SSC CGL Tier 1 2013 Quantitative Aptitude Questions with Solutions

Explanation

The total sum of all nine numbers is found by multiplying the average by the quantity: 9 × 50 = 450. The sum of the first five numbers is 5 × 54 = 270, and the sum of the last three numbers is 3 × 52 = 156. To isolate the sixth number, subtract the sums of these two groups from the total sum of the nine numbers. The calculation is 450 - (270 + 156), which simplifies to 450 - 426 = 24.

Explanation

Using the fundamental trigonometric identity sec²θ - tan²θ = 1, we can deduce that (sec θ + tan θ)(sec θ - tan θ) = 1. Given that sec θ + tan θ = √5 + 2, the value of sec θ - tan θ must be its reciprocal, which is 1/(√5 + 2). By rationalizing the denominator, we get sec θ - tan θ = √5 - 2(5 - 4) = √5 - 2.

By adding the two equations, sec θ + tan θ = √5 + 2 and sec θ - tan θ = √5 - 2, we find 2 sec θ = 2√5, which gives sec θ = √5 and consequently cos θ = 1/√5. Similarly, by subtracting the second equation from the first, we find 2 tan θ = 4, which gives tan θ = 2. Since tan θ = sin θ / cos θ, it follows that sin θ = tan θ · cos θ = 2 · 1/√5 = 2/√5.

The value of sin θ + cos θ is therefore 2/√5 + 1/√5 = 3/√5.

Explanation

The value of the expression is determined using the trigonometric identity for complementary angles, which states that sin(90° - θ) = cos θ. Given that 25° + 65° = 90°, we can express sin 65° in terms of its complement: sin 65° = sin(90° - 25°) = cos 25°

Substituting this value back into the original expression: sin² 25° + sin² 65° = sin² 25° + (cos 25°)² sin² 25° + sin² 65° = sin² 25° + cos² 25°

Using the fundamental trigonometric identity sin² θ + cos² θ = 1 for any angle θ, we find: sin² 25° + cos² 25° = 1

Explanation

In a right-angled triangle formed by the tower, the ground, and the line of sight, the height of the tower h and the distance from the foot d are related to the angle of elevation θ by the tangent function:

tan(θ) = (Opposite)/(Adjacent) = (h)/(d)

Given that the angle of elevation θ is 30° and the distance from the foot d is 100 m, the formula is applied as follows:

tan(30°) = ʰ⁄₁₀₀

The trigonometric value for tan(30°) is (1)/(√(3)). Substituting this value into the equation:

(1)/(√(3)) = ʰ⁄₁₀₀

Rearranging the equation to solve for the height h:

h = (100)/(√(3)) m

Explanation

The problem can be solved using the work-rate formula (M₁ D₁)/(W₁) = (M₂ D₂)/(W₂), where M is the number of men, D is the number of days, and W is the amount of work completed. In the first phase, M₁ = 200 men worked for D₁ = 50 days to complete W₁ = 1/4 of the total work.

The remaining work to be completed is W₂ = 1 - 1/4 = 3/4. The time remaining to finish the work as per the 150-day deadline is D₂ = 150 - 50 = 100 days. Let the total number of men working in the second phase be M₂. Substituting the values into the formula:

(200 × 50)/(1/4) = (M₂ × 100)/(3/4)

¹⁰⁰⁰⁰⁄₁ = (100 × M₂)/(3)

100 = (M₂)/(3)

M₂ = 300

The total number of men required to complete the remaining work in 100 days is 300. Since 200 men were already employed, the number of additional men needed is 300 - 200 = 100.

Explanation

The first nine integral multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, and 27. These numbers form an arithmetic progression (AP) where the first term a = 3, the common difference d = 3, and the number of terms n = 9.

For any arithmetic progression, the average is equal to the middle term when the number of terms is odd. In this series of 9 terms, the middle term is the (ⁿ⁺¹⁄₂)-th term, which is the 5th term. The 5th term is calculated as 3 × 5 = 15.

Alternatively, the average of an arithmetic progression can be calculated using the formula Average = (First Term + Last Term)/(2). Substituting the values, we get:

Average = (3 + 27)/(2) = ³⁰⁄₂ = 15

Explanation

First, calculate the cost price (CP) of the article using the formula CP = (SP × 100)/(100 + P%). Given the initial selling price (SP) is Rs. 300 and the profit is 20%, the calculation is CP = (300 × 100)/(120) = 250.

