SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

We first check if the triangle is right-angled by applying the Pythagorean theorem to the side lengths 48, 55, and 73. Calculating the squares of the two shorter sides gives 48² = 2304 and 55² = 3025. Their sum is 2304 + 3025 = 5329. Since 73² = 5329, the condition a² + b² = c² is satisfied, proving the triangle is a right-angled triangle where the longest side (73 cm) is the hypotenuse. In a right-angled triangle, the length of the median drawn to the hypotenuse is always equal to half the length of the hypotenuse. This is because the midpoint of the hypotenuse is the circumcenter of the triangle, making the median equal to the circumradius. Thus, the length of the median is 73 / 2 = 36.5 cm.

Explanation

The smallest number to be added to a dividend to make it perfectly divisible by a divisor is calculated using the formula Divisor - (Dividend Divisor). First, calculate the remainder when 4567 is divided by 7: 4567 = (652 × 7) + 3 The remainder is 3. Substituting the values into the formula: Number to be added = 7 - 3 = 4. Adding 4 to 4567 results in 4571, which is exactly divisible by 7 as 4571 = 7 × 653.

Explanation

When a merchant cheats by a specific percentage during both the purchase and the sale of goods, the total profit percentage is determined using the successive percentage increase formula. Let x be the cheating percentage while buying and y be the cheating percentage while selling. The net profit percentage is expressed as:

Total Profit % = x + y + ˣʸ⁄₁₀₀

In this problem, the businessman cheats by 12% each time, so x = 12 and y = 12. Substituting these values into the formula:

Profit % = 12 + 12 + (12 × 12)/(100)

Profit % = 24 + ¹⁴⁴⁄₁₀₀

Profit % = 24 + 1.44

Profit % = 25.44%

This formula provides the effective percentage gain resulting from the two separate fraudulent actions. The first 12% gain occurs because he acquires more material than he pays for, and the second 12% gain occurs because he delivers less material than the customer pays for, leading to a combined effect calculated through successive addition.

Explanation

From the given table, the income values for Adam are as follows: Income from Bonus = 90 Income from Arrears = 200

To calculate the percentage of income from bonus relative to the income from arrears for Adam, we use the formula: Percentage = ( (Income from Bonus)/(Income from Arrears) ) × 100

Substituting the respective values into the formula: Percentage = ( ⁹⁰⁄₂₀₀ ) × 100 Percentage = ⁹⁰⁄₂ = 45%

Explanation

The expression ab - a - b + 1 can be factored using the method of grouping. By grouping the first two terms and the last two terms, we get (ab - a) - (b - 1). Factoring out the common factor a from the first group yields a(b - 1) - 1(b - 1). Since (b - 1) is now a common factor for both terms, it can be factored out to give (a - 1)(b - 1). Alternatively, the correctness can be verified by expanding the product (a - 1)(b - 1) using the distributive property, where a(b - 1) - 1(b - 1) = ab - a - b + 1, confirming that the two sides are algebraically identical.

Explanation

The expression is a series of differences of squares of consecutive integers. Using the algebraic identity a² - b² = (a - b)(a + b), and noting that for any two consecutive integers a and b, the term (a - b) is equal to 1, each pair simplifies as follows: 100² - 99² = (100 - 99)(100 + 99) = 100 + 99 98² - 97² = (98 - 97)(98 + 97) = 98 + 97 ... 12² - 11² = (12 - 11)(12 + 11) = 12 + 11

The entire series simplifies to the sum of all natural numbers from 11 to 100. The sum of the first n natural numbers is given by the formula Sₙ = (n(n + 1))/(2). To calculate the sum from 11 to 100, we subtract the sum of the first 10 natural numbers from the sum of the first 100 natural numbers: Sum = ¹⁰⁰⁽¹⁰¹⁾⁄₂ - ¹⁰⁽¹¹⁾⁄₂ Sum = 5050 - 55 Sum = 4995

Explanation

The initial savings of Mr. X are determined by the difference between his monthly income and expenditure, which is 26,500 - 20,500 = 6,000. To find the increase in savings, we calculate the absolute increase in his income and expenditure. The increase in income is 12% of 26,500, calculated as 0.12 × 26,500 = 3,180. The increase in expenditure is 6% of 20,500, calculated as 0.06 × 20,500 = 1,230.

The net increase in savings is the difference between the increase in income and the increase in expenditure, which is 3,180 - 1,230 = 1,950. The percentage increase in savings is then calculated by dividing this increase by the original savings and multiplying by 100. Using the formula for percentage increase, we get (1,950 / 6,000) × 100. This simplifies to 195 / 6, which equals 32.5%.

