SSC CHSL Tier 1 2015 Quantitative Aptitude Questions with Solutions

Explanation

Let d be the original number of days required to finish the work by 150 workers. Total work = 150d worker-days

When workers drop out daily, the number of workers working on consecutive days forms an arithmetic progression with first term a = 150 and common difference d_diff = -4: 150, 146, 142, ...

Let n be the total number of days taken to finish the work. Since it takes 8 more days, n = d + 8, which gives d = n - 8.

The sum of this arithmetic progression for n terms represents the total work done: Total work = (n/2) × [2a + (n - 1)d_diff] Total work = (n/2) × [2 × 150 + (n - 1)(-4)] Total work = (n/2) × [304 - 4n] Total work = n(152 - 2n)

Equating the two expressions for the total work: 150(n - 8) = n(152 - 2n) 150n - 1200 = 152n - 2n² 2n² - 2n - 1200 = 0 n² - n - 600 = 0 (n - 25)(n + 24) = 0

Solving for the positive value of n gives n = 25.

Explanation

Let the initial ratio of milk and water be 5 : 1, which means there are 5 parts of milk and 1 part of water.

When 5 litres of water is added to the mixture, the new ratio of milk and water becomes 5 : 2.

Since only water is added, the quantity of milk remains constant at 5 parts in both ratios. The water content increases from 1 part to 2 parts, which is an increase of 2 - 1 = 1 part.

This increase of 1 part corresponds directly to the 5 litres of water added. Therefore, 1 part is equal to 5 litres.

The quantity of milk in the mixture is represented by 5 parts. Quantity of milk = 5 × 5 = 25 litres.

Explanation

The odd integers between 2 and 22 are 3, 5, 7, 9, 11, 13, 15, 17, 19, and 21. These numbers form an arithmetic progression where the first term a is 3 and the last term l is 21.

For any arithmetic progression, the average of the terms is equal to the average of the first and the last term, given by the formula:

Average = (a + l) / 2

Substituting the values:

Average = (3 + 21) / 2 = 24 / 2 = 12

Explanation

To find the least number to be subtracted, we calculate the square root of 32146 using the long division method.

Pairing the digits from right to left, we get 3, 21, and 46. First, the largest square number less than or equal to 3 is 1, with a remainder of 2. Bringing down 21 gives 221.

Doubling the quotient 1 gives 2. Since 27 × 7 = 189, we subtract 189 from 221 to get 32. Bringing down 46 gives 3246.

Doubling the quotient 17 to get 34, we find the next digit. Since 349 × 9 = 3141, we subtract 3141 from 3246, which leaves a remainder of 105.

The remainder 105 represents the excess value that prevents 32146 from being a perfect square. Subtracting 105 from 32146 results in 32041, which is the perfect square of 179.

Explanation

The roller is a cylinder where the length (h) is 1.2 m and the radius (r) is half of the 84 cm diameter, which is 42 cm or 0.42 m.

The area covered by the roller in one complete revolution is equal to its curved surface area: Area per revolution = 2πrh Area per revolution = 2 × (22/7) × 0.42 × 1.2 = 3.168 m²

The total area leveled in 500 revolutions is: Total area = 500 × 3.168 = 1584 m²

The total cost of leveling the ground at the rate of ₹1.50 per m² is: Total cost = 1584 × 1.50 = ₹2376

Explanation

Let P be the amount deposited in the post office and B be the amount deposited in the bank. The total amount deposited is: P + B = 3,00,000

The monthly interest income is ₹2,000. Thus, the annual interest income is: 2,000 × 12 = ₹24,000

Since the annual interest rates for the post office and the bank are 10% and 6% respectively, we can express this as: 0.10 P + 0.06 B = 24,000

Multiplying this equation by 100 to clear decimals: 10 P + 6 B = 24,00,000 5 P + 3 B = 12,00,000

Now, multiplying the first equation (P + B = 3,00,000) by 3: 3 P + 3 B = 9,00,000

Subtracting this from the simplified interest equation: (5 P + 3 B) - (3 P + 3 B) = 12,00,000 - 9,00,000 2 P = 3,00,000 P = 1,50,000

Substituting P back into the total deposit equation gives: B = 1,50,000

The difference between the two deposits is: 1,50,000 - 1,50,000 = 0 (Nil)

Explanation

The volume V of a right circular cylinder with base radius r and height h is given by the formula V = πr²h.