When the article is sold for the new price of Rs. 235, the loss is the difference between the cost price and the new selling price: Loss = 250 - 235 = 15. The loss percentage is then calculated as (Loss)/(CP) × 100, which gives ¹⁵⁄₂₅₀ × 100 = 6%.

Explanation

The initial count of balls in the box is 100 blue, 50 red, and 50 black, totaling 200 balls. When 25% of the blue balls are removed, the remaining blue balls are 100 - (0.25 × 100) = 75. When 50% of the red balls are removed, the remaining red balls are 50 - (0.50 × 50) = 25. The number of black balls remains unchanged at 50. The current total number of balls in the box is the sum of the remaining balls: 75 + 25 + 50 = 150. The percentage of black balls is determined by the ratio of the number of black balls to the total number of balls currently present, multiplied by 100. Using the formula (Black balls)/(Total balls) × 100, we get ⁵⁰⁄₁₅₀ × 100 = ⅓ × 100, which simplifies to 100 / 3%.

Explanation

To avoid a collision at the crossing, the two cars must reach the intersection point at different times. The time taken by an object to cover a distance d at a constant speed v is given by the formula t = ᵈ⁄ᵥ.

For the first car, the distance to the crossing is d₁ = 40 m and its speed is v₁, so the time taken to reach the crossing is t₁ = (40)/(v₁). For the second car, the distance is d₂ = 50 m and its speed is v₂, so the time taken is t₂ = (50)/(v₂).

A collision will occur if both cars reach the crossing at the exact same time, which is represented by the condition t₁ = t₂. Substituting the expressions for time, we get (40)/(v₁) = (50)/(v₂), which simplifies to (v₁)/(v₂) = ⁴⁰⁄₅₀ or v₁:v₂ = 4:5. Consequently, to ensure that the cars do not collide, their speeds must be such that they do not reach the crossing simultaneously, which means t₁ ≠ t₂ and thus v₁:v₂ ≠ 4:5.

Explanation

Let the initial distance be D and initial time be T, so the initial speed S1 = D/T. The new distance is D/2 and the new time is 2T, so the new speed S2 = (D/2) / (2T) = D/4T. The ratio of the two speeds S1:S2 is (D/T) : (D/4T), which simplifies to 4:1.

Explanation

Using the simple interest formula SI = (P × R × T)/(100), we equate the interest from both scenarios:

(4000 × x × 3)/(100) = (5000 × 12 × 2)/(100)

By canceling the denominators and simplifying the constants: 12000x = 5000 × 24 12000x = 120000

Solving for x: x = ¹²⁰⁰⁰⁰⁄₁₂₀₀₀ x = 10

The value of x is 10%.

Explanation

The given equation is x² - 3x + 1 = 0. Dividing the entire equation by x, we get x - 3 + ¹⁄ₓ = 0, which simplifies to x + ¹⁄ₓ = 3. The expression to be evaluated can be grouped as (x² + (1)/(x²)) + (x + ¹⁄ₓ). To find the value of the first group, we use the identity (x + ¹⁄ₓ)² = x² + (1)/(x²) + 2. Substituting x + ¹⁄ₓ = 3 into this identity gives 3² = x² + (1)/(x²) + 2, which simplifies to x² + (1)/(x²) = 9 - 2 = 7. Finally, by adding the values of the two groups together, we get 7 + 3 = 10.

Explanation

The given equation is (4x - 3)/(x) + (4y - 3)/(y) + (4z - 3)/(z) = 0. By splitting each term into individual fractions, we get:

(⁴ˣ⁄ₓ - ³⁄ₓ) + ((4y)/(y) - (3)/(y)) + ((4z)/(z) - (3)/(z)) = 0

Simplifying the linear terms:

(4 - ³⁄ₓ) + (4 - (3)/(y)) + (4 - (3)/(z)) = 0

Rearranging the constants and the reciprocal terms:

4 + 4 + 4 - (³⁄ₓ + (3)/(y) + (3)/(z)) = 0

12 - 3(¹⁄ₓ + (1)/(y) + (1)/(z)) = 0

Transposing the negative term to the right side:

12 = 3(¹⁄ₓ + (1)/(y) + (1)/(z))

Dividing both sides by 3:

¹⁄ₓ + (1)/(y) + (1)/(z) = ¹²⁄₃ = 4

Explanation

To find the simplest value of the expression 2√(50) + √(18) - √(72), first simplify each radical by extracting perfect square factors. The term √(50) can be written as √(25 × 2), which simplifies to 5√2. Therefore, 2√(50) becomes 2 × 5√2 = 10√2. The term √(18) can be written as √(9 × 2), which simplifies to 3√2. The term √(72) can be written as √(36 × 2), which simplifies to 6√2.

