SSC CHSL Tier 1 2013 Quantitative Aptitude Questions with Solutions

Explanation

Using the trigonometric double-angle identity cos 2θ = cos²θ − sin²θ, the given equation can be written as:

2 cos 2θ = 1

cos 2θ = ½

For a positive acute angle θ, 2θ lies between 0° and 180°. Since cos 60° = ½, we have:

2θ = 60°

θ = 30°

Now, calculating the value of cot θ:

cot θ = cot 30° = √3

Explanation

A system of two linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has infinitely many solutions when the two lines are coincident, which occurs when the coefficients satisfy the condition: a₁/a₂ = b₁/b₂ = c₁/c₂

Rewriting the given equations in the form ax + by + c = 0: First equation: 4x - y - 2 = 0, where a₁ = 4, b₁ = -1, and c₁ = -2 Second equation: -8x + 2y + 4 = 0, where a₂ = -8, b₂ = 2, and c₂ = 4

Calculating the ratios of the coefficients: a₁/a₂ = 4 / -8 = -½ b₁/b₂ = -1 / 2 = -½ c₁/c₂ = -2 / 4 = -½

Since the ratios are equal (a₁/a₂ = b₁/b₂ = c₁/c₂ = -½), the two equations represent the exact same line. Consequently, every point on the line is a solution, giving the system infinitely many solutions.

Explanation

Given equation: a² + b² + c² = 2(a - b - c) - 3

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

Splitting the constant 3 into 1 + 1 + 1 to group the terms into 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 is zero, each individual squared term must be equal to zero: a - 1 = 0 → a = 1 b + 1 = 0 → b = -1 c + 1 = 0 → c = -1

Substituting these values into the required expression: 2a - 3b + 4c = 2(1) - 3(-1) + 4(-1) 2a - 3b + 4c = 2 + 3 - 4 = 1

Explanation

By calculating the approximate decimal values, we have a = 0 (or if it is a typo for √6 - √5, then a is approximately 0.213), b = √5 - 2 which is approximately 0.236, and c = 2 - √3 which is approximately 0.268. Comparing these values gives the relationship a < b < c, which corresponds to option A.

Explanation

The given algebraic relationship is: a = b² / (b - a)

Multiplying both sides by (b - a): a(b - a) = b² ab - a² = b²

Rearranging the terms of the equation yields: a² - ab + b² = 0

The algebraic expansion for the sum of cubes is: a³ + b³ = (a + b)(a² - ab + b²)

Substituting the value of a² - ab + b² = 0 into the expansion: a³ + b³ = (a + b) × 0 = 0

Explanation

Given the condition xy + yz + zx = 0, we can express -yz as xy + zx.

Adding x² to both sides gives: x² - yz = x² + xy + zx = x(x + y + z)

By symmetry, we can write the other denominators in the same form: y² - zx = y(x + y + z) z² - xy = z(x + y + z)

Substituting these expressions into the given algebraic sum: 1/(x² - yz) + 1/(y² - zx) + 1/(z² - xy) = 1/(x(x + y + z)) + 1/(y(x + y + z)) + 1/(z(x + y + z)) = (1 / (x + y + z)) × (1/x + 1/y + 1/z) = (1 / (x + y + z)) × ((yz + zx + xy) / xyz)

Since xy + yz + zx = 0, the term (yz + zx + xy) in the numerator is 0.

Therefore, the entire expression simplifies to: (1 / (x + y + z)) × (0 / xyz) = 0

Explanation

When two people A and B can complete a work individually in a and b days respectively, the time taken by them to complete the work working together is given by the formula:

Time = (a × b) / (a + b)

Substituting a = 20 and b = 30 into the formula:

Time = (20 × 30) / (20 + 30) Time = 600 / 50 Time = 12 days

Explanation

The volume of a room is calculated using the formula Volume = Floor Area × Height. By rearranging this formula to solve for the floor area, we get Floor Area = Volume / Height. Substituting the given volume of 204 m³ and the height of 6 m into this equation yields Floor Area = 204 m³ / 6 m = 34 m².

Explanation

The total surface area of a cube with side length a is given by the formula 6a².

6a² = 96 a² = 16 a = 4 cm

The volume of a cube is given by the formula a³. Substituting the value of the side length:

Volume = 4³ = 64 cm³

Explanation

Let the radius of the circle be r. Since the length of OP is equal to the diameter of the circle, we have OP = 2r.

