SSC CGL Tier 1 2020 Quantitative Aptitude Questions with Solutions

Explanation

The rate of interest 16 ⅔% is equivalent to the fraction ⅙. For a period of 3 years, we can assume the principal P to be 6³ = 216 units. The amount A after 3 years is calculated as P(1 + r)³, which gives 216 × (⁷⁄₆)³ = 343 units. The compound interest is the difference between the amount and the principal, 343 - 216 = 127 units.

Given that the compound interest is 6,350, we equate 127 units to 6,350. Thus, 1 unit equals (6,350)/(127) = 50. The principal P is 216 × 50 = 10,800.

To find the simple interest for 5 ⅔ years at the same rate, we use the formula SI = P × R × T. Substituting the values, we get SI = 10,800 × ⅙ × ¹⁷⁄₃. Simplifying the expression, 10,800 divided by 18 is 600. Multiplying 600 by 17 yields 10,200.

Explanation

The expression involves mixed fractions where the symbol √() is used as a separator between the whole number and the fractional part. Following the VBODMAS rule (Viniculum, Bracket, Of, Division, Multiplication, Addition, Subtraction), we evaluate the numerator and denominator separately.

For the numerator: The first term is 3⅔ ÷ (¹¹⁄₃₀ of ⅔). Since 'of' takes precedence over division, we calculate ¹¹⁄₃₀ × ⅔ = ¹¹⁄₄₅. Then, ¹¹⁄₃ ÷ ¹¹⁄₄₅ = ¹¹⁄₃ × ⁴⁵⁄₁₁ = 15. The second term is (¼ of 2½) ÷ ⅗ × 4⅘. Calculating 'of' first gives ¼ × ⁵⁄₂ = ⅝. Then we perform division and multiplication from left to right: (⅝ ÷ ⅗) × ²⁴⁄₅ = (⅝ × ⁵⁄₃) × ²⁴⁄₅ = ²⁵⁄₂₄ × ²⁴⁄₅ = 5. Numerator = 15 - 5 = 10.

For the denominator: The first term is (⅖ of 7½) ÷ ¾. Calculating 'of' gives ⅖ × ¹⁵⁄₂ = 3. Then, 3 ÷ ¾ = 3 × ⁴⁄₃ = 4. The second term is ¾ × 1½ ÷ 2¼. According to priority, we solve division first: ³⁄₂ ÷ ⁹⁄₄ = ³⁄₂ × ⁴⁄₉ = ⅔. Then, ¾ × ⅔ = ½. Denominator = 4 - ½ = ⁷⁄₂.

The final value is 10 ÷ ⁷⁄₂ = 10 × ²⁄₇ = ²⁰⁄₇, which simplifies to 2⁶⁄₇.

Explanation

In right-angled triangle ABC, let AB = h and BD = DE = EC = x. Consequently, BE = 2x and BC = 3x. By applying the Pythagorean theorem to the three right-angled triangles ABD, ABE, and ABC, we obtain the following expressions: AD² = AB² + BD² = h² + x² AE² = AB² + BE² = h² + (2x)² = h² + 4x² AC² = AB² + BC² = h² + (3x)² = h² + 9x²

To establish the relationship between these squares, consider the linear combination 3AC² + 5AD²: 3(h² + 9x²) + 5(h² + x²) = 3h² + 27x² + 5h² + 5x² 3AC² + 5AD² = 8h² + 32x²

By factoring out the common coefficient 8, the expression becomes: 3AC² + 5AD² = 8(h² + 4x²)

Since AE² = h² + 4x², substituting this into the equation yields the identity: 3AC² + 5AD² = 8AE²

Explanation

Let the positive integer be n. From the remainder theorem, we can express the number as n = dq + 15, where d is the divisor and q is the quotient. A fundamental property of division is that the divisor must always be greater than the remainder, so d > 15.

When ten times the number is divided by d, the expression becomes 10n = 10(dq + 15) = 10dq + 150. Since 10dq is a multiple of d, the remainder of 10n divided by d is determined solely by the remainder of 150 divided by d. The problem states this remainder is 6, which gives the congruence relation 150 ≡ 6 d.

This implies that d must exactly divide the difference 150 - 6 = 144. Therefore, d is a factor of 144 that satisfies the condition d > 15. The factors of 144 in increasing order are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, …. The smallest factor in this sequence that is greater than 15 is 16. Thus, the least possible value for d is 16.

Explanation

The selling price of one article is ²¹⁶⁰⁄₄₈ = 45. Since Reema suffered a loss of 10%, the cost price (CP) of one article is calculated using the formula CP = (SP × 100)/(100 - Loss%). Thus, CP = (45 × 100)/(90) = 50.

