Notes-Class-9-Mathematics-1-Chapter-4-Ratio and Proportion-Maharashtra Board

Ratio and proportion

Ratio : A ratio is the relation which one quantity bears to another of the same kind and having the same units.

Direct and Inverse Proportion :

(i) Direct Proportion : If the ratio of the two quantities is constant, then it is said that the two quantities are in direct proportion.

Example :

The information about the distance covered by a car using certain quantity of petrol is given in the table.

Petrol : x litres	1	3	4	6
Distance : y km	12	36	48	72
\(\frac{x}{y}\)	\(\frac{1}{12}\)	\(\frac{3}{36}=\frac{1}{12}\)	\(\frac{4}{48}=\frac{1}{12}\)	\(\frac{6}{72}=\frac{1}{12}\)

The ratio of the consumption of petrol (in litres) and the distance covered by the car (in km) is constant. In this case, it is said that the two quantities are in direct

proportion.

(ii) Inverse Proportion : If the product of the two quantities is constant, then it is said that the two quantities are in inverse proportion.

Example :

The information about the speed of the vehicle (in km/h) and the time taken (in hours) to cover 120 km is given in the following table.

Speed (km/h) : x km/h	15	30	45	60
Time (hrs) : y	6	3	2	1.5
x × y km	90	90	90	90

Here, the product of the speed of the vehicle and the time taken to cover 90 km is constant. In such a case, it is said that the quantities are in inverse proportion.

Properties of ratio :

(i) Ratio of numbers a and b is written as a : b or \(\frac{a}{b}\). In this ratio a is called the predecessor (first term) and b is called successor (Second term).

(ii) In the ratio of two numbers, if the second term is 100 then it is known as a percentage.

(iii) The ratio remains unchanged, if its terms are multiplied or divided by non-zero number.

• Example :  3 : 4 = 6 : 8 = 9 : 12, Similarly 2 : 3 : 5 = 8 : 12 : 20. If k is a non-zero number, then a : b = ak : bk a : b : c = ak : bk : ck

(iv) The quantities taken in the ratio must be expressed in the same unit.

(v) The ratio of two quantities is unitless.

• Example : The ratio of 2 kg and 300 g is not 2 : 300 , but it is 2000 : 300 as (2 kg = 2000 gm) i.e. 20 : 3

Comparison of ratios :

The numbers a, b, c, d being positive, comparison of ratios \(\frac{a}{b}\), \(\frac{c}{d}\) can be done

using following rules :

(i) If ad > bc then \(\frac{a}{b}>\frac{c}{d}\)

(ii) If ad < bc then \(\frac{a}{b}<\frac{c}{d}\)

(iii) If ad = bc then \(\frac{a}{b}=\frac{c}{d}\)

Operations on equal ratios :

Using the properties of equality, we can perform some operations on ratios.

a, b, c, d are positive integers.

(1) Invertendo : If \(\frac{a}{b}=\frac{c}{d}\) then \(\frac{b}{a}=\frac{d}{c}\) This result is known as invertendo.

(2) Alternando : If \(\frac{a}{b}=\frac{c}{d}\) then \(\frac{a}{c}=\frac{b}{d}\) This result is known as alternando.

(3) Componendo : If \(\frac{a}{b}=\frac{c}{d}\) then \(\frac{a+b}{b}=\frac{c+d}{d}\)

\(\frac{a}{b}=\frac{c}{d}\) ...add 1 to both side

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

∴ \(\frac{a+b}{b}=\frac{c+d}{d}\)

This result is known as componendo

(4) Dividendo : If \(\frac{a}{b}=\frac{c}{d}\) then \(\frac{a-b}{b}=\frac{c-d}{d}\)

\(\frac{a}{b}=\frac{c}{d}\)  ...subtracting 1 from both side

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

∴ \(\frac{a-b}{b}=\frac{c-d}{d}\)

This result is known as dividendo.

(5) Componendo-dividendo : If \(\frac{a}{b}=\frac{c}{d}\) then \(\frac{a+b}{a-b}=\frac{c+d}{c-d}\)

By componendo, \(\frac{a+b}{b}=\frac{c+d}{d}\)     …..(1)

By dividend, \(\frac{a-b}{b}=\frac{c-d}{d}\)     …..(2)

Dividing (1) by (2),

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

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

∴ \(\frac{a+b}{a-b}=\frac{c+d}{c-d}\)

General form of Componendo and Dividendo :

If \(\frac{a}{b}=\frac{c}{d}\) then \(\frac{a+b}{b}=\frac{c+d}{d}\)      …..... (Performing componendo once)

\(\frac{a+2b}{b}=\frac{c+2d}{d}\)    …..... (Performing componendo twice)

Generally \(\frac{a+mb}{b}=\frac{c+md}{d}\)      …..... (Performing componendo m times)   …(1)

Similarly If \(\frac{a}{b}=\frac{c}{d}\) then \(\frac{a-mb}{b}=\frac{c-md}{d}\)   ... (Performing dividendo m times) ... (2)

And If  \(\frac{a}{b}=\frac{c}{d}\) then \\(\frac{a+mb}{a-mb}=\frac{c+md}{c-md}\)              ... [Dividing (1) by (2)]

Application of properties of equal ratios :

To solve some types of equations, it is convenient to use properties of equal ratios rather than using other methods.

Theorem on equal ratios : 

(i) If \(\frac{a}{b}=\frac{c}{d}\) then (\frac{a}{b}=\frac{a+c}{b+d}=\frac{c}{d}\) . This property is called the theorem of equal ratios.

(ii) If \(\frac{a}{b}=\frac{c}{d}=\frac{e}{f}=......\) (finite terms) and if l, m, n, ... are non-zero numbers such that lb + md + nf + ... # 0,

then each ratio = \(\frac{la+mc+ne+...}{lb+md+nf+...}\)

This is the general form of the above theorem.

Continued Proportion :

(i) If \(\frac{a}{b}=\frac{b}{c}\) then, a, b, c are in continued proportion.

(ii) If ac = b², then dividing both the sides by bc, we get, \(\frac{a}{b}=\frac{b}{c}\)

(iii) If \(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=\frac{d}{e}=\frac{e}{f}....\)   then a, b, c, d, e, and f.... are said to be in continued proportion.

k – method :

(1) Examples on equal ratios can be easily solved by equating each ratio to k. This method is popularly known as the k-method.

(2) If four positive real numbers a, b, c and d are in proportion, we have, \(\frac{a}{b}=\frac{b}{c}\)

If we equate each of these ratios to k, we get, \(\frac{a}{b}=\frac{b}{c}\) = k or a = bk and c = dk

(3) If a, b, c are in continued proportion, we have, \(\frac{a}{b}=\frac{b}{c}\) . Equating each of these ratios to k, we get, \(\frac{a}{b}=\frac{b}{c}\) = k. Thus, b = ck and a = bk = ck.k. Hence, we have, b = ck and a = ck².

(4) If four numbers a, b, c and d are in continued proportion, we have, \(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\)

In this case, c = dk, b = dk² and a = dk³.

(5) If five numbers a, b, c, d and e are in continued proportion, we have, \(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=\frac{d}{e}\) = k. In this case, d = ek, c = ek², b = ek³ and a = ek⁴.