The tangent PA is perpendicular to the radius OA at the point of contact A, making triangle OAP a right-angled triangle at A.

In the right-angled triangle OAP: sin(∠APO) = OA / OP = r / 2r = ½

Since sin(30°) = ½, the angle ∠APO is equal to 30°.

By the symmetry of tangents drawn from an external point, the line OP bisects the angle ∠APB. Therefore, ∠BPO is also equal to 30°.

The total angle ∠APB is the sum of ∠APO and ∠BPO, which is 30° + 30° = 60°.

Explanation

The radius (r) of the circle is half of its diameter: r = 20/2 = 10 cm

A perpendicular drawn from the centre of a circle to a chord bisects the chord. Thus, the perpendicular divides the 12 cm chord into two equal segments of length: 12/2 = 6 cm

This perpendicular, the radius, and half of the chord form a right-angled triangle. Applying the Pythagorean theorem: Radius² = Distance² + (Half of chord)² 10² = Distance² + 6² 100 = Distance² + 36 Distance² = 100 - 36 = 64 Distance = √64 = 8 cm

Explanation

For two similar triangles, the ratio of their areas is equal to the ratio of the squares of their corresponding sides. This relation is expressed as:

Area₁/Area₂ = (Side₁/Side₂)²

By substituting the given values:

360/250 = (8/x)²

Simplifying the fraction on the left:

36/25 = 64/x²

Taking the square root on both sides yields:

6/5 = 8/x

Cross-multiplying to solve for x:

6x = 40 x = 40/6 x = 20/3 x = 6 ⅔ cm

Explanation

According to the angle sum property of a triangle, the sum of all interior angles of a triangle is always equal to 180°.

Let the measure of ∠ACB be x.

Given the relations: ∠ABC = 5x ∠BAC = 3x

Using the angle sum formula: ∠ABC + ∠BAC + ∠ACB = 180°

Substitute the values of the angles into the equation: 5x + 3x + x = 180° 9x = 180° x = 20°

The measure of ∠ABC is: ∠ABC = 5x = 5 × 20° = 100°

Explanation

The perpendicular line drawn from a vertex of a triangle to its opposite side is known as an altitude. In Euclidean geometry, the three altitudes of any triangle always intersect at a single common point of concurrency. This unique point where these three perpendiculars meet is called the orthocentre.

Explanation

The formula for the amount under compound interest is: A = P × (1 + R/100)ⁿ

Substituting the given values of amount, rate, and time: 2916 = P × (1 + 8/100)² 2916 = P × (1 + 2/25)² 2916 = P × (27/25)² 2916 = P × 729/625 P = (2916 × 625) / 729 P = 4 × 625 P = 2500

So, the principal amount is ₹2500.

Now, we calculate the simple interest on this principal at 9% per annum for 3 years using the simple interest formula: SI = (P × R × T) / 100 SI = (2500 × 9 × 3) / 100 SI = 25 × 27 SI = 675

Explanation

The weighted average formula is used to find the average of all the numbers combined:

Weighted Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂)

Here, the first group has n₁ = 30 numbers with an average A₁ = 40, and the second group has n₂ = 40 numbers with an average A₂ = 30.

Sum of the first 30 numbers = 30 × 40 = 1200 Sum of the other 40 numbers = 40 × 30 = 1200

Total sum of all 70 numbers = 1200 + 1200 = 2400 Total count of numbers = 30 + 40 = 70

Average of all the numbers = 2400 ÷ 70 = 240/7

Converting this improper fraction into a mixed fraction: 240 ÷ 7 = 34 with a remainder of 2, which gives 34 2/7.

Explanation

Evaluate the expression step-by-step from the innermost radical outward:

First, evaluate the innermost square root: √81 = 9

Substitute this value back into the expression: √(40 + √(9 × 9))

Next, simplify the term under the middle radical: √(9 × 9) = √81 = 9

Substitute this value into the main expression: √(40 + 9)

Finally, simplify the remaining addition and find the square root: √49 = 7

Explanation

Let the present ages of the two boys be 5x and 6x.

According to the given condition, the ratio of their ages after two years is 7:8. This can be written as:

(5x + 2) / (6x + 2) = 7/8

Cross-multiplying the terms gives:

8(5x + 2) = 7(6x + 2)

40x + 16 = 42x + 14

2x = 2

x = 1

The present ages of the two boys are 5 years and 6 years.

