Properties of Proportion: Types and Solved Examples - Embibe
  • Written By Gurudath
  • Last Modified 24-06-2022
  • Written By Gurudath
  • Last Modified 24-06-2022

Properties of Proportion: Types, Differences, Examples

Properties of Proportion: We frequently encounter situations wherein we compare quantities in terms of their magnitudes/measurements in our daily lives. For example, we compare students’ grades during school admission, but during police recruiting, we compare candidates’ weights/heights. When we divide two amounts of the same sort, we get a ratio of the two quantities. As a result, understanding the Properties of Proportion is essential.

The ratio of two quantities of the same kind and in the same units is a fraction that shows how many times one quantity is of the other. An equality of two ratios is called a proportion. Continue reading the article to know more details like the types of properties of proportions, the difference between the ratio and proportion properties and more.

Ratio

A ratio is a comparison of two quantities. Thus, the ratio of two numbers (a) and (b\left( {b \ne 0} \right)) is (a \div b) or, (\frac{a}{b}) and is denoted by (a:b.)

In the ratio (a:b,) the quantities (numbers) (a) and (b) are called the terms of the ratio. The former (‘a’) is called the first or antecedent, and the latter (‘b’) is the second term or consequent.

Thus, in the ratio (a:b,\, a) is the antecedent and (b) is the consequent.

We know that a fraction does not change when its numerator and denominator are multiplied or divided by the same non-zero number. So, if its first and second term is multiplied or divided by the same non-zero number, a ratio does not alter.

A ratio (a:b) is simplest if it’s antecedent (a) and consequent (b) have no common factor other than (1.) A ratio in the simplest form is also called the ratio in the lowest terms.

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Proportion

Proportions are simple mathematical tools that use ratios to define the relation between multiple quantities. More frequently, the knowledge of ratio and proportion is applied together to solve day to day problems.

Proportion finds application in solving many lifestyle problems like a business while handling transactions or cooking, etc. It set up a relation between two or more quantities and thus helps in their comparison.

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Definition: An equality of two ratios is called a proportion.

Consider two ratios, \(3:24\) and \(4:36.\) We find that \(3:24 = 1:8\) and \(4:32 = 1:8.\) So, \(3:24 = 4:32\) is a proportion.

Four numbers \(a,\,b,\,c,\,d\) are said to be in proportion if the ratio of the first two is equal to the ratio of the last two, i.e., \(a:b = c:d\)

If four numbers \(a,\,b,\,c,\,d\) are in proportion, we write \(a:b::c:d,\) which is read as \(a\) is to \(b\) as \(c\) is to \(d.\) Here, \(a\) is the first term, \(b\) is the second term, \(c\) is the third term, and \(d\) is the fourth term of the proportion.

The first and fourth terms of the proportions are called extreme terms or extremes. The second and third terms are called the middle terms or means.

We observe \(2,\,4,\,16\) and \(64\) are in proportion because \(2:4 = 16:64.\)

Here, \(2\) and \(64\) are extremes, and \(4\) and \(16\) are means.

Also, \(40,\,70,\,200\) and \(350\) are in proportion because \(40:70 = 200:350\)

Consider the numbers\(40,\,70,\,200\,350.\) We find that \(40:70 = 200:350.\) So, the given numbers are in proportion. \(40\) and \(350\) are extremes, and \(70\) and \(200\) are means.

From the above example, we can say that product of extreme terms \( = 40 \times 350 = 14000\) and product of mean terms \( = 70 \times 200 = 14000\)

Therefore, a product of extreme terms \(=\) product of mean terms

Thus, we observed that if four numbers are in proportion, then the product of the extreme terms is equal to the product of the mean terms.

In other words, \(a:b = c:d\) if \(ad = bc.\)

If \(ad \ne bc,\) then \(a,\,b,\,c,\,d\) are not in proportion.

Continued Proportion

Three numbers \(a,\,b,\,c\) are in continued proportion if \(a,\,b,\,b,\,c\) are in proportion.

