Ratio:
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 [latex]\frac{1}{4}[/latex]
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.

Example 1 Solve: 2 : [latex]x[/latex] = 6 : 15
Solution:

2 : [latex]x[/latex] = 6 : 15
[latex]x[/latex] = 2 x [latex]\frac{15}{6}[/latex]
[latex]x[/latex] = 5

Example 2 Solve: 8 : ([latex]x[/latex] - 1) = 2:7
Solution:

8 : ([latex]x[/latex] - 1) = 2:7
[latex]x[/latex] - 1 = 8 x 7/2
[latex]x[/latex] = 29

Example 3 Solve: (7 - [latex]x[/latex]) : 3 = 4[latex]x[/latex] : 9
Solution:

(7 - [latex]x[/latex]) : 3 = 4[latex]x[/latex] : 9
9 x (7 - [latex]x[/latex]) = 3 x 4[latex]x[/latex]
63 - 9[latex]x[/latex] = 12[latex]x[/latex]
21[latex]x[/latex] = 63 (divide by 21)
[latex]x[/latex] = 3

Example 1 The Fourth Proportional to 4, 9, 12 is
Solution:

Let the fourth propotional to 4, 9, 12 be [latex]x[/latex].
Then, 4 : 9 :: 12 : [latex]x[/latex] ⇔ 4 x [latex]x[/latex] = 9 x 12 ⇔ [latex]x[/latex] = [latex]\frac{9 \times 12}{4}[/latex] = 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 [latex]x[/latex].
Then, 16 : 36 :: 36 : [latex]x[/latex] ⇔ 16 x [latex]x[/latex] = 36 x 36 ⇔ [latex]x[/latex] = [latex]\frac{36 \times 36}{16}[/latex] = 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
= [latex]\sqrt{0.08 \times 0.18}[/latex] = [latex]\sqrt{\frac{8}{100} \times \frac{18}{100}}[/latex] = [latex]\sqrt{\frac{144}{100 \times 100}}[/latex] = [latex]\frac{12}{100}[/latex] = 0.12

1. Mean proportional between a and b is [latex]\sqrt{a b}[/latex]
2. Product of means = Product of extremes

Example: If a : b :: c : d then (b x c) = (a x d)

3. Comparison of ratios: (a : b) > (c : d) ⇔ [latex]\frac{a}{b}[/latex] > [latex]\frac{c}{d}[/latex]
4. Compounded ratio: (a : b), (c : d), (e : f) = (ace : bdf)
5. Duplicate ratio of (a : b) = ([latex]a^2 : b^2[/latex])
6. Sub - duplicate ratio of (a : b) = ([latex]\sqrt{a} : \sqrt{b}[/latex])
7. Triplicate ratio of (a : b) = ([latex]a^3 : b^3[/latex])
8. Sub - triplicate ratio of (a : b) = ([latex]a^{\frac{1}{3}}[/latex] : [latex]b^{\frac{1}{3}}[/latex])
9. If [latex]\frac{a}{b} = \frac{c}{d}[/latex], then [latex]\frac{(a + b)}{(a - b)} =\frac{(c + d)}{(c -d)}[/latex] (Componendo and dividendo)

1. If a : b = 4 : 6 and b : c = 5 : 2, then find a : b : c?
Solution:

Given that
a : b = 4 : 6
b : c = 5 : 2 = (5 x [latex]\frac{6}{5}[/latex]) : (2 x [latex]\frac{6}{5}[/latex]) = (6 : [latex]\frac{12}{5}[/latex])
Therefore, a : b : c = 4 : 6 : [latex]\frac{12}{5}[/latex]

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 [latex]x[/latex] Then, 2 : 6 :: 9 : [latex]x[/latex] ⇒ 2 x [latex]x[/latex] = 6 x 9 ⇒ [latex]x[/latex] = [latex]\frac{6 * 9}{2}[/latex] ⇒ [latex]x[/latex] = 27 Therefore, fourth proportional to 2, 6, 9, is 27.
(ii). Given numbers are 12 and 24 Let the third number be [latex]x[/latex] Then, 12 : 24 :: 24 : [latex]x[/latex] ⇒ 12 x [latex]x[/latex] = 24 x 24 ⇒ [latex]x[/latex] = [latex]\frac{24 * 24}{12}[/latex] ⇒ [latex]x[/latex] = 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 [latex]\sqrt{a b}[/latex] ⇒ [latex]\sqrt{0.0016 * 0.16}[/latex] ⇒ [latex]\sqrt{\frac{16}{10000}}[/latex] x [latex]\sqrt{\frac{16}{100}}[/latex] ⇒ [latex]\frac{4}{100}[/latex] x [latex]\frac{4}{10}[/latex] ⇒ [latex]\frac{16}{1000}[/latex] ⇒ 0.0016 Therefore, mean proportional between 0.0016 and 0.16 is 0.016

3. Divide the 2000 rice bags among the farmer - [latex]x[/latex], farmer - [latex]y[/latex], farmer - [latex]z[/latex] 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 - [latex]x[/latex] = 2000 x [latex]\frac{45}{78}[/latex] = 1153.9 rice bags
Share of farmer - [latex]y[/latex] = 2000 x [latex]\frac{24}{78}[/latex] = 615.4 rice bags
Share of farmer - [latex]z[/latex] = 2000 x [latex]\frac{9}{78}[/latex] = 230.8 rice bags

4: If [latex]a[/latex] : [latex]b[/latex] = 4 : 6, find (2[latex]a[/latex] + 5[latex]b[/latex]) : (3[latex]a[/latex] - [latex]b[/latex])?
Solution:

Given that
[latex]a[/latex] : [latex]b[/latex] = 4 : 6
⇒ [latex]\frac{a}{b}[/latex] = [latex]\frac{4}{6}[/latex]
Consider, (2[latex]a[/latex] + 5[latex]b[/latex]) : (3[latex]a[/latex] - [latex]b[/latex])
⇒ (2[latex](\frac{a}{b})[/latex] + 5) : (3[latex](\frac{a}{b})[/latex] - 1)
⇒ (2[latex](\frac{4}{6})[/latex] + 5) : (3[latex](\frac{4}{6})[/latex] - 1)
⇒ ([latex](\frac{4}{3})[/latex] + 5) : [latex](\frac{8}{3})[/latex] -1)
⇒ [latex]\frac{19}{3}[/latex] : [latex]\frac{5}{3}[/latex]
⇒ [latex]\frac{\frac{19}{3}}{\frac{5}{3}}[/latex]
⇒ [latex]\frac{19}{5}[/latex]
Therefore, (2[latex]a[/latex] + 5[latex]b[/latex]) : (3[latex]a[/latex] - [latex]b[/latex]) = 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[latex]x[/latex] and 3[latex]x[/latex]respectively.
Then, [latex]\frac{5x}{3x + 10}[/latex] = [latex]\frac{4}{5}[/latex]
⇒ 4 (3[latex]x[/latex] + 10) = 5 x 5[latex]x[/latex]
⇒ 12[latex]x[/latex] + 40 = 25[latex]x[/latex]
⇒ 25[latex]x[/latex] - 12[latex]x[/latex] = 40
⇒ 13[latex]x[/latex] = 40
⇒ [latex]x[/latex] = [latex]\frac{40}{13}[/latex]
⇒ [latex]x[/latex] = 3.076
Therefore, the quantity of cement in the given mixture is 3.76