After 12 years, their ages will be:

First boy: 5 + 12 = 17 years

Second boy: 6 + 12 = 18 years

The ratio of their ages after 12 years will be 17/18.

Explanation

Let the two parts of 50 be x and y.

According to the given conditions: x + y = 50

The sum of their reciprocals is: 1/x + 1/y = 1/12

Combining the fractions: (x + y) / xy = 1/12

Substituting x + y = 50 into the equation: 50 / xy = 1/12 xy = 50 × 12 xy = 600

We need two numbers whose sum is 50 and product is 600. This can be represented by the quadratic equation: t² - 50t + 600 = 0 (t - 20)(t - 30) = 0

Solving for t yields t = 20 and t = 30. Thus, the two parts are 20 and 30.

Explanation

Let the original length of the rectangle be L and the breadth be B. The original area is A₁ = L × B.

When both the length and breadth are doubled, the new length becomes 2L and the new breadth becomes 2B. The new area is A₂ = 2L × 2B = 4 × L × B = 4A₁.

The increase in area is A₂ - A₁ = 4A₁ - A₁ = 3A₁.

The percentage increase in area is calculated as (Increase in Area / Original Area) × 100% = (3A₁ / A₁) × 100% = 300%.

Explanation

Let the marks obtained by Supriyo be S. Since Mahuya got 10% less marks than Supriyo, Mahuya's marks represent 90% of Supriyo's marks.

We can express this relation as: 0.90 × S = 81

Solving for S: S = 81 / 0.90 S = 90

Supriyo scored 90 marks in Mathematics.

Explanation

Since the distance covered in both cases remains constant, the relation between speed and time is:

s₁ × t₁ = s₂ × t₂

Substituting the given values into the formula:

80 × 4.5 = s₂ × 4

s₂ = (80 × 4.5) / 4

s₂ = 20 × 4.5

s₂ = 90 km/h

Explanation

Using the weighted average formula, the cost price (CP) of 100 gm of the blended tea is: CP = (7 × 18 + 3 × 13) / (7 + 3) CP = (126 + 39) / 10 CP = 165 / 10 = ₹16.5

The selling price (SP) of the blended tea is ₹18.15 per 100 gm.

The percentage gain is calculated using the formula: Percentage Gain = ((SP - CP) / CP) × 100% Percentage Gain = ((18.15 - 16.5) / 16.5) × 100% Percentage Gain = (1.65 / 16.5) × 100% Percentage Gain = 10%

Explanation

Profit ratio of partners in a partnership is given by the formula: Profit of A / Profit of B = (Investment of A × Time period of A) / (Investment of B × Time period of B)

Let B join the business after x months. Since the total period is one year, B's money was invested for (12 - x) months, while A's money was invested for 12 months.

Substituting the given values: 2 / 1 = (64000 × 12) / (48000 × (12 - x))

Simplifying the equation: 2 = (64 × 12) / (48 × (12 - x)) 2 = 768 / (48 × (12 - x)) 96 × (12 - x) = 768 12 - x = 768 / 96 12 - x = 8 x = 12 - 8 x = 4

Therefore, B joined the business after 4 months.

Explanation

Let the label price of the article be L.

A bought the article at a 30% concession, making A's Cost Price (CP): CP = 0.70 × L

A sold the article for ₹8,750 with a 25% profit on his CP: Selling Price (SP) = CP × (1 + 0.25) 8,750 = 1.25 × CP

Calculating the Cost Price: CP = 8,750 ÷ 1.25 CP = 7,000

Substituting CP back to find the label price: 7,000 = 0.70 × L L = 7,000 ÷ 0.70 L = 10,000

Explanation

Using the selling price formula:

Selling Price = Cost Price × (1 + Gain/100)

Substituting Cost Price = ₹150 and Gain = 20%:

Selling Price = 150 × (1 + 20/100) = 150 × 1.2 = 180

Explanation

To find the greatest possible profit, we need to maximize the difference between the selling price and the cost price of each book. This is achieved by purchasing each book at the lowest possible cost price and selling each book at the highest possible selling price.

The given range for the cost price of a book is ₹150 to ₹300. Thus, the minimum cost price of one book is ₹150.

The given range for the selling price of a book is ₹250 to ₹350. Thus, the maximum selling price of one book is ₹350.

