Given the equation C = 2πr + b, make b the subject. A. b = C – 2πr, B. b = C + 2πr, C. b = 2πr (C – r) D. b = C/2πr

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Asked on July 30, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-07-30T08:57:45.807Z

here are the solutions to the mathematics questions: SECTION A: OBJECTIVE TEST QUESTIONS 1. The number 709,045 in words is "Seven hundred nine thousand, forty five". A) Seven hundred nine thousand, forty five 2. In the number 28.356, the digit 5 is in the hundredths place. C) Hundredths 3. To express 0.00205 in standard form, move the decimal point 3 places to the right. This gives 2.05 × 10^-3. D) 2.05 × 10^-3 4. To convert 101101_2 to base ten: 1 × 2^5 + 0 × 2^4 + 1 × 2^3 + 1 × 2^2 + 0 × 2^1 + 1 × 2^0 = 1 × 32 + 0 × 16 + 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1 = 32 + 0 + 8 + 4 + 0 + 1 = 45 A) 45 5. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. 2 is prime. 17 is prime. 39 is not prime because 39 = 3 × 13. 1 is not considered a prime number. In this context, 39 is the composite number, and thus not prime. C) 39 6. To approximate 14.897 to 2 decimal places, look at the third decimal place (7). Since it is 5 or greater, round up the second decimal place. The 9 in the hundredths place rounds up to 10, so the 8 in the tenths place becomes 9, and the hundredths place becomes 0. 14.897 ≈ 14.90 B) 14.90 7. To find the L.C.M of 9, 15, and 20: Prime factorization: 9 = 3^2 15 = 3 × 5 20 = 2^2 × 5 L.C.M = 2^2 × 3^2 × 5 = 4 × 9 × 5 = 180 D) 180 8. Simplify ((1)/(2) + (1)/(3)) ÷ ((1)/(2) - (1)/(3)): First, calculate the sum: (1)/(2) + (1)/(3) = (3)/(6) + (2)/(6) = (5)/(6) Next, calculate the difference: (1)/(2) - (1)/(3) = (3)/(6) - (2)/(6) = (1)/(6) Now, divide: (5/6)/(1/6) = (5)/(6) × (6)/(1) = 5 A) 5 9. To express 15 m as a percentage of 1.20 km: Convert 1.20 km to meters: 1.20 km = 1.20 × 1000 m = 1200 m Percentage = 15 m1200 m × 100\% Percentage = (15)/(1200) × 100 = (15)/(12) = (5)/(4) = 1.25\% A) 1.25% 10. To arrange (1)/(3), (2)/(5), (3)/(4), (1)/(2) in ascending order, convert them to decimals: (1)/(3) ≈ 0.333 (2)/(5) = 0.4 (3)/(4) = 0.75 (1)/(2) = 0.5 In ascending order: 0.333, 0.4, 0.5, 0.75, which corresponds to (1)/(3), (2)/(5), (1)/(2), (3)/(4). A) (1)/(3), (2)/(5), (1)/(2), (3)/(4) 11. The perimeter of a rectangle with length 15 cm and width 8 cm is: Perimeter = 2 × (length + width) Perimeter = 2 × (15 cm + 8 cm) = 2 × 23 cm = 46 cm C) 46 cm 12. The area of a triangle is given by (1)/(2) × base × height. Given Area = 60 cm^2 and height = 12 cm. 60 = (1)/(2) × base × 12 60 = 6 × base base = (60)/(6) = 10 cm B) 10 cm 13. The total surface area of a cube with side length 3 cm is: TSA = 6 × (side)^2 TSA = 6 × (3 cm)^2 = 6 × 9 cm^2 = 54 cm^2 A) 54 cm^2 14. A triangular prism has 2 triangular faces, 3 rectangular faces, 6 vertices, and 9 edges. C) 9 15. A parallelogram, in general, has no lines of symmetry. A) 0 16. Simplify (a^2b^3 - ab^2)/(ab(b-1)) Factor out ab^2 from the numerator: ab^2(ab-1) The expression becomes (ab^2(ab-1))/(ab(b-1)) = (b(ab-1))/(b-1). Note: Assuming there might be a typo in the question and the denominator was intended to be ab(ab-1) or a^2b^2 - ab, the expression would simplify to b. Given the options, this is the most likely intended answer if a typo exists.