Substituting the given values, where radius r = 5 cm, height h = 21 cm, and π = 22/7:

V = (22/7) × 5² × 21 V = (22/7) × 25 × 21 V = 22 × 25 × 3 V = 1650 cm³

Explanation

The volume V of a cone is given by the formula V = 1/3 × π × r² × h, where r is the radius of the base and h is the height.

Taking the side of 9 cm as the base radius (r) and the side of 12 cm as the height (h), we substitute these values into the formula:

V = 1/3 × π × 9² × 12 V = 1/3 × π × 81 × 12 V = 27 × 12 × π V = 324π cm³

Explanation

The total work is constant and can be expressed using the formula M₁ × D₁ × H₁ = M₂ × D₂ × H₂, where M is the number of men, D is the number of days, and H is the number of hours worked per day.

Substituting the given values: M₁ = 12, D₁ = 10, H₁ = 8 M₂ = 16, H₂ = 7 ½ = 7.5 D₂ = x

12 × 10 × 8 = 16 × x × 7.5 960 = 120 × x x = 960 ÷ 120 x = 8

Explanation

The total number of family members initially is 7 (1 mother and 6 children). Since their average age is 12 years, the sum of their ages is 7 × 12 = 84 years.

When the mother is excluded, the remaining 6 children have an average age that is reduced by 5 years, making the new average age 12 - 5 = 7 years. The sum of the children's ages is 6 × 7 = 42 years.

Subtracting the sum of the children's ages from the total sum gives the age of the mother: 84 - 42 = 42 years.

Explanation

Dividing the given equations by a and b respectively, we get: x/a = sin θ + cos θ y/b = sin θ - cos θ

Squaring both equations on both sides: x²/a² = (sin θ + cos θ)² = sin² θ + cos² θ + 2 sin θ cos θ y²/b² = (sin θ - cos θ)² = sin² θ + cos² θ - 2 sin θ cos θ

Adding these two squared equations: x²/a² + y²/b² = (sin² θ + cos² θ + 2 sin θ cos θ) + (sin² θ + cos² θ - 2 sin θ cos θ) x²/a² + y²/b² = 2(sin² θ + cos² θ)

Using the standard trigonometric identity sin² θ + cos² θ = 1: x²/a² + y²/b² = 2 × 1 = 2

Explanation

The given equation is: a² + b² + c² = 2(a - b - c) - 3

Rearranging all terms to the left side: a² + b² + c² - 2a + 2b + 2c + 3 = 0

By splitting the constant 3 into 1 + 1 + 1, we can group the terms to form perfect squares: (a² - 2a + 1) + (b² + 2b + 1) + (c² + 2c + 1) = 0 (a - 1)² + (b + 1)² + (c + 1)² = 0

Since the sum of squares of real numbers must be zero, each independent term must equal zero: a - 1 = 0 → a = 1 b + 1 = 0 → b = -1 c + 1 = 0 → c = -1

Calculating the required value: a + b + c = 1 + (-1) + (-1) = -1

Explanation

Using the identity (x + y)² = x² + y² + 2xy, we substitute the given values: 4² = 14 + 2xy 16 = 14 + 2xy 2xy = 2 xy = 1

Next, we use the identity (x - y)² = (x + y)² - 4xy to find the difference between x and y: (x - y)² = 4² - 4(1) (x - y)² = 16 - 4 = 12 x - y = ±√12 = ±2√3

