SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

In the right-angled triangle Δ ABC, given that ∠ B = 90° and ∠ A = 30°, the side AC functions as the hypotenuse with a length of 10 cm. To find the length of side AB, which is the base relative to ∠ A, we apply the trigonometric cosine ratio:

cos A = (Adjacent Side)/(Hypotenuse) = (AB)/(AC)

Substituting the given values into the formula:

cos 30° = (AB)/(10)

Using the standard trigonometric value cos 30° = √3{2}, we have:

√3{2} = (AB)/(10)

By rearranging the equation to solve for AB:

AB = 10 × √3{2} = 5√3 cm

This result is also consistent with the properties of a 30°-60°-90° special right triangle, where the sides follow the ratio 1 : √3 : 2. In such a triangle, the side opposite the 60° angle (AB) is always √3{2} times the hypotenuse. Therefore, AB = √3{2} × 10 = 5√3 cm.

Explanation

In a right-angled triangle, cot θ is defined as the ratio of the base (B) to the perpendicular (P). Given cot θ = ²⁴⁄₇, we have B = 24 and P = 7. The hypotenuse (H) can be calculated using the Pythagorean theorem:

H = √(P²⁾⁺^B^^^²

H = √(7²⁾⁺²⁴^^^²

H = √(49 + 576)

H = √(625) = 25

The sine of an angle is the ratio of the perpendicular to the hypotenuse:

sin θ = (P)/(H) = ⁷⁄₂₅

Explanation

The relationship between the selling price (SP), marked price (MP), and discount percentage (d) is given by the formula SP = MP × (100 - d)/(100).

Given that the selling price is Rs 1050 at a 30% discount, we first determine the marked price of the article: 1050 = MP × (100 - 30)/(100) 1050 = MP × ⁷⁰⁄₁₀₀ MP = (1050 × 100)/(70) = 1500

With the marked price identified as Rs 1500, the selling price at a 15% discount is calculated as follows: SP = 1500 × (100 - 15)/(100) SP = 1500 × ⁸⁵⁄₁₀₀ SP = 15 × 85 = 1275

Using a direct proportional approach, since the marked price remains constant, the ratio of the selling prices is equal to the ratio of the remaining percentages: (SP₁)/(100 - d₁) = (SP₂)/(100 - d₂) ¹⁰⁵⁰⁄₇₀ = (SP₂)/(85) 15 = (SP₂)/(85) SP₂ = 15 × 85 = 1275

Explanation

The fourth proportional d to three numbers a, b, and c is defined by the ratio a:b = c:d, which can be expressed as d = (b × c)/(a). In this problem, a = 336, b = 288, and c = 161. Substituting these values into the formula gives:

d = (288 × 161)/(336)

The fraction ²⁸⁸⁄₃₃₆ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 48. This yields:

²⁸⁸⁄₄₈ = 6 ³³⁶⁄₄₈ = 7

So, the expression becomes:

d = (6 × 161)/(7)

Next, divide 161 by 7:

¹⁶¹⁄₇ = 23

Finally, multiply the results:

d = 6 × 23 = 138

Explanation

The numbers between 9 and 90 that are divisible by 8 form an arithmetic progression. The first multiple of 8 in this range is a = 16 (since 8 × 2 = 16), and the last multiple is l = 88 (since 8 × 11 = 88). For any sequence in an arithmetic progression, the average is the mean of the first and last terms, calculated using the formula (a + l)/(2). Substituting the values, we get (16 + 88)/(2) = ¹⁰⁴⁄₂ = 52.

Explanation

The ratio of the quantities can be calculated using the rule of alligation. Let the quantity sold at 15% profit be Q₁ and the quantity sold at 8% profit be Q₂. The mean profit is 12%.

The ratio Q₁ : Q₂ is given by: (12 - 8) : (15 - 12) = 4 : 3

The total quantity is 630 kgs. To find the amount sold at 8% profit, we use the proportion of Q₂ in the total ratio: Quantity at 8% profit = (3)/(4 + 3) × 630 Quantity at 8% profit = ³⁄₇ × 630 = 3 × 90 = 270 kgs

Explanation

Given the relation 50% of a = b, we can express b as 0.5a. The problem asks to find the factor x such that b% of 40 = x × a. We can substitute the expression for b into the percentage formula:

b% of 40 = ᵇ⁄₁₀₀ × 40

Substituting b = 0.5a into this equation:

(0.5a)/(100) × 40 = (0.5 × 40)/(100) × a = ²⁰⁄₁₀₀ × a = 0.2a

The coefficient of a is 0.2, which identifies the relationship as 0.2 of a.

