SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

The BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction) is applied to simplify the given expression.

First, evaluate the operation inside the bracket: 11³ ÷ 11 = 11³⁻¹ = 11² = 121

Substitute this value back into the expression: 40 × 7 - 2 × 121 ÷ 22 + 30

Next, perform the multiplication and division operations: 40 × 7 = 280 2 × 121 ÷ 22 = 242 ÷ 22 = 11

Now, substitute these simplified terms back into the expression: 280 - 11 + 30

Finally, perform the addition and subtraction: 280 - 11 + 30 = 269 + 30 = 299

Explanation

In the parallelogram PQRS, the side PQ is parallel to SR, and PQ = SR. Since PQ is produced to T, the line PT is parallel to SR.

In the triangles TQM and SRM, we have: Angle TMQ = Angle SMR because they are vertically opposite angles. Angle TQM = Angle SRM because they are alternate interior angles for the parallel lines PT and SR with transversal QR. QM = MR because M is the midpoint of QR.

By the Angle-Side-Angle congruence criterion, triangle TQM is congruent to triangle SRM. This congruence means their corresponding sides are equal, so TQ = SR.

The length of PT is the sum of PQ and TQ. Substituting PQ = SR and TQ = SR into this relation gives: PT = PQ + TQ = SR + SR = 2SR

Explanation

To find the amount of water to be added, we calculate the initial quantities of water and milk as 445/6 litres and 623/6 litres respectively. Setting up the equation for the new ratio of water to milk as 3:4 after adding x litres of water, we get (445/6 + x) / (623/6) = 3/4, which simplifies to 24x = 89, giving x = 3 17/24 litres.

Explanation

Using the relation: Loss = Cost Price - Selling Price

For 90 articles, this is expressed as: Loss = CP of 90 articles - SP of 90 articles

Given that the loss is equal to the selling price of 8 articles, and the total cost price of 90 articles is ₹13,230, let SP be the selling price of one article: 8 × SP = 13,230 - 90 × SP

Rearranging the terms to solve for SP: 98 × SP = 13,230 SP = 13,230 / 98 SP = 135

Explanation

Since A is an acute angle, we use the trigonometric identity cos A = √(1 - sin² A) to find the value of cos A: cos A = √(1 - (2/5)²) = √(1 - 4/25) = √21/5

The secant value is the reciprocal of the cosine value: sec A = 1/cos A = 5/√21

Substituting the values of sin A, cos A, and sec A into the given expression: (5 sin A + 2 cos A) / (√21 sec A)

Calculate the numerator: 5(2/5) + 2(√21/5) = 2 + 2√21/5 = (10 + 2√21)/5

Calculate the denominator: √21(5/√21) = 5

Dividing the numerator by the denominator gives: ((10 + 2√21)/5) / 5 = (10 + 2√21)/25

Explanation

Using the standard algebraic identity (a + b)² - (a - b)² = 4ab, we can substitute a = 3x and b = 2y into the formula.

This directly yields: (3x + 2y)² - (3x - 2y)² = 4 × (3x) × (2y) = 24xy

Alternatively, expanding both squared terms: (3x + 2y)² = 9x² + 12xy + 4y² (3x - 2y)² = 9x² - 12xy + 4y²

Subtracting the second expression from the first: (9x² + 12xy + 4y²) - (9x² - 12xy + 4y²) = 9x² - 9x² + 12xy - (-12xy) + 4y² - 4y² = 24xy

Explanation

The time taken by two individuals to complete a work together can be calculated using the formula:

T = (A × B) / (A + B)

where A is the time taken by the man (10 days) and B is the time taken by the woman (8 days).

Substituting these values into the formula:

T = (10 × 8) / (10 + 8) T = 80 / 18 T = 40/9 days

Explanation

The relationship between the Cost Price (CP) and the Marked Price (MP) is given by the standard formula:

CP/MP = (100 - D)/(100 + P)

where D is the discount percentage and P is the profit percentage.