Since it is given that x > y, the value of x - y must be positive: x - y = 2√3

Now we solve the system of linear equations: x + y = 4 x - y = 2√3

Adding these two equations: 2x = 4 + 2√3 x = 2 + √3

Subtracting the second equation from the first: 2y = 4 - 2√3 y = 2 - √3

Explanation

Given: cos θ + sin θ = m sec θ + cosec θ = n

Expressing n in terms of sine and cosine: n = 1/cos θ + 1/sin θ n = (sin θ + cos θ) / (sin θ cos θ)

Substituting cos θ + sin θ = m: n = m / (sin θ cos θ) sin θ cos θ = m/n

Now, squaring both sides of the first equation: m² = (cos θ + sin θ)² m² = cos² θ + sin² θ + 2 sin θ cos θ m² = 1 + 2 sin θ cos θ m² - 1 = 2 sin θ cos θ

Substituting this expression into n(m² - 1): n(m² - 1) = n(2 sin θ cos θ) n(m² - 1) = n × 2(m/n) n(m² - 1) = 2m

Explanation

Let a = 1 + √3 and b = √2.

The given expression can be written in terms of a and b as: 1/(a + b) + 1/(a - b)

By taking the common denominator, we simplify this expression to: ((a - b) + (a + b)) / ((a + b)(a - b)) = 2a / (a² - b²)

Next, we calculate the terms in the denominator: a² = (1 + √3)² = 1 + 2√3 + 3 = 4 + 2√3 b² = (√2)² = 2 a² - b² = 4 + 2√3 - 2 = 2 + 2√3

Since 2a = 2(1 + √3) = 2 + 2√3, we have: a² - b² = 2a

Substituting this back into the simplified expression gives: 2a / 2a = 1

Explanation

The surface area of a sphere of radius r is given by the formula S = 4πr².

When the radius is increased by 2 cm, the new radius is r + 2. The change in the surface area is: 4π(r + 2)² - 4πr² = 352

Expanding and simplifying this equation: 4π(r² + 4r + 4 - r²) = 352 16π(r + 1) = 352

Using π = 22/7: 16 × (22/7) × (r + 1) = 352 (22/7) × (r + 1) = 22 r + 1 = 22 × (7/22) r + 1 = 7 r = 6 cm

Explanation

Let the two numbers be 3x and 4x, where x is their highest common factor. Since 3 and 4 are co-prime, their highest common factor is indeed x.

Given that the highest common factor is 15, we have x = 15.

The sum of the two numbers is 3x + 4x = 7x.

Substituting the value of x into the sum formula: Sum = 7 × 15 = 105

Explanation

Let the cost price of the article be ₹100.

The shopkeeper marks the price 30% higher than the cost price, so the marked price is: 100 + 30% of 100 = ₹130

He then offers a 10% discount on this marked price: Discount = 10% of 130 = ₹13

The selling price of the article is: 130 - 13 = ₹117

The profit earned is: Selling Price - Cost Price = 117 - 100 = ₹17

Since the cost price is ₹100, this profit corresponds to a profit percentage of 17%.

Alternatively, using the successive percentage change formula: Net percentage change = x + y + (xy / 100) Here, x = +30% and y = -10%. Net percentage change = 30 - 10 + (30 × -10 / 100) = 20 - 3 = 17% profit.

Explanation

Let the four successive angles of the cyclic quadrilateral be A, B, C, and D. In a cyclic quadrilateral, the sum of opposite angles is 180°, which gives A + C = 180° and B + D = 180°.

The first three successive angles A, B, and C are in the ratio 1:3:4. Let A = k, B = 3k, and C = 4k.

Using the relation for opposite angles A and C: A + C = 180° k + 4k = 180° 5k = 180° k = 36°

Angle B is calculated as: B = 3 × 36° = 108°

Using the relation for opposite angles B and D to find the fourth angle D: B + D = 180° 108° + D = 180° D = 180° - 108° = 72°

Explanation

Let the Cost Price of one chair be CP and the Selling Price of one chair be SP.