* If the denominator was a^2b^2 - ab, then (ab^2(ab-1))/(ab(ab-1)) = b. B) b 17. To find the coefficient of x in the expansion of (2x-3)(x-5): Expand the expression: (2x-3)(x-5) = 2x(x) + 2x(-5) - 3(x) - 3(-5) = 2x^2 - 10x - 3x + 15 = 2x^2 - 13x + 15 The coefficient of x is -13. C) -13 SECTION B: THEORY 1. (a) (i) Simplify 2(1)/(2) + 1(1)/(4) - (1)/(10) Step 1: Convert mixed numbers to improper fractions. 2(1)/(2) = (2 × 2 + 1)/(2) = (5)/(2) 1(1)/(4) = (1 × 4 + 1)/(4) = (5)/(4) Step 2: Rewrite the expression with improper fractions. (5)/(2) + (5)/(4) - (1)/(10) Step 3: Find the least common multiple (LCM) of the denominators (2, 4, 10), which is 20. Step 4: Convert each fraction to an equivalent fraction with the denominator 20 and perform the operations. (5 × 10)/(2 × 10) + (5 × 5)/(4 × 5) - (1 × 2)/(10 × 2) = (50)/(20) + (25)/(20) - (2)/(20) = (50 + 25 - 2)/(20) = (75 - 2)/(20) = (73)/(20) Step 5: Convert the improper fraction back to a mixed number. (73)/(20) = 3(13)/(20) The simplified expression is 3(13)/(20). 1. (a) (ii) Convert 73 to a binary number. Step 1: Divide 73 by 2 and record the remainder. 73 ÷ 2 = 36 remainder 1 Step 2: Continue dividing the quotient by 2 until the quotient is 0. 36 ÷ 2 = 18 remainder 0 18 ÷ 2 = 9 remainder 0 9 ÷ 2 = 4 remainder 1 4 ÷ 2 = 2 remainder 0 2 ÷ 2 = 1 remainder 0 1 ÷ 2 = 0 remainder 1 Step 3: Read the remainders from bottom to top. The binary number is 1001001_2. 1. (b) Solve the simultaneous equations: 2x + 3y = 12; x - y = 1. Step 1: Label the equations. 2x + 3y = 12 (Equation 1) x - y = 1 (Equation 2) Step 2: From Equation 2, express x in terms of y. x = y + 1 (Equation 3) Step 3: Substitute Equation 3 into Equation 1. 2(y + 1) + 3y = 12 2y + 2 + 3y = 12 Step 4: Combine like terms and solve for y. 5y + 2 = 12 5y = 12 - 2 5y = 10 y = (10)/(5) = 2 Step 5: Substitute the value of y back into Equation 3 to find x. x = 2 + 1 = 3 The solution is x=3, y=2. 2. (a) N45,000.00 was shared by Aisha, Bolanle and Chidima in the ratio 2:3:4. How much did each person get? Step 1: Find the total number of ratio parts. Total parts = 2 + 3 + 4 = 9 Step 2: Calculate Aisha's share. Aisha's share = (2)/(9) × N45,000 = 2 × N5,000 = N10,000 Step 3: Calculate Bolanle's share. Bolanle's share = (3)/(9) × N45,000 = 3 × N5,000 = N15,000 Step 4: Calculate Chidima's share. Chidima's share = (4)/(9) × N45,000 = 4 × N5,000 = N20,000 Aisha received N10,000, Bolanle received N15,000, and Chidima received N20,000. 2. (b) If 5 men can build a wall in 12 days, how many days will it take 8 men working at the same rate? Step 1: Recognize that this is an inverse proportion problem (more men means fewer days). Let M_1 = 5 men, D_1 = 12 days. Let M_2 = 8 men, D_2 = ? days. Step 2: Use the inverse proportion formula M_1 D_1 = M_2 D_2. 5 men × 12 days = 8 men × D_2 days 60 = 8 D_2 Step 3: Solve for D_2. D_2 = (60)/(8) = (15)/(2) = 7.5 It will take 8 men 7.5 days to build the wall. 3. (a) The ages of 20 students in a JSS3 class are 12, 13, 14, 12, 15, 13, 14, 14, 13, 12, 15, 14, 11, 12, 14, 13, 15, 12, and 13. (i) Prepare the frequency table for the data. (ii) Find the median (iii) Calculate the mean. Note: There are 19 ages listed in the question, despite stating "20 students". The calculations below are based on the 19 provided ages. Step 1: List the ages in ascending order. 