# Ratio and Proportion

Ratio and proportion are fundamental concepts in mathematics. Ratios express relationships between quantities, while proportions establish equality between ratios. They are essential for solving various mathematical problems and real-world applications.

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## Ratio and Proportion

- Ratio: A ratio represents the quantitative relationship between two quantities, often expressed as a fraction or in the form of “a to b”.
**Proportion**: Proportion establishes the equality of two ratios, indicating that the relative sizes of the quantities involved are consistent.

**Importance**

**Measurement**: Ratios and proportions are vital in measurement, whether for recipes in cooking, dimensions in construction, or mixing ingredients.**Finance**: Understanding ratios is crucial in financial planning, such as calculating interest rates, analyzing investments, or managing budgets.**Scaling**: Ratios and proportions help in scaling objects accurately, such as maps, architectural designs, and models.**Problem-Solving**: They provide a framework for solving various mathematical and real-world problems, from basic arithmetic to complex scenarios.**Comparison**: Ratios and proportions facilitate comparison between different quantities or situations, aiding decision-making processes.

## Explanation of Ratio

A ratio represents the quantitative relationship between two or more quantities, typically expressed as a comparison or fraction.

**Examples**

**Recipes**: A recipe might require a ratio of 2 cups of flour to 1 cup of sugar, indicating the relative amounts of ingredients.**Maps**: On a map, the scale might be 1 inch representing 10 miles, illustrating the relationship between map distance and real-world distance.**Financial Statements**: In financial statements, a company’s debt-to-equity ratio of 2:1 indicates that for every $2 of debt, there is $1 of equity.

##### Expressing Ratios in Different Forms

**Fraction**: Ratios can be represented as fractions, such as 2/3, where the numerator and denominator represent the quantities being compared.**Decimal**: Ratios can also be expressed as decimals, like 0.5, which is equivalent to the ratio 1:2.**Percentage**: Ratios can be converted into percentages, for instance, a ratio of 3:5 can be expressed as 60%, indicating 3 out of 5 parts.

## Types of Ratios

##### Duplicate Ratio:

**Definition**: Represents a ratio where the antecedent and consequent are both doubled.**Example**: If the ratio is 2:3, the duplicate ratio would be 4:6.

##### Triplicate Ratio:

**Definition**: Indicates a ratio where both parts are tripled.**Example**: For a ratio of 1:2, the triplicate ratio becomes 3:6.

##### Sub-Duplicate Ratio:

**Definition**: Denotes a ratio where both parts are halved.**Example**: If the ratio is 4:5, the sub-duplicate ratio would be 2:2.5.

##### Sub-Triplicate Ratio:

**Definition**: Represents a ratio where both parts are divided into thirds.**Example**: With a ratio of 6:7, the sub-triplicate ratio becomes 2:2.33 (approximately).

##### Inverse Ratio:

**Definition**: Indicates the reciprocal of a given ratio.**Example**: If the ratio is 3:4, the inverse ratio would be 4:3.

##### Compound Ratio:

**Definition**: Comprises two or more ratios multiplied together.**Example**: Combining ratios like 2:3 and 4:5 results in a compound ratio of 8:15.

## Proportion

**Meaning**: Proportion is a mathematical concept that establishes equality between two ratios.**Representation**: It expresses the relationship between different quantities in a balanced manner.

## Basic Formulas of Proportions

**1. If four quantities are in proportion, then Product of Means = Product of Extremes**

For example, in the proportion a:b::c:d, we have, bc = ad.

**2. Fourth proportional: If a:b::c:x, then x is called the fourth proportional of a, b, c.**

**Illustration**: Find a fourth proportional to the numbers 2, 5, 4.**Solution**: So, letting x be the fourth proportional, we have:

2 * x = 5 * 4

Solving for x:

x = (5 * 4) / 2

x = 10

**3. Third Proportional: If a:b::b:x, then x is called the third proportional of a, b.**

x = bยฒ/a

Illustration: Find a third proportional to the numbers

2.5, 1.5

Absolutely! The third proportional to the numbers 2.5 and 1.5 is **0.9**.

We can find the third proportional using the same logic as the previous example. Here’s how:

**Square the second number (b):**b = 1.5, so bยฒ = 1.5ยฒ = 2.25**Divide the squared value by the first number (a):**a = 2.5, so 2.25 / 2.5 = 0.9

Therefore, the third proportional to 2.5 and 1.5 is 0.9.

**4. Mean Proportional: If a:x::x:b, then x is called the mean or second proportional of a, b.**

x = ab

**Illustration**: Find the mean proportional between 48 and 12.

**Solution**: Here’s how to find it:

- Let
**x**be the mean proportional. - Set up the proportion:
**48 : x = x : 12**ย - Since the product of extremes equals the square of the middle term, we can write:
**48 * 12 = x^2** - Solve for x:
**x = โ576**;**x = 24**

**5. If a/b = c/d, then**

**Question 1**: If a recipe calls for 2 cups of flour and 3 cups of sugar to make 12 cookies, how much flour is needed for 18 cookies?

