 8:30 p.m. and 9:00 p.m. Converting the police pursuit time to hours gives 30 minutes = ½ hour. The minimum speed required for the police jeep is calculated as:

Speed = (Distance)/(Time) = (45)/(1/2) = 90 km/h

Explanation

The number of officers working in company Q is 250 and the number of officers working in company R is 350. The ratio of the number of officers in Q to the number of officers in R is calculated using the formula:

Ratio = (Officers in Q)/(Officers in R)

Substituting the values from the table:

Ratio = ²⁵⁰⁄₃₅₀

To simplify the ratio, divide both terms by their greatest common divisor, which is 50:

(250 ÷ 50)/(350 ÷ 50) = ⁵⁄₇

The simplified ratio is 5:7.

Explanation

The expression is solved using the algebraic identity (x + ¹⁄ₓ)² = x² + (1)/(x²) + 2. Given the initial relationship K + (1)/(K) = 3, squaring both sides of the equation yields (K + (1)/(K))² = 3². Expanding the left side of the equation results in K² + (1)/(K²) + 2(K)((1)/(K)), which simplifies to K² + (1)/(K²) + 2. By substituting the squared value of the right side, the equation becomes K² + (1)/(K²) + 2 = 9. To isolate the required terms, 2 is subtracted from both sides, leading to K² + (1)/(K²) = 9 - 2, which equals 7.

Explanation

The value of the expression is found using the standard algebraic identity for the square of a trinomial: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). By rearranging this formula to solve for the sum of squares, we get a² + b² + c² = (a + b + c)² - 2(ab + bc + ca).

Substituting the given values a + b + c = 9 and ab + bc + ca = 23 into the equation: a² + b² + c² = 9² - 2(23) a² + b² + c² = 81 - 46 a² + b² + c² = 35