Componendo-Dividendo, Ratio Expressions and Maxima-Minima
Componendo-Dividendo, Ratio Expressions and Maxima-Minima
This topic is CBT-2 level in difficulty but each question is short once you know the trick. Componendo-dividendo turns (a + b)/(a − b) back into a/b; a ratio x : y lets you evaluate any homogeneous expression; and the smallest or largest value of an expression comes from completing the square or from AM ≥ GM.
1. Rules Box
Componendo-dividendo: if a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d), and the reverse also holds
Homogeneous expression in x and y: put x = ak, y = bk; k cancels
ax² + bx + c with a > 0 has minimum c − b²/(4a) at x = −b/(2a); with a < 0 it has a maximum there
For positive x: x + k/x ≥ 2√k, equality at x = √k
Fixed sum: a product is largest when the parts are equal
| Expression | Least or greatest value | At |
|---|---|---|
| x² − 6x + 14 | Least 5 | x = 3 |
| −2x² + 8x + 3 | Greatest 11 | x = 2 |
| x + 9/x (x > 0) | Least 6 | x = 3 |
| ab with a + b = 20 | Greatest 100 | a = b = 10 |
✗ x/y = 5/3, so (x² − y²)/(x² + y²) = (5 − 3)/(5 + 3) | ✓ Square the ratio first: (25 − 9)/(25 + 9) = 8/17
✗ Least value of x + 9/x (x > 0) is 10, at x = 1 | ✓ AM ≥ GM gives 2√9 = 6, at x = 3
हिंदी नोट: योगांतरानुपात (कम्पोनेंडो-डिविडेंडो): a/b = c/d हो तो (a + b)/(a − b) = (c + d)/(c − d)। द्विघात व्यंजक का न्यूनतम या अधिकतम मान x = −b/2a पर मिलता है। धनात्मक x के लिए x + k/x का न्यूनतम मान 2√k होता है।
Exam Pointer: These patterns had no verified NTPC shift in our check and are tagged syllabus-based. They belong to the CBT-2 Mathematics syllabus and appear regularly in SSC and other RRB papers; expect one such question in a harder shift.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Componendo-dividendo with plain ratios
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. If (a + b)/(a − b) = 7/3, find a : b.
Componendo-dividendo gives a/b = (7 + 3)/(7 − 3) = 10/4 = 5/2.
Answer: 5 : 2
EXAMATLAS LEVEL
Q. If (5x + 3y)/(5x − 3y) = 7/3, find x : y and (x² + y²)/(x² − y²).
Componendo-dividendo gives 5x/3y = 10/4 = 5/2, so x/y = 3/2. Then (9 + 4)/(9 − 4) = 13/5.
Answer: 3 : 2; 13/5
Pattern 2: Componendo-dividendo with square roots
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. If (√a + √b)/(√a − √b) = 3, find a : b.
√a/√b = (3 + 1)/(3 − 1) = 2, so a/b = 4.
Answer: 4 : 1
EXAMATLAS LEVEL
Q. Solve (√(x + 3) + √(x − 3))/(√(x + 3) − √(x − 3)) = 2.
Componendo-dividendo gives √(x + 3)/√(x − 3) = 3/1. Squaring, (x + 3)/(x − 3) = 9, so x + 3 = 9x − 27 and x = 30/8 = 15/4. Check: √6.75 ≈ 2.598 and √0.75 ≈ 0.866, and 3.464/1.732 = 2.
Answer: x = 15/4
Pattern 3: Ratio given, value of an expression
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. If x : y = 3 : 4, find (3x + 2y)/(5x − y).
Put x = 3, y = 4: (9 + 8)/(15 − 4) = 17/11.
Answer: 17/11
EXAMATLAS LEVEL
Q. If x/y = 5/3, find (x² − y²)/(x² + y²) and (x³ + y³)/(x³ − y³).
Put x = 5, y = 3: (25 − 9)/(25 + 9) = 16/34 = 8/17, and (125 + 27)/(125 − 27) = 152/98 = 76/49.
Answer: 8/17 and 76/49
Pattern 4: Least or greatest value of a quadratic
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Find the least value of x² − 6x + 14.
x² − 6x + 14 = (x − 3)² + 5, which is least when x = 3. Least value = 5.
Answer: 5
EXAMATLAS LEVEL
Q. Find the greatest value of −2x² + 8x + 3 and the value of x where it occurs.
x = −b/(2a) = −8/(−4) = 2. Value = −8 + 16 + 3 = 11. (Completed square: 11 − 2(x − 2)².)
Answer: 11 at x = 2
Pattern 5: Least value by AM ≥ GM and greatest product
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. For x > 0, find the least value of x + 9/x.
x + 9/x ≥ 2√9 = 6, with equality when x = 9/x, that is x = 3.
Answer: 6
EXAMATLAS LEVEL
Q. For positive x and y with 2x + 3y = 24, find the greatest value of xy. Also find the least value of (x² + 3x + 9)/x for x > 0.
2x and 3y have a fixed sum 24, so their product is greatest when 2x = 3y = 12: 6xy ≤ 144, xy ≤ 24 (at x = 6, y = 4). The second is x + 3 + 9/x ≥ 3 + 6 = 9, at x = 3.
Answer: 24; 9
60-Second Revision
- (a + b)/(a − b) = m/n means a/b = (m + n)/(m − n).
- For a ratio x : y, substitute the ratio numbers directly into a homogeneous expression.
- Quadratic extreme at x = −b/(2a).
- x + k/x ≥ 2√k for x > 0; equal parts give the largest product.
Next Step: Algebra done. Practise identity and quadratic questions in the ExamAtlas RRB NTPC 2026 mock tests; in most of them, substituting a simple value such as x = 1 or the ratio numbers checks your answer in seconds.