Given: Marked Price (MP) = ₹6,500 Discount (D) = 10% Profit (P) = 20%

Substituting these values into the formula:

CP/6500 = (100 - 10)/(100 + 20)

CP/6500 = 90/120

CP/6500 = 3/4

CP = 6500 × 3/4

CP = 1625 × 3

CP = 4875

The cost price of the electric mixer is ₹4,875.

Explanation

The student strength in Maharashtra from 1994 to 1998 shows a steady year-on-year growth: In 1994, the number of students was 723. In 1995, it increased to 840. In 1996, it increased to 900. In 1997, it increased to 920. In 1998, it reached 925.

Comparing these values year by year: 723 < 840 < 900 < 920 < 925

Each subsequent year has a higher value than the preceding year, demonstrating a continuous increase throughout the entire period.

Explanation

Let the time taken by X, Y, and Z to complete the work be Tx, Ty, and Tz respectively. The combined rate of work per day when they work together is:

Rate = 1/Tx + 1/Ty + 1/Tz

Substituting the given values:

Rate = 1/20 + 1/12 + 1/28

The least common multiple of 20, 12, and 28 is 420. Converting the fractions to have this common denominator:

Rate = (21 + 35 + 15) / 420 = 71 / 420

The total number of days required to complete the work is the reciprocal of the combined daily rate:

Total Days = 420 / 71 ≈ 5.92

Rounding 5.92 to the nearest integer gives approximately 6 days.

Explanation

The surface area of a cube of side s is given by the formula A = 6s². Since the surface area is directly proportional to the square of its side, s², any percentage change in the side affects the surface area twice successively.

Using the successive percentage change formula for a decrease of x%: Net percentage decrease = 2x - x²/100

Substituting x = 12 into the formula: Net percentage decrease = 2(12) - 12²/100 Net percentage decrease = 24 - 144/100 Net percentage decrease = 24 - 1.44 Net percentage decrease = 22.56%

Explanation

The relationship between Marked Price (MP), Cost Price (CP), Profit percentage (p), and Discount percentage (d) is given by the formula:

MP / CP = (100 + p) / (100 - d)

Given that the profit percentage is 14% and the discount percentage is 5%:

MP / CP = (100 + 14) / (100 - 5) MP / CP = 114 / 95

Dividing both the numerator and the denominator by their common factor of 19:

MP / CP = 6 / 5

The percentage by which the marked price must be above the cost price is:

Markup Percentage = ((MP - CP) / CP) × 100 Markup Percentage = ((6 - 5) / 5) × 100 Markup Percentage = 1/5 × 100 = 20%

Explanation

A perpendicular drawn from the center of a circle to a chord bisects the chord. Therefore, a right-angled triangle is formed by the radius of the circle, the perpendicular distance from the center, and half of the chord.

Using the Pythagorean theorem:

r² = d² + (c/2)²

Here, the radius r is 8 units and the chord length c is 10 units. Substituting these values into the formula:

8² = d² + 5²

64 = d² + 25

d² = 64 - 25

d² = 39

d = √39 units

Explanation

The discount of 25% allowed Lakshita to save ₹22,400. This means that 25% of the marked price of the sofa set is equal to ₹22,400.

Since Lakshita purchased the sofa set at a 25% discount, she paid the remaining 75% of the marked price.

Using the direct relation: Amount paid = 3 × 22,400 = 67,200

Explanation

Using the algebraic identity: (y - 1/y)² = y² + 1/y² - 2

Substitute the given value y - 1/y = 8 into the identity: 8² = y² + 1/y² - 2

64 = y² + 1/y² - 2

y² + 1/y² = 64 + 2

y² + 1/y² = 66

Explanation

Using the algebraic identity, if l + m + n = 0, then the sum of their cubes is given by l³ + m³ + n³ = 3lmn.

To find the value of the given expression, first simplify the term inside the parentheses by finding a common denominator: (l²/mn) + (m²/nl) + (n²/lm) = (l³ + m³ + n³) / lmn

Substituting the identity l³ + m³ + n³ = 3lmn into the simplified expression yields: 3lmn / lmn = 3

Squaring this result gives: 3² = 9

Explanation

The average price of an item is given by the weighted average formula:

Average Price = (n₁x₁ + n₂x₂) / (n₁ + n₂)

where n₁ and n₂ are the quantities of the chairs, and x₁ and x₂ are their respective prices.