Explanation

The stoppage time per hour is determined by calculating the reduction in distance covered due to stops and finding how long it would take to cover that distance at the original speed. Without stops, the train covers 120 km in one hour. With stops, it covers only 50 km in one hour. The difference in distance is 120 - 50 = 70 km.

The time spent at stops is equivalent to the time the train would have taken to travel this 70 km at its non-stop speed of 120 kmph. Using the formula for stoppage time per hour:

Stoppage time per hour = (Speed without stoppages - Speed with stoppages)/(Speed without stoppages) × 60 minutes

Substituting the given values:

Stoppage time = (120 - 50)/(120) × 60 = ⁷⁰⁄₁₂₀ × 60 = ⁷⁄₁₂ × 60 = 35 minutes

Explanation

Simple interest is earned uniformly over the period. Since the simple interest for two years is Rs 4000, the interest for the first year is 4000 / 2 = 2000 and the interest for the second year is also Rs 2000. For compound interest, the interest earned in the first year is identical to simple interest, which is Rs 2000. The total compound interest for two years is given as Rs 4180, which means the interest earned in the second year is 4180 - 2000 = 2180.

The difference between the second year's compound interest and the second year's simple interest is 2180 - 2000 = 180. This difference of Rs 180 is the interest earned on the first year's interest of Rs 2000. The rate of interest r per annum can be calculated using the relation: r = ((Interest on first year's interest)/(First year's interest)) × 100 r = (¹⁸⁰⁄₂₀₀₀) × 100 r = ¹⁸⁄₂ = 9%

Explanation

The remainder of 2468 divided by 37 is calculated using the division algorithm Dividend = (Divisor × Quotient) + Remainder. Dividing 246 by 37 gives a quotient of 6 since 37 × 6 = 222. The difference 246 - 222 = 24 is then combined with the last digit to form 248. Dividing 248 by 37 again gives a quotient of 6 as 37 × 6 = 222. Subtracting 222 from 248 leaves a final remainder of 26. This can also be solved quickly by noting that 999 is a multiple of 37 (37 × 27 = 999), which means 1000 leaves a remainder of 1 when divided by 37. Therefore, 2468 = 2(1000) + 468 ≡ 2(1) + 468 37. Dividing 468 by 37 gives 468 = 37 × 12 + 24, so the total remainder is 2 + 24 = 26.

Explanation

If A completes 75% of the work in 9 days, the time taken to complete the full work is T_A = 9 ÷ ¾ = 9 × ⁴⁄₃ = 12 days. If B completes half the work in 8 days, the time taken to complete the full work is T_B = 8 × 2 = 16 days.

The combined rate of A and B working together is the sum of their individual rates: Rate_A₊_B = ¹⁄₁₂ + ¹⁄₁₆ = (4 + 3)/(48) = ⁷⁄₄₈ job per day.

To find the time taken to complete half of the job, we divide the required work by the combined rate: Time = (Work)/(Rate) = (1/2)/(7/48) = ½ × ⁴⁸⁄₇ = ²⁴⁄₇ days.

Explanation

To solve the equation ²ˣ⁄₃ - (5(4x)/(5) - ⁴⁄₃){2} = ⅓, first simplify the expression inside the brackets. Distributing the factor 5 into the parentheses gives 4x - ²⁰⁄₃. The equation then becomes: ²ˣ⁄₃ - (4x - 20)/(3){2} = ⅓

Next, divide the terms in the numerator of the second fraction by 2: ²ˣ⁄₃ - (2x - ¹⁰⁄₃) = ⅓

Distribute the negative sign to remove the parentheses: ²ˣ⁄₃ - 2x + ¹⁰⁄₃ = ⅓

To eliminate the fractions, multiply every term in the equation by 3: 2x - 6x + 10 = 1