**Solution**:

- Given: 2 cups of flour for $12$ cookies
- $2$ cups of flour is to $12$ cookies as ๐ฅย cups of flour is to 18ย cookies
- 2/12 = ๐ฅ/18
- $=3$ cups of flour

**Question 2**: If a car travels 240 miles in 4 hours, how far will it travel in 6 hours at the same speed?

**Solution**:

- Given: 240ย miles in 4 hours
- 240ย miles is to 4ย hours as ๐ฅย miles is to 6ย hours
- 240/4=๐ฅ/6
- $360$ miles

**Question 3**: If a bag contains 3 red balls and 5 blue balls, what is the ratio of red balls to blue balls?

**Solution**:

- Ratio of red balls to blue balls = 3:5

**Question 4**: A recipe calls for a ratio of milk to water as 4:1. If you need 8 cups of milk, how many cups of water are needed?

**Solution**:

- Given: $4$ parts milk for $1$ part water
- $4$ parts milk is to $1$ part water as $8$ cups milk is to ๐ฅย cups water
- 4/1=8/๐ฅ
- $2$ cups water

**Question 5**: If the ratio of boys to girls in a class is 3:5, and there are 24ย students in total, how many girls are there?

**Solution**:

- Total parts = 3+5=8
- Girls parts = (5/8)ร24=15ย girls

**Question 6**: If ๐ฅ:๐ฆ=4:7ย and ๐ฆ:๐ง=3:5, what is ๐ฅ:๐ง?

**Solution**:

- Given: ๐ฅ:๐ฆ=4:7ย and ๐ฆ:๐ง=3:5
- To find ๐ฅ:๐ง, multiply the given ratios: ๐ฅ:๐ฆ:๐ง=4:7:5
- ๐ฅ:๐ง=4:5

**Question 7**: If ๐:๐=2:3ย and ๐:๐=5:4, what is $a:b:c$?

**Solution**:

- Given: ๐:๐=2:3ย and ๐:๐=5:4
- To find ๐:๐:๐,ย combine the given ratios: ๐:๐:๐=2:3:4

**Question 8**: The ages of A and B are in the ratio 5:7. If the sum of their ages is 48ย years, find the age of each.

**Solution**:

- Total parts = 5+7=12
- Age of A = (5/12)ร48=20ย years
- Age of B = (7/12)ร48=28ย years

**Question 9**: If 3ย pens cost 15 dollars, what is the cost of $8$ pens?

**Solution**:

- Cost of $1$ pen = $5$ dollars
- Cost of $8$ pens = $=40$ dollars

**Question 10**: The perimeter of a rectangle is $30$ cm, and its length to width ratio is 3:2.ย Find the length and width of the rectangle.

**Solution**:

- Let the length be 3๐ฅย and the width be 2๐ฅ
- Perimeter = 2(๐๐๐๐๐กโ+๐ค๐๐๐กโ)ย = 30 cm
- 2(3๐ฅ+2๐ฅ)=30
- 2(5๐ฅ)=30
- 10๐ฅ=30
- ๐ฅ=3
- Length = 3ร3= 9ย cm, Width = $=6$ cm

## FAQ’s

A ratio compares two quantities of the same kind. It can be written in a few ways:

**a:b**(colon notation)**a to b****a/b**(fraction notation, not always recommended as it can be confused with division)

For example, a ratio of 3:2 for apples to oranges means there are 3 apples for every 2 oranges.

A proportion is a statement that two ratios are equal. It’s written as:

**a:b = c:d**

This means the relationship between a and b is the same as the relationship between c and d.

The fourth proportional (x) is a number that completes a proportion where the product of the means equals the product of the extremes. Given numbers a, b, and c:

**a : b = c : x**

Then:

**a * x = b * c**

Solve for x using this equation.

The third proportional (x) is a number that creates a proportion where the square of the middle term equals the product of the extremes. Given numbers a and b:

**a : x = x : b**

Then:

**x^2 = a * b**

Solve for x using this equation.

**How do I find the mean proportional?**

The mean proportional (x) is a number that creates a proportion where the product of the extremes equals the square of the middle term. Given numbers a and b:

**a : x = x : b**

Then:

**a * b = x^2**

Solve for x using this equation.

Absolutely! Ratios and proportions are used in many fields, including:

**Mixing paints or ingredients in a recipe****Scaling up or down a recipe****Calculating dosages in medicine****Creating similar geometric shapes****Solving distance and speed problems**