Substituting these simplified forms back into the original expression gives 10√2 + 3√2 - 6√2. Combining these like terms by adding and subtracting their coefficients results in (10 + 3 - 6)√2 = 7√2. Using the provided value of √2 = 1.414, the final calculation is 7 × 1.414 = 9.898.

Explanation

The given equation can be rearranged to group the terms for completing the square. Transferring the terms from the right side to the left side: a² - 2a + b² - 2b + 4c² + 4c + 3 = 0

The constant 3 can be split as 1 + 1 + 1 to form perfect squares for each variable: (a² - 2a + 1) + (b² - 2b + 1) + (4c² + 4c + 1) = 0 (a - 1)² + (b - 1)² + (2c + 1)² = 0

Since a, b, and c are real numbers, the sum of squares can only be zero if each individual square term is zero. Setting each part to zero gives: a - 1 = 0 ⟹ a = 1 b - 1 = 0 ⟹ b = 1 2c + 1 = 0 ⟹ c = -½

Substituting these values into the expression a² + b² + c²: a² + b² + c² = (1)² + (1)² + (-½)² a² + b² + c² = 1 + 1 + ¼ a² + b² + c² = 2 + ¼ = 9⁄4

Explanation

The number x can be expressed in the form x = 289k + 18, where k is an integer quotient. To find the remainder when x is divided by 17, we observe that 289 is a perfect square of 17, meaning 289 = 17 × 17. Because the divisor 289 is a multiple of the new divisor 17, the first part of the expression 289k is perfectly divisible by 17 and leaves no remainder. Consequently, the remainder y is simply the remainder obtained when the original remainder 18 is divided by 17. Since 18 = (17 × 1) + 1, the value of y is 1.

Explanation

The general equation of a straight line is given by ax + by + c = 0. When the constant term c is equal to zero, the equation simplifies to ax + by = 0. To determine if a line passes through the origin, we substitute the coordinates of the origin, (0, 0), into the equation. Substituting x = 0 and y = 0 into ax + by = 0 yields a(0) + b(0) = 0, which results in 0 = 0. Since the coordinates of the origin satisfy the equation regardless of the non-zero values of a and b, the straight line necessarily passes through the origin.

Explanation

Let the denominator of the fraction be x. Since the numerator is 4 less than the denominator, it can be represented as x - 4. The original fraction is (x - 4)/(x).

According to the given condition, if the numerator is decreased by 2 and the denominator is increased by 1, the denominator becomes eight times the numerator. This relationship is expressed by the equation: x + 1 = 8(x - 4 - 2) x + 1 = 8(x - 6) x + 1 = 8x - 48 7x = 49 x = 7

Substituting x = 7 into the expressions for the numerator and denominator, the denominator is 7 and the numerator is 7 - 4 = 3. Therefore, the fraction is 3/7.

Explanation

Given the equations x² = y + z, y² = z + x, and z² = x + y, we can evaluate the expression S = (1)/(1 + x) + (1)/(1 + y) + (1)/(1 + z) by manipulating each term.

Multiply the numerator and denominator of the first term by x: (x)/(x(1 + x)) = (x)/(x + x²) Substitute x² = y + z into the denominator: (x)/(x + y + z)

Similarly, multiply the numerator and denominator of the second term by y and the third term by z: (y)/(y(1 + y)) = (y)/(y + y²) = (y)/(y + z + x) (z)/(z(1 + z)) = (z)/(z + z²) = (z)/(z + x + y)

Now, sum the three transformed terms: S = (x)/(x + y + z) + (y)/(x + y + z) + (z)/(x + y + z) S = (x + y + z)/(x + y + z) = 1

Alternatively, using the method of substitution for symmetric equations, let x = y = z. The given equations become x² = 2x, which implies x = 2 (for x ≠ 0). Substituting x = y = z = 2 into the target expression gives: (1)/(1 + 2) + (1)/(1 + 2) + (1)/(1 + 2) = ⅓ + ⅓ + ⅓ = 1

Explanation

In the right-angled triangle ABC, the sum of angles is 180°. Given ∠A = 90° and ∠C = 55°, we find ∠B using the angle sum property: ∠B = 180° - (∠A + ∠C) = 180° - (90° + 55°) = 35°.

