In mathematics, a **Ratio** is a relationship or comparison between two quantities. In general , the ratio between two numbers indicates how many times the first number contains the second and specifies how much of one thing there is in comparison to the second thing. In a ratio, the two quantities ‘a’ and ‘b’ are separated by a symbol “**:**” and written as a fraction in the same units. Since the two quantities are in same units, ratio does not have any units.

If the two ratios are equal, then they are said to be in **Proportion**.

A ratio is simply a fraction. The following notations all express the ratio of a to b

- a : b, a ÷ b, or a/b

**1**. ‘a’ is termed as first term or antecedent, ‘b’ as second term or consequent in the ratio a : b.

**Example**:

- 1 : 4 represents \(\frac{1}{4}\)

where 1 is antecedent, 4 is consequent

**2**. The multiplication or division of each term of a ratio by the same non – zero number does not affect the ratio.

**Example**:

- 2 : 4 = 12 : 10

⇒2 x 12 : 4 x 10

⇒24 : 40

**Proportion:**

**1**. If a, b, c, d are in proportion then,

- a : b = c : d

a : b :: c : d

Here, a and d -> extremes

b and c -> mean terms

**2**. If a : b = c : d, then d is the fourth proportional to a, b, c.

**3**. If a : b = b : c, then c is the third proportional to a and b.

(i) \(x\) is directly proportional to \(y\), if \(x\) = k\(y\) for some constant k and we write, \(x\) ∝ \(y\).

(ii) \(x\) is inversely proportional to \(y\), if \(xy\) = k for some constant k and we write, \(x\) ∝ \(\frac{1}{y}\).

- Thus, a : b :: c : d ⇔ (b x c) = (a x d).

**Example 1**

Solve: 2 : \(x\) = 6 : 15

**Solution**:

- 2 : \(x\) = 6 : 15

\(x\) = 2 x \(\frac{15}{6}\)

\(x\) = 5

**Example 2**

Solve: 8 : (\(x\) – 1) = 2:7

**Solution**:

- 8 : (\(x\) – 1) = 2:7

\(x\) – 1 = 8 x 7/2

\(x\) = 29

**Example 3**

Solve: (7 – \(x\)) : 3 = 4\(x\) : 9

**Solution**:

- (7 – \(x\)) : 3 = 4\(x\) : 9

9 x (7 – \(x\)) = 3 x 4\(x\)

63 – 9\(x\) = 12\(x\)

21\(x\) = 63 (divide by 21)

\(x\) = 3

- (i)

(ii)

(iii)

**Example 1**

The Fourth Proportional to 4, 9, 12 is

**Solution**:

- Let the fourth propotional to 4, 9, 12 be \(x\).

Then, 4 : 9 :: 12 : \(x\) ⇔ 4 x \(x\) = 9 x 12 ⇔ \(x\) = \(\frac{9 \times 12}{4}\) = 27.

Fourth Proportional to 4, 9, 12 is 27.

**Example 2**

The Third Propotional to 16 and 36 is

**Solution**:

- Let the third Propotional to 16 and 36 be \(x\).

Then, 16 : 36 :: 36 : \(x\) ⇔ 16 x \(x\) = 36 x 36 ⇔ \(x\) = \(\frac{36 \times 36}{16}\) = 81.

Third proportional to 16 and 36 is 81.

**Example 3**

The mean proportional between 0.08 and 0.18.

**Solution**:

- Mean proportional between 0.08 and 0.18

= \(\sqrt{0.08 \times 0.18}\) = \(\sqrt{\frac{8}{100} \times \frac{18}{100}}\) = \(\sqrt{\frac{144}{100 \times 100}}\) = \(\frac{12}{100}\) = 0.12

2. Product of means = Product of extremes

3. Comparison of ratios: (a : b) > (c : d) ⇔ \(\frac{a}{b}\) > \(\frac{c}{d}\)

4. Compounded ratio: (a : b), (c : d), (e : f) = (ace : bdf)

5. Duplicate ratio of (a : b) = (\(a^2 : b^2\))

6. Sub – duplicate ratio of (a : b) = (\(\sqrt{a} : \sqrt{b}\))

7. Triplicate ratio of (a : b) = (\(a^3 : b^3\))

8. Sub – triplicate ratio of (a : b) = (\(a^{\frac{1}{3}}\) : \(b^{\frac{1}{3}}\))

9. If \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{(a + b)}{(a – b)} =\frac{(c + d)}{(c -d)}\) (Componendo and dividendo)

- Given that

a : b = 4 : 6

b : c = 5 : 2 = (5 x \(\frac{6}{5}\)) : (2 x \(\frac{6}{5}\)) = (6 : \(\frac{12}{5}\))

Therefore, a : b : c = 4 : 6 : \(\frac{12}{5}\)

**2. Calculate:**

(i) The fourth proportional to 2, 6, 9,?

