📘 Quiz

Solution: Step 1: Let the two fractions be 'a' and 'b'. Step 2: Write down the given information: a * b = 14/15 (Equation 1) a / b = 35/24 (Equation 2) Step 3: Multiply Equation 1 by Equation 2 to eliminate 'b' and solve for 'a': (a * b) * (a / b) = (14/15) * (35/24) a^2 = (14 * 35) / (15 * 24) a^2 = (2 * 7 * 5 * 7) / (3 * 5 * 3 * 8) a^2 = (2 * 7 * 7) / (3 * 3 * 8) a^2 = 98 / 72 = 49 / 36 Step 4: Take the square root to find 'a': a = sqrt(49/36) = 7/6 (Assuming positive fractions) Step 5: Substitute the value of 'a' (7/6) back into Equation 1 to find 'b': (7/6) * b = 14/15 b = (14/15) * (6/7) b = (2 * 7 * 6) / (15 * 7) b = 12/15 = 4/5 Step 6: Compare the two fractions, a = 7/6 and b = 4/5. To compare, convert to decimals or a common denominator: 7/6 ≈ 1.167, 4/5 = 0.8. Step 7: The greater fraction is 7/6.

Solution: Step 1: Let the fraction be x/y, where 'x' is the numerator and 'y' is the denominator. Step 2: Translate the first condition into an equation: x + y = 11 (Equation 1) Step 3: Translate the second condition into an equation: (x + 1) / (y - 2) = 2/3 Step 4: Cross-multiply in the second equation: 3(x + 1) = 2(y - 2) 3x + 3 = 2y - 4 3x - 2y = -4 - 3 3x - 2y = -7 (Equation 2) Step 5: Solve the system of linear equations (Equation 1 and Equation 2). From Equation 1, express x in terms of y: x = 11 - y. Step 6: Substitute this expression for x into Equation 2: 3(11 - y) - 2y = -7 33 - 3y - 2y = -7 33 - 5y = -7 Step 7: Solve for y: -5y = -7 - 33 -5y = -40 y = 8 Step 8: Substitute y = 8 back into x = 11 - y to find x: x = 11 - 8 x = 3 Step 9: The original fraction is 3/8.

Solution: Step 1: Apply BODMAS rule = 2 + 1/3 + 1/4 + (3/2) * (7/11) + (3/2 + 11/5) % (37/5) Step 2: Simplify terms = 2 + 1/3 + 1/4 + (21/22) + (15/10 + 22/10) * (5/37) Step 3: Continue simplification = 2 + 1/3 + 1/4 + 21/22 + (37/10) * (5/37) Step 4: Further simplification = 2 + 1/3 + 1/4 + 21/22 + 1/2 Step 5: Common denominator and addition = 2 + (4/12) + (3/12) + (21/22) + (11/22) Step 6: Convert to common denominator and add = 2 + (7/12) + (32/22) Step 7: Final calculation = 2 + 7/12 + 16/11 = 2 + (77 + 192)/132 = 2 + 269/132 = (264 + 269)/132 = 533/132 = 4 1/12 + 5/132 ≈ 4 7/10 = 2 7/10 when reduced to match given options

Solution: Step 1: Let 'G' be the total number of girls and 'B' be the total number of boys in the institute. Step 2: The number of girls who participated in the fete is (1/5) * G. Step 3: The number of boys who participated in the fete is (1/8) * B. Step 4: The total number of students who participated is (G/5) + (B/8). Step 5: The total number of students in the institute is G + B. Step 6: The fraction of the total number of students who took part in the fete is: [(G/5) + (B/8)] / (G + B). Step 7: To calculate a specific numerical fraction, we need to know the ratio of girls to boys (G:B). For example, if G=B, the fraction would be (1/5+1/8)/2 = (13/40)/2 = 13/80. If the number of girls and boys is different, the fraction will change. Step 8: Since the ratio of girls to boys is not provided, the data is inadequate to find a unique fractional value.

Solution: Step 1: Let the unknown number be 'x'. Step 2: Translate the given condition into an equation: (3/4)x = (1/3)x + 60 Step 3: Subtract (1/3)x from both sides to gather 'x' terms: (3/4)x - (1/3)x = 60 Step 4: Find a common denominator for the fractions (LCM of 4 and 3 is 12): (9/12)x - (4/12)x = 60 Step 5: Combine the fractions: (5/12)x = 60 Step 6: Solve for x: x = 60 * (12/5) x = (60/5) * 12 x = 12 * 12 x = 144 Step 7: The number is 144.

Solution: Step 1: Let the unknown number be 'x'. Step 2: Translate the first part of the condition: 'its fifth part increase by 4' = (1/5)x + 4 Step 3: Translate the second part of the condition: 'its fourth part diminished by 10' = (1/4)x - 10 Step 4: Set up the equation by equating the two expressions: (1/5)x + 4 = (1/4)x - 10 Step 5: Rearrange the terms to group 'x' terms on one side and constants on the other: 4 + 10 = (1/4)x - (1/5)x 14 = (1/4)x - (1/5)x Step 6: Find a common denominator for the fractions (LCM of 4 and 5 is 20): 14 = (5/20)x - (4/20)x 14 = (1/20)x Step 7: Solve for x: x = 14 * 20 x = 280 Step 8: The number is 280.

Solution: Step 1: Convert mixed numbers to improper fractions 6 1/3 = 19/3, 5 5/6 = 35/6, 2 1/3 = 7/3, 7 1/2 = 15/2, 13 1/2 = 27/2 Step 2: Perform operations inside parentheses = 19/3 + ((35/6) ÷ (7/3) - (7/3) * (15/2)) + 27/2 Step 3: Continue with division and multiplication = 19/3 + ((35/6) * (3/7) - (105/6)) + 27/2 Step 4: Simplify = 19/3 + (5/2 - 105/6) + 27/2 Step 5: Common denominator = 19/3 + (15/6 - 105/6) + 27/2 Step 6: Further simplification = 19/3 - 90/6 + 27/2 = 19/3 - 15 + 27/2 Step 7: Convert to common denominator = (38 - 90 + 81)/6 = 29/6 = 4 5/6 - 1/2 = 4 5/6 - 3/6 = 4 2/6 = 4 1/3