Explanation

To determine the number of triangles with integer side lengths a, b, c and perimeter P = 13, we use the triangle inequality which states that the sum of any two sides must be greater than the third side. Given a + b + c = 13, the condition a + b > c implies 13 - c > c, leading to 2c < 13 or c < 6.5. Assuming c is the longest side, it must also satisfy 3c ≥ 13, meaning c ≥ 5. Thus, the possible integer values for the largest side c are 5 and 6.

When c = 6, the other two sides must satisfy a + b = 13 - 6 = 7. The integer pairs (a, b) that satisfy a ≤ b ≤ 6 are (1, 6), (2, 5), and (3, 4), yielding three triangles. When c = 5, the other two sides must satisfy a + b = 13 - 5 = 8. The integer pairs (a, b) that satisfy a ≤ b ≤ 5 are (3, 5) and (4, 4), yielding two triangles. Summing these results gives a total of 5 possible triangles. Alternatively, for an odd perimeter P, the number of triangles can be calculated using the formula round((P+3)² / 48). Substituting P = 13 gives 16² / 48 = 256 / 48 ≈ 5.33, which rounds to 5.

Explanation

Let the Principal be P. The Amount A is given as ¹⁷⁄₁₀ of the principal, so A = ¹⁷⁄₁₀P. The Simple Interest SI earned during the period is the difference between the amount and the principal: SI = A - P = ¹⁷⁄₁₀P - P = ⁷⁄₁₀P. The time period T is 2 years and 6 months, which is equivalent to 2.5 years.

Using the standard formula for Simple Interest SI = (P × R × T)/(100), where R is the rate of interest per annum:

⁷⁄₁₀P = (P × R × 2.5)/(100)

By canceling P from both sides, the equation becomes:

0.7 = (2.5R)/(100)

70 = 2.5R

R = (70)/(2.5) = 28

Therefore, the rate of interest per annum is 28 percent.

Explanation

The person's speed from P to Q is v₁ = 50 km/h. The return speed from Q to P is increased by 60%, which means v₂ = 50 × (1 + 0.60) = 50 × 1.6 = 80 km/h. When a person travels equal distances at different speeds, the average speed for the entire journey is the harmonic mean of the speeds, calculated using the formula Vₐᵥ_g = (2v₁v₂)/(v₁ + v₂). Substituting the known values into the formula gives Vₐᵥ_g = (2 × 50 × 80)/(50 + 80) = ⁸⁰⁰⁰⁄₁₃₀ = ⁸⁰⁰⁄₁₃ ≈ 61.538 km/h. Rounded to two decimal places, this results in an average speed of 61.53 km/h.

Explanation

To compare speeds in different units, we convert kilometers per hour to meters per second using the formula 1 km/h = ⁵⁄₁₈ m/s. For 99 km/h, the calculation is: 99 × ⁵⁄₁₈ = 27.5 m/s This result shows that 99 km/h is equivalent to 27.5 m/s, which is a greater speed than 24 m/s.

Explanation

In △ ABC, the point where the internal bisectors of ∠ B and ∠ C intersect is the incenter of the triangle. The angle formed at this intersection point, denoted as ∠ BIC, is related to the opposite vertex angle ∠ A by the standard geometric formula:

∠ BIC = 90° + (∠ A)/(2)

Given that the internal bisectors intersect at an angle of 105°, we substitute this value into the formula to solve for ∠ A:

105° = 90° + (∠ A)/(2)

Subtracting 90° from both sides of the equation:

15° = (∠ A)/(2)

Multiplying both sides by 2:

∠ A = 30°

This property is derived from the sum of angles in a triangle. In △ BIC, the sum of angles is ∠ BIC + (∠ B)/(2) + (∠ C)/(2) = 180°. In △ ABC, the sum of angles is ∠ A + ∠ B + ∠ C = 180°, which can be rearranged as (∠ B + ∠ C)/(2) = 90° - (∠ A)/(2). Substituting this into the first equation confirms the relation ∠ BIC = 180° - (90° - (∠ A)/(2)) = 90° + (∠ A)/(2).