Combine the terms containing x and constants: -4x + 10 = 1 -4x = 1 - 10 -4x = -9

Divide both sides by -4 to find the value of x: x = ⁻⁹⁄₋₄ = ⁹⁄₄

Explanation

The area of the four walls of a cuboid is equal to its lateral surface area, which is calculated using the formula 2h(l + b), where l is the length, b is the breadth, and h is the height. By substituting the given values of 400 cm² for the area, 15 cm for the length, and 8 cm for the height into this formula, we get the following equation:

400 = 2 × 8 × (15 + b)

Simplifying the equation: 400 = 16 × (15 + b)

Dividing both sides by 16: 25 = 15 + b

Subtracting 15 from both sides to solve for b: b = 25 - 15 = 10

The breadth of the cuboid is 10 cm.

Explanation

The selling price of an article is determined by the formula SP = MP × (100 - D)/(100), where MP is the marked price and D is the discount percentage. Since the marked price remains constant, the selling price is directly proportional to (100 - D).

In the first case, a 20% discount results in a selling price that is 80% of the marked price. 80% of MP = 2400 1% of MP = ²⁴⁰⁰⁄₈₀ = 30

In the second case, a 32.5% discount results in a selling price that is 100% - 32.5% = 67.5% of the marked price. We can calculate the new selling price by multiplying this percentage by the value of 1% of the marked price: New SP = 67.5 × 30 New SP = 2025

Explanation

The fourth proportional d to three numbers a, b, and c is calculated using the property of proportions a:b = c:d, which gives the formula d = (b × c)/(a). Substituting the given values a = 24, b = 120, and c = 22:

d = (120 × 22)/(24)

By simplifying the fraction, 120 divided by 24 results in 5. The calculation then becomes:

d = 5 × 22 = 110

Explanation

The numbers between 8 and 74 that are divisible by 7 form an arithmetic progression. The first number in this sequence is 14, and the last number is 70.

For any set of numbers in an arithmetic progression, the average is simply the arithmetic mean of the first and last terms, calculated by the formula: Average = (a + l)/(2)

In this formula, a represents the first term (14) and l represents the last term (70). Substituting these values into the formula: Average = (14 + 70)/(2) = ⁸⁴⁄₂ = 42

Explanation

By applying the rule of alligation, we can determine the ratio in which the two parts of the rice were sold. The first part was sold at a 5% profit, the second part at an 11% profit, and the overall mean profit was 7%.

The ratio of the quantity sold at 5% profit to the quantity sold at 11% profit is: (11 - 7) : (7 - 5) = 4 : 2 = 2 : 1

The total weight of the rice is 1200 kg. Using the ratio 2:1, the total parts are 2 + 1 = 3. The quantity sold at 5% profit corresponds to 2 parts out of 3: Quantity at 5% profit = ⅔ × 1200 = 800 kg

Explanation

The given condition 20% of a = b can be expressed mathematically as b = 0.2a. The target expression is b% of 20, which is defined as ᵇ⁄₁₀₀ × 20. By substituting the value of b in terms of a into this expression, we get:

b% of 20 = (0.2a)/(100) × 20 b% of 20 = (0.2 × 20)/(100) × a b% of 20 = ⁴⁄₁₀₀ × a

This result is equivalent to 4% of a. Alternatively, using the percentage identity x% of y = y% of x, the term b% of 20 is the same as 20% of b. Since b is 20% of a, the expression becomes 20% of (20% of a). In decimal form, this is 0.20 × 0.20 × a = 0.04a, which represents 4% of a.

Explanation

The time lost due to stoppages per hour is calculated by the difference between the speed without stoppages and the speed with stoppages, relative to the speed without stoppages. In this case, the bus covers 60 km in one hour without stops, but only 45 km when stops are included. The distance lost due to stopping is 60 - 45 = 15 km.

The time it would have taken to cover this 15 km distance at the actual running speed of 60 kmph is the total stoppage time per hour. We can use the following formula:

Stop time per hour = (Excluding speed - Including speed)/(Excluding speed) × 60 minutes

Substituting the values into the formula: Stop time = (60 - 45)/(60) × 60 = ¹⁵⁄₆₀ × 60 = 15 minutes

By covering 15 km less every hour due to stops, the bus effectively remains stationary for 15 minutes during that hour.