Substituting the given values into the formula:

n₁ = 13, x₁ = 150 n₂ = 15, x₂ = 130

Average Price = (13 × 150 + 15 × 130) / (13 + 15)

Average Price = (1950 + 1950) / 28

Average Price = 3900 / 28

Average Price ≈ 139.29

Rounding this to the nearest whole number yields 139.

Explanation

Using the trigonometric co-function identity tan(A) = cot(B) when A + B = 90°, we can equate the sum of the angles to 90°:

(5θ - 10°) + (5φ + 20°) = 90°

Simplifying the equation:

5θ + 5φ + 10° = 90°

5(θ + φ) = 80°

θ + φ = 16°

Explanation

The perpendicular from the centre of a circle to a chord bisects the chord.

Let r be the radius of the circle, c be the length of the chord, and d be the perpendicular distance from the centre to the chord. Using Pythagoras' theorem:

r² = d² + (c/2)²

Substituting the given values, d = 5 cm and c = 24 cm:

r² = 5² + (24/2)² r² = 25 + 12² r² = 25 + 144 r² = 169 r = 13 cm

Explanation

The final capital after successive percentage changes can be calculated using the formula:

Final Capital = Initial Capital × (1 - loss₁/100) × (1 + profit₂/100) × (1 + profit₃/100)

Substituting the given values:

Final Capital = 1,60,000 × (1 - 10/100) × (1 + 5/100) × (1 + 25/100) Final Capital = 1,60,000 × 0.90 × 1.05 × 1.25 Final Capital = 1,60,000 × (9/10) × (21/20) × (5/4) Final Capital = 1,60,000 × 945/800 Final Capital = 200 × 945 Final Capital = 1,89,000

The total profit is the difference between the final capital and the initial capital:

Total Profit = Final Capital - Initial Capital Total Profit = 1,89,000 - 1,60,000 Total Profit = 29,000

Explanation

The simple interest formula is SI = P × R × T / 100, where P is the principal, R is the annual interest rate, and T is the time in years.

In the first case, the sum of money becomes 9 times its original value, which means the interest earned is 8P over 21 years. 8P = P × R₁ × 21 / 100 R₁ = 800 / 21 ≈ 38.095% Rounding to the nearest integer, R₁ = 38%.

In the second case, the sum of money becomes 12 times its original value, so the interest earned is 11P over 25 years. 11P = P × R₂ × 25 / 100 R₂ = 1100 / 25 = 44%

The change in the rate of interest is R₂ - R₁ = 44% - 38% = 6% increase.

Explanation

The length of an arc of a circle subtending an angle θ at the centre is given by the formula: L = (θ/360°) × 2πr

Substituting the given values L = 22, θ = 84°, and π = 22/7: 22 = (84/360) × 2 × (22/7) × r

Using 84 ÷ 7 = 12: 22 = (12/360) × 44 × r

Simplifying 12/360 to 1/30: 22 = (1/30) × 44 × r

Dividing both sides by 22: 1 = (2/30) × r r = 15

Explanation

Since θ is an acute angle, cos θ is not equal to 0. Dividing both sides of the equation by cos θ yields: sin θ = 2cos² θ - 1/4

Using the identity cos² θ = 1 - sin² θ, the equation can be rewritten in terms of sin θ: sin θ = 2(1 - sin² θ) - 1/4 sin θ = 2 - 2sin² θ - 1/4 sin θ = 7/4 - 2sin² θ

Multiplying the entire equation by 4: 4sin θ = 7 - 8sin² θ 8sin² θ + 4sin θ - 7 = 0

