4.2 Reverse Application (finding a ratio from a componendo-dividendo value)
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If (a+b)/(a−b) = p/q, then reversing the logic: a/b = (p+q)/(p−q).
Example 28: If a/b = 5/3, find the value of (a+b)/(a−b). Direct substitution: a = 5, b = 3 → (5+3)/(5−3) = 8/2 = 4. Verify using the rule: (a+b)/(a−b) = (5+3)/(5−3) = 4. Matches.
Example 29: If (a+b)/(a−b) = 3, find a : b. Using the reverse rule with p = 3, q = 1: a/b = (3+1)/(3−1) = 4/2 = 2 : 1. Check: a = 2, b = 1 → (2+1)/(2−1) = 3/1 = 3. ✔
Example 30: If (5x + 3y)/(5x − 3y) = 3/1, find x : y. Treat 5x as “a” and 3y as “b”: 5x/3y = (3+1)/(3−1) = 4/2 = 2 → 5x = 6y → x : y = 6 : 5. Check: x=6,y=5 → 5x=30, 3y=15 → (30+15)/(30−15) = 45/15 = 3. ✔
Example 31A: Verify the componendo-dividendo identity using x/y = 5/3, i.e. confirm (x+y)/(x−y) = (5+3)/(5−3). Substitute x=5, y=3: LHS = (5+3)/(5−3) = 8/2 = 4. RHS (formula) = (5+3)/(5−3) = 8/2 = 4. Both sides match, confirming the identity holds for any x/y = p/q, not just for the specific numbers chosen — this is why the rule can be applied to algebraic expressions (like 5x and 3y above) and not just plain numbers.
Componendo-dividendo is most valuable when a variable is buried inside both a sum and a difference of two expressions set equal to a numeric ratio — instead of cross-multiplying (which often creates an unwanted quadratic or a messy expansion), applying the rule immediately isolates the ratio of the two original expressions in one step.
Example 31B (reverse application, additional): If (7x + 4y)/(7x − 4y) = 5/3, find x : y. Treat 7x as “a” and 4y as “b”: 7x/4y = (5+3)/(5−3) = 8/2 = 4 → 7x = 16y → x : y = 16 : 7. Check: x=16, y=7 → 7x=112, 4y=28 → (112+28)/(112−28) = 140/84 = 5/3. ✔
Example 31C (reverse application combined with solving for actual values, additional): If (a+b)/(a−b) = 7/3 and a − b = 12, find the values of a and b. From the given ratio, a + b = (7/3) × (a − b) = (7/3) × 12 = 28. Now solve the pair a + b = 28 and a − b = 12: adding the two equations, 2a = 40 → a = 20; then b = 28 − 20 = 8. a = 20, b = 8. Check: (20+8)/(20−8) = 28/12 = 7/3. ✔