Explanation

The rate of interest r for a two-year period when both simple interest SI and compound interest CI are known is determined by the formula: r = (2 × (CI - SI))/(SI) × 100

By substituting the given values: r = (2 × (1040 - 1000))/(1000) × 100 r = (2 × 40)/(10) r = 8

Alternatively, since simple interest is constant every year, the interest for the first year is 1000 / 2 = 500. The difference between CI and SI over two years, which is 1040 - 1000 = 40, represents the interest earned on the first year's interest. Thus, r = (40 / 500) × 100 = 8%.

Explanation

The given equation is 5/2 - (6/5)(x - 15/2) = -x/5. First, distribute the fraction -6/5 into the parentheses:

5/2 - 6x/5 + (6/5)(15/2) = -x/5

Simplify the product (6/5)(15/2) by canceling terms:

5/2 - 6x/5 + 9 = -x/5

Next, combine the constant terms on the left side of the equation. To add 5/2 and 9, convert 9 to a fraction with a denominator of 2:

5/2 + 18/2 - 6x/5 = -x/5 23/2 - 6x/5 = -x/5

Now, isolate the variable x by adding 6x/5 to both sides of the equation:

23/2 = 6x/5 - x/5 23/2 = 5x/5 23/2 = x

Explanation

The value of a³ - b³ can be determined using the standard algebraic identity a³ - b³ = (a - b)³ + 3ab(a - b). Given the values a - b = 2 and ab = 24, substitute them directly into the formula:

a³ - b³ = (2)³ + 3(24)(2) a³ - b³ = 8 + 144 a³ - b³ = 152

Alternatively, by inspection of the equations a - b = 2 and ab = 24, we can identify that the numbers are 6 and 4, since 6 - 4 = 2 and 6 × 4 = 24. Therefore, a³ - b³ = 6³ - 4³ = 216 - 64 = 152.

Explanation

The relationship between the given values can be expressed using the algebraic identity (a + b)³ = a³ + b³ + 3ab(a + b). Substituting the known values a + b = 4 and a³ + b³ = 28 into this formula allows for a direct solution.

4³ = 28 + 3ab(4)

64 = 28 + 12ab

Subtracting 28 from both sides:

36 = 12ab

Dividing by 12:

ab = 3

Alternatively, using the identity a³ + b³ = (a + b)((a + b)² - 3ab):

28 = 4(4² - 3ab)

7 = 16 - 3ab

3ab = 16 - 7

3ab = 9

ab = 3

Explanation

Let the fraction be x. According to the question, the sum of twice the fraction and its reciprocal is ¹⁷⁄₆. This is expressed by the equation 2x + ¹⁄ₓ = ¹⁷⁄₆.

Multiplying the entire equation by 6x to eliminate the denominators results in 12x² + 6 = 17x. Rearranging this into a standard quadratic equation gives 12x² - 17x + 6 = 0. To solve for x, we factorize the quadratic by splitting the middle term into -9x and -8x, as their product is 72x² (matching 12 × 6).

12x² - 9x - 8x + 6 = 0 3x(4x - 3) - 2(4x - 3) = 0 (3x - 2)(4x - 3) = 0

Solving for x gives two possible values: x = ⅔ or x = ¾. Comparing these results with the given options, the fraction is ¾. Substituting ¾ back into the original expression verifies the result: 2(¾) + ⁴⁄₃ = ³⁄₂ + ⁴⁄₃ = (9 + 8)/(6) = ¹⁷⁄₆.

Explanation

In an arithmetic progression, the nᵗʰ term is given by aₙ = a + (n-1)d, where a is the first term and d is the common difference.

Given a₃ = -9 and a₇ = 11, we can find the common difference d using the formula for the difference between any two terms aₘ and aₖ: d = (aₘ - aₖ)/(m - k)

Substituting the known values: d = (a₇ - a₃)/(7 - 3) = (11 - (-9))/(4) = ²⁰⁄₄ = 5

To find the 15th term (a₁₅), we can project from the 7th term: a₁₅ = a₇ + (15 - 7)d a₁₅ = 11 + 8(5) a₁₅ = 11 + 40 = 51

Explanation

The reflection of a point (x, y) in a horizontal line y = k results in a point (x', y') where the x-coordinate remains the same (x' = x) and the y-coordinate is determined by the formula y' = 2k - y. This is because the line y = k is the perpendicular bisector of the segment connecting the original point and its reflection.

