Important Ratio and Proportion Formulas, Tricks and Examples

Important Ratio and Proportion Formulas, Tricks & Examples

Ratio and proportions are foundational concepts in quantitative aptitude, with applications across topics such as numbers, geometry, speed, distance and time, and time and work. Mastery of these concepts can make problem-solving quicker and more intuitive. Let’s explore their definitions, formulas, shortcuts, and examples to boost your understanding and confidence.

Concepts of Ratio and Proportions

1. Ratio

A ratio compares two or more quantities of the same kind. It is dimensionless and has no units. For instance, in the ratio 1:3, the first element is called the antecedent and the second is the consequent.Ratios can have multiple elements, such as 1:3:7.

Key Rules for Ratios

Adding/Subtracting a quantity to each element:
- For a ratio a:ba:b and a positive number xx:
  - If $a>ba > b$ , then $(a+x):(b+x)<a:b(a+x):(b+x) < a:b$
  - If $a<ba < b$ , then $(a+x):(b+x)>a:b(a+x):(b+x) > a:b$
  - If $a=ba = b$ , then $(a+x):(b+x)=a:b=1(a+x):(b+x) = a:b = 1$
- For a negative $xx$ , the inequalities reverse.
Duplicate and Triplicate Ratios:
- Duplicate ratio: $a2:b2a^2:b^2$
- Triplicate ratio: $a3:b3a^3:b^3$
- Sub-duplicate: $a:b\sqrt{a}:\sqrt{b}$
- Sub-triplicate: $a3:b3\sqrt[3]{a}:\sqrt[3]{b}$
Comparison of Ratios with Unity:
- If $a>ba > b$ , $a:b>1a:b > 1$
- If $a<ba < b$ , $a:b<1a:b < 1$
- If $a=ba = b$ , $a:b=1a:b = 1$

Common Mistakes in Ratios

Always ensure that quantities have the same units before calculating ratios.

2. Proportion

When two ratios are equal, they form a proportion. For example,

a:b=c:da:b = c:d

implies

a,b,c,a, b, c,

and

dd

are in proportion, denoted as

a:b::c:da:b::c:d

Proportion Formulas

Componendo: $a+ba−b=c+dc−d\frac{a+b}{a-b} = \frac{c+d}{c-d}$
Dividendo: $a−ba+b=c−dc+d\frac{a-b}{a+b} = \frac{c-d}{c+d}$
Componendo and Dividendo Combined: $a+ba−b=c+dc−d\frac{a+b}{a-b} = \frac{c+d}{c-d}$

Additional Key Proportionality Concepts

Direct Proportion: $x∝y⇒x=k⋅yx \propto y \Rightarrow x = k \cdot y$
Inverse Proportion: $x⋅y=kx \cdot y = k$

Important Formulas for Ratio and Proportions

$a:b=c:d⇒a⋅d=b⋅ca:b = c:d \Rightarrow a \cdot d = b \cdot c$ (Product of means = Product of extremes)
If $(a:b)>(c:d)(a:b) > (c:d)$ , then $ab>cd\frac{a}{b} > \frac{c}{d}$ .
Compounded Ratios: For $(a:b),(c:d),(e:f)(a:b), (c:d), (e:f)$ : $(ace):(bdf)(ace):(bdf)$
Duplicate Ratio: $a2:b2a^2:b^2$
Sub-duplicate Ratio: $a:b\sqrt{a}:\sqrt{b}$
Triplicate Ratio: $a3:b3a^3:b^3$
Sub-triplicate Ratio: $a3:b3\sqrt[3]{a}:\sqrt[3]{b}$
Proportional Distribution of Liquids: $Pure liquid quantity after n operations=x⋅(1−yx)n\text{Pure liquid quantity after } n \text{ operations} = x \cdot \left(1 – \frac{y}{x}\right)^n$

Solved Examples

Example 1: Food Sufficiency for Students

Question: If food for 50 students lasts for 45 days, how many days will it last for 75 students?Solution:

For 1 student, food lasts $45×5045 \times 50$ days.
For 75 students, food lasts: $45×5075=30 days\frac{45 \times 50}{75} = 30 \text{ days}$

Example 2: Alloy Mixing

Question: Two alloys A and B have gold and copper in ratios 7:2 and 7:11, respectively. If equal quantities of both alloys are mixed, what is the ratio of gold to copper in the mixture?Solution:

Gold: $79+718=2118=76\frac{7}{9} + \frac{7}{18} = \frac{21}{18} = \frac{7}{6}$
Copper: $29+1118=1518=56\frac{2}{9} + \frac{11}{18} = \frac{15}{18} = \frac{5}{6}$
Final Ratio: $7:57:5$

Example 3: Proportional Distribution of Money

Question: P, Q, R, and S share money in the ratio 4:3:5:2. If R gets Rs. 1000 more than S, what is Q’s share?Solution:

Let the shares be $4x,3x,5x,2x4x, 3x, 5x, 2x$ .
$5x−2x=1000⇒x=100035x – 2x = 1000 \Rightarrow x = \frac{1000}{3}$ .
Q’s share = $3x=30003=Rs.10003x = \frac{3000}{3} = Rs. 1000$ .

Example 4: Increasing Seats in Classes

Question: Chemistry, Physics, and Math seats are in the ratio 6:5:7. If they increase by 30%, 45%, and 60% respectively, what is the new ratio?Solution:

Original seats: $6x,5x,7x6x, 5x, 7x$
Increased seats: $6x⋅130100:5x⋅145100:7x⋅1601006x \cdot \frac{130}{100} : 5x \cdot \frac{145}{100} : 7x \cdot \frac{160}{100}$
Simplify: $39:29:5639:29:56$

Conclusion

Mastering ratio and proportion is key to solving a wide range of mathematical problems, particularly in competitive exams. By understanding the formulas, applying effective tricks, and practicing examples, you can improve your speed and accuracy. With consistent practice, these concepts will become second nature, enhancing your overall problem-solving abilities.