Explanation

Using the trigonometric identity cot A = (cos A)/(sin A), the given expression cot² A - cos² A can be rewritten by substituting for the cotangent term:

(cos² A)/(sin² A) - cos² A

Factoring out the common term cos² A from the expression yields:

cos² A ( (1)/(sin² A) - 1 )

Taking the common denominator inside the parentheses, the expression becomes:

cos² A ( (1 - sin² A)/(sin² A) )

According to the Pythagorean identity sin² A + cos² A = 1, we have 1 - sin² A = cos² A. Substituting this into the numerator:

cos² A ( (cos² A)/(sin² A) )

Since (cos² A)/(sin² A) = cot² A, the final simplified product is:

cos² A · cot² A

Explanation

The volume V of a hemisphere is calculated using the formula V = ⅔π r³, where r represents the radius. Substituting the given volume of 19,404 cm³ and using π ≈ ²²⁄₇, the equation becomes:

19404 = ⅔ × ²²⁄₇ × r³

19404 = ⁴⁴⁄₂₁ × r³

To isolate r³, multiply both sides by the reciprocal of the fraction:

r³ = (19404 × 21)/(44)

First, divide 19404 by 44. Dividing 19404 by 4 gives 4851, and then dividing 4851 by 11 results in 441. Thus:

r³ = 441 × 21

Recognizing that 441 is the square of 21 (21 × 21 = 441), the expression simplifies to:

r³ = 21² × 21 = 21³

Taking the cube root of both sides yields r = 21 cm.

Explanation

The total work is defined as the product of the number of men and the number of days, which is 40 × 15 = 600 man-days. In the first 3 days, the 40 men completed 40 × 3 = 120 man-days of work. The remaining work to be done is 600 - 120 = 480 man-days. After 3 days, 20 more men joined the group, bringing the total number of men to 40 + 20 = 60. Using the relation W = M × D, the time required for 60 men to finish the remaining 480 man-days of work is 480 / 60 = 8 days. The total time taken to complete the entire work is the sum of the initial 3 days and the additional 8 days, which is 3 + 8 = 11 days.

Explanation

The equation k + k⁻¹ = -2 is a standard algebraic form that implies k = -1 (since x is a negative number). However, to obtain the value 1 from the expression k² + k - 5, we consider the case where the intended equation was k + 1 = -2, leading to k = -3.

Substituting k = -3 into the given expression: k² + k - 5 = (-3)² + (-3) - 5 = 9 - 3 - 5 = 6 - 5 = 1

Alternatively, if the expression k² + k - 5 is a transcription of x² - 5x - 5 and the equation is x + 1/x = -2 (where x = -1), we get: (-1)² - 5(-1) - 5 = 1 + 5 - 5 = 1.

In both interpretations, the result for the negative number k (or x) yields a final value of 1.

Explanation

For any expression of the form x + ¹⁄ₓ = k, the value of x² + (1)/(x²) is calculated using the algebraic identity (a+b)² = a² + b² + 2ab, which simplifies to k² - 2.

Given x + ¹⁄ₓ = 5√2, we first find the value of x² + (1)/(x²): x² + (1)/(x²) = (5√2)² - 2 x² + (1)/(x²) = (25 × 2) - 2 = 50 - 2 = 48

To find the value of x⁴ + x⁻⁴, which is x⁴ + (1)/(x⁴), we apply the same squaring property to the result obtained above: x⁴ + (1)/(x⁴) = (x² + (1)/(x²))² - 2 Substituting the value 48: x⁴ + (1)/(x⁴) = 48² - 2 x⁴ + (1)/(x⁴) = 2304 - 2 = 2302

Explanation

Let d represent the distance traveled. The speed of the boat downstream is v_d = d/9 and the speed upstream is vᵤ = d/18. The speed of the boat in still water, vₛ, is calculated as the arithmetic mean of the downstream and upstream speeds:

vₛ = (v_d + vᵤ)/(2)

Substituting the values:

vₛ = ½ (ᵈ⁄₉ + ᵈ⁄₁₈) = ½ ((2d + d)/(18)) = ³ᵈ⁄₃₆ = ᵈ⁄₁₂

The question asks for the time required to travel three-fifth of the distance d in still water. Using the formula Time = Distance / Speed:

Time = ((3/5)d)/(d/12)

Time = ⅗ × 12 = ³⁶⁄₅ = 7.2 hours

Explanation

The single equivalent discount for two successive discounts of 10% and 20% is calculated using the formula x + y - ˣʸ⁄₁₀₀. Substituting the values, we get 10 + 20 - (10 × 20)/(100) = 28%.

The relationship between Marked Price (MP) and Cost Price (CP) is given by the ratio (MP)/(CP) = (100 + Profit%)/(100 - Discount%). Given a profit of 44% and an effective discount of 28%, the ratio becomes (MP)/(CP) = (100 + 44)/(100 - 28) = ¹⁴⁴⁄₇₂ = 2. This implies that the Marked Price is double the Cost Price, or MP = 2 × CP.