Given the point (4, 7) and the line y = -1, we have x = 4, y = 7, and k = -1. Substituting these values into the formula: x' = 4 y' = 2(-1) - 7 = -2 - 7 = -9

The coordinates of the reflected point are (4, -9).

Explanation

The slope m of a line passing through two points (x₁, y₁) and (x₂, y₂) is determined by the formula m = y₂ - y₁{x₂ - x₁}. Substituting the given slope m = -2/3 and the coordinates of points A (x, -3) and B (5, 2) into this formula yields:

-2/3 = (2 - (-3))/(5 - x)

Simplifying the numerator on the right side:

-2/3 = (5)/(5 - x)

Applying cross-multiplication to solve for x:

-2(5 - x) = 3 × 5 -10 + 2x = 15

Isolating the term with x:

2x = 15 + 10 2x = 25 x = 25 / 2 = 12.5

Explanation

In △ ABC, since DE is parallel to BC, the smaller triangle △ ADE is similar to the larger triangle △ ABC by the AA similarity criterion. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Given the ratio AD : DB = 2 : 3, the length of the side AB of the larger triangle is the sum of its segments: AB = AD + DB = 2 + 3 = 5 units

The ratio of the corresponding sides of △ ADE and △ ABC is: (AD)/(AB) = ⅖

Using the property of similar triangles: (Area(△ ADE))/(Area(△ ABC)) = ((AD)/(AB))² = (⅖)² = ⁴⁄₂₅

Let the area of △ ADE be 4 units and the area of △ ABC be 25 units. The area of the quadrilateral BDEC is calculated by subtracting the area of the smaller triangle from the larger one: Area(BDEC) = Area(△ ABC) - Area(△ ADE) = 25 - 4 = 21 units

The ratio of the area of △ ADE to the area of quadrilateral BDEC is 4 : 21.

Explanation

The standard trigonometric values for the given angles are cot 60° = (1)/(√(3)) and cos 45° = (1)/(√(2)). Substituting these values into the expression cot 60° - cos 45° yields:

(1)/(√(3)) - (1)/(√(2))

To solve this, we find a common denominator by multiplying the two denominators: √3 × √2 = √6. Adjusting the numerators accordingly, the expression becomes:

√2 - √3{√6}

Explanation

In right-angled △ PQR where ∠ Q = 90°, the side RQ is the base (adjacent side) relative to ∠ R, and PR is the hypotenuse. The relationship between the adjacent side and the hypotenuse is defined by the cosine function:

cos R = (RQ)/(PR)

Given that m∠ R = 60° and RQ = 4√3 cm, substitute these values into the formula:

cos 60° = 4√3{PR}

Using the standard trigonometric value cos 60° = ½:

½ = 4√3{PR}

By cross-multiplying to solve for the length of PR:

PR = 2 × 4√3 = 8√3 cm

Explanation

Using the trigonometric identity 1 + tan² θ = sec² θ, we can find the value of sec θ directly. Given that tan θ = 9/40, we substitute this into the formula:

sec² θ = 1 + (9/40)² sec² θ = 1 + 81/1600 sec² θ = (1600 + 81)/1600 sec² θ = 1681/1600

Taking the square root of both sides: sec θ = √(1681/1600) = 41/40

Alternatively, in a right-angled triangle, tan θ is the ratio of the perpendicular to the base (P/B). Here, P = 9 and B = 40. Using the Pythagorean theorem, the hypotenuse H = √(P²⁾⁺^B^^^² = √(9²⁾⁺⁴⁰^^^² = √(81 + 1600) = √(1681) = 41. Since sec θ is the ratio of the hypotenuse to the base (H/B), we have sec θ = 41/40.

Explanation

Let the fraction be x. According to the problem, the relationship is expressed as 6x - 7(1/x) = 11. To solve for x, multiply the entire equation by x to eliminate the fraction, resulting in the quadratic equation 6x² - 7 = 11x.