Thus, if \(a,\,b,\,c\) are in continued proportion, then \(a,\,b,\,b,\,c\) are in proportion, i.e., \(a:b::b:c\)

\(\Rightarrow \) product of extreme terms \(=\) product of mean terms

\( \Rightarrow a \times c = b \times b\)

\( \Rightarrow ac = {b^2}\)

\( \Rightarrow {b^2} = ac\)

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Mean Proportional

If \(a,\,b,\,c\) are in continued proportion, then \(b\) is the mean proportional between \(a\) and \(c.\)

If \(b\) is the mean proportional between \(a\) and \(c,\) then \({b^2} = ac.\)

Properties of Proportion PDF

We will study the properties of proportion calculator or how to calculate proportion. We all know that a proportion is an expression which states that the two ratios are equal.

In general, four numbers are in proportion if the ratio of the primary two quantities is equal to the last two.

  1. The numbers \(a,\,b,\,c\) and \(d\) are proportional if the ratio of the first two quantities is equal to the ratio of the last two quantities, i.e., \(a:b::c:d\) and is read as \(‘a\) is to \(b\) is as \(c\) is to \(d’.\) The symbol \(‘::’\) stands for ‘is as’.<br>
  2. Each quantity in a proportion is called its term, or it is proportional. <br>
  3. In proportion, the first and the last terms are called the extremes, whereas the second and the third terms are called the means. <br>
  4. If four numbers \(p,\,q,\,r\) and \(s\) are proportional (i.e., \(p:q::r:s\)), then \(p\) and \(s\) are extreme terms, and \(q\) and \(r\) are called middle terms. <br>
  5. The fourth term of a proportion is called the fourth proportional. <br>
  6. For every proportion, the product of the extremes is always equal to the product of the means, i.e., \(p:q::r:s\) if and only if\(ps = qr.\) <br>
  7. From the terms of a given proportion, we can make three more proportions. <br>
  8. If \(p:q = q:r,\) then \(p,\,q,\,r\) are said to be in continued proportion. <br>
  9. If \(p,\,q,\,r\) are in continued proportion (i.e., \(p:q::q:r\)), then \(q\) is the mean proportional between \(p\) and \(r.\) <br>
  10. If \(p,\,q,\,r\) are in continued proportion (i.e., \(p:q::q:r\)), then the third quantity is the third proportional to the first and second, i.e., \(r\) is the third proportional to \(p\) and \(q.\)

Types of Proportion

There are two basic types of proportion, namely

  1. Direct proportion
  2. Indirect or inverse proportion

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1. Direct Proportion

Direct proportion describes the direct relationship between two quantities. In simple words, if one quantity increases, the opposite quantity also increases and vice-versa. For instance, if the speed of a car is increased, it covers more distance during a fixed amount of your time.

If \(a\) is directly proportional to \(b,\) then it is denoted as \(a \propto b\)

2. Inverse Proportion

This type describes the indirect relationship between two quantities. In simple words, if one quantity increases, the opposite quantity decreases and vice-versa. In notation, an inverse proportion is written as \(y \propto \frac{1}{x}\). For instance, increasing the car’s speed will cover a hard and fast distance in less time.

Difference between Ratio and Proportion Properties

Ratio

The ratio is employed to match the dimensions of two things with an equivalent unit. The ratio is expressed employing a colon \(\left( : \right)\) or slash \(\left( / \right).\) The ratio is an expression.

Proportion

The proportion is used to precise the relation of two ratios. Proportion is expressed using the double colon (\left( {::} \right)) or adequate to the symbol (\left( = \right).) Proportion is an equation.

Study About Direct Proportion

Solved Examples – Properties of Proportions Worksheet

Q.1: Are 36,49,6,7 in proportion?
Ans:
We have the product of extreme terms \( = 36 \times 7 = 252\)
Product of means \( = 49 \times 6 = 294\)
Therefore, the product of extreme terms \( \ne \) product of mean terms
Hence, \(36,49,6,7\) are not in proportion.