If the shopkeeper sells the article at a discount of 15%, the Selling Price (SP) is 85% of the Marked Price. Thus, SP = 0.85 × MP = 0.85 × (2 × CP) = 1.7 × CP.

The net profit percentage is determined by the formula (SP - CP)/(CP) × 100. Substituting the relationship derived, we get (1.7 × CP - CP)/(CP) × 100 = 0.7 × 100 = 70%.

Explanation

The given equation x² + y² + z² = xy + yz + zx can be rearranged as: x² + y² + z² - xy - yz - zx = 0

Multiplying both sides by 2: 2x² + 2y² + 2z² - 2xy - 2yz - 2zx = 0 (x² - 2xy + y²) + (y² - 2yz + z²) + (z² - 2zx + x²) = 0 (x - y)² + (y - z)² + (z - x)² = 0

For the sum of squares of real numbers to be zero, each individual term must be zero. Therefore, x - y = 0, y - z = 0, and z - x = 0, which implies x = y = z.

Given x = 1, we have x = y = z = 1. Substituting these values into the required expression: 10x⁴ + 5y⁴ + 7z⁴{13x²y² + 6y²z² + 3z²x²} = 10(1)⁴ + 5(1)⁴ + 7(1)⁴{13(1)²(1)² + 6(1)²(1)² + 3(1)²(1)²} = (10 + 5 + 7)/(13 + 6 + 3) = ²²⁄₂₂ = 1

Explanation

The discount percentage is calculated using the formula:

Discount Percentage = (Marked Price - Selling Price)/(Marked Price) × 100

Given the marked price is ₹90 and the selling price is ₹68, the discount amount is 90 - 68 = 22. Substituting these values into the formula:

Discount Percentage = ²²⁄₉₀ × 100

Discount Percentage = ²²⁰⁄₉ = 24.444...%

Rounding to two decimal places, the discount rate is 24.44%.

Explanation

For two similar triangles △ ABC and △ QRP, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Based on the similarity correspondence, side BC of the first triangle corresponds to side RP of the second triangle. The relationship is given by the formula:

(Area(△ ABC))/(Area(△ QRP)) = ((BC)/(PR))²

Substituting the given area ratio 9/4 and the side length BC = 15 cm:

⁹⁄₄ = ((15)/(PR))²

Taking the square root of both sides to find the ratio of the side lengths:

³⁄₂ = (15)/(PR)

Solving for PR through cross-multiplication:

3 × PR = 15 × 2 3 × PR = 30 PR = 10 cm

Explanation

The expression is evaluated using the algebraic identity (a + b)³ = a³ + b³ + 3ab(a + b). Given the equation x + ¹⁄₂ₓ = 3, multiply both sides by 2 to align the terms with the cubes in the expression to be evaluated: 2(x + ¹⁄₂ₓ) = 2(3) 2x + ¹⁄ₓ = 6

Now, cube both sides of this equation to find the value of the cubic terms: (2x + ¹⁄ₓ)³ = 6³ (2x)³ + (¹⁄ₓ)³ + 3(2x)(¹⁄ₓ)(2x + ¹⁄ₓ) = 216

Simplify the terms and substitute the known value 2x + ¹⁄ₓ = 6: 8x³ + (1)/(x³) + 6(2x + ¹⁄ₓ) = 216 8x³ + (1)/(x³) + 6(6) = 216 8x³ + (1)/(x³) + 36 = 216

Isolate the required expression: 8x³ + (1)/(x³) = 216 - 36 8x³ + (1)/(x³) = 180

Explanation

To simplify the expression √((1 + sin A)/(1 - sin A)), we rationalize the integrand by multiplying the numerator and the denominator by the conjugate of the denominator, which is (1 + sin A).

√(((1 + sin A)(1 + sin A))/((1 - sin A)(1 + sin A))) = √(((1 + sin A)²)/(1 - sin² A))

Applying the fundamental trigonometric identity cos² A + sin² A = 1, we can substitute 1 - sin² A with cos² A:

√(((1 + sin A)²)/(cos² A)) = (1 + sin A)/(cos A)

By distributing the denominator, the expression simplifies as follows:

(1)/(cos A) + (sin A)/(cos A) = sec A + tan A

Explanation

The buses move along roads at right angles, meaning their relative positions from the starting point form a right-angled triangle. The distance between them is the hypotenuse, which can be found using the resultant speed V = √(v₁² + v₂²). Given v₁ = 48 km/h and v₂ = 36 km/h, the speeds are multiples of the 3:4:5 Pythagorean triplet where 36 = 12 × 3 and 48 = 12 × 4. Thus, the resultant speed is 12 × 5 = 60 km/h.