Rearranging the terms into the standard form ax² + bx + c = 0, we get: 6x² - 11x - 7 = 0

Using the factorization method, we split the middle term -11x into two terms whose coefficients multiply to 6 × -7 = -42 and add up to -11. These numbers are -14 and 3: 6x² - 14x + 3x - 7 = 0 2x(3x - 7) + 1(3x - 7) = 0 (2x + 1)(3x - 7) = 0

Solving for x gives two possible values: x = -1/2 and x = 7/3. Substituting x = 7/3 back into the original condition, 6 times the fraction is 6 × (7/3) = 14, and 7 times its reciprocal is 7 × (3/7) = 3. The difference is 14 - 3 = 11, which satisfies the given condition.

Explanation

The sum of the first n terms of an arithmetic progression can be calculated using the formula Sₙ = ⁿ⁄₂(a + l), where n is the number of terms, a is the first term, and l is the last term. Substituting the given values n = 11, a = -31, and l = 29 into the formula, we get S₁₁ = ¹¹⁄₂(-31 + 29). Simplifying the expression inside the parentheses results in S₁₁ = ¹¹⁄₂(-2). Multiplying these values yields 11 × (-1), which equals -11.

Explanation

The reflection of a point (x, y) in a vertical line x = k results in an image (x', y') where the y-coordinate remains unchanged and the x-coordinate is transformed such that the line x = k is the perpendicular bisector of the segment joining the point and its image. The formula for the new x-coordinate is x' = 2k - x.

Applying this to the point (-1, 5) with the line of reflection x = 1: x' = 2(1) - (-1) = 2 + 1 = 3 y' = 5

The resulting coordinates for the reflected point are (3, 5).

Explanation

The centroid (G) of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is calculated using the formula G = ((x₁ + x₂ + x₃)/(3), (y₁ + y₂ + y₃)/(3)).

Substituting the given vertices A(1, -5), B(-4, 0), and C(3, -4) into the formula:

x-coordinate = (1 + (-4) + 3)/(3) = ⁰⁄₃ = 0

y-coordinate = (-5 + 0 + (-4))/(3) = ⁻⁹⁄₃ = -3

The coordinates of the centroid are (0, -3).

Explanation

The slope m of a line passing through two points (x₁, y₁) and (x₂, y₂) is determined by the formula m = (y₂ - y₁)/(x₂ - x₁). Substituting the given slope m = 4/3 and the coordinates of points A(x, -5) and B(2, -3) into this equation, we get:

⁴⁄₃ = (-3 - (-5))/(2 - x)

Simplifying the expression in the numerator:

⁴⁄₃ = (-3 + 5)/(2 - x) ⁴⁄₃ = (2)/(2 - x)

To find the value of x, we apply cross-multiplication:

4(2 - x) = 3 × 2 8 - 4x = 6 8 - 6 = 4x 2 = 4x x = ²⁄₄ x = ½

Explanation

In Δ ABC, since DE is parallel to BC, Δ ADE is similar to Δ ABC by the AA similarity criterion. The ratio of the corresponding sides of these two triangles is given by AD/AB. Given the ratio AD:DB = 1:4, the total length AB is the sum of AD and DB, which results in AD/AB = 1/(1+4) = 1/5.

The property of similar triangles states that the ratio of their areas is equal to the square of the ratio of their corresponding sides. Thus, the ratio Area(Δ ADE) / Area(Δ ABC) = (AD/AB)² = (1/5)² = 1/25.

By setting the Area(Δ ADE) as 1 unit of area, the Area(Δ ABC) becomes 25 units. The area of the quadrilateral BDEC is obtained by subtracting the area of the smaller triangle from the larger triangle: Area(BDEC) = Area(Δ ABC) - Area(Δ ADE) = 25 - 1 = 24 units. Therefore, the ratio of the area of Δ ADE to the area of the quadrilateral BDEC is 1:24.

Explanation

The trigonometric values for the given angles are cot 30° = √3 and csc 60° = (2)/(√(3)). Substituting these values into the expression:

(1)/(√(2)) cot 30° + (1)/(√(3)) csc 60° = (1)/(√(2)) (√3) + (1)/(√(3)) ( (2)/(√(3)) )

This simplifies to:

√3{√2} + ⅔

To add these fractions, find the common denominator, which is 3√2. Multiplying the first term by 3/3 and the second term by √2/√2:

3√3{3√2} + 2√2{3√2} = 3√3 + 2√2{3√2}

Explanation

In a right-angled triangle ABC where ∠ B = 90°, the sum of the remaining angles A and C must be 90°. Given m∠ A = 60°, it follows that m∠ C = 90° - 60° = 30°.

