2.5 Continued Ratio

A continued ratio compares three or more quantities together, written as a : b : c. To combine two separate ratios a : b and b : c into a single continued ratio, make the common term (b) equal in both by taking the LCM.

Example 10: If a : b = 2 : 3 and b : c = 4 : 5, find a : b : c. LCM of the b-terms (3 and 4) = 12. a : b = 2 : 3 = 8 : 12 (multiplied by 4) b : c = 4 : 5 = 12 : 15 (multiplied by 3) So a : b : c = 8 : 12 : 15.

Example 11: If a : b = 3 : 4, b : c = 5 : 6, c : d = 7 : 8, find a : d. a : b : c: LCM of 4 and 5 = 20. a:b = 3:4 = 15:20, b:c = 5:6 = 20:24. So a:b:c = 15:20:24. Now bring c:d = 7:8 to match c = 24: multiply 7:8 by (24/7)… better to scale c:d so its first term equals 24. c:d = 7:8, multiply by 24/7: this gets messy with fractions, so instead find LCM of 24 and 7 = 168. Scale a:b:c by 7 (168/24=7): 15:20:24 → 105:140:168. Scale c:d by 24 (168/7=24): 7:8 → 168:192. So a:b:c:d = 105:140:168:192. a : d = 105 : 192 = 35 : 64 (dividing by 3).

Example 11A (continued-ratio word problem, additional): The ratio of monthly incomes of A and B is 5 : 6, and the ratio of the incomes of B and C is 4 : 3. If the sum of the incomes of A and C is ₹19,000, find B’s income. Combine the ratios: LCM of the B-terms (6 and 4) = 12. A : B = 5 : 6 = 10 : 12 (multiplied by 2) B : C = 4 : 3 = 12 : 9 (multiplied by 3) So A : B : C = 10 : 12 : 9. Sum of A’s and C’s parts = 10 + 9 = 19 parts = ₹19,000 → 1 part = ₹1,000. B’s income = 12 × 1,000 = ₹12,000.