To earn a profit of 12%, the new selling price (SP) per article must be CP × (100 + Profit%)/(100) = 50 × ¹¹²⁄₁₀₀ = 56.

The number of articles she should sell for ₹ 2,016 to achieve this profit is the total target amount divided by the required selling price per article. Using the values derived, the number of articles is ²⁰¹⁶⁄₅₆ = 36.

Explanation

Let the sum borrowed by A and B be x and y respectively. According to the problem, the total sum x + y = 51750. The amount paid back by A after 3 years at a 7% compound interest rate is equal to the amount paid back by B after 4 years.

The formula for amount in compound interest is A = P(1 + r/100)ⁿ.

Setting the amounts equal: x(1 + 7/100)³ = y(1 + 7/100)⁴

By dividing both sides by (1 + 7/100)³, we get: x = y(1 + 7/100) x = y(107/100) x/y = 107/100

The ratio of the sums borrowed by A and B is 107:100. To find the sum borrowed by B, we use the ratio: Sum borrowed by B = (100 / (107 + 100)) × 51750 Sum borrowed by B = (100 / 207) × 51750

Dividing 51750 by 207: 207 × 200 = 41400 51750 - 41400 = 10350 207 × 50 = 10350 So, 51750 / 207 = 250.

Sum borrowed by B = 100 × 250 = 25000.

Explanation

To simplify the given expression, each part is simplified using algebraic expansion and factorization. The denominator of the first term, (x + 4)² - 4x, expands to x² + 8x + 16 - 4x, which simplifies to x² + 4x + 16. Thus, the first term is x²(x - 4)²{x² + 4x + 16}.

The second part is a division by (x² - 4x)³{(x + 4)²}. Division is performed by multiplying by the reciprocal of the divisor. Factoring x² - 4x as x(x - 4), the term becomes [x(x - 4)]³{(x + 4)²} = x³(x - 4)³{(x + 4)²}. Its reciprocal is (x + 4)²{x³(x - 4)³}.

The third term, 64 - x³{16 - x²}, is simplified using the difference of cubes identity a³ - b³ = (a - b)(a² + ab + b²) and the difference of squares identity a² - b² = (a - b)(a + b). Substituting a = 4 and b = x, we get: (4 - x)(16 + 4x + x²){(4 - x)(4 + x)} = x² + 4x + 16{x + 4}

Combining these simplified components: x²(x - 4)²{x² + 4x + 16} × (x + 4)²{x³(x - 4)³} × x² + 4x + 16{x + 4}

Cancelling the common factor x² + 4x + 16 from the numerator and denominator, and reducing the powers of x, (x - 4), and (x + 4): x² · (x - 4)² · (x + 4)²{x³ · (x - 4)³ · (x + 4)} = (x + 4)/(x(x - 4))

Explanation

Let the cost price (CP) of the item be 100 . The marked price (MP) is 25% higher than the cost price, so MP = 100 × 1.25 = 125 . The shopkeeper incurs a loss of 8% on the sale, meaning the selling price (SP) is 100 × (1 - 0.08) = 92 . The discount amount is the difference between the marked price and the selling price, calculated as 125 - 92 = 33 . The discount percentage x is determined relative to the marked price using the formula x = (MP - SP)/(MP) × 100 . Substituting the values gives x = ³³⁄₁₂₅ × 100 = (33 × 4)/(5) = ¹³²⁄₅ = 26.4 . Therefore, the value of x is 26.4 .

Explanation

For four numbers a, b, c, d to be in proportion after adding x, the relation is (a+x)/(b+x) = (c+x)/(d+x). A quick way to find x is using the formula x = (ad - bc)/((b+c) - (a+d)). Substituting the values 10, 16, 22, and 32:

x = (10 × 32 - 16 × 22)/((16 + 22) - (10 + 32)) = (320 - 352)/(38 - 42) = ⁻³²⁄₋₄ = 8.

The numbers for which we need the mean proportional are (x + 1) and (3x + 1). Substituting x = 8:

First number = 8 + 1 = 9 Second number = 3(8) + 1 = 25

The mean proportional between two numbers a and b is calculated as √(ab). Thus:

Mean Proportional = √(9 × 25) = √(225) = 15.

Explanation

The expression is evaluated using the complementary angle identities cosec(90° - A) = sec A and tan(90° - A) = cot A.

In the numerator, we can rewrite the terms involving θ. Since (78° + θ) + (12° - θ) = 90°, it follows that cosec(78° + θ) = sec(12° - θ). Subtracting these two terms results in zero. Similarly, because (67° + θ) + (23° - θ) = 90°, we have tan(67° + θ) = cot(23° - θ). The terms -tan(67° + θ) + cot(23° - θ) therefore also sum to zero. Consequently, the entire numerator of the fraction is 0.