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15. (i) Prepare the frequency table for the data. Step 2: Create a frequency table. | Age (x) | Tally | Frequency (f) | fx | | :------ | :---- | :------------ | :--- | | 11 | I | 1 | 11 | | 12 | IIIII | 5 | 60 | | 13 | IIIII | 5 | 65 | | 14 | IIIII | 5 | 70 | | 15 | III | 3 | 45 | | Total | | 19 | 251 | (ii) Find the median. Step 3: Since there are 19 data points (an odd number), the median is the (N+1)/(2)-th term. Median position = (19+1)/(2) = (20)/(2) = 10-th term Step 4: From the sorted list, the 10th term is 13. The median age is 13. (iii) Calculate the mean. Step 5: Calculate the sum of (frequency × age) and the total frequency from the table. fx = 251 f = 19 Step 6: Calculate the mean using the formula Mean = ( fx)/( f). Mean = (251)/(19) ≈ 13.2105 The mean age is approximately 13.21 (to 2 decimal places). 3. (b) A bag contains 5 red balls, 3 blue balls and 2 green balls. If a ball is picked at random from the bag, what is the probability that it is: (i) Red? (ii) Not blue? Step 1: Calculate the total number of balls in the bag. Total balls = 5 (red) + 3 (blue) + 2 (green) = 10 balls (i) Probability of picking a Red ball. Step 2: Calculate the probability of picking a red ball. P(Red) = Number of red ballsTotal number of balls = (5)/(10) = (1)/(2) The probability of picking a red ball is (1)/(2). (ii) Probability of picking a ball that is Not blue. Step 3: Calculate the number of balls that are not blue. Not blue balls = Red balls + Green balls = 5 + 2 = 7 Step 4: Calculate the probability of picking a ball that is not blue. P(Not blue) = Number of not blue ballsTotal number of balls = (7)/(10) Alternatively, P(Not blue) = 1 - P(Blue) = 1 - (3)/(10) = (7)/(10). The probability of picking a ball that is not blue is (7)/(10). 4. Using a ruler and pair of compasses only, construct triangle ABC with AB = 8.5 cm, BC = 11.5 cm and angle ABC = 60°. Measure (i) AC, (ii) angle BAC. ✂️ _That answer was long and got cut off. Reply continue and I'll finish it._

Accepted Answer · 2026-07-30T08:57:45.807Z

here are the solutions to the mathematics questions: SECTION A: OBJECTIVE TEST QUESTIONS 1. The number 709,045 in words is "Seven hundred nine thousand, forty five". A) Seven hundred nine thousand, forty five 2. In the number 28.356, the digit 5 is in the hundredths place. C) Hundredths 3. To express 0.00205 in standard form, move the decimal point 3 places to the right. This gives 2.05 × 10^-3. D) 2.05 × 10^-3 4. To convert 101101_2 to base ten: 1 × 2^5 + 0 × 2^4 + 1 × 2^3 + 1 × 2^2 + 0 × 2^1 + 1 × 2^0 = 1 × 32 + 0 × 16 + 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1 = 32 + 0 + 8 + 4 + 0 + 1 = 45 A) 45 5. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. 2 is prime. 17 is prime. 39 is not prime because 39 = 3 × 13. 1 is not considered a prime number. In this context, 39 is the composite number, and thus not prime. C) 39 6. To approximate 14.897 to 2 decimal places, look at the third decimal place (7). Since it is 5 or greater, round up the second decimal place. The 9 in the hundredths place rounds up to 10, so the 8 in the tenths place becomes 9, and the hundredths place becomes 0. 14.897 ≈ 14.90 B) 14.90 7. To find the L.C.M of 9, 15, and 20: Prime factorization: 9 = 3^2 15 = 3 × 5 20 = 2^2 × 5 L.C.M = 2^2 × 3^2 × 5 = 4 × 9 × 5 = 180 D) 180 8. Simplify ((1)/(2) + (1)/(3)) ÷ ((1)/(2) - (1)/(3)): First, calculate the sum: (1)/(2) + (1)/(3) = (3)/(6) + (2)/(6) = (5)/(6) Next, calculate the difference: (1)/(2) - (1)/(3) = (3)/(6) - (2)/(6) = (1)/(6) Now, divide: (5/6)/(1/6) = (5)/(6) × (6)/(1) = 5 A) 5 9. To express 15 m as a percentage of 1.20 km: Convert 1.20 km to meters: 1.20 km = 1.20 × 1000 m = 1200 m Percentage = 15 m1200 m × 100\% Percentage = (15)/(1200) × 100 = (15)/(12) = (5)/(4) = 1.25\% A) 1.25% 10. To arrange (1)/(3), (2)/(5), (3)/(4), (1)/(2) in ascending order, convert them to decimals: (1)/(3) ≈ 0.333 (2)/(5) = 0.4 (3)/(4) = 0.75 (1)/(2) = 0.5 In ascending order: 0.333, 0.4, 0.5, 0.75, which corresponds to (1)/(3), (2)/(5), (1)/(2), (3)/(4). A) (1)/(3), (2)/(5), (1)/(2), (3)/(4) 11. The perimeter of a rectangle with length 15 cm and width 8 cm is: Perimeter = 2 × (length + width) Perimeter = 2 × (15 cm + 8 cm) = 2 × 23 cm = 46 cm C) 46 cm 12. The area of a triangle is given by (1)/(2) × base × height. Given Area = 60 cm^2 and height = 12 cm. 60 = (1)/(2) × base × 12 60 = 6 × base base = (60)/(6) = 10 cm B) 10 cm 13. The total surface area of a cube with side length 3 cm is: TSA = 6 × (side)^2 TSA = 6 × (3 cm)^2 = 6 × 9 cm^2 = 54 cm^2 A) 54 cm^2 14. A triangular prism has 2 triangular faces, 3 rectangular faces, 6 vertices, and 9 edges. C) 9 15. A parallelogram, in general, has no lines of symmetry. A) 0 16. Simplify (a^2b^3 - ab^2)/(ab(b-1)) Factor out ab^2 from the numerator: ab^2(ab-1) The expression becomes (ab^2(ab-1))/(ab(b-1)) = (b(ab-1))/(b-1). Note: Assuming there might be a typo in the question and the denominator was intended to be ab(ab-1) or a^2b^2 - ab, the expression would simplify to b. Given the options, this is the most likely intended answer if a typo exists.* If the denominator was a^2b^2 - ab, then (ab^2(ab-1))/(ab(ab-1)) = b. B) b 17. To find the coefficient of x in the expansion of (2x-3)(x-5): Expand the expression: (2x-3)(x-5) = 2x(x) + 2x(-5) - 3(x) - 3(-5) = 2x^2 - 10x - 3x + 15 = 2x^2 - 13x + 15 The coefficient of x is -13. C) -13 SECTION B: THEORY 1. (a) (i) Simplify 2(1)/(2) + 1(1)/(4) - (1)/(10) Step 1: Convert mixed numbers to improper fractions. 2(1)/(2) = (2 × 2 + 1)/(2) = (5)/(2) 1(1)/(4) = (1 × 4 + 1)/(4) = (5)/(4) Step 2: Rewrite the expression with improper fractions. (5)/(2) + (5)/(4) - (1)/(10) Step 3: Find the least common multiple (LCM) of the denominators (2, 4, 10), which is 20. Step 4: Convert each fraction to an equivalent fraction with the denominator 20 and perform the operations. (5 × 10)/(2 × 10) + (5 × 5)/(4 × 5) - (1 × 2)/(10 × 2) = (50)/(20) + (25)/(20) - (2)/(20) = (50 + 25 - 2)/(20) = (75 - 2)/(20) = (73)/(20) Step 5: Convert the improper fraction back to a mixed number. (73)/(20) = 3(13)/(20) The simplified expression is 3(13)/(20). 1. (a) (ii) Convert 73 to a binary number. Step 1: Divide 73 by 2 and record the remainder. 73 ÷ 2 = 36 remainder 1 Step 2: Continue dividing the quotient by 2 until the quotient is 0. 36 ÷ 2 = 18 remainder 0 18 ÷ 2 = 9 remainder 0 9 ÷ 2 = 4 remainder 1 4 ÷ 2 = 2 remainder 0 2 ÷ 2 = 1 remainder 0 1 ÷ 2 = 0 remainder 1 Step 3: Read the remainders from bottom to top. The binary number is 1001001_2. 1. (b) Solve the simultaneous equations: 2x + 3y = 12; x - y = 1. Step 1: Label the equations. 2x + 3y = 12 (Equation 1) x - y = 1 (Equation 2) Step 2: From Equation 2, express x in terms of y. x = y + 1 (Equation 3) Step 3: Substitute Equation 3 into Equation 1. 2(y + 1) + 3y = 12 2y + 2 + 3y = 12 Step 4: Combine like terms and solve for y. 5y + 2 = 12 5y = 12 - 2 5y = 10 y = (10)/(5) = 2 Step 5: Substitute the value of y back into Equation 3 to find x. x = 2 + 1 = 3 The solution is x=3, y=2. 2. (a) N45,000.00 was shared by Aisha, Bolanle and Chidima in the ratio 2:3:4. How much did each person get? Step 1: Find the total number of ratio parts. Total parts = 2 + 3 + 4 = 9 Step 2: Calculate Aisha's share. Aisha's share = (2)/(9) × N45,000 = 2 × N5,000 = N10,000 Step 3: Calculate Bolanle's share. Bolanle's share = (3)/(9) × N45,000 = 3 × N5,000 = N15,000 Step 4: Calculate Chidima's share. Chidima's share = (4)/(9) × N45,000 = 4 × N5,000 = N20,000 Aisha received N10,000, Bolanle received N15,000, and Chidima received N20,000. 