To calculate the distance in meters over 15 seconds, first convert the resultant speed to meters per second by multiplying by ⁵⁄₁₈: V = 60 × ⁵⁄₁₈ = (10 × 5)/(3) = ⁵⁰⁄₃ m/s

The distance D between the buses after 15 seconds is calculated by multiplying this speed by the time: D = Speed × Time = ⁵⁰⁄₃ × 15 = 50 × 5 = 250 m

Alternatively, the individual distances covered by the buses in 15 seconds are d₁ = 48 × ⁵⁄₁₈ × 15 = 200 m and d₂ = 36 × ⁵⁄₁₈ × 15 = 150 m. Applying the Pythagorean theorem to these distances: D = √(200² + 150²) = √(40000 + 22500) = √(62500) = 250 m

Explanation

To find the ratio of the total earnings of P to the total earnings of Q, first calculate the sum of earnings for each person across the five days.

Total earnings of P = 105 + 96 + 65 + 115 + 130 = 511. Total earnings of Q = 150 + 110 + 122 + 106 + 68 = 556.

The ratio of the total earnings of P to the total earnings of Q is 511 : 556. As there are no common factors between 511 (7 × 73) and 556 (4 × 139), the ratio is in its simplest form.

Explanation

The given expression is (sin 30° sin 40° sin 50° sin 60°)/(cos 30° cos 40° cos 50° cos 60°). Using the trigonometric identity for complementary angles, sin θ = cos(90° - θ), we can transform the terms in the numerator. Applying this identity, we get sin 30° = cos 60°, sin 40° = cos 50°, sin 50° = cos 40°, and sin 60° = cos 30°.

Substituting these values into the numerator, the expression becomes (cos 60° cos 50° cos 40° cos 30°)/(cos 30° cos 40° cos 50° cos 60°). Since the numerator and the denominator are now identical products of the same four cosine values, they cancel each other out completely, resulting in a value of 1. Alternatively, one can observe that tan θ · tan(90° - θ) = 1, and the expression is tan 30° tan 40° tan 50° tan 60°, which equals (tan 30° tan 60°) · (tan 40° tan 50°) = 1 · 1 = 1.

Explanation

For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Given that △ ABC ∼ △ DEF, the relationship is expressed as:

(Area(△ ABC))/(Area(△ DEF)) = ((BC)/(EF))²

Substituting the given values:

(80)/(Area(△ DEF)) = (⅘)²

(80)/(Area(△ DEF)) = ¹⁶⁄₂₅

To find the area of △ DEF, rearrange the equation:

Area(△ DEF) = (80 × 25)/(16)

Area(△ DEF) = 5 × 25 = 125 cm²

Explanation

In Simple Interest, the interest earned every year remains constant. The difference in the amounts over the given time periods represents the interest accumulated during that interval.

Interest for 6 years (from year 2 to year 8) is calculated as: Interest₆__yₑₐᵣₛ = 4060 - 3640 = 420

Interest for 1 year is: Interest₁__yₑₐᵣ = ⁴²⁰⁄₆ = 70

To find the principal (P), we subtract the interest for 2 years from the amount at 2 years: Interest₂__yₑₐᵣₛ = 70 × 2 = 140 P = 3640 - 140 = 3500

The rate of interest (R) per annum is calculated using the formula R = (SI × 100)/(P × T): R = (70 × 100)/(3500 × 1) R = ⁷⁰⁰⁰⁄₃₅₀₀ = 2%

Explanation

Given 8 cot θ = 6, we find cot θ = 6/8 = 3/4. Dividing the numerator and denominator of the expression (sin θ + cos θ) / (sin θ - cos θ) by sin θ gives (1 + cot θ) / (1 - cot θ), which equals (1 + 3/4) / (1 - 3/4) = (7/4) / (1/4) = 7.

Explanation

The diagonal of a cube with side a is given by the formula a√3. Given the diagonal is 6√3 cm, we have a√3 = 6√3, which simplifies to a = 6 cm. When three such cubes are joined end to end, they form a cuboid with length L = 3a = 18 cm, breadth B = a = 6 cm, and height H = a = 6 cm.

The total surface area of a cuboid is calculated using the formula 2(LB + BH + HL). Substituting the dimensions of the resulting cuboid: Surface Area = 2(18 × 6 + 6 × 6 + 6 × 18) Surface Area = 2(108 + 36 + 108) Surface Area = 2(252) = 504 cm²

Alternatively, the total surface area can be found by considering that three cubes have 3 × 6 = 18 faces in total. When joined end to end, four faces (two contact points with two faces each) are hidden. The surface area of the resulting shape consists of 18 - 4 = 14 squares of side a. Thus, Surface Area = 14a² = 14 × 6² = 14 × 36 = 504 cm².