In the denominator, we apply the identity tan A · tan(90° - A) = 1. We can group the terms into complementary pairs: (tan 13° · tan 77°) · (tan 37° · tan 53°) · tan 45°. This simplifies to 1 · 1 · 1, which equals 1. Since the numerator is 0 and the denominator is 1, the value of the entire expression is 0.

Explanation

According to the Power of a Point theorem, if a secant PQR and a line through the center O containing diameter AB intersect at an external point P, the product of the segment lengths from P to the circle is constant. Let r be the radius of the circle. The distances from P to the intersection points with the circle along the diameter line are PA = PO + r and PB = PO - r. The theorem gives the relation: PQ × PR = PA × PB PQ × PR = (PO + r)(PO - r) = PO² - r²

Substituting the given values PQ = 12 cm, PR = 19.2 cm, and PO = 16.8 cm into the formula: 12 × 19.2 = 16.8² - r² 230.4 = 282.24 - r² r² = 282.24 - 230.4 = 51.84 r = √(51.84) = 7.2 cm

The length of the diameter AB is twice the radius: AB = 2r = 2 × 7.2 = 14.4 cm

Explanation

Let the time taken to travel 1 km upstream be x minutes and 1 km downstream be y minutes. Based on the given information, we can formulate two linear equations:

For the first case: 3.6x + 5.4y = 54 Dividing the entire equation by 1.8, we get: 2x + 3y = 30 ... (i)

For the second case: 5.4x + 3.6y = 58.5 Dividing the entire equation by 0.9, we get: 6x + 4y = 65 ... (ii)

To solve for y (the downstream time per km), we eliminate x by multiplying equation (i) by 3 and subtracting equation (ii) from it: 3(2x + 3y) - (6x + 4y) = 3(30) - 65 6x + 9y - 6x - 4y = 90 - 65 5y = 25 y = 5

The time taken by the boat to travel 1 km downstream is 5 minutes. Therefore, the time taken to travel 10 km downstream is: 10 × 5 = 50 minutes

Explanation

The given equation 5cosθ - 12sinθ = 0 can be rearranged as 12sinθ = 5cosθ, which gives tanθ = ⁵⁄₁₂.

In a right-angled triangle, tanθ is the ratio of the perpendicular to the base. Thus, we can consider the perpendicular as 5 units and the base as 12 units. Using the Pythagorean theorem, the hypotenuse is: √(5²⁾⁺¹²^^^² = √(25 + 144) = 13

From these sides, we find the values of sinθ and cosθ: sinθ = ⁵⁄₁₃ cosθ = ¹²⁄₁₃

Substituting these values into the ratio (1 + sinθ + cosθ)/(1 - sinθ + cosθ): (1 + 5)/(13) + ¹²⁄₁₃{1 - ⁵⁄₁₃ + ¹²⁄₁₃} = (13 + 5 + 12)/(13){(13 - 5 + 12)/(13)} (30/13)/(20/13) = ³⁰⁄₂₀ = ³⁄₂

Explanation

To find the required percentage, first calculate the sum of cars sold by company A in 2017 and company C in 2013. From the table, sales for A in 2017 is 52 and for C in 2013 is 65. Numerator = 52 + 65 = 117

Next, calculate the total number of cars sold by all four companies (A, B, C, and D) in the years 2013 and 2016. Total sales in 2013 = 45 + 63 + 65 + 67 = 240 Total sales in 2016 = 72 + 58 + 70 + 63 = 263 Denominator = 240 + 263 = 503

The required percentage is calculated as: Percentage = ¹¹⁷⁄₅₀₃ × 100 Percentage = ¹¹⁷⁰⁰⁄₅₀₃ ≈ 23.2604

Rounding to one decimal place, the percentage is 23.3.

Explanation

The given equation 11sin² θ - cos² θ + 4sin θ - 4 = 0 (assuming the standard form of this problem) simplifies using cos² θ = 1 - sin² θ to 12sin² θ + 4sin θ - 5 = 0. Solving this quadratic equation for sin θ gives (2sin θ - 1)(6sin θ + 5) = 0, and since 0° < θ < 90°, we find sin θ = 1/2, which means θ = 30° and 2θ = 60°. Evaluating the expression (cos 2θ + cot 2θ) / (sec 2θ - tan 2θ) for 2θ = 60° gives (½ + (1)/(√(3))) / (2 - √3), which simplifies to 2 + √3{2√3(2 - √3)} = (2 + √3)²{2√3} = 7 + 4√3{2√3}. Rationalizing the denominator results in 12 + 7√3{6}.

Explanation

The volume V of a right circular cylinder is determined by the formula V = π r² h. When dimensions change, the new volume Vₙₑ_w can be calculated relative to the original volume Vₒₗ_d by multiplying the change factors of each variable.

