### Ratio and Proportion

Read till taught in 8th

## Some properties of Ratio and Proportion

### Invertendo

Invertendo implies a:b::c:d then b:a::d:c.
Proof:
Given, a:b::c:d
implies a/b = c/d
⇒ ad = bc
⇒ bc = ad
Making RHS LHS and LHS RHS ⇒ b/a = d/c
⇒ b:a :: d:c

### Alternendo

Alternendo implies a:b::c:d then a:c::b:d.
Proof:
Given, a:b::c:d
implies a/b = c/d
⇒ ad = bc
⇒ a/c = b/d
⇒ a:c :: b:d

### Componendo

Componendo implies a:b::c:d then (a+b):b::(c+d):d.
Proof:
Given, a:b::c:d
implies a/b = c/d
Adding one two doth sides
⇒ a/b + 1 = c/d + 1
⇒ (a+b)/b = (c+d)/d
⇒ (a+b):b :: (c+d):d

### Dividendo

Dividendo implies a:b::c:d then (a-b):b::(c-d):d.
Proof:
Given, a:b::c:d
implies a/b = c/d
Subtracting 1 from both sides
⇒ a/b - 1 = c/d - 1
⇒ (a-b)/b = (c-d)/d
⇒ (a-b):b :: (c-d):d

### Componendo and Dividendo

Componendo and Dividendo implies a:b::c:d then (a+b):(a-)b::(c+d):(c-d).
Proof:
Given, a:b::c:d
By componendo, (a+b)/b = (c+d)/d
⇒ (a+b)/(c+d) = b/d
By dividendo, (a-b)/b = (c-d)/d
⇒ (a-b)/(c-d)= b/d
Equating the end results
⇒(a+b)/(c+d) = (a-b)/(c-d)
By alternendo
⇒(a+b)/(a-b)=(c+d)/(c-d)
⇒(a+b):(a-b)::(c+d):(c-d)

### Convertendo

Convertendo implies a:b::c:d then a:(a-b)::c:(c-d).
Proof:
Given, a:b::c:d
implies a/b = c/d
Taking reciprocal
b/a = d/c
Applying dividendo
⇒ (b-a)/a = (d-c)/c
Multiplying by -1
⇒ (a-b):a :: (c-d):c
Taking reciprocal
⇒a/(a-b)= c/(c-d)
⇒a:(a-b):: c:(c-d)

### If a/b = c/d = e/f, then each ratio = (a+c+e)/(b+d+f) = (sum of antecedents)/(sum of consequents)

If a/b = c/d = e/f = k then a = bk, c = dk, e = fk
(a + c + e)/(b + d + f) = (bk + dk + fk)/(b + d + f) = k
⇒ a/b = c/d = e/f = (a + c + e)/(b + d + f)
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