Ratio and Proportion

Ratio

The ratio of two quantities of the same type ( let a and b) is used to express how many times bigger or smaller, one quantity is compared to other. For example , if a = 3 and b = 4, then we can write \(\frac{a}{b}\) = \(\frac{3}{4}\). Also, we can write a = \(\frac{3}{4}\) b or a is three-fourth of b. The ratios 2:5 and 8:20.

In a ratio a : b or \(\frac{a}{b}\), a is called the antecedent and b is called the consequent. The ratio b : a is the inverse ratio of a:b and vice-versa.

Compound ratio
 If a : b and c : d be any two ratios. then a : b \(\times\) c : d = \(\frac{a}{b}\) \(\times\) \(\frac{c}{d}\) = \(\frac{ac}{bd}\) = ac : bd is called compound ratio.

Duplicate and sub -duplicate ratio
If a : b be a ratio, then the duplicate ratio of \(\frac{a}{b}\) = (\(\frac{a}{b}\))² = \(\frac{a^2}{b^2}\) = a²: b²
And, the sup-duplicate ratio of \(\frac{a}{b}\) = √ \(\frac{a}{b}\)

Triplicate and sub - triplicate ratio
If a : b be a ratio, then the triplicate ratio of \(\frac{a}{b}\) = (\(\frac{a}{b}\))³ and sub-triplicate vratio of \(\frac{a}{b}\) =³√ \(\frac{a}{b}\)

Proportion

Similarly, if two or more than two ratios are equal, those quantities which make ratios are proportional.
Two ratios a : b and c : d equal or \(\frac{a}{b}\) = \(\frac{c}{d}\), then a, b, c and d are in proportion.
Now, let us study some related examples of proportion.
Example: \(\frac{8}{20}\) = \(\frac{2}{5}\)
Or, \(\frac{20}{8}\) = \(\frac{5 \times 4}{2 \times 4}\) = \(\frac{5}{2}\)
\(\therefore\) \(\frac{20}{8}\) = \(\frac{5}{2}\)

Continued proportion
If a , b and c be any three number such that the ratio of the a and b is equal to the ratio of b and c, then such ratio is known as a compound proportion.

\(\therefore\) \(\frac{a}{b}\) = \(\frac{b}{c}\) is said to be continued proportion. Then, ac= b²

a : b = b : c

Here, a is 1^st proportion

b is mean proportion

c is 3^rd proportion

Mean proportion (b) =√ac

Properties of proportion

If a, b, c and d are in proportion, then we can verify the following six properties of proportion.

1. Invertendo
2. Alternendo
3. Componendo
4. Dividendo
5. Componendo and Dividendo
6. Addendo

a) Invertendo
If \(\frac{a}{b}\) =\(\frac{c}{d}\), then \(\frac{b}{a}\) =\(\frac{d}{c}\) is known as invertendo properties of proportion.

Proof:

Here, \(\frac{a}{b}\) = \(\frac{c}{d}\)

Then, 1 \(\div\)\(\frac{a}{b}\) = 1\(\div\)\(\frac{c}{d}\) (1 is divided by both ratio)

1 \(\times\)\(\frac{b}{a}\) = 1\(\times\)\(\frac{d}{c}\)

\(\therefore\) \(\frac{b}{a}\) =\(\frac{d}{c}\)

Hence, if \(\frac{a}{b}\) = \(\frac{c}{d}\), then \(\frac{b}{a}\) = \(\frac{d}{c}\)

b) Alternendo
If \(\frac{a}{b}\) = \(\frac{c}{d}\), then \(\frac{a}{c}\) = \(\frac{b}{d}\) is known as alternendo property of proportion.

Proof:

Here, \(\frac{a}{b}\) = \(\frac{c}{d}\)

Multiplying both by \(\frac{b}{c}\), we get \(\frac{a}{b}\) \(\times\) \(\frac{b}{c}\) = \(\frac{c}{d}\) \(\times\) \(\frac{b}{c}\)

or, \(\frac{a}{c}\) = \(\frac{b}{d}\)

\(\therefore\) \(\frac{a}{c}\) = \(\frac{b}{d}\)

c) Componendo
If \(\frac{a}{b}\) = \(\frac{c}{d}\), then \(\frac{a + b}{b}\) = \(\frac{c + d}{d}\) is known as componendo property of proportion.