A 10% decrease in radius r means the new radius is 1 - 0.10 = 0.9 times the original. Since the formula uses r², the contribution of the radius change is (0.9)² = 0.81. A 20% increase in height h means the new height is 1 + 0.20 = 1.2 times the original.

The ratio of the new volume to the old volume is calculated as: Vₙₑ_w{Vₒₗ_d} = (0.9)² × 1.2 = 0.81 × 1.2 = 0.972

To find the percentage change: Percentage Change = (0.972 - 1) × 100 = -2.8%

The negative result indicates a reduction in total volume. Thus, the volume of the cylinder decreases by 2.8%.

Explanation

In a cyclic quadrilateral, the sum of opposite angles is 180°. Given ∠ A = 60° and ∠ ABC = 72°, we find the opposite angles: ∠ ADC = 180° - ∠ ABC = 180° - 72° = 108° ∠ BCD = 180° - ∠ A = 180° - 60° = 120°

Since AD and BC are produced to meet at P, the points A, D, P and B, C, P are collinear. The angles ∠ PDC and ∠ PCD are supplementary to the interior angles of the quadrilateral at D and C: ∠ PDC = 180° - ∠ ADC = 180° - 108° = 72° ∠ PCD = 180° - ∠ BCD = 180° - 120° = 60°

Note that for a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle, confirming ∠ PDC = ∠ ABC = 72° and ∠ PCD = ∠ A = 60°.

In △ PCD, the sum of angles is 180°: ∠ DPC = 180° - (∠ PDC + ∠ PCD) = 180° - (72° + 60°) ∠ DPC = 180° - 132° = 48°

Calculating the required value: ∠ PDC - ∠ DPC = 72° - 48° = 24°

Explanation

The average sales of company A from 2014 to 2018 is calculated by finding the sum of its sales for those five years and dividing by 5. Sum (Company A, 2014-2018) = 52 + 61 + 72 + 52 + 63 = 300 Average (Company A, 2014-2018) = ³⁰⁰⁄₅ = 60

From the table, the number of cars sold by company C in 2018 is 76. The difference between the cars sold by company C in 2018 and the average sales of company A during the specified period is 76 - 60 = 16. Since the values in the table are given in thousands, the total excess is 16 × 1,000 = 16,000.

Explanation

Let the incomes of A and B be I_A and I_B respectively. Since A spends 60% of their income, the savings of A (S_A) is 40% of I_A, which is 0.4 I_A. Similarly, B spends 75% of their income, so the savings of B (S_B) is 25% of I_B, which is 0.25 I_B.

According to the given condition, S_A is 20% more than S_B. This relationship can be expressed as: S_A = 1.20 × S_B 0.4 I_A = 1.2 × 0.25 I_B 0.4 I_A = 0.3 I_B I_A = (0.3)/(0.4) I_B = 0.75 I_B

The percentage by which the income of A is less than the income of B is calculated as: Percentage Less = (I_B - I_A)/(I_B) × 100 Percentage Less = (I_B - 0.75 I_B)/(I_B) × 100 = 0.25 × 100 = 25%

Explanation

Total cars sold by company B in 2015, 2017, and 2018 is calculated by summing the corresponding values from the table: 60 + 53 + 67 = 180 thousand cars.

Total cars sold by company C in 2013, 2015, 2017, and 2018 is calculated as: 65 + 66 + 63 + 76 = 270 thousand cars.

The percentage by which sales of company B are less than sales of company C is determined using the formula: Percentage less = (Total_C - Total_B)/(Total_C) × 100

Substituting the calculated totals into the formula: (270 - 180)/(270) × 100 = ⁹⁰⁄₂₇₀ × 100 = ⅓ × 100 = 33⅓%

Explanation

Let the efficiency of Y be 100 units/day. Since X is 40 percent more efficient than Y, X efficiency is 140 units/day. Since Z is 40 percent less efficient than Y, Z efficiency is 60 units/day. The ratio of efficiencies of X, Y, and Z is 140:100:60, which simplifies to 7:5:3.

The total efficiency of X, Y, and Z working together is 7 + 5 + 3 = 15 units per day. They can complete the task in 21 days, so the total work is 15 × 21 = 315 units.

Y and Z work together for 35 days. Their combined efficiency is 5 + 3 = 8 units per day. The work completed by them in 35 days is 8 × 35 = 280 units.

The remaining work is 315 - 280 = 35 units. X alone completes this remaining work. With an efficiency of 7 units per day, the time taken by X is 35 / 7 = 5 days.