2. (b) If 5 men can build a wall in 12 days, how many days will it take 8 men working at the same rate? Step 1: Recognize that this is an inverse proportion problem (more men means fewer days). Let M_1 = 5 men, D_1 = 12 days. Let M_2 = 8 men, D_2 = ? days. Step 2: Use the inverse proportion formula M_1 D_1 = M_2 D_2. 5 men × 12 days = 8 men × D_2 days 60 = 8 D_2 Step 3: Solve for D_2. D_2 = (60)/(8) = (15)/(2) = 7.5 It will take 8 men 7.5 days to build the wall. 3. (a) The ages of 20 students in a JSS3 class are 12, 13, 14, 12, 15, 13, 14, 14, 13, 12, 15, 14, 11, 12, 14, 13, 15, 12, and 13. (i) Prepare the frequency table for the data. (ii) Find the median (iii) Calculate the mean. Note: There are 19 ages listed in the question, despite stating "20 students". The calculations below are based on the 19 provided ages. Step 1: List the ages in ascending order. 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15. (i) Prepare the frequency table for the data. Step 2: Create a frequency table. | Age (x) | Tally | Frequency (f) | fx | | :------ | :---- | :------------ | :--- | | 11 | I | 1 | 11 | | 12 | IIIII | 5 | 60 | | 13 | IIIII | 5 | 65 | | 14 | IIIII | 5 | 70 | | 15 | III | 3 | 45 | | Total | | 19 | 251 | (ii) Find the median. Step 3: Since there are 19 data points (an odd number), the median is the (N+1)/(2)-th term. Median position = (19+1)/(2) = (20)/(2) = 10-th term Step 4: From the sorted list, the 10th term is 13. The median age is 13. (iii) Calculate the mean. Step 5: Calculate the sum of (frequency × age) and the total frequency from the table. fx = 251 f = 19 Step 6: Calculate the mean using the formula Mean = ( fx)/( f). Mean = (251)/(19) ≈ 13.2105 The mean age is approximately 13.21 (to 2 decimal places). 3. (b) A bag contains 5 red balls, 3 blue balls and 2 green balls. If a ball is picked at random from the bag, what is the probability that it is: (i) Red? (ii) Not blue? Step 1: Calculate the total number of balls in the bag. Total balls = 5 (red) + 3 (blue) + 2 (green) = 10 balls (i) Probability of picking a Red ball. Step 2: Calculate the probability of picking a red ball. P(Red) = Number of red ballsTotal number of balls = (5)/(10) = (1)/(2) The probability of picking a red ball is (1)/(2). (ii) Probability of picking a ball that is Not blue. Step 3: Calculate the number of balls that are not blue. Not blue balls = Red balls + Green balls = 5 + 2 = 7 Step 4: Calculate the probability of picking a ball that is not blue. P(Not blue) = Number of not blue ballsTotal number of balls = (7)/(10) Alternatively, P(Not blue) = 1 - P(Blue) = 1 - (3)/(10) = (7)/(10). The probability of picking a ball that is not blue is (7)/(10). 4. Using a ruler and pair of compasses only, construct triangle ABC with AB = 8.5 cm, BC = 11.5 cm and angle ABC = 60°. Measure (i) AC, (ii) angle BAC. ✂️ _That answer was long and got cut off. Reply continue and I'll finish it._

Given the equation C = 2πr + b, make b the subject. A. b = C – 2πr, B. b = C + 2πr, C. b = 2πr (C – r) D. b = C/2πr

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Given the equation C = 2πr + b, make b the subject. A. b = C – 2πr, B. b = C + 2πr, C. b = 2πr (C – r) D. b = C/2πr

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