The expression to evaluate is 2 sec C × ½ sin A. The coefficients 2 × ½ simplify to 1, leaving the expression as sec C × sin A. Using the trigonometric values for the calculated angles: sec 30° = (2)/(√(3)) sin 60° = √3{2}

Substituting these values into the expression: (2)/(√(3)) × √3{2} = 1

Alternatively, since C = 90° - A, we can use the co-function identity sec(90° - A) = csc A. The expression then becomes csc A × sin A. Because csc A is the reciprocal of sin A, the product csc A × sin A is equal to 1 for any valid angle A.

Explanation

The largest 4-digit number is 9999. To find the largest multiple of 81 within this range, we divide 9999 by 81 and subtract the resulting remainder from 9999.

Dividing 9999 by 81: 99 ÷ 81 = 1 with a remainder of 18. Bringing down the next 9 gives 189. 189 ÷ 81 = 2 (81 × 2 = 162) with a remainder of 27. Bringing down the final 9 gives 279. 279 ÷ 81 = 3 (81 × 3 = 243) with a remainder of 36.

The remainder is 36. Subtracting this from the largest 4-digit number: 9999 - 36 = 9963

Alternatively, multiplying the divisor by the integer quotient obtained from the division gives the result: 81 × 123 = 9963

Since adding another 81 would result in a 5-digit number (9963 + 81 = 10044), 9963 is the largest 4-digit number exactly divisible by 81.

Explanation

The time taken by two people working together is related to their individual times by the formula 1/Tₜₒₜₐₗ = 1/T_A + 1/T_B. Given that A and B together complete the job in 7.5 days (Tₜₒₜₐₗ = 7.5) and A alone completes it in 10 days (T_A = 10), the equation is:

1/7.5 = 1/10 + 1/T_B

Rearranging to solve for the rate of B:

1/T_B = 1/7.5 - 1/10

Converting 7.5 to the fraction 15/2 simplifies the calculation:

1/T_B = 2/15 - 1/10

Using a common denominator of 30:

1/T_B = 4/30 - 3/30 = 1/30

This yields T_B = 30 days. Using the efficiency method, if the total work is assumed to be 30 units (the LCM of 7.5 and 10), then the combined efficiency of A and B is 30 / 7.5 = 4 units per day. Since A's efficiency is 30 / 10 = 3 units per day, B's efficiency is 4 - 3 = 1 unit per day. Therefore, B takes 30 / 1 = 30 days to complete the work alone.

Explanation

In a right-angled triangle, tan θ is the ratio of the perpendicular (P) to the base (B). Given tan θ = ⁷⁄₂₄, we can take P = 7 and B = 24. Using the Pythagorean triple (7, 24, 25) or the Pythagorean theorem, the hypotenuse (H) is calculated as H = √(P² + B²) = √(7² + 24²) = √(49 + 576) = √(625) = 25. The value of cosec θ is the ratio of the hypotenuse to the perpendicular, which is expressed as cosec θ = (H)/(P) = ²⁵⁄₇.

Explanation

The perimeter P of a rectangle is given by the formula P = 2(l + b), where l is the length and b is the breadth. Given the perimeter is 48 cm and the breadth is 10 cm, we substitute these values into the formula to find the length:

48 = 2(l + 10) 24 = l + 10 l = 14 cm

The area A of a rectangle is determined by the formula A = l × b. Substituting the length and the given breadth:

A = 14 × 10 = 140 cm²

Explanation

The selling price (SP) is calculated by applying the discount to the marked price (MP) using the formula SP = MP × (1 - (Discount%)/(100)). Given that the selling price is 1000 and the discount is 20%, the equation becomes 1000 = MP × (1 - 0.20), which simplifies to 1000 = MP × 0.80. To find the marked price, we divide the selling price by 0.8: MP = (1000)/(0.8) = 1250. This calculation shows that the original price before the 20% reduction was 1250.