Q.2: Are 4,12,36 in continued proportion?
Ans:
We know that three numbers \(a,b,c\) are in continued proportion if \(a,b,b,c\) are in proportion.
Therefore, \(4,12,36\) will be in continued proportion if \(4,12,12,36\) are in proportion.
We have the product of extreme terms \( = 4 \times 36 = 144\)
Product of mean \( = 12 \times 12 = 144\)
Therefore, the product of extreme terms \(=\) product of means
\( \Rightarrow 4,12,12,36\) are in proportion.
\( \Rightarrow 4,12,36\) are in continued proportion.

Q.3: The first three terms of a proportion are 3,5 and 21, respectively. Find its fourth term.
Ans:
Let the fourth term be \(x.\)
Then, \(3,5,21,\,x\) are in proportion.
\( \Rightarrow 3 \times x = 5 \times 21\)
\( \Rightarrow 3x = 5 \times 21\)
\( \Rightarrow x = \frac{{5 \times 21}}{3}\)
\( \Rightarrow x = 5 \times 7 = 35.\)

Q.4: What must be added to the numbers 6,10,14 and 22 to be in proportion?
Ans:
Let the required number be \(x.\)
Then, \(6 + x,\,10 + x,\,14 + x,\,22 + x\) are in proportion.
\( \Rightarrow \) Product of extreme terms \(=\) product of mean terms.
\( \Rightarrow \left( {6 + x} \right)\left( {22 + x} \right) = \left( {10 + x} \right)\left( {14 + x} \right)\)
\( \Rightarrow 132 + 6x + 22x + {x^2} = 140 + 10x + 14x + {x^2}\)
\( \Rightarrow 132 + 28x = 140 + 24x\)
\( \Rightarrow 28x – 24x = 140 – 132\)
\( \Rightarrow 4x = 8\)
\( \Rightarrow x = 2\)
Hence, the required number is \(2.\)

NCERT Solutions for Chapter-Ratio and Proportion

Q.5: For every 12 mangoes that I buy, 3 turn out to be rotten. At this rate, how many rotten mangoes will I have if l buy 100 mangoes?
Ans:
Suppose I have \(x\) rotten mangoes if I buy \(100\) mangoes. Then, the ratio of rotten mangoes to mangoes bought is \(x:100.\)
Since for every \(12\) mangoes that I buy, \(3\) turns out to be rotten. Therefore, the ratio of rotten mangoes to the mangoes bought is \(3:12.\)
Therefore, \(x:100 = 3:12\)
\( \Rightarrow x \times 12 = 100 \times 3\)
\( \Rightarrow 12x = 300\)
\( \Rightarrow x = 25\)
Thus, there will be \(25\) rotten mangoes if I buy \(100\) mangoes.

Summary

In this article, we studied the definition of ratio and proportion properties, addition property of proportion, properties of proportion calculator, properties of proportions worksheet, and more. Also, we have studied the properties of proportion, and different types of proportion and solved some example problems on the same.

PRACTICE QUESTIONS RELATED TO PROPORTION

FAQs on Properties of Proportion

We have provided some frequently asked questions about Properties of Proportion here:

Q.1: What is the difference between ratio and proportion?
Ans:
Ratio: The ratio is employed to match the dimensions of two things with an equivalent unit. The ratio is expressed employing a colon \(\left( : \right)\) or slash \(\left( / \right).\) The ratio is an expression.
Proportion: The proportion is used to precise the relation of two ratios. Proportion is expressed using the double colon \(\left( {::} \right)\) or adequate to the symbol \(\left( = \right).\) Proportion is an equation.

Q.2: What are the types of proportion?
Ans:
There are two basic types of proportion, namely
1. Direct proportion
2. Indirect or inverse proportion

Q.3: What is continued and mean proportion?
Ans:
Three numbers \(a,\,b,\,c\) are in continued proportion if \(a,b,b,c\) are in proportion. If \(a,b,c\) are in continued proportion, then \(b\) is the mean proportional between \(a\) and \(c.\)

Q.4: What is proportion?
Ans:
An equality of two ratios is called a proportion.

Q.5: What is the ratio?
Ans:
The ratio of two quantities of the same kind and in the same units is a fraction that shows how many times one quantity is of the other.

We hope you find this article on ‘Properties of Proportion helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.

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