Proportion
2026-02-28 11:29 Diff
https://brightchamps.com/en-us/math/commercial-math/proportion

Proportions follow specific properties that make it easier to solve ratio-related problems. The properties of proportions are: 

  • Cross Multiplication Property
  • Invertendo Property
  • Alternendo Property
  • Componendo and Dividendo Property
  • Mean Proportional Property

Cross Multiplication Property:

In a proportion a/b = c/d, the product of the means equals the product of the extremes: \(a × d = b × c\). 


For example, If \({4 \over 6} = {6 \over 9} \), then \(4 \times 9 = 6 \times 6 = 36\)

Invertendo Property:

Invertendo property states that if two ratios are equal, then their reciprocal is also equal. If  \({{a \over b} = {c \over d}} \implies {{b \over a} = {d \over c}} ​\)

For example, \({8 \over 10} = {16 \over 20} \implies {10\over 8} = {20\over 16}\)

Componendo and Dividendo Property:

According to the componendo property, when we add the numerator and denominator of each ratio to forms a new ratio. If \({a \over b} = {c \over d}\) then \({{a + b} \over b} = {{c + d} \over d}\). 

​

 For example, if \({5 \over 7} = {15\over 21}\), then \({{5 + 7} \over 7} = {{15 + 21} \over 21} \implies {12 \over 7} = {36 \over 21}\)

Dividendo property states that, subtract the denominator from the numerator of each ratio to form a new ratio. If \({a \over b} = {c \over d} \implies {{a - b} \over b} = {{c - d} \over d}\)

For example, if  \({5 \over 7} = {15\over 21}\), then \({{5 - 7} \over 7} = {{15 - 21} \over 21} \implies {-2 \over 7} = {-6 \over 21}\)

Mean Proportional Property:

The mean proportional property states that, if there are three quantities a, b, and c are in continued proportion, then b is the mean proportional between a and c. 

If \({a \over b} = {b \over c} \implies b^2 = a \times c\)
 

For example, if 4, \(x\), and 9 are in continued proportion, then:
 


\({4 \over x} = {x \over 9}\)

\(x^2 = 4 \times 9 = 36\)

\(x\) \(=\) \(6\)