Percentage, Ratio and Proportion

Percentage

Percentage is a fundamental concept in mathematics that represents a part of a whole as a fraction of 100. The word "percent" literally means "per hundred." It's a widely used tool for comparing quantities, expressing changes, and understanding financial data.

To express any fraction or decimal as a percentage, you multiply it by 100 and add the '%' symbol. For example, the fraction 1/4 is equal to 0.25. As a percentage, it is 0.25 * 100% = 25%. Similarly, 3/5 is 0.6, which is 0.6 * 100% = 60%.

Conversely, to convert a percentage back to a fraction or decimal, you divide by 100. For instance, 50% is 50/100 = 0.5 or 1/2. And 12.5% is 12.5/100 = 0.125 or 1/8.

Key Concepts and Formulas

The basic formula for calculating percentage is:
Percentage = (Part / Whole) * 100

Calculating Percentage of a Number: To find 'x%' of a number 'N', you calculate (x/100) * N.
Example: To find 20% of 500, it's (20/100) * 500 = 0.20 * 500 = 100.

Expressing One Number as a Percentage of Another: To find what percentage is number 'A' of number 'B', you calculate (A / B) * 100.
Example: What percentage is 25 of 100? It's (25 / 100) * 100 = 25%.
Example: What percentage is 30 of 150? It's (30 / 150) * 100 = (1/5) * 100 = 20%.

Percentage Increase/Decrease:
Percentage Increase = ((New Value - Original Value) / Original Value) * 100
Percentage Decrease = ((Original Value - New Value) / Original Value) * 100
Example: If a price increased from $50 to $60, the percentage increase is (($60 - $50) / $50) * 100 = ($10 / $50) * 100 = (1/5) * 100 = 20%.
Example: If a population decreased from 1000 to 800, the percentage decrease is (($1000 - $800) / $1000) * 100 = ($200 / $1000) * 100 = (1/5) * 100 = 20%.

Successive Percentage Changes

When a quantity undergoes multiple percentage changes one after another, the net change is not simply the sum of individual changes. The formula for two successive changes is:
Net Percentage Change = (A + B + (A * B) / 100) %
Here, 'A' and 'B' are the percentage changes. If there's an increase, the value is positive; if there's a decrease, the value is negative.
Example: A price is increased by 10% and then by 20%.
Net Change = (10 + 20 + (10 * 20) / 100) % = (30 + 200 / 100) % = (30 + 2) % = 32% increase.
Example: A price is increased by 20% and then decreased by 10%.
Net Change = (20 + (-10) + (20 * -10) / 100) % = (10 - 200 / 100) % = (10 - 2) % = 8% increase.

For more than two successive changes, you can apply this formula iteratively or use a chain method.
Example: A number is increased by 10%, then by 20%, then by 30%.
First two changes: 10% and 20%. Net change = (10 + 20 + (10*20)/100)% = 32%.
Now, apply the third change (30%) to the result of the first two (32%).
Net change = (32 + 30 + (32 * 30) / 100)% = (62 + 960 / 100)% = (62 + 9.6)% = 71.6% increase.

Applications of Percentage

Profit and Loss: Profit % = ((Selling Price - Cost Price) / Cost Price) * 100. Loss % = ((Cost Price - Selling Price) / Cost Price) * 100.
Simple Interest and Compound Interest: Interest calculations are based on percentages of the principal amount.
Discounts: Discounts are usually expressed as a percentage of the marked price.
Population Growth/Decline: Changes in population are often reported as percentage changes.
Taxation: Income tax, sales tax, etc., are calculated as percentages.

Common Percentage Equivalents

Memorizing common fractional and decimal equivalents of percentages can save a lot of time.

Knowing these can help you quickly calculate things like 33.33% of 900, which is simply (1/3) * 900 = 300, instead of (33.33/100) * 900.

Ratio and Proportion

Ratio and proportion are closely related concepts used to compare quantities. A ratio indicates how many times one number contains another. A proportion is a statement that two ratios are equal.

Ratio

A ratio is a comparison of two quantities by division. It is usually expressed in the form a:b or as a fraction a/b. The numbers 'a' and 'b' are called the terms of the ratio. 'a' is the antecedent, and 'b' is the consequent.
Example: If there are 5 boys and 10 girls in a class, the ratio of boys to girls is 5:10, which can be simplified to 1:2. The ratio of girls to boys is 10:5, simplified to 2:1. The ratio of boys to the total number of students is 5:(5+10) = 5:15, simplified to 1:3.

Ratios should always be simplified to their lowest terms. You do this by dividing both terms by their greatest common divisor (GCD).
Example: The ratio 24:36 can be simplified by dividing both numbers by their GCD, which is 12. So, 24/12 : 36/12 = 2:3.

Types of Ratios

1. Continued Ratio: A ratio of three or more quantities, like a:b:c.
Example: If the ratio of ages of A, B, and C is 2:3:4, it means for every 2 years of A's age, B is 3 years old, and C is 4 years old.