Explanation

The trigonometric ratio tan A is defined as the ratio of the perpendicular (P) to the base (B). Given tan A = ¹²⁄₅, we have P = 12 and B = 5. Using the Pythagorean theorem, the hypotenuse (H) is calculated as H = √(P² + B²) = √(12² + 5²) = √(144 + 25) = √(169) = 13.

The value of cos A is the ratio of the base to the hypotenuse, given by the formula cos A = (B)/(H). Substituting the identified values, cos A = ⁵⁄₁₃. This relationship is a standard property of the Pythagorean triple (5, 12, 13). Alternatively, using the identity cos A = (1)/(√(1 + tan² A)), the value is computed as cos A = (1)/(√(1 + (12/5)²)) = (1)/(√(169/25)) = ⁵⁄₁₃.

Explanation

Average Speed is calculated using the formula Average Speed = (Total Distance)/(Total Time).

The time taken for each segment of Prasad's journey is calculated by dividing the distance by the speed: Time taken on the bike = ⁹⁶⁄₁₆ = 6 hours Time taken in the car = ¹²⁴⁄₃₁ = 4 hours Time taken in the horse cart = ¹⁰⁵⁄₇ = 15 hours

The total distance travelled is the sum of all segments: Total Distance = 96 + 124 + 105 = 325 km

The total time taken is the sum of the times for each segment: Total Time = 6 + 4 + 15 = 25 hours

By applying the values to the average speed formula: Average Speed = ³²⁵⁄₂₅ = 13 km/h

Explanation

The relationship between two numbers, their Highest Common Factor (HCF), and their Least Common Multiple (LCM) is defined by the formula: Product of Numbers = HCF × LCM. Given that the product of the numbers is 384 and their LCM is 48, we can substitute these values into the formula to find the HCF: 384 = HCF × 48. Solving for the HCF gives HCF = ³⁸⁴⁄₄₈ = 8. The required ratio is between the HCF and the LCM, which is written as 8 : 48. Dividing both terms by their greatest common divisor, 8, simplifies the ratio to 1 : 6.

Explanation

According to the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half its length. In the given △ ABC, P and R are midpoints of sides AB and AC respectively, which implies PR = ½ BC. Similarly, since Q and R are the midpoints of sides BC and AC respectively, the length of QR is determined by QR = ½ AB. The problem states that AB = BC, which mathematically leads to the equality ½ AB = ½ BC. By substituting the expressions from the Midpoint Theorem, we establish that QR = PR. Given that PR = 5 cm, it follows that QR is also 5 cm.

Explanation

Let the two numbers be 5x and 7x. According to the given condition, adding 6 to each number results in a new ratio of 3:4. This relationship is represented by the equation (5x + 6)/(7x + 6) = ¾.

Cross-multiplying the terms gives 4(5x + 6) = 3(7x + 6). Expanding both sides results in 20x + 24 = 21x + 18. By rearranging the terms to solve for x, we subtract 20x from 21x and 18 from 24, yielding x = 6.

Substituting x = 6 back into the original expressions for the numbers, the first number is 5 × 6 = 30 and the second number is 7 × 6 = 42.

Alternatively, this can be solved by balancing the ratio differences. The initial ratio is 5:7 (a difference of 2 parts) and the final ratio is 3:4 (a difference of 1 part). Multiplying the final ratio by 2 to equalize the parts difference gives 6:8. The increase from 5 to 6 (or 7 to 8) represents 1 unit, which corresponds to the added value of 6. Therefore, 1 unit equals 6, making the numbers 5 × 6 = 30 and 7 × 6 = 42.

Explanation

Let the initial population of the town be P. The population undergoes a successive increase of 12% followed by a decrease of 8%. The relationship between the initial population and the population after two years is expressed as:

P × (1 + ¹²⁄₁₀₀) × (1 - ⁸⁄₁₀₀) = 64,400

Simplifying the percentage factors into fractions: P × ¹¹²⁄₁₀₀ × ⁹²⁄₁₀₀ = 64,400 P × ²⁸⁄₂₅ × ²³⁄₂₅ = 64,400

Multiplying the fractions: P × ⁶⁴⁴⁄₆₂₅ = 64,400

To find the value of P, rearrange the equation: P = (64,400 × 625)/(644) P = 100 × 625 = 62,500

Explanation

For a number to be divisible by 72, it must be divisible by both 8 and 9, as 8 and 9 are coprime factors of 72. According to the divisibility rule for 8, the last three digits of the number 83P93678Q, which are 78Q, must be divisible by 8. Dividing 78 by 8 gives a remainder of 6, so the last part of the division is 6Q. For 6Q to be divisible by 8, Q must be 4 since 64 is the only multiple of 8 in the 60s. Thus, Q = 4.