Proof:

Here, \(\frac{a}{b}\) = \(\frac{c}{d}\)

Then, adding one on both side, we get

\(\frac{a}{b}\) + 1 = \(\frac{c}{d}\) + 1

\(\frac{a + b}{b}\) = \(\frac{c + d}{d}\)

Hence, if \(\frac{a}{b}\) = \(\frac{c}{d}\), then \(\frac{a + b}{b}\) = \(\frac{c + d}{d}\)

d) Dividendo
If \(\frac{a}{b}\) = \(\frac{c}{d}\), then \(\frac{a - b}{b}\) = \(\frac{c - d}{d}\) is known as dividendo property of proportion.

Proof:

Here, \(\frac{a}{b}\) = \(\frac{c}{d}\)

subtracting 1 from both sides, we get

\(\frac{a}{b}\) - 1 = \(\frac{c}{d}\) - 1

or, \(\frac{a - b}{b}\) = \(\frac{c - d}{d}\)

\(\therefore\) \(\frac{a - b}{b}\) = \(\frac{c - d}{d}\)

e) Componendo and dividendo
If \(\frac{a}{b}\) = \(\frac{c}{d}\), then \(\frac{a + b}{a - b}\) = \(\frac{c + d}{c - d}\) is known as componendo and dividendo property of proportion.

Proof:

Here, \(\frac{a}{b}\) = \(\frac{c}{d}\)

By compendendo we have,

\(\frac{a + b}{b}\) = \(\frac{c + d}{d}\)................. (1)

Again, by dividendo, we have

\(\frac{a - b}{b}\) = \(\frac{c - d}{d}\) ..................... (2)

Now, dividing equation (1) by (2), we get

\(\frac {\frac {a+b}b}{\frac {a-b}b}\) = \(\frac {\frac {c+d}d}{\frac {c-d}d}\)

or, \(\frac{a + b}{b}\) \(\times\) \(\frac{b}{a - b}\) = \(\frac{c + d}{d}\) \(\times\) \(\frac{d}{c - d}\)

or, \(\frac{a + b}{a - b}\) = \(\frac{c + d}{c - d}\)

\(\therefore\) \(\frac{a + b}{a - b}\) = \(\frac{c + d}{c - d}\)

f) Addendo
If \(\frac{a}{b}\) = \(\frac{c}{d}\), then, \(\frac{a}{b}\) = \(\frac{c}{d}\) = \(\frac{a + c}{b + d}\) is known as addendo property of proportion.

Proof:

Here, \(\frac{a}{b}\) = \(\frac{c}{d}\)

By alternendo, we get,

\(\frac{a}{c}\) = \(\frac{b}{d}\)

By alternendo we get,

\(\frac{a + c}{c}\) = \(\frac{b + d}{d}\)

Again, by alternendo we get,

\(\frac{a + c}{b + d}\) = \(\frac{c}{d}\)

\(\therefore\)\(\frac{a}{b}\) = \(\frac{c}{d}\) = \(\frac{a + c}{b + d}\)

Hence, if \(\frac{a}{b}\) = \(\frac{c}{d}\) then, \(\frac{a}{b}\) = \(\frac{c}{d}\) = \(\frac{a + c}{b + d}\)

Similarly, if \(\frac{a}{b}\) = \(\frac{c}{d}\) = \(\frac{e}{f}\) then, \(\frac{a}{b}\) = \(\frac{c}{d}\) = \(\frac{e}{f}\) = \(\frac{a + c + e}{b + d + f}\) and so on.

Solution of discontinued and continued proportion (k-method):

Discontinued proportion

If a, b, c and d are in discontinued proportion,

Let, \(\frac ab\) = \(\frac cd\) = k 

Then,

\(\frac ab\) = k,

∴ a = bk....................(i)

\(\frac cd\) = k,

∴ c = dk.....................(ii)

In terms of the denominator with 'k' is a constant number, express the two numerators. We solve the problems related to proportion.

Continued proportion

If a, b, c and d are in a continued proportion, 

Let, \(\frac ab\) = \(\frac bc\) = \(\frac cd\)= k 

or, \(\frac cd\) = k

∴ c = dk...................(i)

or, \(\frac bc\) = k

∴ b = ck = d.k.k = dk^{2..............(II)}

or, \(\frac ab\) = k,

∴ a = bk = dk².k = dk³.............(iii)

∴ a = dk³, b = dk² and c = dk

So, if a, b, c and d are in contiuned proportion, we express a, b, c in terms of d with 'k' constant and solve the problem.