Ratio and proportion are fundamental concepts in mathematics that help us understand the relationship between quantities. The concept of ratio dates back to ancient civilizations, where it was used for trade, construction, and measurement. The Egyptians, Babylonians, and Greeks contributed significantly to the development of ratio and proportion.
The understanding of ratios and proportions expanded, with contributions from mathematicians like Euclid and Archimedes. The development of algebraic notation further enhanced their application and problem-solving capabilities.
Today, ratio and proportion have general applications in diverse fields, such as geometry, statistics, finance, and everyday life. They are crucial for scaling, mixing ingredients, calculating percentages, analyzing data, and solving a wide range of mathematical problems.
In this article, we will discuss the basic definition of ratio and proportion, formulas, and their main difference.
Ratio and Proportion
The concept of ratio involves comparing two or more quantities, expressed as a fraction. It represents the relative sizes or values of the quantities being compared. Proportion extends the idea of ratio to compare multiple ratios and state their equality.
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The formula of ratio and proportion
Ratio formula:
The formula for a ratio is:
Ratio = a/b or a: b
In this formula,
- “a” and “b” represent the quantities being compared.
- The ratio is calculated by dividing the value of “a” by the value of “b”.
- The result is typically expressed as a fraction or using a colon (:).
Proportion formula:
The formula for proportion, which compares two or more ratios to establish equivalence is:
Proportion = a/b = c/d or a: b: c: d
In this formula,
- a, b, c, and d represent the quantities or values in the ratios being compared.
- The proportion is formed by equating the product of the means (a & d) to the product of the extremes (b & c).
Types of proportion
Several types of proportions can be faced in mathematics. Here are the commonly discussed types:
- Direct Proportion
- Inverse Proportion
Direct Proportion:
Direct proportion also known as, direct variation, this type of proportion occurs when two quantities change in the same ratio. If one quantity rises, the other quantity also increases, and as one quantity decreases, the other quantity decreases. It can be represented as “y = kx,” where “y” and “x” are the two quantities and “k” is the constant of proportionality.
Inverse Proportion:
In inverse to direct proportion, inverse proportion occurs when two quantities change in opposite ratios. If one quantity grows, the other quantity decreases, and vice versa. It can be represented as “y = k/x” or “xy = k,” where “y” and “x” are the two quantities, and “k” is the constant of proportionality.
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How to Calculate Ratio and Proportion?
Below are a few examples to understand how to find ratios and proportions.
Example 1: For Ratio
In a wooden box, there are 23 fruits. Of these 23 fruits, 7 are mango, 5 are apple, 3 are orange, and 8 are banana. Evaluate the ratio of:
- Mango to apple
- Orange to total fruits
- Banana to apple
- Orange to apple
Solution:
Step 1:
First, write the number of fruits.
Total fruits = 23
Mango = 7
Apple = 5
Orange = 3
Banana = 8
Step 2:
Find the ratio of mango to apple.
Number of mango = 7
Number of apple = 5
The ratio is
⇒ 7: 5
Or
⇒ 7/5
Step 3:
Ratio of Oranges with total fruits
Number of total fruits = 23
Number of orange = 3
Subtract the total fruits from orange to find the difference = 23 – 3 = 20
Now, the ratio is
⇒ 3: 20
or
⇒ 3/20
Step 4:
Evaluate the ratio of banana to apple.
Number of banana = 8
Number of apple = 5
The ratio is
⇒ 8: 5
or
⇒ 8/5
Step 5:
Find the ratio of orange to apple.
Number of orange = 3
Number of apple = 5
The ratio is
⇒ 3: 5
Or
⇒ 3/5
Example 2: For Proportion
If 15 buses are required to bring the workers of a factory in two weeks, how many buses are required to bring an equal number of workers in eight days?
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Solution:
Step 1:
Suppose that we write the given data of buses and days.
So, buses required to bring the workers of a factory in two weeks = 15
Buses required to bring the workers of a factory in eight days = x
Step 2:
We write the formula of proportion and write the data of buses and days allowing the formula.
⇒ a: b :: c : d
Buses: days:: buses: days
⇒ 15: 14: : x : 8
Step 3:
We write the ratios in the form of fractions and put an equality sign between them.
⇒ 15/14 = x/8
Step 4:
Calculate the above expression to find the value of x.
⇒ 15/14 = x/8
⇒ (15/14) * 8 = x
⇒ (15/7) * 4 = x
⇒ 15*4/7 = x
⇒ x = 15*4/7
⇒ x = 60/7 = 8.57 ≈ 9
⇒ x = 9
Therefore, 9 more buses are required to bring the same number of workers. Buses required bringing the workers of a factory in eight days = 15 + 9 = 24.
You can also explore online tools to solve proportions to find the missing value “x” to ease up the calculations.
Conclusion
In this article, we have discussed the basic definition of ratio and proportion, the formulas of ratio and proportion, the type of proportion, and the main difference between ratio and proportion in detail. Also with the help of an illustration topic will be clarified. After completing this article anyone can simply solve this topic.
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Frequently Asked Questions
Question 1:
What is the ratio?
Solution:
The ratio is a comparison of two quantities or numbers, typically expressed as a fraction or with a colon (:). It represents the relative sizes or values of the quantities being compared.
Question 2:
What is proportion?
Solution:
Proportion is an extension of the concept of a ratio and establishes an equivalence between two or more ratios. It states that the fractions on either side of the proportion sign (=) are equal
Question 3:
The basic concept of ratio and proportion
Solution:
Ratio and proportion are fundamental concepts in mathematics that help us understand the relationship between quantities. In ratio and proportion, you can find an unknown value by setting up equivalent fractions or using cross-multiplication to solve for the missing value.
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