**1. RATIO**

A ratio is a relation between two quantities of the same kind.

Ratio between the two numbers say 3 and 4 may be written as a

(i) fraction, $\frac{3}{4}$

(ii) division, 3 ÷ 4 or

(iii) with the ratio sign (:), 3 : 4

It means, $\frac{3}{4}=3\xf74=3:4$

(Read as 3 to 4 or 3 ‘is to’ 4)

In the ratio 3 : 4

(i) 3 and 4 are called terms of the ratio

(ii) 3 is called the first term or antecedent

(iii)4 is called the second term or consequent.**Example**

Varun has 6 balloons and Tarun has 8 balloons

(i) We say the ratio of their balloons is 3 : 4 (The common factor is eliminated)

(ii) We can compare their balloons in two ways:**Comparison by Difference**

Tarun has 2(8–6) more balloons than Varun. This method is called comparison by difference.**Comparison by Division**

If we divide 6 by 8, we get $\frac{6}{8}=\frac{3}{4}$ (lowest term). This method is called comparison by division.

Thus the ratio of the balloons of Varun and Tarun is 3 : 4**Other examples:**

(A) The ratio between Rs. 2 and Rs. 3 = 2 : 3

(B) The ratio between 3 cm and 4 cm = 3 : 4

(C) The ratio between 1 cm and 20 mm = 1 : 2 (since 20 mm = 2 cm)

(D) The ratio between 10 kg and 15 kg = 2 : 3

(E) The ratio between 3 kg and 1000 g = 3 : 1 (since 1000 g = 1 kg)

**2. FINDING RATIO**

To find a ratio between two quantities, we must have

The quantities of the same kind, for example 2 books can be compared with 5 books.

The quantities must be expressed in the same units. Example to compare Rs. 2 with 50 p, the ratio is 200 :50 or 4 : 1.

Usually ratio is expressed in its simplest form, i.e. the form in which its terms have no common factor except 1 Example 4 : 8 is expressed as 1 : 2.

Multiplying or dividing both terms of a ratio by the same number does not change the value of the ratio**e.g.** $\frac{5}{9}=\frac{5}{9}\times \frac{2}{2}=\frac{10}{18}$

The order of the ratio is also very important, Example the ratio 3 : 7 is different from the ratio 7 : 3.

**3. COMPARISON OF RATIOS**

Let us take the two ratios 3 : 5 and 4 : 7 and compare them

$3:5=\frac{3}{5}\text{and}4:7=\frac{4}{7}$

Now we compare the two fractions by making their denominators equal L.C.M. of 5 and 7 is 35.

$\therefore \frac{3}{5}=\frac{3}{5}\times \frac{7}{7}=\frac{21}{35}$

and $\frac{4}{7}=\frac{4}{7}\times \frac{5}{5}=\frac{20}{35}$

since 21 > 20

$\therefore \frac{21}{35}\frac{20}{35}$

or $\frac{3}{5}>\frac{4}{7}$

Hence 3 : 5 > 4 : 7**I. Equivalent Ratios**

To determine whether two ratios are equivalent or not, either multiply or divide the numerator and denominator of both parts of one ratio by the same number. The choice of the number will depend on the other ratio to be compared.**II. Simplify ratios to the Lowest Terms**

The ratio h : k is in the lowest terms if h and k do not have any common factor and h and k are whole numbers.

To reduce a ratio to the lowest terms, divide h and k by their common factor. If h or k is a fraction or a decimal number, then multiply by a suitable factor to make it a whole number.**III. Ratios related to a given Ratio**

In general, if A : B = x : y then

(A) B : A = y : x

(B) A : (A + B) = x : (x + y)

(C) B : (A + B) = y : (x + y)

(D) A : (A – B) = x : (x – y) where x > y where x > y

(E) B : (A – B) = y : (x – y) where x > y where x > y

(f) (A + B) : (A – B) = (x + y) : (x – y)

**4. PROPORTION**

An equality relation between two ratios is called the proportion.**Example:**(i) If $\frac{2}{3}=\frac{4}{6}$, Here the four quantities 2, 3, 4 and 6 are said to be in proportion.

(ii) If $\frac{1}{2}=\frac{x}{3}$, Here the four quantities 1, 2, x and 3 are in proportion.

