In most ratio questions you are never told the real numbers — you only know how the quantities compare, such as a:b=2:3. The k-method turns that comparison into ordinary algebra you can substitute and simplify.
Substitute k, then cancel
• If a:b=2:3, write a=2k and b=3k, where k is one shared, non-zero “size” both quantities are built from.
• Every term carries the same k. So in a homogeneous expression (every term has the same degree) the k is common to top and bottom and cancels, leaving a pure number.
• Example: to evaluate p2+q2p2−q2 when p:q=3:1, put p=3k,q=k. Then 9k2+k29k2−k2=10k28k2. Cancel the k2 and reduce: 108=54.
• Componendo–Dividendo: if ba=dc then a−ba+b=c−dc+d. Reach for it when a fraction already equals a number and you must find an a+b or a−b expression — no k needed, just add and subtract.
• It also runs backwards: from x−yx+y=nm you get x:y=(m+n):(m−n) — add for the first part, subtract for the second.
📊Ratio ScalerEnter A and B to explore the ratio — add C for compound ratio
A
:
B
+
C (optional)
A3
B5
Simplified Ratio
3 : 5
3rd Proportional (x where A:B = B:x)
x = 8.3333
Mean Proportional (√A×B)
√15≈ 3.873
Proportion Checker — Is a:b = c:d ?
:=:
If a:b=3:1 then a=3k,b=k, so a−ba+b=2k4k=.
📝Practice Questions
Q1If x:y = 3:1, find the value of (2x + 3y) ÷ (x − 2y).
Q2If (3a − 2b) ÷ (3a + 2b) = 1 ÷ 3, find a : b.
Q3If p:q = 5:2, find the value of (p² − q²) ÷ (p² + q²).
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Real exam questions — k-Method & Componendo
1 question types · 5 solved examples from real SSC papers
When a question gives only a ratio like x:y=4:3 and asks for the value of a messy expression, you do not need the real numbers. Write each letter as a multiple of a shared unknown k, substitute, and watch k cancel. For a±b style fractions, Componendo & Dividendo gets the answer in one line.
How to solve this type
If a:b=m:n, set a=mk and b=nk and substitute; the shared k (or k2) cancels, leaving a pure number. When a fraction equals a value and you must find an a±b expression, use Componendo–Dividendo: ba=dc⇒a−ba+b=c−dc+d.