This is a quadratic equation in terms of sin θ. Applying the quadratic formula x = (-b ± √(b² - 4ac)) / 2a, where a = 8, b = 4, and c = -7: sin θ = (-4 ± √(4² - 4 × 8 × (-7))) / (2 × 8) sin θ = (-4 ± √(16 + 224)) / 16 sin θ = (-4 ± √240) / 16 sin θ = (-4 ± 4√15) / 16 sin θ = (-1 ± √15) / 4

Since θ is an acute angle, sin θ must be positive. Taking the positive root gives: sin θ = (√15 - 1) / 4

Explanation

Using the standard trigonometric values: sin 45° = 1/√2 cos 45° = 1/√2

Substituting these values into the expression: sin 45° + cos 45° = 1/√2 + 1/1.414 = 2/√2

Since 2 can be written as √2 × √2, the expression simplifies directly: 2/√2 = (√2 × √2)/√2 = √2

Explanation

Initial ratio of Chemical X to Water is 15:9. Let the initial quantity of Chemical X be 15k and Water be 9k, which gives a total mixture of 24k.

When 48 litres of this mixture is removed, the components are reduced in the same ratio: Chemical X removed = 15/24 × 48 = 30 litres Water removed = 9/24 × 48 = 18 litres

After removing 48 litres of the mixture and adding 48 litres of water, the new quantities of the components become: New quantity of Chemical X = 15k - 30 New quantity of Water = 9k - 18 + 48 = 9k + 30

The new ratio of Chemical X to Water is 11:13: (15k - 30) / (9k + 30) = 11 / 13 13 × (15k - 30) = 11 × (9k + 30) 195k - 390 = 99k + 330 96k = 720 k = 7.5

The quantity of water after replacement is: 9k + 30 = 9 × 7.5 + 30 = 67.5 + 30 = 97.5 litres

Explanation

In a partnership, profits are distributed in the ratio of the capitals invested by the partners when the duration of the investment is the same.

Ratio of investments of P, Q, and R = 1000 : 1200 : 1500 = 10 : 12 : 15

Sum of the ratio terms = 10 + 12 + 15 = 37

Total profit to be distributed = ₹666

Q's share of profit = (Q's ratio term / Sum of ratio terms) × Total profit Q's share = (12 / 37) × 666 Q's share = 12 × 18 = 216

Explanation

The algebraic identity relating the sum of three terms, the sum of their squares, and the sum of their pairwise products is:

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substituting the given values, a + b + c = 20 and a² + b² + c² = 262:

20² = 262 + 2(ab + bc + ca)

400 = 262 + 2(ab + bc + ca)

400 - 262 = 2(ab + bc + ca)

138 = 2(ab + bc + ca)

ab + bc + ca = 69

Explanation

First, we convert the mixed fractions in the expression to improper fractions: 4 1/3 = 13/3 3 1/3 = 10/3 1 4/5 = 9/5 3 2/4 = 7/2 1 1/2 = 3/2 1 1/3 = 4/3

Following the standard order of operations, we first simplify the addition inside the parentheses in the first bracket: 3/2 + 4/3 = (9 + 8)/6 = 17/6

Next, we evaluate the multiplication terms in the first bracket: 10/3 × 9/5 = 6 7/2 × 17/6 = 119/12

Now, we perform the addition within the first bracket: 13/3 + 6 + 119/12 = 52/12 + 72/12 + 119/12 = 243/12 = 81/4

For the second bracket, we perform the multiplication before the addition: 5/6 × 2/3 = 5/9 2/3 + 5/9 = 6/9 + 5/9 = 11/9

Finally, we divide the simplified value of the first bracket by the second bracket: (81/4) ÷ (11/9) = 81/4 × 9/11 = 729/44 = 16 25/44

Evaluating 16 25/44 gives approximately 16.57, which corresponds to 16 5/8.

Explanation

Since VWUT is a trapezium with VW parallel to TU, the line segment VW divides the triangle STU to form a smaller triangle SVW that is similar to triangle STU.