Since AD is perpendicular to BC, △ABD is also a right-angled triangle with ∠ADB = 90°. Applying the angle sum property to △ABD: ∠BAD + ∠ADB + ∠B = 180° ∠BAD + 90° + 35° = 180° ∠BAD = 180° - 125° = 55°.

Alternatively, in any right triangle, the altitude to the hypotenuse creates two smaller triangles that are similar to the original triangle. In this configuration, △BAD is similar to △BCA, which implies ∠BAD = ∠BCA = 55°.

Explanation

The centroid G of a triangle is the point where its three medians intersect. A fundamental geometric property of the centroid is that it divides the triangle ABC into three smaller triangles of equal area: Δ ABG, Δ BGC, and Δ CGA. This occurs because the centroid lies at a distance from the base that is exactly one-third of the total altitude of the triangle from that base. Therefore, the area of Δ BGC is exactly one-third of the area of the entire triangle ABC.

Using the given total area: Area(Δ BGC) = ⅓ × Area(Δ ABC) Area(Δ BGC) = ⅓ × 48 cm² = 16 cm²

Explanation

In a cyclic quadrilateral, the diagonals AC and BD are chords of the circumcircle that intersect at point P. According to the Intersecting Chords Theorem, the product of the segments of one chord is equal to the product of the segments of the other, which means AP * CP = BP * DP.

Explanation

In a circle where AB is the diameter, any point P on the circumference creates a right-angled triangle APB because the angle subtended by a diameter at the circumference is 90°. Given the radius is 7 cm, the diameter AB is 2 × 7 = 14 cm. When the perpendicular PN is dropped from P to AB, N is the foot of the perpendicular on the hypotenuse. In the right-angled triangle APB, the square of the leg PB is equal to the product of its projection BN on the hypotenuse and the hypotenuse AB itself. Therefore, PB² = BN · AB. Substituting the given values PB = 12 cm and AB = 14 cm into the formula, we get 12² = BN · 14, which simplifies to 144 = 14 · BN. Solving for BN, we find BN = ¹⁴⁴⁄₁₄ = ⁷²⁄₇ = 10 ²⁄₇ cm. This value represents the distance of point N from point B.

Explanation

In a triangle where O is the circumcenter, the distances from the center to each vertex are equal, so OP = OQ = OR. This property creates isosceles triangles within the larger triangle. In △OPR, since OP = OR, the base angles are equal, which gives ∠ORP = ∠OPR = 25°. Similarly, in △OQR, since OQ = OR, the base angles ∠ORQ and ∠OQR are equal. Applying the angle sum property to △OQR, we have 2 × ∠ORQ + ∠QOR = 180°. Given ∠QOR = 110°, it follows that 2 × ∠ORQ = 180° - 110° = 70°, so ∠ORQ = 35°. The total measure of ∠PRQ is the sum of the two adjacent angles ∠ORP and ∠ORQ, resulting in 25° + 35° = 60°.

Explanation

At any given time, the angle of elevation of the sun is the same for all vertical objects in the same vicinity. This implies that the ratio of the height of an object to the length of its shadow is constant, as the objects and their shadows form similar right-angled triangles.

Using the property of similar triangles, the relationship can be expressed as: (Height of Stick)/(Shadow of Stick) = (Height of Tower)/(Shadow of Tower)

Substituting the provided values into the equation: (12 cm)/(8 cm) = (Height of Tower)/(40 m)

Simplifying the ratio on the left side: ³⁄₂ = (Height of Tower)/(40)

Solving for the height of the tower: Height of Tower = ³⁄₂ × 40 Height of Tower = 3 × 20 Height of Tower = 60

Since the shadow of the tower is measured in meters, the height of the tower is 60 meters.