According to the question: 25 × CP = 30 × SP

This can be rewritten to find the ratio of Selling Price to Cost Price: SP / CP = 25 / 30 = 5 / 6

The formula for loss percentage is: Loss % = (CP - SP) / CP × 100 Loss % = (1 - SP / CP) × 100 Loss % = (1 - 5/6) × 100 = 1/6 × 100 = 16 ⅔%

Explanation

The selling price of the mixture is ₹11 per litre at a profit of 10%. The cost price of the mixture is calculated using the relation:

Cost Price of mixture = Selling Price / (1 + Profit Rate) = 11 / (1 + 0.10) = 11 / 1.1 = ₹10 per litre

Let the cost price of liquid Y be ₹y per litre. Since liquid X costs ₹2 more per litre than Y, the cost price of liquid X is ₹(y + 2) per litre.

Since the liquids X and Y are mixed in the ratio 3:2, the weighted average of their individual costs yields the cost price of the mixture:

(3 × (y + 2) + 2 × y) / (3 + 2) = 10

(3y + 6 + 2y) / 5 = 10

5y + 6 = 50

5y = 44

y = 8.80

The cost price of liquid X is y + 2:

8.80 + 2 = ₹10.80 per litre

Explanation

Using the complementary angle identity sin(90° - θ) = cos θ, we have sin² θ + sin² (90° - θ) = sin² θ + cos² θ = 1.

By pairing the terms whose angles sum to 90°, we can simplify the expression: (sin² 1° + sin² 89°) + (sin² 11° + sin² 79°) + (sin² 21° + sin² 69°) + (sin² 31° + sin² 59°) + (sin² 41° + sin² 49°) + sin² 45°

Substituting the identity for each of the five pairs and using the standard value sin 45° = 1/√2: 1 + 1 + 1 + 1 + 1 + (1/√2)² = 5 + ½ = 5 ½

Explanation

If a set of numbers has an average of A, and each number in the set is multiplied by a constant k, then the average of the new set of numbers is also multiplied by the same constant k.

New Average = Original Average × k

Given that the original average is 21 and each number is multiplied by 8, we substitute these values into the relation:

New Average = 21 × 8 = 168

This linear property of the arithmetic mean ensures that scaling every data point by a factor scales the entire mean by that same factor.

Explanation

Let S = x + y + z. The given equations are:

xS = 20 yS = 30 zS = 50

Adding these three equations together:

xS + yS + zS = 20 + 30 + 50 S(x + y + z) = 100 S² = 100 S = ±10

Therefore, x + y + z = ±10.

Multiplying by 2 gives:

2(x + y + z) = 2(±10) = ±20

Taking the positive value present in the options, the result is 20.

Explanation

Using the standard trigonometric identity: (cos A + sin A)² + (cos A - sin A)² = 2(cos² A + sin² A)

Since cos² A + sin² A = 1, this simplifies to: (cos A + sin A)² + (cos A - sin A)² = 2

Substitute the given equation cos A + sin A = √2 cos A into the identity: (√2 cos A)² + (cos A - sin A)² = 2 2 cos² A + (cos A - sin A)² = 2 (cos A - sin A)² = 2 - 2 cos² A (cos A - sin A)² = 2(1 - cos² A)

Using the identity 1 - cos² A = sin² A: (cos A - sin A)² = 2 sin² A

Taking the positive square root on both sides for the acute angle A gives: cos A - sin A = √2 sin A

Explanation

A point lies on a given line if its coordinates satisfy the equation of the line. For the line y = 3x, substituting the coordinates of the origin, x = 0 and y = 0, we get 0 = 3 × 0, which is a true statement. This confirms that the line passes through the point (0,0).

Explanation

The given sequence is 40960, 10240, 2560, 640, 200, 40, 10.

Analyzing the relationship between consecutive terms shows that each number is obtained by dividing the preceding term by 4:

40960 ÷ 4 = 10240 10240 ÷ 4 = 2560 2560 ÷ 4 = 640

Following this consistent pattern, the next term should be:

640 ÷ 4 = 160

Instead of 160, the sequence lists 200. Applying the rule of division by 4 to the correct term 160 continues to yield the rest of the sequence correctly:

160 ÷ 4 = 40 40 ÷ 4 = 10

Thus, 200 is the incorrect number in the sequence.