(iii) If 2, 3, 4 and 6 are in proportion, then $\frac{2}{3}=\frac{4}{6}$ (or 2 : 3 : : 4 : 6)

(iv) If x : 3 : : 4 : 6, then $\frac{x}{3}=\frac{4}{6}$

(v) If a, b, c and d are in proportion or if a : b = c : d or if $\frac{a}{b}=\frac{c}{d}$, then ad = bc (by cross multiplication)

Also, a and d are called extremes or end terms and b and c are called means or middle terms.

We can say that four terms or number are said to be in proportion when the product of extreme terms (end terms) is equal to the product of mean terms (middle terms).

**In a proportion, if the middle terms are repeated, then each of the middle term is called the mean proportional.**

For example, in a proportion 3, 9, 9, 27; 9 is called the mean proportional.

Product of extremes = 3 $\times $ 27 = 81

Product of means = 9 $\times $ 9 = 81

3, 9, 27 are said to be in proportion and the middle term 9 is called the mean proportional.

Hence if a : b = b : c

then $ac={b}^{2}$

(product of extremes = product of means)

The middle term b is called the means proportional between a and c.

Now let us see the other proportions involving the terms of the given proportion.

Let us take a proportion 1 : 2 = 4 : 8

(A) 1 : 2 = $\frac{1}{2}$ and 4 : 8 = $\frac{4}{8}=\frac{1}{2}$

thus 1 : 2 = 4 : 8

(B) We can also write the above proportion as 2 : 1 = 8 : 4

2 : 1 = $\frac{2}{1}$ = 2 and 8 : 4 = $\frac{2}{1}$ = 2

Thus 2 : 1 = 8 : 4

(C) Second way of writing the above proportion is: 1 : 4 = 2 : 8

1 : 4 = $\frac{1}{4}$ and 2 : 8 = $\frac{2}{8}=\frac{1}{4}$

Thus 1 : 4 = 2 : 8

(D) Third way of writing the same proportion is: 4 : 1 = 8 : 2

4 : 1 = $\frac{1}{4}$ = 4 and 8 : 2 = $\frac{8}{2}$ = 4

Thus 4 : 1 = 8 : 2

Hence there are three more proportions which can be obtained by just changing the positions of the terms of proportion.

**5. RATIO OF THREE QUANTITIES**

Ratios can also be used to compare more than two quantities.**Example:** The ages of three children are 10 years, 11 years and 13 years. The ratio of their ages is 10 : 11 : 13.**Note:** The three quantities must be in the same unit.

The ratio of three quantities can be simplified:

(A) by converting all quantities of different units to the same unit;

(B) if some or all the quantities are fractions, multiply all of them by the L.CM. of the denominators of the fractions. If any of them is a mixed number like $3\frac{1}{2}$, change it to an improper fraction first.

(C) if the quantities have a common factor, divide all of them by the common factor.

(D) if some or all the quantities are decimals, convert all of them to whole numbers by multiplying all of them by a suitable power of 10, i.e., 10,100, 1000, etc.

**6. UNITARY METHOD**

When two quantities are related such that an increase or decrease in one causes a corresponding increase or decrease in the other. This is called direct proportion.**Example:** Cost of 2 apples is Rs. 12. What will be the cost of 6 apples?**Solution:** In solving the problems of this kind, we first find the value of one (unit) and then proceed to find the value of the required quantity.

Clearly, cost of 6 apples will be more

Cost of 2 apples = Rs. 12

Cost of 1 apple = Rs. $\frac{12}{2}$= Rs. 6

$\therefore $ Cost of 6 apples = Rs. 6 $\times $ 6 = Rs. 36**Alternative:**

$\begin{array}{cc}\mathrm{Apples}& \mathrm{cost}\mathrm{of}\mathrm{Rs}.\\ 2& 12\\ 6& \mathrm{x}\end{array}$

$\frac{2}{6}=\frac{12}{\mathrm{x}}$

$2\mathrm{x}=6\times 12$

$\mathrm{x}=\frac{6\times 12}{2}$

x = Rs. 36

Cost of 6 apples is Rs. 36.

The method of finding first the value of one (unit) quantity from the value of given quantities and then finding the value of the required quantities is called the Unitary Method of Method of One.

In this method, the first line of statement be arranged in such a way that the quantity which is of same denomination as the answer (unknown quantity) comes last.