Explanation

In the circle where A, B, C, and D are points on the circumference, the chords AC and BD intersect at point E. According to the exterior angle theorem for triangle CED, the exterior angle at vertex E, which is ∠ BEC, is equal to the sum of the two opposite interior angles, ∠ ECD and ∠ EDC.

Given that ∠ BEC = 130° and ∠ ECD = 20°, we substitute these values into the exterior angle relation: ∠ BEC = ∠ ECD + ∠ EDC 130° = 20° + ∠ EDC ∠ EDC = 130° - 20° = 110°

Since the points E, D, and B are collinear, ∠ EDC is the same as ∠ BDC. Furthermore, ∠ BAC and ∠ BDC are angles subtended by the same arc BC at the circumference of the circle. By the property that angles in the same segment of a circle are equal, ∠ BAC = ∠ BDC. Therefore, ∠ BAC = 110°.

Explanation

The area of a triangle is calculated using the formula Area = ½ × base × height. Let the three sides of the triangle be a, b, and c, and their corresponding altitudes be hₐ, h_b, and h_c. The area A of the triangle can be expressed in three ways as A = ½ a hₐ = ½ b h_b = ½ c h_c. Given that the three altitudes are equal, we have hₐ = h_b = h_c = h. Substituting this into the area equations gives ½ a h = ½ b h = ½ c h. Since the altitude h is a non-zero value, we can divide the entire expression by ½ h, which yields a = b = c. Because all three sides of the triangle are equal, the triangle is equilateral.

Explanation

Let the two poles be represented by their heights h₁ = 36 m and h₂ = 24 m. When a horizontal line is drawn from the top of the shorter pole to the taller pole, it forms a right-angled triangle where the vertical side is the difference in height between the two poles.

Difference in height d = 36 - 24 = 12 m Angle with the horizontal θ = 60° Length of the wire L = Hypotenuse

Using the trigonometric relation sin θ = (Opposite)/(Hypotenuse): sin 60° = (12)/(L) √3{2} = (12)/(L) L = (24)/(√(3))

By rationalizing the denominator: L = 24 × √3{3} = 8√3

Explanation

The fundamental trigonometric identity states that csc² θ - cot² θ = 1. This identity can be factored using the difference of squares formula, a² - b² = (a - b)(a + b), which gives (csc θ - cot θ)(csc θ + cot θ) = 1. By rearranging this, we find that 1/(csc θ - cot θ) = csc θ + cot θ.

Substituting this identity into the given expression: 1/(csc θ - cot θ) - 1/sin θ = (csc θ + cot θ) - 1/sin θ

Since the reciprocal of sin θ is csc θ, the expression becomes: (csc θ + cot θ) - csc θ = cot θ

The positive and negative csc θ terms cancel each other, resulting in the final value of cot θ.

Explanation

The given equation is cos Θ + sin Θ = √2 cos Θ. By rearranging this expression, we get: sin Θ = √2 cos Θ - cos Θ sin Θ = (√2 - 1) cos Θ

To solve for cos Θ - sin Θ, we multiply both sides of the equation by (√2 + 1): (√2 + 1) sin Θ = (√2 + 1)(√2 - 1) cos Θ √2 sin Θ + sin Θ = ((√2)² - 1²) cos Θ √2 sin Θ + sin Θ = (2 - 1) cos Θ √2 sin Θ + sin Θ = cos Θ

By transposing sin Θ to the right side of the equation, we obtain: √2 sin Θ = cos Θ - sin Θ

Alternatively, this can be solved using the trigonometric identity (a cos θ + b sin θ)² + (b cos θ - a sin θ)² = a² + b². For a = 1 and b = 1, the relation becomes: (cos Θ + sin Θ)² + (cos Θ - sin Θ)² = 1² + 1² = 2

Substituting the given value cos Θ + sin Θ = √2 cos Θ: (√2 cos Θ)² + (cos Θ - sin Θ)² = 2 2 cos² Θ + (cos Θ - sin Θ)² = 2 (cos Θ - sin Θ)² = 2 - 2 cos² Θ (cos Θ - sin Θ)² = 2(1 - cos² Θ) (cos Θ - sin Θ)² = 2 sin² Θ

Taking the square root of both sides results in cos Θ - sin Θ = √2 sin Θ.