Explanation

Let O be the center of the circle, and let the diameter AB lie along the horizontal line. For any point P on the circle, the inscribed angle ∠BAP is equal to half of the central angle ∠BOP. Similarly, for any point Q on the circle, the inscribed angle ∠ABQ is equal to half of the central angle ∠AOQ.

Given that ∠BAP = ∠ABQ, it follows that their corresponding central angles are equal, meaning ∠BOP = ∠AOQ. Since O is the midpoint of the diameter AB, this equality of central angles means that the points P and Q are symmetric with respect to the perpendicular bisector of AB.

Due to this symmetry, the perpendicular distance from P to AB is equal to the perpendicular distance from Q to AB. Therefore, the chord PQ is parallel to the diameter AB. A quadrilateral with one pair of opposite sides parallel is a trapezium, and since all its vertices lie on the circle, ABQP is a cyclic trapezium.

Explanation

When water is added to milk and the mixture is sold at the cost price of milk, the percentage gain is given by the formula:

Gain% = (Quantity of Water / Quantity of Milk) × 100

Substituting the given gain of 20%:

20 = (Quantity of Water / Quantity of Milk) × 100

Quantity of Water : Quantity of Milk = 20 : 100 = 1 : 5

Explanation

For an equilateral triangle of side a, the altitude h is given by the formula: h = (√3/2) × a

The centroid G divides the altitude/median in the ratio 2:1 from the vertex. Therefore, the distance from a vertex to the centroid AG is: AG = (2/3) × h = (2/3) × (√3/2) × a = a/√3

Given side a = 10 cm, we substitute the value: AG = 10/√3 cm

Rationalizing the denominator by multiplying the numerator and denominator by √3 gives: AG = (10√3)/3 cm

Explanation

To find the value of x, substitute the standard trigonometric values into the given equation:

tan 30° = 1/√3 sin 30° = ½ cot 45° = 1 sec 60° = 2

Substituting these values into the left-hand side (LHS) of the equation: LHS = (x - x(1/√3)²) / (1 + (1/√3)²) LHS = (x - x/3) / (1 + 1/3) LHS = (2x/3) / (4/3) LHS = 2x/4 LHS = x/2

Substituting the values into the right-hand side (RHS) of the equation: RHS = (½)² + 4(1)² - (2)² RHS = ¼ + 4 - 4 RHS = ¼

Equating LHS and RHS: x/2 = ¼ x = 2/4 x = ½

Explanation

To find the speed of the train, we use the formula:

Speed = Distance / Time

When a train crosses a tree, the distance covered is equal to the length of the train.

Distance = 250 m Time = 50 s

Speed = 250 / 50 = 5 m/s

To convert the speed from meters per second (m/s) to kilometers per hour (km/hr), we multiply by the conversion factor 18/5:

Speed = 5 × 18/5 = 18 km/hr

Explanation

The rate of discount is calculated using the formula:

Discount = Marked Price - Selling Price

Given that the Marked Price is ₹250 and the Selling Price is ₹225:

Discount = 250 - 225 = ₹25

The discount percentage is:

Discount % = (Discount / Marked Price) × 100

Discount % = (25 / 250) × 100 = 10%

Explanation

Let the labelled price of the bag be ₹100.

Since Mohan purchased the bag at a 20% discount on the labelled price, his cost price is: Cost Price = 100 - (20% of 100) = ₹80

He then sold the bag with a 40% profit on his cost price, so the selling price is: Selling Price = 80 + (40% of 80) = 80 + 32 = ₹112

The profit made relative to the labelled price is: Profit = Selling Price - Labelled Price = 112 - 100 = ₹12

The percentage of profit calculated on the labelled price is: Percentage Profit = (Profit / Labelled Price) × 100 = (12 / 100) × 100 = 12%

Explanation

The given six-digit number is of the form 5416x6. Since it ends in 6, its square root must end in either 4 or 6.