The maximum profit that can be made on a single book is: Maximum Profit per book = Maximum Selling Price - Minimum Cost Price Maximum Profit per book = ₹350 - ₹150 = ₹200

For 15 books, the greatest possible profit is: Total Greatest Profit = 15 × ₹200 = ₹3,000

Explanation

Let the ten's digit of the original number be t and the unit's digit be u.

Based on the first condition, the unit's digit is 1 less than twice the ten's digit: u = 2t - 1

The original number can be written as: 10t + u = 10t + (2t - 1) = 12t - 1

When the digits are interchanged, the new number becomes: 10u + t = 10(2t - 1) + t = 21t - 10

The difference between the new and the original number is: (21t - 10) - (12t - 1) = 9t - 9

We are given that this difference is less than the original number by 20: (12t - 1) - (9t - 9) = 20 3t + 8 = 20 3t = 12 t = 4

Substituting t = 4 into the unit's digit equation: u = 2(4) - 1 = 7

This gives the original number as 47.

We can easily verify this: the new number is 74. The difference between the new and original number is 74 - 47 = 27. This difference is exactly 20 less than the original number, as 47 - 20 = 27.

Explanation

The volume of a right prism is calculated using the formula: Volume = Area of the base × height

The base of this prism is a triangle. The area of a triangle with a given inradius (r) and semi-perimeter (s) is given by the formula: Area of base = r × s

The perimeter of the triangle is 28 cm, so its semi-perimeter is: s = 28 / 2 = 14 cm

Using the given inradius of 4 cm, the area of the base is: Area of base = 4 × 14 = 56 cm²

Now, substituting the volume and the base area into the volume formula of the prism: 366 = 56 × height height = 366 / 56 = 183 / 28 ≈ 6.54 cm

Since this height does not match 4 cm, 8 cm, or 6 cm, the correct option is indeed None of these.

Explanation

For any real angle θ, the value of cos²θ satisfies the inequality: cos²θ ≤ 1

Substituting the given expression into this inequality: (x + y)²/(4xy) ≤ 1

Assuming x and y have the same sign so that xy > 0, we can multiply both sides by 4xy: (x + y)² ≤ 4xy

Expanding the left side: x² + 2xy + y² ≤ 4xy

Subtracting 4xy from both sides: x² - 2xy + y² ≤ 0

This simplifies to: (x - y)² ≤ 0

Since the square of any real number is always non-negative, the inequality (x - y)² ≤ 0 can only be satisfied when: (x - y)² = 0

Solving this equation gives: x = y

Explanation

To compare ∛2 and √3, raise both quantities to the sixth power, which is the least common multiple of their root indices 3 and 2.

Evaluating ∛2 raised to the power of 6: (2⁽1/3))⁶ = 2² = 4

Evaluating √3 raised to the power of 6: (3⁽1/2))⁶ = 3³ = 27

Comparing the results, 27 is greater than 4. Since both original numbers are positive, the square root of 3 is greater than the cube root of 2.

Explanation

By the Cauchy-Schwarz inequality for real numbers a, b, and c: (a² + b² + c²)(1² + 1² + 1²) ≥ (a + b + c)²

Substituting the given value a + b + c = 9: 3(a² + b² + c²) ≥ 9² 3(a² + b² + c²) ≥ 81 a² + b² + c² ≥ 27

The equality holds when a = b = c = 3, yielding the minimum value of 27.

Explanation

The least common multiple (LCM) of a set of fractions is determined using the standard relation:

LCM of fractions = LCM of Numerators ÷ HCF of Denominators

For the given fractions 2/3, 4/9, 5/6, and 2/18:

The numerators are 2, 4, 5, and 2. The LCM of the numerators is LCM(2, 4, 5, 2) = 20.

The denominators are 3, 9, 6, and 18. The HCF of the denominators is HCF(3, 9, 6, 18) = 3.

Substituting these values into the relation:

LCM of fractions = 20/3

Explanation

We simplify the given expression step by step. Let the expression be: E = 3 + 1/√3 + 1/(√3 + 3) + 1/(√3 - 3)

First, combine the third and fourth terms by finding a common denominator: 1/(√3 + 3) + 1/(√3 - 3) = ((√3 - 3) + (√3 + 3)) / ((√3 + 3)(√3 - 3))

Simplify the numerator and the denominator: Numerator: √3 - 3 + √3 + 3 = 2√3 Denominator: (√3)² - 3² = 3 - 9 = -6

This gives: 2√3 / -6 = -√3 / 3 = -1 / √3

Now, substitute this result back into the original expression: E = 3 + 1/ √3 - 1/ √3 E = 3

Explanation

When a divisor a divides 228 leaving a remainder of 18, we can express this relation as: 228 = a × q + 18

Here, q is the quotient and the divisor a must be strictly greater than the remainder, meaning a > 18.

