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.
ScanSolve AI Answer
Here are the solutions to the problems. 1. Write down the reciprocal of (i) (6)/(7) The reciprocal of a fraction (a)/(b) is (b)/(a). Reciprocal of (6)/(7) = (7)/(6) (ii) 9 The reciprocal of a number x is (1)/(x). Reciprocal of 9 = (1)/(9) (iii) (1)/(14) Reciprocal of (1)/(14) = (14)/(1) = 14 (iv) 12(3)/(5) Step 1: Convert the mixed number to an improper fraction. 12(3)/(5) = (12 × 5 + 3)/(5) = (60 + 3)/(5) = (63)/(5) Step 2: Find the reciprocal. Reciprocal of (63)/(5) = (5)/(63) 2. Simplify. (i) (4)/(7) + (9)/(14) Step 1: Find a common denominator, which is 14. (4)/(7) = (4 × 2)/(7 × 2) = (8)/(14) Step 2: Add the fractions. (8)/(14) + (9)/(14) = (8+9)/(14) = (17)/(14) (ii) (7)/(10) + (3)/(5) Step 1: Find a common denominator, which is 10. (3)/(5) = (3 × 2)/(5 × 2) = (6)/(10) Step 2: Add the fractions. (7)/(10) + (6)/(10) = (7+6)/(10) = (13)/(10) (iii) (8)/(9) + 16 Step 1: Write 16 as a fraction with denominator 9. 16 = (16 × 9)/(9) = (144)/(9) Step 2: Add the fractions. (8)/(9) + (144)/(9) = (8+144)/(9) = (152)/(9) (iv) 9 + (1)/(3) Step 1: Write 9 as a fraction with denominator 3. 9 = (9 × 3)/(3) = (27)/(3) Step 2: Add the fractions. (27)/(3) + (1)/(3) = (27+1)/(3) = (28)/(3) (v) 24 ÷ (6)/(7) Step 1: To divide by a fraction, multiply by its reciprocal. 24 ÷ (6)/(7) = 24 × (7)/(6) Step 2: Perform the multiplication. 24 × (7)/(6) = (24)/(6) × 7 = 4 × 7 = 28 (vi) 3(3)/(5) + (4)/(5) Step 1: Convert the mixed number to an improper fraction. 3(3)/(5) = (3 × 5 + 3)/(5) = (15+3)/(5) = (18)/(5) Step 2: Add the fractions. (18)/(5) + (4)/(5) = (18+4)/(5) = (22)/(5) (vii) 3(3)/(7) + (8)/(21) Step 1: Convert the mixed number to an improper fraction. 3(3)/(7) = (3 × 7 + 3)/(7) = (21+3)/(7) = (24)/(7) Step 2: Find a common denominator, which is 21. (24)/(7) = (24 × 3)/(7 × 3) = (72)/(21) Step 3: Add the fractions. (72)/(21) + (8)/(21) = (72+8)/(21) = (80)/(21) (viii) 5(4)/(7) + 1(3)/(10) Step 1: Convert mixed numbers to improper fractions. 5(4)/(7) = (5 × 7 + 4)/(7) = (35+4)/(7) = (39)/(7) 1(3)/(10) = (1 × 10 + 3)/(10) = (10+3)/(10) = (13)/(10) Step 2: Find a common denominator for 7 and 10, which is 70. (39)/(7) = (39 × 10)/(7 × 10) = (390)/(70) (13)/(10) = (13 × 7)/(10 × 7) = (91)/(70) Step 3: Add the fractions. (390)/(70) + (91)/(70) = (390+91)/(70) = (481)/(70) (ix) 15(3)/(7) ÷ 1(23)/(49) Step 1: Convert mixed numbers to improper fractions. 15(3)/(7) = (15 × 7 + 3)/(7) = (105+3)/(7) = (108)/(7) 1(23)/(49) = (1 × 49 + 23)/(49) = (49+23)/(49) = (72)/(49) Step 2: To divide by a fraction, multiply by its reciprocal. (108)/(7) ÷ (72)/(49) = (108)/(7) × (49)/(72) Step 3: Simplify by canceling common factors. (108)/(7) × (49)/(72) = (108 × 49)/(7 × 72) = (108)/(72) × (49)/(7) (108)/(72) = (3 × 36)/(2 × 36) = (3)/(2) (49)/(7) = 7 (3)/(2) × 7 = (21)/(2) (21)/(2)