Explanation

To find the ratio A : B : C from the equation A = 2B = 4C, divide each term by the Least Common Multiple (LCM) of the numerical coefficients. The coefficients are 1, 2, and 4. Their LCM is 4. Dividing the entire equation by 4 yields:

A/4 = 2B/4 = 4C/4

A/4 = B/2 = C/1

In an equality expressed in the form A/x = B/y = C/z, the ratio A : B : C is directly given by the denominators x : y : z. Substituting the values from the simplified equation, the ratio A : B : C is 4 : 2 : 1. Alternatively, setting the entire expression equal to the LCM (4) provides A = 4, 2B = 4 (which means B = 2), and 4C = 4 (which means C = 1).

Explanation

The total cost of the initial 4 items is calculated by multiplying the average cost by the number of items: 4 × 1250 = 5000. When a fifth item costing 2000 is added, the new total cost for the 5 items becomes 5000 + 2000 = 7000. The new average is obtained by dividing this total cost by the new number of items: ⁷⁰⁰⁰⁄₅ = 1400. Alternatively, using the deviation method, the new item brings an excess of 2000 - 1250 = 750 over the previous average. Distributing this excess equally among all 5 items adds an increment of ⁷⁵⁰⁄₅ = 150 to the original average, resulting in 1250 + 150 = 1400.

Explanation

Let CP be the cost price. A loss of 20% means the selling price is 100% - 20% = 80% of the cost price. Using the formula SP = CP × (1 - Loss%), we find the cost price: 24 = CP × 0.80 CP = (24)/(0.8) = 30

When the selling price is Rs 36, the profit is calculated as: Profit = Selling Price - Cost Price = 36 - 30 = 6 Profit Percentage = (Profit)/(CP) × 100 = ⁶⁄₃₀ × 100 = 20%

Alternatively, using the direct ratio of selling prices to their percentage multipliers of cost price: SP₁{100 - L₁} = SP₂{100 + P₂} ²⁴⁄₈₀ = (36)/(100 + P₂₎ ³⁄₁₀ = (36)/(100 + P₂₎ 3 × (100 + P₂) = 360 100 + P₂ = 120 P₂ = 20%

Explanation

Let the unknown value be represented by x. The problem states that 0.25% of x is 20, which translates into the mathematical equation:

(0.25)/(100) × x = 20

To solve for x, rearrange the equation to isolate the variable:

x = (20 × 100)/(0.25)

Since 0.25 is equivalent to the fraction ¼, the division can be simplified:

x = (2000)/(1/4)

x = 2000 × 4

x = 8000

Alternatively, using a percentage-based shortcut, if 0.25% of a number is 20, then 1% of that number is four times as much because 0.25 × 4 = 1. Therefore, 1% is equal to 20 × 4 = 80. To find the full value (100%), multiply the value of 1% by 100. This yields 80 × 100 = 8000. This method allows for rapid mental calculation by breaking the percentage down into easily manageable parts.

Explanation

When the distance is constant, speed and time are inversely proportional (v₁ t₁ = v₂ t₂). Given the initial speed v₁ = 40 km/hr and initial time t₁ = 25 minutes, we can find the required speed v₂ for the new time t₂ = 20 minutes using the equation:

40 × 25 = v₂ × 20

1000 = v₂ × 20

v₂ = ¹⁰⁰⁰⁄₂₀ = 50 km/hr

To find the necessary increase in speed, subtract the original speed from the new speed:

Increase in speed = v₂ - v₁ = 50 - 40 = 10 km/hr

Explanation

In compound interest, the interest for any given year is equal to the interest of the preceding year plus the interest earned on that interest at the specified rate. The relationship between the compound interest of two consecutive years is defined by the formula CIₙ₊₁ = CIₙ(1 + ʳ⁄₁₀₀).

Given that the compound interest for the 3rd year CI₃ is Rs. 1,240 and the rate of interest r is 9%, the interest for the 4th year is calculated as follows:

CI₄ = 1240 × (1 + ⁹⁄₁₀₀) CI₄ = 1240 × 1.09 CI₄ = 1240 + (1240 × 0.09) CI₄ = 1240 + 111.6 CI₄ = 1351.6