Rearranging the equation: a × q = 228 - 18 a × q = 210

This shows that a must be a factor of 210. The two-digit factors of 210 are 10, 14, 15, 21, 30, 35, 42, and 70. The largest two-digit factor of 210 is 70, which is also greater than 18, making it the highest possible two-digit value for a.

Explanation

Any integer lying between 100 and 1000 is a three-digit number. Any three-digit number can be represented algebraically as 100a + 10b + c, where a, b, and c are the digits in the hundreds, tens, and units places respectively.

The sum of the digits of this number is a + b + c.

Subtracting the sum of the digits from the original number gives: (100a + 10b + c) - (a + b + c) = 99a + 9b = 9(11a + b)

Since 11a + b is an integer, the resulting expression 9(11a + b) is always a multiple of 9, which means the result is always divisible by 9.

Explanation

Let the required number be x.

The condition that the number is as much greater than 31 as it is less than 75 means that the difference between the number and 31 is equal to the difference between 75 and the number. This is represented by the formula for the arithmetic mean (midpoint) of two numbers:

x = (31 + 75) / 2

Calculating the value:

x = 106 / 2 x = 53

Explanation

Using the algebraic identity for the cube of a binomial: (x - y)³ = x³ - y³ - 3xy(x - y)

Substitute the given condition x - y = 1 into this identity: 1³ = x³ - y³ - 3xy(1) 1 = x³ - y³ - 3xy

The expression to evaluate is: x³ - y³ - 3(xy - 2)

Expanding this expression gives: x³ - y³ - 3xy + 6

Substituting the relation x³ - y³ - 3xy = 1 into the expanded expression yields: 1 + 6 = 7

Alternatively, substituting a simple set of values satisfying x - y = 1, such as y = 0 and x = 1, gives: 1³ - 0³ - 3(0 - 2) = 1 + 6 = 7

Explanation

The marked price of the book is 20% above the cost price (CP). Therefore, the relationship between marked price and cost price is given by the formula: Marked Price = CP × (1 + Markup Percentage)

Substituting the given values: 180 = CP × (1 + 0.20) 180 = 1.20 × CP CP = 180 / 1.20 = 150

The total cost price of publishing a single copy of the book is ₹150.

According to the expenditure chart, the cost of paper accounts for 15% of the total cost price of publishing the book.

Cost of paper = 15% of CP Cost of paper = 0.15 × 150 = 22.50

Explanation

Let the number be x. It is given that 20% of this number is 120: 0.20 × x = 120 x = 120 ÷ 0.20 = 600

To find 120% of this number: 1.20 × 600 = 720

Alternatively, since percentage is directly proportional to the value, if 20% corresponds to 120, then 120% corresponds to: 120 × (120 ÷ 20) = 120 × 6 = 720

Explanation

According to the division algorithm, the relation between the components of a division sum is: Dividend = (Divisor × Quotient) + Remainder

From the given information: Remainder = 2

Since the divisor is 6 times the remainder: Divisor = 6 × 2 = 12

Since the divisor is also 3 times the quotient: Quotient = Divisor ÷ 3 = 12 ÷ 3 = 4

Substituting these values into the division formula: Dividend = (12 × 4) + 2 = 48 + 2 = 50

Explanation

Write the mixed number as 999 plus 692/693, so the bracket becomes (999 x 693 + 692)/693, which simplifies to 692999/693. Since 693 equals 7 times 99, multiplying this by 99 cancels the 99 and leaves 692999/7. Adding the 1/7 outside the bracket gives 693000/7, which works out exactly to 99000.

Explanation

To find the range of a, we can estimate the value of its cube root: ∛a = √26 + √7 + √63

We can lower-bound each term under the square root using perfect squares: √26 > √25 = 5 √7 > √4 = 2 √63 > √49 = 7

Adding these inequalities gives: ∛a > 5 + 2 + 7 = 14

Since 14 is greater than 9, it follows that: ∛a > 9

Cubing both sides of the inequality yields: a > 9³ a > 729

Explanation

To find the total number of prime factors in the product 6³³³ × 7²²² × 8¹¹¹, we first express each base as a product of its prime factors.