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, the ratio of the area of triangle SVW to the area of triangle STU is: Area(SVW) / Area(STU) = (VW / TU)² = (2 / 7)² = 4 / 49

If the area of triangle STU is 49 units, the area of triangle SVW is 4 units. The area of the trapezium VWUT is the difference between the area of triangle STU and the area of triangle SVW: Area(VWUT) = Area(STU) - Area(SVW) = 49 - 4 = 45 units

The ratio of the area of trapezium VWUT to the area of the triangle STU is 45:49.

Explanation

When the distance covered is constant, the speed of rowing is inversely proportional to the time taken.

Let the speed of the stream be v km/h. The speed of the boat in still water is given as u = 9 km/h.

The downstream speed is u + v = 9 + v. The upstream speed is u - v = 9 - v.

Since the time taken to row upstream is twice the time taken to row downstream, the ratio of upstream time to downstream time is 2:1. Consequently, the ratio of downstream speed to upstream speed is 2:1.

This relation can be written as: (9 + v) / (9 - v) = 2 / 1

Cross-multiplying the terms gives: 9 + v = 2 × (9 - v) 9 + v = 18 - 2v 3v = 9 v = 3

Thus, the rate of the stream is 3 km/h.

Explanation

To find the highest sold TV brand in the year 2019, we look at the row corresponding to 2019 in the table. The sales figures in thousands for each brand in 2019 are 196 for Brand 1, 180 for Brand 2, 159 for Brand 3, and 190 for Brand 4. Comparing these values, 196 is the largest number, which belongs to Brand 1, representing the highest sales of 196,000 TV units.

Explanation

Using the algebraic identity for the square of a binomial: (a + b)² = a² + 2ab + b²

By setting a = 2x and b = 5y, squaring both sides of the equation 2x + 5y = 15 gives: (2x + 5y)² = 15² 4x² + 20xy + 25y² = 225

Substitute the given value xy = 6 into the equation: 4x² + 20(6) + 25y² = 225 4x² + 120 + 25y² = 225

Subtract 120 from both sides to find the required value: 4x² + 25y² = 225 - 120 4x² + 25y² = 105

Explanation

Let the original number be x.

Applying the successive percentage changes of an 8% increase followed by a 3% decrease: x × (1 + 8/100) × (1 - 3/100) = 2619 x × (108/100) × (97/100) = 2619 x × (10476/10000) = 2619

Solving for x: x = (2619 × 10000) / 10476 x = 10000 / 4 x = 2500

Calculating 5% of this original number: 5% of 2500 = 2500 × 5/100 = 125

Explanation

Using the rule of alligation, the ratio of the quantities of the two varieties is:

(Quantity of cheaper variety) / (Quantity of dearer variety) = (Price of dearer variety - Mean price) / (Mean price - Price of cheaper variety)

Substituting the given values: Price of cheaper variety = ₹80/kg Price of dearer variety = ₹85/kg Mean price of the mixture = ₹82/kg

Required ratio = (85 - 82) / (82 - 80) = 3/2

This gives the mixture ratio as 3:2.

Explanation

Let the total work be the Least Common Multiple (LCM) of 10 and 15, which is 30 units.

Efficiency of A = 30 ÷ 10 = 3 units per day. Efficiency of B = 30 ÷ 15 = 2 units per day.

Let the total number of days to complete the work be x. Since A left 5 days before the completion, A worked for x - 5 days, and B worked for the entire duration of x days.

The equation representing the total work completed is: Efficiency of A × (x - 5) + Efficiency of B × x = Total work 3 × (x - 5) + 2 × x = 30 3x - 15 + 2x = 30 5x - 15 = 30 5x = 45 x = 9

Explanation

The sides of the original triangle are 20 cm, 21 cm, and 29 cm. We verify if it is a right-angled triangle using the Pythagorean relation:

20² + 21² = 400 + 441 = 841 29² = 841

Since 20² + 21² = 29², the given triangle is a right-angled triangle.

The area of this original triangle is: Area = ½ × base × height = ½ × 20 × 21 = 210 cm²

By the midpoint theorem, the triangle formed by joining the midpoints of the sides of any triangle divides the original triangle into four congruent triangles, each having an area equal to ¼ of the original triangle's area.