Explanation

Given the trigonometric relations sin α = m sin β and tan α = n tan β. From the first relation, we have sin β = (sin α)/(m). From the second relation, we have tan β = (tan α)/(n).

To find the value of cos² α, we use the identity cos β = (sin β)/(tan β). Substituting the expressions for sin β and tan β gives: cos β = (sin α / m)/(tan α / n) = (n sin α)/(m tan α) = (n cos α)/(m)

We then substitute these values for sin β and cos β into the fundamental identity sin² β + cos² β = 1: ((sin α)/(m))² + ((n cos α)/(m))² = 1 sin² α + n² cos² α = m²

By substituting sin² α = 1 - cos² α, we obtain: 1 - cos² α + n² cos² α = m² cos² α (n² - 1) = m² - 1 cos² α = (m² - 1)/(n² - 1)

Explanation

The expression cos⁴ Θ - sin⁴ Θ can be factored using the algebraic identity a² - b² = (a + b)(a - b), where a = cos² Θ and b = sin² Θ.

cos⁴ Θ - sin⁴ Θ = (cos² Θ + sin² Θ)(cos² Θ - sin² Θ)

Applying the fundamental trigonometric identity cos² Θ + sin² Θ = 1:

1 × (cos² Θ - sin² Θ) = 2/3 cos² Θ - sin² Θ = 2/3

The double-angle identity for cosine states that cos 2Θ = cos² Θ - sin² Θ = 1 - 2sin² Θ. Therefore, the value of the required expression is:

1 - 2sin² Θ = 2/3

Explanation

The trigonometric ratio tan Θ represents the ratio of the perpendicular (P) to the base (B) in a right-angled triangle. Given tan Θ = 3/4, we can assign P = 3 and B = 4. Using the Pythagorean theorem, the hypotenuse (H) is determined by the formula H = √(P² + B²). Substituting the given values, H = √(3² + 4²) = √(9 + 16) = √(25) = 5. Since csc Θ is the ratio of the hypotenuse to the perpendicular (H/P), substituting the calculated values yields csc Θ = 5/3.

Explanation

The expression is 1/(1 + tan² θ) + 1/(1 + cot² θ). Using the fundamental trigonometric identities 1 + tan² θ = sec² θ and 1 + cot² θ = csc² θ, the expression simplifies to:

1/sec² θ + 1/csc² θ

Since 1/sec θ = cos θ and 1/csc θ = sin θ, we can rewrite the terms as:

cos² θ + sin² θ

According to the Pythagorean identity, sin² θ + cos² θ = 1 for any value of θ. Thus, the value of the given expression is 1.

Explanation

Let the cost price (CP) of one article be x. The problem states that the selling price (SP) of 20 articles is equal to the cost price of 22 articles. Therefore, SP of 20 articles = 22x. The cost price of these same 20 articles is 20x.

The profit earned on selling these 20 articles is the difference between their selling price and their cost price: Profit = SP of 20 articles - CP of 20 articles = 22x - 20x = 2x

The gain percentage is calculated using the formula: Gain % = (Profit)/(Cost Price) × 100 Gain % = ²ˣ⁄₂₀ₓ × 100 = ⅒ × 100 = 10%

A quicker approach for problems where the selling price of m items equals the cost price of n items is to use the formula Gain % = (n - m)/(m) × 100. Substituting n = 22 and m = 20, the calculation becomes (22 - 20)/(20) × 100 = ²⁄₂₀ × 100 = 10%.

Explanation

Let the cost price of the article be CP. Since the man gains 10% at the initial selling price (SP₁), we have: SP₁ = CP + 10% of CP = 1.1CP

If the article is sold at double the price, the new selling price (SP₂) is: SP₂ = 2 × SP₁ = 2 × 1.1CP = 2.2CP

The profit made at this new price is the difference between the new selling price and the cost price: Profit = SP₂ - CP = 2.2CP - CP = 1.2CP

To find the profit percentage, we use the formula: Profit Percentage = (Profit / CP) × 100 Profit Percentage = (1.2CP / CP) × 100 = 120%

Alternatively, if we assume the cost price is 100, the initial selling price is 110. Doubling this price gives a new selling price of 220. the profit is 220 - 100 = 120, which is a 120% profit on the original cost price of 100.