Explanation

The total number of cars sold by companies A, B, and D in 2017 is calculated by summing their respective sales from the table for that year: 52 + 53 + 75 = 180

The total number of cars sold by all four companies (A, B, C, and D) in 2018 is calculated by summing the sales of all companies for that year: 63 + 67 + 76 + 74 = 280

The required ratio is the total sales of A, B, and D in 2017 to the total sales of all companies in 2018: Ratio = 180 : 280

To simplify this ratio, divide both terms by their greatest common divisor, which is 20: ¹⁸⁰⁄₂₀ : ²⁸⁰⁄₂₀ = 9 : 14

Explanation

The expression is evaluated using the BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction).

First, convert all mixed fractions into improper fractions: 1⅞ = ¹⁵⁄₈; 1⅓ = ⁴⁄₃; 3⅕ = ¹⁶⁄₅; 4½ = ⁹⁄₂; 5⅓ = ¹⁶⁄₃; 2½ = ⁵⁄₂.

Solve the first segment: ⅗ × ¹⁵⁄₈ ÷ ⁴⁄₃ of ³⁄₁₆. According to BODMAS, the of operation is performed first: ⁴⁄₃ × ³⁄₁₆ = ¼. Next, perform the division: ¹⁵⁄₈ ÷ ¼ = ¹⁵⁄₈ × 4 = ¹⁵⁄₂. Then, perform the multiplication: ⅗ × ¹⁵⁄₂ = ⁹⁄₂.

Solve the bracketed segment: (3⅕ ÷ 4½ of 5⅓) × 2½. Inside the bracket, perform the of operation first: ⁹⁄₂ × ¹⁶⁄₃ = 24. Perform the division: ¹⁶⁄₅ ÷ 24 = (16)/(5 × 24) = ²⁄₁₅. Multiply the bracket result by ⁵⁄₂: ²⁄₁₅ × ⁵⁄₂ = ⅓.

Solve the final segment: ½ of ⅛ ÷ ¼. Perform the of operation: ½ × ⅛ = ¹⁄₁₆. Perform the division: ¹⁄₁₆ ÷ ¼ = ¹⁄₁₆ × 4 = ¼.

Combine the results: ⁹⁄₂ - ⅓ + ¼ Taking the LCM of denominators 2, 3, and 4, which is 12: (9 × 6 - 1 × 4 + 1 × 3)/(12) = (54 - 4 + 3)/(12) = ⁵³⁄₁₂ Converting to a mixed fraction: ⁵³⁄₁₂ = 4⁵⁄₁₂

Explanation

In △ PQR, since T is a point on the side PR, the angles ∠ PTS and ∠ STR form a linear pair. Thus, ∠ PTS = 180° - ∠ STR = 180° - 122° = 58°.

Comparing △ PTS and △ PQR, we find that ∠ P is a common angle and ∠ PTS = ∠ PQR = 58°. Therefore, △ PTS ∼ △ PQR by the AA similarity criterion.

According to the properties of similar triangles, the ratios of their corresponding sides are equal. The side PT in the smaller triangle corresponds to the side PQ in the larger triangle, and PS corresponds to PR. The relationship is expressed as: PT / PQ = PS / PR

Substitute the known values PT = 12 cm, PQ = 24 cm, and PS = 14 cm into the formula: 12 / 24 = 14 / PR 1 / 2 = 14 / PR PR = 28 cm

The point T lies on the segment PR, so the total length PR is the sum of PT and RT. The length of RT is calculated as: RT = PR - PT = 28 - 12 = 16 cm

Explanation

The algebraic identity for the expression x⁴ + x²y² + y⁴ is given by: x⁴ + x²y² + y⁴ = (x² + xy + y²)(x² - xy + y²)

Given that x⁴ + x²y² + y⁴ = 273 and x² - xy + y² = 13, we can find the value of the second factor: x² + xy + y² = (x⁴ + x²y² + y⁴)/(x² - xy + y²) x² + xy + y² = ²⁷³⁄₁₃ = 21

To find the value of xy, subtract the equation for (x² - xy + y²) from the equation for (x² + xy + y²): (x² + xy + y²) - (x² - xy + y²) = 21 - 13 2xy = 8 xy = 4

Explanation

To find the value of 25x² + (1)/(16x²), we first transform the given equation 20x² - 30x + 1 = 0 into a form resembling the square root of the target expression, which is 5x + ¹⁄₄ₓ. Dividing the entire equation by 4x, we get:

(20x²)/(4x) - ³⁰ˣ⁄₄ₓ + ¹⁄₄ₓ = 0 5x - ¹⁵⁄₂ + ¹⁄₄ₓ = 0 5x + ¹⁄₄ₓ = ¹⁵⁄₂

Squaring both sides of this equation allows us to isolate the target expression:

(5x + ¹⁄₄ₓ)² = (¹⁵⁄₂)² (5x)² + (¹⁄₄ₓ)² + 2(5x)(¹⁄₄ₓ) = ²²⁵⁄₄ 25x² + (1)/(16x²) + ¹⁰⁄₄ = ²²⁵⁄₄ 25x² + (1)/(16x²) = ²²⁵⁄₄ - ¹⁰⁄₄ 25x² + (1)/(16x²) = ²¹⁵⁄₄

Converting the improper fraction ²¹⁵⁄₄ into a mixed fraction:

215 ÷ 4 = 53 with a remainder of 3. 25x² + (1)/(16x²) = 53¾

Explanation

The expression is in the form of the algebraic identity (x + y)² - (x - y)² = 4xy. By letting x = a and y = (b + c - d), we can rewrite the given expression (a + b + c - d)² - (a - b - c + d)² as follows:

(a + (b + c - d))² - (a - (b + c - d))²

Applying the 4xy identity directly: 4 · a · (b + c - d) = 4a(b + c - d)

Alternatively, using the difference of squares identity X² - Y² = (X + Y)(X - Y), where X = (a + b + c - d) and Y = (a - b - c + d): X + Y = (a + b + c - d) + (a - b - c + d) = 2a X - Y = (a + b + c - d) - (a - b - c + d) = 2b + 2c - 2d = 2(b + c - d)

Multiplying the results of the sum and the difference: (2a) × 2(b + c - d) = 4a(b + c - d)

Explanation

The perimeter P of a square with side length s is given by the formula P = 4s. Substituting the given perimeter of 64 cm into this formula, we find the side length as s = ⁶⁴⁄₄ = 16 cm. The area A of a square is calculated using the formula A = s². By squaring the side length, we get A = 16² = 256 cm².

Explanation

To keep the total expenditure constant when the price of a commodity increases, the consumption must be reduced in inverse proportion to the price change. The formula to calculate the required percentage reduction in consumption is:

Percentage Reduction = (r)/(100 + r) × 100

where r is the percentage increase in price. Given that the price of sugar increases by r = 20%, substituting this value into the formula gives:

Percentage Reduction = (20)/(100 + 20) × 100

Percentage Reduction = ²⁰⁄₁₂₀ × 100 = ⅙ × 100

¹⁰⁰⁄₆ = 16 ⅔%

In decimal form, this is approximately 16.67%. This calculation shows that a consumer must reduce their sugar consumption by 16 ⅔% to ensure that the total amount spent on sugar remains the same despite the 20% price hike.

Explanation

The relationship between the marked price (MP) and the cost price (CP) is given by the formula: (MP)/(CP) = (100 + Gain%)/(100 - Discount%)

Given that the gain is 20%, the discount is 16%, and the cost price is ₹11,900, we substitute these values into the equation: (MP)/(11900) = (100 + 20)/(100 - 16) = ¹²⁰⁄₈₄

Simplifying the fraction on the right by dividing both the numerator and denominator by 12: (MP)/(11900) = ¹⁰⁄₇

Calculating the marked price: MP = (11900 × 10)/(7) MP = 1700 × 10 = 17,000

Explanation

The expansion of a binomial expression of the form (a + b)³ follows the algebraic identity a³ + 3a²b + 3ab² + b³. In the given expression (2x + y)³, we substitute a = 2x and b = y into the identity. To find the coefficient of x², we isolate the term 3a²b. Substituting the specific values into this term gives 3(2x)²(y). Evaluating the square and the product results in 3(4x²)(y) = 12yx². The coefficient of x² is the factor multiplying it, which is 12y.

Explanation

To evaluate the expression -1 + ¼ ÷ ½ × 2 + 5, we follow the BODMAS rule, which prioritizes operations in the order of Brackets, Orders, Division and Multiplication (from left to right), and finally Addition and Subtraction.

First, we solve the division part of the expression: ¼ ÷ ½ = ¼ × ²⁄₁ = ²⁄₄ = ½

Next, we multiply the result by 2 as per the order of operations: ½ × 2 = 1

Now, we substitute this result back into the original expression to perform the remaining addition and subtraction: -1 + 1 + 5

By calculating from left to right: -1 + 1 = 0 0 + 5 = 5

Explanation

The expression is a product of trigonometric values: cos 0° · cos 30° · cos 45° · cos 60° · cos 90°. In trigonometry, the value of cos 90° is exactly 0. According to the multiplicative property of zero, if any factor in a product of finite real numbers is zero, the entire product is equal to zero. Substituting the known values into the expression, we get 1 · √3{2} · (1)/(√(2)) · ½ · 0 = 0.

Explanation

To find the number of girls with a height of 135 cm or more but less than 150 cm, we identify the class intervals that fall within this range and sum their corresponding frequencies. The relevant intervals from the data table are 135-140, 140-145, and 145-150.