2. Duplicate Ratio: The ratio a²:b² is the duplicate ratio of a:b.
Example: The duplicate ratio of 2:3 is 2²:3² = 4:9.

3. Sub-duplicate Ratio: The ratio √a : √b is the sub-duplicate ratio of a:b.
Example: The sub-duplicate ratio of 9:16 is √9 : √16 = 3:4.

4. Triplicate Ratio: The ratio a³:b³ is the triplicate ratio of a:b.
Example: The triplicate ratio of 2:3 is 2³:3³ = 8:27.

5. Sub-triplicate Ratio: The ratio ³√a : ³√b is the sub-triplicate ratio of a:b.
Example: The sub-triplicate ratio of 8:27 is ³√8 : ³√27 = 2:3.

6. Inverse Ratio: The ratio 1/a : 1/b is the inverse ratio of a:b.
Example: The inverse ratio of 2:3 is 1/2 : 1/3. To simplify this, multiply both terms by the LCM of the denominators (which is 6): (1/2)*6 : (1/3)*6 = 3:2.

Proportion

A proportion is an equation stating that two ratios are equal. If a:b and c:d are two ratios, and they are equal (a:b = c:d), then a, b, c, and d are said to be in proportion.
This can be written as:
a/b = c/d
In a proportion a:b = c:d, 'a' and 'd' are called the 'extremes' (or extreme terms), and 'b' and 'c' are called the 'means' (or mean terms).

The fundamental property of proportion is that the product of the means is equal to the product of the extremes.
So, for a:b = c:d, we have a * d = b * c.
Example: If 5:10 = x:20, then using the property, 5 * 20 = 10 * x. This gives 100 = 10x, so x = 10.

Types of Proportion

1. Continued Proportion: Three quantities a, b, and c are said to be in continued proportion if a:b = b:c. In this case, 'b' is called the mean proportional between 'a' and 'c'.
From a/b = b/c, we get b² = a * c, which means b = √(a*c).
Example: Find the mean proportional between 4 and 9.
Mean proportional = √(4 * 9) = √36 = 6. So, 4:6 = 6:9.

2. Third Proportional: If a:b = b:c, then 'c' is called the third proportional to 'a' and 'b'.
From a/b = b/c, we get c = b² / a.
Example: Find the third proportional to 4 and 6.
Here, a=4, b=6. We need to find c such that 4:6 = 6:c.
c = 6² / 4 = 36 / 4 = 9. So, the third proportional is 9.

Applications of Ratio and Proportion

1. Distribution of Money/Assets: When a sum of money or assets is to be divided among people in a specific ratio.
Example: If $1000 is to be divided between A and B in the ratio 2:3, A gets (2/(2+3)) * 1000 = (2/5) * 1000 = $400, and B gets (3/(2+3)) * 1000 = (3/5) * 1000 = $600.

2. Mixtures and Alligations: Ratios are used to determine the proportions of different ingredients in a mixture.
Example: In a mixture of milk and water, if the ratio is 3:1, it means for every 3 parts of milk, there is 1 part of water.

3. Map Scales: Maps use ratios to represent distances on the ground. A scale of 1:100,000 means 1 unit on the map represents 100,000 of the same units on the ground.

4. Speed, Distance, Time: The relationship between speed, distance, and time is proportional. If time is constant, distance is proportional to speed. If distance is constant, speed is inversely proportional to time.

5. Partnership: In business partnerships, profits are shared among partners in the ratio of their capital investment and the duration for which they invested.

Solving Problems Involving Both Concepts

Many problems require combining percentage and ratio concepts.
Example: A sum of money is divided among A, B, and C such that A's share is 25% more than B's share, and B's share is 20% less than C's share. If the total sum is $1160, find the share of each.
Step 1: Express relationships as ratios.
A's share is 25% more than B's: A = B + 0.25B = 1.25B. So, A:B = 1.25:1 = 5:4.
B's share is 20% less than C's: B = C - 0.20C = 0.80C. So, B:C = 0.80:1 = 4:5.
Step 2: Combine the ratios.
We have A:B = 5:4 and B:C = 4:5. Since the 'B' term is the same in both ratios (4), we can directly combine them: A:B:C = 5:4:5.
Step 3: Distribute the total sum according to the combined ratio.
Total parts = 5 + 4 + 5 = 14 parts.
Total sum = $1160.
Value of one part = $1160 / 14 = $82.857 (approx). Let's recheck the calculation.
Ah, let's assume the total sum leads to cleaner numbers or re-check the problem statement.
Let's assume the total sum was $1400 for easier calculation.
Total parts = 14. Total sum = $1400.
Value of one part = $1400 / 14 = $100.
A's share = 5 parts * $100/part = $500.
B's share = 4 parts * $100/part = $400.
C's share = 5 parts * $100/part = $500.
Let's check: Is A (500) 25% more than B (400)? 400 * 1.25 = 500. Yes.
Is B (400) 20% less than C (500)? 500 * 0.80 = 400. Yes.
Total sum = 500 + 400 + 500 = $1400. Correct.
*(Note: If the sum is indeed $1160, the shares would be A = 5 * (1160/14) ≈ $414.29, B = 4 * (1160/14) ≈ $331.43, C = 5 * (1160/14) ≈ $414.29. The sum would be approximately $1160.)*