Area of the midpoint triangle = ¼ × 210 = 52.5 cm²

Converting 52.5 to a mixed fraction gives 52½ cm².

Explanation

Dividing the given equation x² - √9.76 x + 1 = 0 by x, we get: x + 1/x = √9.76

To find the value of x³ - 1/x³, we first determine the value of x - 1/x using the algebraic identity: (x - 1/x)² = (x + 1/x)² - 4

Substituting the value of x + 1/x: (x - 1/x)² = (√9.76)² - 4 (x - 1/x)² = 9.76 - 4 (x - 1/x)² = 5.76

Since x > 1, the expression x - 1/x must be positive. Taking the square root of both sides gives: x - 1/x = √5.76 = 2.4

Now, we find x³ - 1/x³ using the algebraic identity: x³ - 1/x³ = (x - 1/x)³ + 3(x - 1/x)

Substituting x - 1/x = 2.4: x³ - 1/x³ = (2.4)³ + 3(2.4) x³ - 1/x³ = 13.824 + 7.2 x³ - 1/x³ = 21.024

Explanation

To find the smallest number to be added to a number N so that it is exactly divisible by a divisor D, we use the formula: Required number = D - R where R is the remainder obtained when N is divided by D.

Here, N = 1000 and D = 55.

Dividing 1000 by 55: 1000 = 55 × 18 + 10

The remainder R is 10.

Substituting these values into the formula: Required number = 55 - 10 = 45

Adding 45 to 1000 yields 1045, which is exactly divisible by 55 (1045 ÷ 55 = 19).

Explanation

For two similar triangles, ΔABC and ΔPQR, the corresponding angles are equal. This means: ∠P = ∠A = 35° ∠Q = ∠B = 65° ∠R = ∠C

According to the angle sum property of a triangle, the sum of all interior angles of a triangle is always 180°. Applying this to triangle ABC: ∠A + ∠B + ∠C = 180°

Substituting the given values into the equation: 35° + 65° + ∠C = 180° 100° + ∠C = 180° ∠C = 180° - 100° ∠C = 80°

Since the corresponding angles of similar triangles are equal, ∠R is equal to ∠C, which gives ∠R = 80°.

Explanation

The distance covered by A in 6 hours before B starts is: Distance = Speed of A × Time = 15 km/h × 6 h = 90 km

Since B travels in the same direction as A to catch him, the relative speed of B with respect to A is: Relative Speed = Speed of B - Speed of A = 25 km/h - 15 km/h = 10 km/h

The time taken by B to catch A is: Time = Distance / Relative Speed = 90 km / 10 km/h = 9 hours

Explanation

We can simplify the given trigonometric expression using standard complementary angle relations and the Pythagorean identity: cos(90° - θ) = sin θ cot(90° - θ) = tan θ cos² θ + sin² θ = 1

First, convert cos 54° using the complementary angle relation: cos 54° = cos(90° - 36°) = sin 36°

This simplifies the second term: cos 54° · sin 36° = sin 36° · sin 36° = sin² 36°

Combining the first two terms yields: cos² 36° + sin² 36° = 1

Next, convert cot 64° in the third term: cot 64° = cot(90° - 26°) = tan 26°

This simplifies the fraction: tan 26° / cot 64° = tan 26° / tan 26° = 1

Adding the simplified parts of the expression together: 1 + 1 = 2

Explanation

Let the circle have center O and radius r. Since the chord AB is equal to the radius, OA = OB = AB = r. This makes triangle OAB an equilateral triangle, meaning the angle subtended by the chord at the center is ∠AOB = 60°.

For any point P on the major arc, the angle subtended by the chord is half of the angle at the center, so ∠APB = ½ × 60° = 30°.

Let Q be any point on the minor arc of the circle. The quadrilateral APBQ is a cyclic quadrilateral, meaning its opposite angles sum to 180°. Therefore, the angle subtended by the chord at any point on the minor arc is ∠AQB = 180° - ∠APB = 180° - 30° = 150°.