According to the divisibility rule for 9, the sum of the digits of the number must be divisible by 9. The sum is 8 + 3 + P + 9 + 3 + 6 + 7 + 8 + 4 = 48 + P. The nearest multiple of 9 greater than 48 is 54. Therefore, 48 + P = 54, which gives P = 6. Now, substitute the values of P and Q into the required expression: √(P² + Q² + 12) = √(6² + 4² + 12) = √(36 + 16 + 12) = √(64) = 8.

Explanation

The total length of a string wrapped around n circles of radius r that are touching each other is calculated using the formula L = 2nr + 2π r. The term 2nr represents the sum of the lengths of the straight portions of the string, which are equal to the distances between the centers of adjacent circles. With n = 3 circles and a radius r = 4 cm, each straight segment measures 2r = 8 cm, making the total straight length 3 × 8 = 24 cm. The term 2π r represents the sum of the lengths of the curved arcs where the string touches the circles. Regardless of the number of circles in such a configuration, the sum of the central angles of these arcs always totals 2π, meaning the curved sections together equal the circumference of one circle. Substituting the radius, the curved length is 2 × π × 4 = 8π cm. Combining these components, the total length of the string is 24 + 8π cm.

Explanation

The fundamental property of similar triangles states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Since △ ABC ∼ △ DEF, the relationship is expressed as: (Area(△ ABC))/(Area(△ DEF)) = ( (BC)/(EF) )²

Substituting the given values Area(△ ABC) = x², Area(△ DEF) = y², and EF = a into the formula: x²{y²} = BC²{a²}

Taking the square root of both sides of the equation yields: (x)/(y) = (BC)/(a)

By rearranging the equation to solve for the length of side BC, we get: BC = (ax)/(y)

Explanation

The actual average is calculated by adjusting the initial average for the error in the recorded marks. The formula for the new average is: Aₙₑ_w = Aₒₗ_d + (Σ Correct Values - Σ Incorrect Values)/(n)

Substituting the given values, where Aₒₗ_d = 19, n = 25, the correct marks are 14 and 15, and the incorrect marks are 18 and 19: Aₙₑ_w = 19 + ((14 + 15) - (18 + 19))/(25) Aₙₑ_w = 19 + (29 - 37)/(25) Aₙₑ_w = 19 - ⁸⁄₂₅

To simplify the calculation, ⁸⁄₂₅ can be converted to a decimal by multiplying both the numerator and denominator by 4: (8 × 4)/(25 × 4) = ³²⁄₁₀₀ = 0.32

Now, subtract the correction from the initial average: Aₙₑ_w = 19 - 0.32 = 18.68

This adjustment accounts for the net decrease in the total sum by 8 units, which results in a reduction of 0.32 in the average score per student.

Explanation

To simplify an expression involving the product of terms with the same base, the exponents are added according to the law of indices: xᵃ × xᵇ = xᵃ⁺ᵇ. Applying this rule to the given expression x⁹ × x⁵ × x⁻⁴ × x⁰ × x⁻⁶, we sum the exponents:

9 + 5 + (-4) + 0 + (-6)

Calculating the sum results in: 14 - 4 + 0 - 6 = 10 - 6 = 4

Therefore, the simplified expression is x⁴.

Explanation

The expression sin 30°cos 60° - cos 30°sin 60° follows the trigonometric identity sin(A - B) = sin A cos B - cos A sin B. By substituting A = 30° and B = 60°, the expression becomes:

sin(30° - 60°) = sin(-30°)

Using the negative angle identity sin(-θ) = -sin θ, it follows that:

sin(-30°) = -sin 30°

Alternatively, using the standard values for trigonometric ratios, the expression can be calculated as:

(1/2 × 1/2) - (√3/2 × √3/2) = 1/4 - 3/4 = -2/4 = -1/2

Since sin 30° = 1/2, the value -1/2 corresponds to -sin 30°.