The base 6 can be factorized into prime numbers as 2 × 3. The base 7 is already a prime number. The base 8 can be factorized into prime numbers as 2³.

Substituting these prime factorizations back into the expression gives:

(2 × 3)³³³ × 7²²² × (2³)¹¹¹

Applying the laws of exponents, we expand the terms:

2³³³ × 3³³³ × 7²²² × 2³³³

Combining the powers of the base 2:

2⁶⁶⁶ × 3³³³ × 7²²²

The total number of prime factors is the sum of the exponents of all the prime bases in this prime factorization:

666 + 333 + 222 = 1221

Explanation

To find the value of (x + y + z) / (3z), first express the ratios with a common term for y.

Given: x : y = 3 : 4 y : z = 3 : 4

To align the ratio of y, multiply the terms of the first ratio by 3 and the second ratio by 4: x : y = 9 : 12 y : z = 12 : 16

This gives the combined ratio: x : y : z = 9 : 12 : 16

Let x = 9k, y = 12k, and z = 16k, where k is a non-zero constant. Substituting these values into the given expression:

(x + y + z) / (3z) = (9k + 12k + 16k) / (3 × 16k) = 37k / 48k = 37/48

Explanation

To find the value of the given expression, we simplify each square root in the numerator and denominator by factoring out the perfect squares:

Numerator: √72 = √(36 × 2) = 6√2 √363 = √(121 × 3) = 11√3 √175 = √(25 × 7) = 5√7

Multiplying these terms together: 6√2 × 11√3 × 5√7 = (6 × 11 × 5) × √(2 × 3 × 7) = 330√42

Denominator: √32 = √(16 × 2) = 4√2 √147 = √(49 × 3) = 7√3 √252 = √(36 × 7) = 6√7

Multiplying these terms together: 4√2 × 7√3 × 6√7 = (4 × 7 × 6) × √(2 × 3 × 7) = 168√42

Dividing the simplified numerator by the simplified denominator: (330√42) / (168√42) = 330 / 168

Dividing both the numerator and the denominator by their greatest common divisor, which is 6: 330 ÷ 6 = 55 168 ÷ 6 = 28

This simplifies exactly to 55/28.

Explanation

In the initial mixture of 400 gm, the ratio of zinc to copper is 5:3, which gives a total of 5 + 3 = 8 parts.

The weight of 1 part is 400 ÷ 8 = 50 gm.

When copper is added to make the ratio 5:4, the zinc parts remain unchanged at 5, while the copper parts increase from 3 to 4.

The increase in copper is 4 - 3 = 1 part.

Since 1 part corresponds to 50 gm, the amount of copper that must be added is 50 gm.

Explanation

From the given averages, we can formulate equations for the sum of the ages: A + B = 20 × 2 = 40 B + C = 19 × 2 = 38 C + A = 21 × 2 = 42

Adding these three equations together: 2(A + B + C) = 40 + 38 + 42 = 120 A + B + C = 60

By subtracting each two-person sum from the total sum of 60: A = 60 - (B + C) = 60 - 38 = 22 years B = 60 - (C + A) = 60 - 42 = 18 years C = 60 - (A + B) = 60 - 40 = 20 years

Explanation

The original price P of a commodity when a reduction of r% in price enables a person to buy q kg more for an expenditure E is given by the formula:

P = (r × E) ÷ ((100 - r) × q)

Substituting the given values: r = 10 E = 225 q = 25

P = (10 × 225) ÷ ((100 - 10) × 25) P = 2250 ÷ (90 × 25) P = 2250 ÷ 2250 P = 1

Explanation

Using the weighted average formula, the percentage of water in the mixture is calculated as:

Water percentage = (w₁p₁ + w₂p₂) / (w₁ + w₂)

where w₁ and w₂ represent the parts of each liquid, and p₁ and p₂ represent their respective water percentages.

Substituting the given values:

w₁ = 10, p₁ = 20% w₂ = 4, p₂ = 35%

Water percentage = (10 × 20 + 4 × 35) / (10 + 4) Water percentage = (200 + 140) / 14 Water percentage = 340 / 14 Water percentage = 170 / 7 Water percentage = 24 2/7 %