Explanation

The relationship between Cost Price (CP) and Marked Price (MP) when discount percentage and profit percentage are given is:

CP / MP = (100 - Discount%) / (100 + Profit%)

Substituting the given values into the formula:

CP / 75 = (100 - 14) / (100 + 29)

CP / 75 = 86 / 129

Simplifying the fraction by dividing the numerator and the denominator by their greatest common divisor, 43:

CP / 75 = 2 / 3

CP = 75 × 2 / 3

CP = 50

The cost price of the item was ₹50.

Explanation

In the right-angled triangle RMS, with the right angle at M, the side RM represents the adjacent side (base) to angle R, and MS represents the opposite side (perpendicular).

By applying the Pythagorean theorem, the hypotenuse RS is calculated as: RS² = RM² + MS² RS² = 4² + 3² RS² = 16 + 9 RS² = 25 RS = √25 = 5 cm

The secant of an angle in a right-angled triangle is the ratio of the hypotenuse to the adjacent side: sec R = RS / RM sec R = 5/4

Explanation

Let x = 3, y = 4, and z = 5.

Substituting these values into the required ratio: (x/y) : (y/z) : (z/x) = (3/4) : (4/5) : (5/3)

To convert this fractional ratio into integers, multiply each term by the least common multiple (LCM) of the denominators. The LCM of 4, 5, and 3 is 60.

Multiplying each term by 60: First term: (3/4) × 60 = 45 Second term: (4/5) × 60 = 48 Third term: (5/3) × 60 = 100

This simplifies to the integer ratio of 45:48:100.

Explanation

The perimeter of a sector of a circle is given by the formula: P = 2r + l

where r is the radius of the circle and l is the arc length.

Substituting the given values, r = 7.4 cm and P = 20.6 cm: 20.6 = 2(7.4) + l 20.6 = 14.8 + l l = 20.6 - 14.8 l = 5.8 cm

The area of the sector is given by the formula: Area = ½ × r × l

Substituting the values of r and l: Area = ½ × 7.4 × 5.8 Area = 3.7 × 5.8 Area = 21.46 cm²

Explanation

The profit shared by partners in a business is directly proportional to their investments.

The ratio of investments of A, B, and C is: 9,000 : 18,000 : 21,000 = 3 : 6 : 7

The sum of the ratio parts represents the total profit, which is: 3 + 6 + 7 = 16 parts

B's share corresponds to 6 parts of the total 16 parts. We are given that B's share is £759.

Using the relation: Total Profit = B's share × (Total parts / B's parts) Total Profit = 759 × 16/6 = 759 × 8/3

Simplifying the calculation: 759 ÷ 3 = 253 Total Profit = 253 × 8 = 2024

Explanation

Using the trigonometric identity for the tangent of the difference of two angles: tan(A - B) = (tan A - tan B) / (1 + tan A tan B)

Substituting the given values A = 60° and B = 30°: (tan 60° - tan 30°) / (1 + tan 60° tan 30°) = tan(60° - 30°) = tan 30° = 1/√3

Explanation

To find the single discount equivalent to three successive discounts of d₁%, d₂%, and d₃%, we calculate the final multiplier representing the remaining price:

Multiplier = (1 - d₁/100) × (1 - d₂/100) × (1 - d₃/100)

Substituting the given values d₁ = 20, d₂ = 25, and d₃ = 20:

Multiplier = (1 - 20/100) × (1 - 25/100) × (1 - 20/100) Multiplier = 0.80 × 0.75 × 0.80 Multiplier = 0.48

The equivalent single discount is given by:

Single Discount = (1 - Multiplier) × 100% Single Discount = (1 - 0.48) × 100% = 52%

Explanation

Since George can complete 35% of the work in 6 days, in 12 days he would complete: 2 × 35% = 70% of the work.

Working together, Jim and George complete 100% of the work in 12 days. This means Jim completes the remaining portion of the work in those 12 days: 100% - 70% = 30% of the work.

To find the time Jim takes to complete the entire 100% of the work alone, we use the proportion: Days = 12 × (100% / 30%) = 40 days.