2.6 Dividing a Given Quantity in a Given Ratio

Two-part division: If a sum S is divided in the ratio m : n, then: - First part = S × m/(m+n) - Second part = S × n/(m+n)

Three-part division: If S is divided in ratio l : m : n, each part = S × (respective term)/(l+m+n).

Example 12: Divide ₹1200 in the ratio 2 : 3 : 5. Total parts = 2+3+5 = 10. Value of 1 part = 1200/10 = 120. Shares = 2×120, 3×120, 5×120 = ₹240, ₹360, ₹600.

Example 13: A sum of money is divided among A, B and C in the ratio 3 : 4 : 5. If C gets ₹150 more than A, find the total sum. Difference in parts between C and A = 5 − 3 = 2 parts = ₹150 → 1 part = ₹75. Total parts = 3+4+5 = 12. Total sum = 12 × 75 = ₹900.

Example 14: Divide 585 into three parts such that the first part : second part = 2 : 3 and second part : third part = 4 : 5. Combine: first:second = 2:3 = 8:12, second:third = 4:5 = 12:15. So first:second:third = 8:12:15, total parts = 35. 1 part = 585/35 = 16.71…— this doesn’t divide evenly, so let’s use a cleaner total: 560 in ratio 8:12:15 (total 35): 1 part = 16, parts = 128, 192, 240.

Example 14A (three-way “difference chain” division, additional): A sum of ₹1,930 is divided among A, B and C such that A gets ₹30 more than B, and B gets ₹50 more than C. Find each share. Let C = x. Then B = x + 50, and A = B + 30 = x + 80. Sum: x + (x + 50) + (x + 80) = 1930 → 3x + 130 = 1930 → 3x = 1800 → x = 600. C = ₹600, B = ₹650, A = ₹680. Check: 600 + 650 + 680 = 1930 ✔

Example 14B (four-part division, additional): ₹4,500 is divided among P, Q, R and S in the ratio 2 : 3 : 4 : 6. Find R’s share, and the amount by which S’s share exceeds Q’s share. Total parts = 2+3+4+6 = 15. Value of 1 part = 4500/15 = 300. R’s share = 4 × 300 = ₹1,200. S = 6 × 300 = ₹1,800, Q = 3 × 300 = ₹900. S exceeds Q by ₹1,800 − ₹900 = ₹900.