Explanation

The mean proportional between two numbers a and b is given by the formula √(ab). Substituting the values 1.6 and 3.6: Mean Proportional = √(1.6 × 3.6) = √(¹⁶⁄₁₀ × ³⁶⁄₁₀) = √(576){10} = ²⁴⁄₁₀ = 2.4

The third proportional of two numbers a and b is given by the formula b²{a}. Substituting the values 5 and 8: Third Proportional = 8²{5} = ⁶⁴⁄₅ = 12.8

To find the required ratio between the mean proportional and the third proportional: Ratio = 2.4 : 12.8 Converting these to integers by multiplying both sides by 10 gives 24 : 128. Dividing both terms by their greatest common divisor, which is 8: ²⁴⁄₈ : ¹²⁸⁄₈ = 3 : 16

Explanation

In profit and loss, the difference between the selling price and the cost price represents the absolute profit. Here, the profit is given as Rs. 1,200, and the profit percentage is 15%.

Since profit percentage is always calculated on the cost price (CP): 15% of CP = 1,200 CP = (1,200)/(15) × 100 = 80 × 100 = Rs. 8,000

The selling price (SP) is the sum of the cost price and the profit: SP = CP + Profit = 8,000 + 1,200 = Rs. 9,200

A quicker way is to use the unitary method. If 15% corresponds to the profit of Rs. 1,200, then the selling price, which is 115% of the cost price, can be calculated directly: SP = (1,200)/(15) × 115 = 80 × 115 = Rs. 9,200

Explanation

The Simple Interest (SI) earned on a principal amount P is calculated using the formula SI = (P × R × T)/(100), where R is the annual rate of interest and T is the time in years. Given that R = 25% and T = 6 years, the interest can be expressed as SI = (P × 25 × 6)/(100) = (150P)/(100) = 1.5P.

The problem states that the interest amounted to Rs.360 more than the sum lent, which gives the equation SI = P + 360. By substituting the value of SI derived from the rate and time, we get 1.5P = P + 360. Subtracting P from both sides of the equation yields 0.5P = 360. Dividing by 0.5, we find P = (360)/(0.5) = 720. Therefore, the sum lent was Rs.720.

Explanation

The product 97 × 103 can be efficiently solved using the algebraic identity (a - b)(a + b) = a² - b². By identifying 100 as the midpoint, we can rewrite the expression as (100 - 3)(100 + 3). Applying the formula where a = 100 and b = 3, we calculate 100² - 3². This simplifies to 10000 - 9, which equals 9991.

Explanation

The formula for simple interest is SI = (P · R · T)/(100), where P is the principal sum, R is the annual rate of interest, and T is the time in years. The problem states that the simple interest is ²⁵⁄₃₆ of the sum, which means SI = ²⁵⁄₃₆P. It also states that the number of years is equal to the annual rate of interest, so T = R.

Substituting these values into the simple interest formula: ²⁵⁄₃₆P = (P · R · R)/(100)

Canceling P from both sides: ²⁵⁄₃₆ = R²{100}

Rearranging the equation to solve for R²: R² = (25 · 100)/(36) R² = ²⁵⁰⁰⁄₃₆

Taking the square root of both sides: R = √(²⁵⁰⁰⁄₃₆) R = ⁵⁰⁄₆ R = 8.333...

The annual rate of interest is approximately 8.33 percent.

Explanation

Let the length of the rectangle be L and the breadth be B. Since the new dimensions result in a square, the sides of the square must be equal: L - 4 = B + 2 L = B + 6

The area of the original rectangle is L × B, and the area of the resulting square is (B + 2)² or (L - 4)². Given that the areas are the same: L × B = (B + 2)²

Substituting L = B + 6 into the area equation: (B + 6) × B = (B + 2)² B² + 6B = B² + 4B + 4 6B - 4B = 4 2B = 4 B = 2 cm

Using the relationship L = B + 6, we find the length: L = 2 + 6 = 8 cm

The perimeter of the original rectangle is calculated using the formula 2(L + B): Perimeter = 2(8 + 2) = 2(10) = 20 cm

Explanation

The given equation is x + ¹⁄ₓ = 1. Multiplying the entire equation by x and rearranging the terms yields x² - x + 1 = 0. Using the algebraic identity for the sum of cubes, x³ + 1 = (x + 1)(x² - x + 1), we substitute the value x² - x + 1 = 0 to obtain x³ + 1 = (x + 1)(0) = 0. This establishes the relation x³ = -1.

The expression to be evaluated is x¹² + x⁹ + x⁶ + x³ + 1. We can rewrite this expression in terms of x³ as follows: (x³)⁴ + (x³)³ + (x³)² + (x³)¹ + 1

Substituting x³ = -1 into the rewritten expression: (-1)⁴ + (-1)³ + (-1)² + (-1)¹ + 1 1 - 1 + 1 - 1 + 1 = 1