(ii) The third proportional to 12 and 24?

(iii) The mean proportional between 0.0016 and 0.16?

**Solution**:

- (i). Given numbers are 2, 6, 9

Let the fourth number be \(x\)

Then, 2 : 6 :: 9 : \(x\)

⇒ 2 x \(x\) = 6 x 9

⇒ \(x\) = \(\frac{6 * 9}{2}\)

⇒ \(x\) = 27

Therefore, fourth proportional to 2, 6, 9, is 27.

(ii). Given numbers are 12 and 24

Let the third number be \(x\)

Then, 12 : 24 :: 24 : \(x\)

⇒ 12 x \(x\) = 24 x 24

⇒ \(x\) = \(\frac{24 * 24}{12}\)

⇒ \(x\) = 48

Therefore, third proportional to 12, 24, is 24.

(iii). Given numbers are 0.0016 and 0.16

Mean proportional between a = 0.0016 and b = 0.16 is

\(\sqrt{a b}\)

⇒ \(\sqrt{0.0016 * 0.16}\)

⇒ \(\sqrt{\frac{16}{10000}}\) x \(\sqrt{\frac{16}{100}}\)

⇒ \(\frac{4}{100}\) x \(\frac{4}{10}\)

⇒ \(\frac{16}{1000}\)

⇒ 0.0016

Therefore, mean proportional between 0.0016 and 0.16 is 0.016

**3. Divide the 2000 rice bags among the farmer – \(x\), farmer – \(y\), farmer – \(z\) in the ratio 45 : 24 : 9?**

**Solution**:

- Given that

The ratios are 45 : 24 : 9

Number of rice bags = 2000

Then, Sum of ratio terms = (45 + 24 + 9) = 78

Therefore,

Share of farmer – \(x\) = 2000 x \(\frac{45}{78}\) = 1153.9 rice bags

Share of farmer – \(y\) = 2000 x \(\frac{24}{78}\) = 615.4 rice bags

Share of farmer – \(z\) = 2000 x \(\frac{9}{78}\) = 230.8 rice bags

**4: If \(a\) : \(b\) = 4 : 6, find (2\(a\) + 5\(b\)) : (3\(a\) – \(b\))?**

**Solution**:

- Given that

\(a\) : \(b\) = 4 : 6

⇒ \(\frac{a}{b}\) = \(\frac{4}{6}\)

Consider, (2\(a\) + 5\(b\)) : (3\(a\) – \(b\))

⇒ (2\((\frac{a}{b})\) + 5) : (3\((\frac{a}{b})\) – 1)

⇒ (2\((\frac{4}{6})\) + 5) : (3\((\frac{4}{6})\) – 1)

⇒ (\((\frac{4}{3})\) + 5) : \((\frac{8}{3})\) -1)

⇒ \(\frac{19}{3}\) : \(\frac{5}{3}\)

⇒ \(\frac{\frac{19}{3}}{\frac{5}{3}}\)

⇒ \(\frac{19}{5}\)

Therefore, (2\(a\) + 5\(b\)) : (3\(a\) – \(b\)) = 19 : 5

**5. A cement concrete mixture contains cement and water in the ratio of 5 : 3. If 10 litres of water is added to the mixture, the ratio becomes 4 : 5. Find the quantity of cement in the given concrete mixture?**

**Solution**:

- Given that

Let the quantity of cement and water be 5\(x\) and 3\(x\)respectively.

Then, \(\frac{5x}{3x + 10}\) = \(\frac{4}{5}\)

⇒ 4 (3\(x\) + 10) = 5 x 5\(x\)

⇒ 12\(x\) + 40 = 25\(x\)

⇒ 25\(x\) – 12\(x\) = 40

⇒ 13\(x\) = 40

⇒ \(x\) = \(\frac{40}{13}\)

⇒ \(x\) = 3.076

Therefore, the quantity of cement in the given mixture is 3.76