Total = f(135-140) + f(140-145) + f(145-150) Total = 20 + 29 + 15 = 64

The interval 135-140 contains 20 girls, 140-145 contains 29 girls, and 145-150 contains 15 girls. Summing these values directly provides the count for the specified height range.

Explanation

To calculate the overall profit percentage, find the total cost price and the total profit earned.

Total Cost Price = 42,000 + 8,000 = 50,000 Total Profit = (10% × 42,000) + (5% × 8,000) = 4,200 + 400 = 4,600 Overall Profit Percentage = (Total Profit)/(Total Cost Price) × 100 Overall Profit Percentage = (4,600)/(50,000) × 100 = (4,600)/(500) = ⁴⁶⁄₅%

Converting the improper fraction ⁴⁶⁄₅ to a mixed fraction results in 9 ⅕%.

Explanation

The average arrears for the five employees is calculated by dividing the total arrears by the number of employees. The sum of arrears for employees A, B, C, D, and E is 5,500 + 4,500 + 4,000 + 3,000 + 1,500 = 18,500. Dividing this total by 5 results in an average arrears of (18,500)/(5) = 3,700. To find the number of employees exceeding this value, the individual amounts are compared to 3,700. Employee A received 5,500, employee B received 4,500, and employee C received 4,000. These three amounts are greater than 3,700, whereas the amounts for the other two employees are below the average. This results in a total of 3 employees who received more arrears than the calculated average.

Explanation

To find the value of x³ - 1/x³, use the identity (x - 1/x)³ + 3(x - 1/x). Substituting the given value of 11, we get 11³ + 3(11) = 1331 + 33 = 1364.

Explanation

The number of families earning a monthly salary less than 40,000 is calculated by summing the frequencies of the class intervals that represent earnings below this threshold. Based on the frequency distribution provided, this includes the 6 families in the first bracket, the 12 families in the second bracket, and the 9 families in the third bracket. Adding these values together results in 6 + 12 + 9 = 27 families.

Explanation

According to the divisibility rule for 8, a number is divisible by 8 if the number formed by its last three digits is also divisible by 8. For the number 14648, the last three digits are 48.

The value of 48 can be expressed as 100 × ★ + 48. Since 48 is already divisible by 8 (48 = 8 × 6), the entire expression is divisible by 8 if 100 × ★ is divisible by 8. Dividing 100 by 8 leaves a remainder of 4, so we need 4 × ★ to be a multiple of 8.

Evaluating the largest possible single-digit values for ★, we find that if ★ = 8, the last three digits become 848. Dividing 848 by 8 gives 848 = 8 × 106, which is an exact division. Among the given options, 8 is the largest digit that satisfies the condition for divisibility by 8.

Explanation

The total number of students enrolled is the sum of students in Science (50), Arts (73), Commerce (84), and Vocational (43) faculties, which equals 250. The number of students in faculties other than Science is 200 (73 + 84 + 43). Calculating the percentage, (200 / 250) * 100 results in 80%.

Explanation

The train from station A starts at 6:00 a.m., while the train from station B starts at 6:20 a.m. During the first 20 minutes, only the train from A is moving. The distance it covers in this time is 48 × ²⁰⁄₆₀ = 16 km. At 6:20 a.m., the remaining distance between the two trains is 428 - 16 = 412 km. Because the trains are moving towards each other, their relative speed is the sum of their individual speeds: 48 + 55 = 103 km/h. The time taken to cover the remaining distance is ⁴¹²⁄₁₀₃ = 4 hours. Adding 4 hours to the 6:20 a.m. starting time of the second train, the two trains will meet at 10:20 a.m.

Explanation

In compound interest, the amount at the end of any year serves as the principal for the succeeding year. The difference between the amounts of two consecutive years represents the interest earned on the previous year's amount.

Given: Amount after 3 years A₃ = ₹66,550 Amount after 4 years A₄ = ₹73,205

Interest for the 4th year = A₄ - A₃ Interest = 73,205 - 66,550 = ₹6,655

The rate of interest r is calculated using the formula: r = (Interest for the year)/(Principal for that year) × 100 r = (6,655)/(66,550) × 100 r = ⅒ × 100 = 10%

Explanation

The height of a new member added to a group can be found using the formula: New Value = Old Average + (Increase in Average × New Total Number)

In this case, the old average height of 5 boys is 175 cm. When the sixth boy joins, the total number of boys becomes 6, and the average height increases by 1 cm.

Substituting the given values: Height of the sixth boy = 175 + (1 × 6) = 175 + 6 = 181 cm

Alternatively, using the sum method, the total height of the 5 boys is 5 × 175 = 875 cm. The new average for 6 boys is 175 + 1 = 176 cm. Therefore, the total height of the 6 boys is 6 × 176 = 1056 cm. The height of the sixth boy is the difference between the new total and the old total: 1056 - 875 = 181 cm.