Explanation

If the number is x, increasing it by 37.5% (75/2%) results in a new value of x + 0.375x = 1.375x. Setting this result equal to 33 gives x = 33 / 1.375, which simplifies to x = 24. Although the word "increment" technically refers to the amount added (which would make the number 88), the absence of 88 in the options and the presence of 24 indicate that 33 was intended to be the final result after the increase.

Explanation

Let the radius of the circle be r. The problem states that the radius of the circle is equal to the side of the square, so the side of the square s = r. The area of a circle is calculated using the formula A_cᵢᵣ_cₗₑ = π r², and the area of a square is calculated using Aₛ_qᵤₐᵣₑ = s². Substituting the value of the side, we have Aₛ_qᵤₐᵣₑ = r². The ratio of the area of the circle to the area of the square is A_cᵢᵣ_cₗₑ : Aₛ_qᵤₐᵣₑ = π r² : r². Dividing both terms by the common factor r² results in the simplified ratio of π : 1.

Explanation

The required markup percentage is calculated using the relationship between Marked Price (MP) and Cost Price (CP): (MP)/(CP) = (100 + Profit%)/(100 - Discount%). Substituting Profit = 20% and Discount = 25%, the equation becomes (MP)/(CP) = (100 + 20)/(100 - 25) = ¹²⁰⁄₇₅. Dividing both numerator and denominator by 15, the ratio simplifies to ⁸⁄₅. The markup percentage over the cost price is determined by the formula (MP - CP)/(CP) × 100. Inserting the values from the simplified ratio results in (8 - 5)/(5) × 100 = ⅗ × 100 = 60%.

Explanation

The net selling price (SP) after successive discounts of x%, y%, and z% on a marked price (MP) is calculated by reducing the price by each percentage sequentially. The mathematical expression for the final price after these three discounts is:

SP = MP × (1 - ˣ⁄₁₀₀) × (1 - ʸ⁄₁₀₀) × (1 - ᶻ⁄₁₀₀)

Substituting the given value of MP = 5,000 into the formula:

SP = 5000 × (100 - x)/(100) × (100 - y)/(100) × (100 - z)/(100)

By simplifying the numerical constants:

SP = (5000)/(100 × 100 × 100) × (100 - x)(100 - y)(100 - z)

SP = ⁵⁰⁰⁰⁄₁₀₀₀₀₀₀ × (100 - x)(100 - y)(100 - z)

SP = ⁵⁄₁₀₀₀ × (100 - x)(100 - y)(100 - z)

SP = ((100 - x)(100 - y)(100 - z))/(200)

Explanation

The selling price of an article after a discount is calculated using the formula SP = MP - (Discount% × MP). Given the marked price MP = 450 and a discount rate of 10%, the discount amount is 450 × ¹⁰⁄₁₀₀ = 45. Subtracting this discount from the marked price gives the selling price as 450 - 45 = 405. Alternatively, the selling price can be found directly by calculating 90% of the marked price: 450 × 0.9 = 405.

Explanation

Let the initial number of balls in bags x and y be 2k and 3k respectively. According to the condition 2k + 5 = 3k - 5, which gives k = 10. The number of balls in each bag now is 2(10) + 5 = 25.

Explanation

The given ratio for division among A, B, and C is 1/2 : 1/3 : 1/4. To simplify this fractional ratio into integers, multiply each term by the least common multiple (LCM) of the denominators 2, 3, and 4, which is 12.

Simplified ratio = (1/2 × 12) : (1/3 × 12) : (1/4 × 12) = 6 : 4 : 3

The sum of the ratio parts is 6 + 4 + 3 = 13. The total amount to be divided is 2,600. The value of one part is calculated as: One part = 2600 / 13 = 200

The individual shares are calculated by multiplying the ratio parts by the value of one part: Share of A = 6 × 200 = 1,200 Share of B = 4 × 200 = 800 Share of C = 3 × 200 = 600

Explanation

The number can be represented using the division algorithm as N = 425k + 45, where k is an integer quotient. When we divide this expression by 17, we check the divisibility of its components. Since 425 is exactly divisible by 17 (17 × 25 = 425), the term 425k leaves a remainder of 0 when divided by 17. Consequently, the remainder of N divided by 17 is the same as the remainder of 45 divided by 17. Dividing 45 by 17 gives 45 = (17 × 2) + 11, resulting in a remainder of 11.