To locate the range of the square root, we evaluate nearby perfect squares of multiples of 10. Since 700² = 490000 and 800² = 640000, the square root lies in the 700s. More precisely, because 730² = 532900 and 740² = 547600, the square root must lie between 730 and 740.

The only integers in this interval ending in 4 or 6 are 734 and 736. Computing their squares: 734² = 538756 736² = 541696

Matching 541696 with the template 5416x6 gives x = 9.

Explanation

Let the total income of Ram babu be I.

The remaining amount after donating 3% of his income to charity is: I × (1 - 0.03) = 0.97 × I

The remaining amount after depositing 12% of the rest in the bank is: 0.97 × I × (1 - 0.12) = 0.97 × 0.88 × I = 0.8536 × I

According to the question, this final remaining amount is equal to ₹12804: 0.8536 × I = 12804 I = 12804 ÷ 0.8536 I = 15000

Explanation

For any triangle ABC, the angle formed at the intersection of the internal bisectors of ∠B and ∠C (point D, which is the incenter of the triangle) is given by the standard relation:

∠BDC = 90° + ½ ∠A

Substituting the given value of ∠A = 80° into the formula:

∠BDC = 90° + ½ × 80° ∠BDC = 90° + 40° ∠BDC = 130°

Explanation

The relationship between speed, distance, and time is given by: Distance = Speed × Time

Let the speed of the boat in still water be v km/h. The speed of the current is 1.5 km/h.

Downstream speed = v + 1.5 Upstream speed = v - 1.5

Since the downstream and upstream distances are equal: (v + 1.5) × 3 = (v - 1.5) × 3.5

Expanding the terms: 3v + 4.5 = 3.5v - 5.25

Rearranging the equation to solve for v: 3.5v - 3v = 4.5 + 5.25 0.5v = 9.75 v = 19.5

Explanation

By Viviani's theorem, the sum of the perpendicular distances from any interior point to the sides of an equilateral triangle is always equal to the altitude of the triangle.

Let the side length of the equilateral triangle ABC be s, and let AX be the altitude to the side BC. The area of triangle ABC is given by ½ × s × AX.

Let P be any point inside the triangle, and let the perpendicular distances from P to the sides AB, BC, and CA be p₁, p₂, and p₃ respectively. The total area of the triangle ABC is the sum of the areas of the three smaller triangles PAB, PBC, and PCA:

Area(ABC) = Area(PAB) + Area(PBC) + Area(PCA)

½ × s × AX = ½ × s × p₁ + ½ × s × p₂ + ½ × s × p₃

Dividing both sides of the equation by ½ × s yields:

AX = p₁ + p₂ + p₃

This shows that the sum of the perpendicular distances from any point inside the triangle to its sides is equal to AX.

Explanation

Let the lengths of the sides of the triangle be represented as 7x, 9x, and 12x.

The smallest side is 7x and the largest side is 12x. The difference between these two sides is: 12x - 7x = 5x

According to the problem, this difference is 15 cm: 5x = 15 x = 3

The length of the largest side is 12x. Substituting the value of x: 12 × 3 = 36 cm.

Explanation

The time duration for which the money is invested is 18 - 12 = 6 years.

The annual interest earned on the principal amount of ₹100,000 at 6% simple interest per annum is: Interest = 100,000 × 0.06 = ₹6000

The total annual deductions from this interest include the boy's pocket money and the trust expenses: Annual deductions = ₹2500 + ₹500 = ₹3000

The net annual amount added back to the trust is the difference between the annual interest earned and the annual deductions: Net annual addition = ₹6000 - ₹3000 = ₹3000

Over the period of 6 years, the total accumulated addition is: Total addition = 6 × 3000 = ₹18,000

The final amount handed over to the boy when he turns 18 is: Final amount = Initial principal + Total addition = 100,000 + 18,000 = ₹118,000

Explanation

In any triangle ABC, the angle between the altitude (perpendicular AE) and the angle bisector (AD) from the vertex A to the opposite side BC is given by the direct formula:

∠EAD = ½ |∠B - ∠C|

Substituting the given values:

∠EAD = ½ |60° - 40°| = ½ × 20° = 10°

Explanation

The total sum of all 13 results is calculated by multiplying the number of results by their average: 13 × 70 = 910

The sum of the first seven results is: 7 × 65 = 455

The sum of the last seven results is: 7 × 75 = 525

Since the seventh result is included in both the first seven and the last seven results, adding these two sums together includes the seventh result twice. The sum of these 14 results is: 455 + 525 = 980

To find the seventh result, subtract the sum of the 13 results from this combined sum: 980 - 910 = 70

Explanation

Using the work formula (M₁ × D₁) / W₁ = (M₂ × D₂) / W₂, we can determine the required number of labours to complete the remaining work in the stipulated time.

The initial parameters are: Number of initial labours (M₁) = 20 Days worked (D₁) = 12 Work completed (W₁) = 5/8

The remaining parameters are: Remaining work (W₂) = 1 - 5/8 = 3/8 Remaining days (D₂) = 16 - 12 = 4 Total labours required (M₂) = 20 + x, where x is the number of extra labours.

Substituting these values into the formula: (20 × 12) / (5/8) = (M₂ × 4) / (3/8)

Cancelling the denominator 8 from both sides: (20 × 12) / 5 = (M₂ × 4) / 3

Simplifying the equation: 4 × 12 = 4M₂ / 3 48 = 4M₂ / 3 M₂ = (48 × 3) / 4 M₂ = 36

Since the total number of labours needed to complete the work on time is 36 and 20 labours are already employed, the number of extra labours required is: 36 - 20 = 16

Explanation

To compare decimal values, it is easiest to represent them with the same number of decimal places: p = -0.120 q = -0.010 r = -0.015

Comparing their absolute values: 0.120 > 0.015 > 0.010

For negative numbers, a larger absolute value corresponds to a smaller algebraic value. Therefore, reversing the inequality gives: -0.120 < -0.015 < -0.010

Substituting the variables back into the inequality yields: p < r < q

Explanation

Based on the monthly family expenditure pie chart, the central angles representing the respective expenses are: Central angle for money spent on food = 150° Central angle for money spent on house rent = 120°

To find the percentage by which the money spent on food exceeds the money spent on house rent, use the percentage increase formula: Required percentage = ((Angle for Food - Angle for Rent) / Angle for Rent) × 100 Required percentage = ((150 - 120) / 120) × 100 = (30 / 120) × 100 = ¼ × 100 = 25%

Explanation

The pie chart represents the distribution of a family's monthly expenditure across various categories in terms of percentages or degrees. To calculate the absolute monetary value spent on specific categories like clothes and miscellaneous expenses, the total monthly expenditure of the family must be known. Since the total value of the family's expenditure or income is not provided, it is impossible to determine the precise monetary amount spent on these items.

Explanation

The family monthly expenditure pie chart indicates that the central angle corresponding to the expenditure on food is 150°. Since a complete circle represents 360°, the proportion of the total expenditure spent on food is the ratio of these two angles.

Amount spent on food = (Central angle of food ÷ 360°) × Total amount spent

Amount spent on food = (150° ÷ 360°) × 7,200 = 5/12 × 7,200 = 3,000

Explanation

Given the quadratic equation: x² - 4x - 1 = 0

Since x is non-zero, dividing both sides of the equation by x gives: x - 4 - 1/x = 0 x - 1/x = 4

Squaring both sides of this equation yields: (x - 1/x)² = 4² x² - 2 + 1/x² = 16 x² + 1/x² = 16 + 2 x² + 1/x² = 18

Explanation

The area of a trapezium is calculated using the formula:

Area = ½ × (a + b) × h

where a and b are the lengths of the parallel sides, and h is the height.

Substituting the given values into the formula:

175 = ½ × (15 + 20) × h

175 = ½ × 35 × h

350 = 35 × h

h = 350 / 35

h = 10 cm