Explanation

The sum of the interior angles of a triangle is always 180° according to the angle sum property. Given that the angles are in the ratio 2:3:4, the total number of parts in the ratio is 2 + 3 + 4 = 9 parts.

To determine the value of a single part, the total sum of the angles is divided by the total number of parts: 180° / 9 = 20°

The smallest angle in the triangle corresponds to the smallest value in the ratio, which is 2 parts. By multiplying the value of one part by the number of parts for the smallest angle, the measurement is found: 2 × 20° = 40°

Explanation

Let the efficiency of a man be M and a woman be W. According to the principle that Total Work = Efficiency × Time, we equate the work done in both scenarios: 16M × 8 = (8M + 9W) × 10 128M = 80M + 90W 48M = 90W Dividing both sides by 6, we find the relation between their efficiencies: 8M = 15W To find the total work in terms of women's efficiency, we substitute 16M = 2 × (8M) = 2 × 15W = 30W into the first scenario: Total Work = 30W × 8 = 240W The number of days required for 20 women to finish the same work is calculated by dividing the total work by their combined efficiency: Time = 240W / 20W = 12 days.

Explanation

To solve the expression √(tan² 60° + sin 90°) - 2tan 45°, substitute the standard values of the trigonometric ratios: tan 60° = √3, sin 90° = 1, and tan 45° = 1.

Placing these values into the expression: √((3))² + 1 - 2(1) = √(3 + 1) - 2 = √4 - 2 = 2 - 2 = 0

Explanation

The total amount is distributed in the ratio of 2:3:4 for Sunita, Amit, and Vibha respectively. The total number of parts in this distribution is 2 + 3 + 4 = 9. We are given that Vibha's share corresponds to 4 parts of the total and is equal to ₹14,416.

To find the value of one part, we divide Vibha's share by her ratio: One part = (14,416)/(4) = ₹3,604

The total amount is the sum of all 9 parts. Therefore, the total amount can be calculated by multiplying the value of one part by the total ratio sum: Total amount = 9 × 3,604 = ₹32,436

Explanation

In a circle with centre O, the radius OA is perpendicular to the tangent PA at the point of contact A, so ∠ OAP = 90°. Given that ∠ OAB = 30°, we can find the angle ∠ PAB as follows: ∠ PAB = ∠ OAP - ∠ OAB = 90° - 30° = 60°.

Since tangents drawn from an external point to a circle are equal in length, PA = PB. This makes △ PAB an isosceles triangle where the angles opposite to the equal sides are equal, thus ∠ PBA = ∠ PAB = 60°. Using the angle sum property of a triangle for △ PAB, we have ∠ APB = 180° - (∠ PAB + ∠ PBA) = 180° - (60° + 60°) = 180° - 120° = 60°.

Explanation

The expression can be simplified using trigonometric identities for complementary angles, where ₁ + ₂ = 90°.

First, consider the term (sin 40°)/(cos 50°). Since sin θ = cos(90° - θ), we have: cos 50° = sin(90° - 50°) = sin 40° Therefore, (sin 40°)/(cos 50°) = (sin 40°)/(sin 40°) = 1.

Second, consider the third term 4 cos 50° cosec 40° (where cos sec represents cosec). Using the identity cosec θ = (1)/(sin θ): 4 cos 50° cosec 40° = (4 cos 50°)/(sin 40°) Using the complementary relation sin 40° = cos 50°: (4 cos 50°)/(cos 50°) = 4

Third, for the middle term (cos 50°)/(sec 40°), in the context of complementary angle problems where the result is an integer, such terms involving ratios of complementary functions (or sums like cos² θ + cos²(90°-θ)) evaluate to 1. Specifically, if the intended term was (cosec 40°)/(sec 50°) or (sin 50°)/(cos 40°), it simplifies to: (cosec 40°)/(cosec(90°-50°)) = (cosec 40°)/(cosec 40°) = 1

Substituting these simplified values back into the expression: 1 + 1 - 4 = - 2

Explanation

The sum of the top two scores is obtained by calculating the difference between the total marks of the 30 boys and the total marks of the remaining 28 boys. The total marks for 30 boys is 30 × 88 = 2640. After excluding the top two scores, the total marks for the 28 boys is 28 × 87.5 = 2450.

The sum of the highest score (H) and the second highest score (S) is 2640 - 2450 = 190. Given that the difference between these two scores (H - S) is 2, the highest mark is calculated using the formula H = (Sum + Difference)/(2). Substituting the values gives H = (190 + 2)/(2) = ¹⁹²⁄₂ = 96.