Explanation

According to the laws of indices, when multiplying terms with the same base, the exponents are added: aᵐ × aⁿ = aᵐ⁺ⁿ. In this expression, the base is 256, so the calculation becomes (256)⁰^.¹⁶⁺⁰^.⁰⁹.

Summing the exponents results in 0.25. The expression simplifies to (256)⁰^.²⁵. The decimal exponent 0.25 is equivalent to the fraction 1/4. Therefore, the expression is (256)¹^/⁴, which represents the fourth root of 256.

The base 256 can be expressed as a power of 4, since 4 × 4 × 4 × 4 = 256, or 4⁴. Substituting this into the expression, we get (4⁴)¹^/⁴. Using the power of a power rule (aᵐ)ⁿ = aᵐ^×ⁿ, the exponents multiply to 4 × 1/4 = 1, resulting in 4¹ = 4.

Explanation

Efficiency and time are inversely proportional for a constant amount of work. If the efficiency of A is taken as 100%, then the efficiency of B is 125%. The ratio of their efficiencies A:B is 100 : 125, which simplifies to 4 : 5.

The ratio of time taken by A and B is the inverse of their efficiency ratio, which is 5 : 4. Given that A takes 6 days to finish the work, the time taken by B is calculated as: Time_B = ⅘ × Time_A Time_B = ⅘ × 6 Time_B = ²⁴⁄₅

Converting the improper fraction ²⁴⁄₅ into a mixed fraction, we get 4 ⅘ days.

Explanation

Let the total work be 60 units, which is the Least Common Multiple of the individual time periods 20, 30, and 60. The daily efficiency of A is 60 / 20 = 3 units, B is 60 / 30 = 2 units, and C is 60 / 60 = 1 unit. According to the problem, A works alone on the first and second days, while on the third day, A is assisted by B and C.

The work done on the first day is 3 units, and on the second day is also 3 units. On the third day, the combined work of A, B, and C is 3 + 2 + 1 = 6 units. Therefore, in one cycle of three days, the total work completed is 3 + 3 + 6 = 12 units. To find the total time required for 60 units of work, we divide the total work by the work done per cycle: 60 / 12 = 5 cycles. Since each cycle represents 3 days, the total duration is 5 × 3 = 15 days.

Explanation

The total work done remains constant, so we equate the total man-boy days from both scenarios. Let M and B represent the work rate of a man and a boy respectively. Using the formula W = (n₁ M + n₂ B) × D₁: 10(2M + 3B) = 8(3M + 2B) 20M + 30B = 24M + 16B 4M = 14B ⟹ 2M = 7B This indicates that the work capacity of 2 men is equal to that of 7 boys. By substituting 2M = 7B into the first scenario, the total work W in terms of boy-days is: W = 10(7B + 3B) = 10(10B) = 100B To find the time for 2 men and 1 boy, we determine their combined work rate: Rate = 2M + 1B = 7B + 1B = 8B The time taken T is the total work divided by the daily rate: T = (100B)/(8B) = 12.5 days

Explanation

The perimeter P of a rectangle is given by P = 2(l + w) and the area A is given by A = l × w. Given P = 48 m, the sum of the dimensions is l + w = 48 / 2 = 24 m. The product of the dimensions is A = 108 m².

We need to find two numbers whose sum is 24 and whose product is 108. These dimensions are the roots of the quadratic equation x² - (sum)x + (product) = 0:

x² - 24x + 108 = 0

Factoring the quadratic equation by finding two numbers that multiply to 108 and add to -24:

(x - 18)(x - 6) = 0

Solving for x gives the dimensions as 18 m and 6 m. Verifying these values, the sum is 18 + 6 = 24 m and the product is 18 × 6 = 108 m².

Explanation

The total surface area of a hemisphere is the sum of its curved surface area and the area of its circular base. The formula for the total surface area (TSA) is: TSA = 2π r² + π r² = 3π r² where r represents the radius of the base. Given that the total surface area is 27π square cm, we can set up the following equation: 3π r² = 27π Dividing both sides of the equation by 3π simplifies the relation to: r² = (27π)/(3π) r² = 9 Taking the square root of both sides gives the radius: r = √9 = 3 cm