NCERT Solutions for Class 9 Political Science Chapter 2

August 8, 202239 Insightful Publications

**Equivalent ratios** are ratios that can be simplified to the same value by dividing both the values (antecedent and consequent) by a common fraction. In other words, two ratios are considered to be equivalent if the bigger one in value can be expressed as a multiple of the other. Hence, to get the equivalent ratio of another ratio, we simply have to multiply the two quantities with the same number. The concept of equivalent ratios is similar to the concept of equivalent fractions. Examples of equivalent ratios are 1:2 and 4:8, 3:5 and 12:20, 9:4 and 18:8, etc. read the complete article to learn the concept of equivalent ratios in detail along with solved examples, methods to find them and common questions regarding them.

Before going into the details of equivalent ratios, we first need to understand what ratios are.

Their ratio is the relationship between two quantities of the same kind and in the same unit that is obtained by dividing one quantity by the other. Both the quantities must be of the same kind means, if one quantity is the number of students, the other quantity must also be the number of students. The ratio between two unlike quantities has no meaning.

For example, out of every \(20\) students, \(8\) of them passed in Mathematics. Therefore, in ratio, we can represent as \(8:20,\) which means from \(20\) students \(8\) students passed in Mathematics. We use the ratio while comparing two quantities. The general form to express a ratio between two quantities say \(a\) and \(b\) is \(a:b,\) which is read as \(a\) is to \(b.\)

A ratio can be represented as a fraction. The concept of an equivalent ratio is similar to the concept of equivalent fractions. A ratio that we get either by multiplying or dividing by the same number, other than zero, to the antecedent and the consequent of a ratio is called an equivalent ratio.

To get a ratio equivalent to a given ratio, we first represent the ratio in fraction form. Then, multiplying or dividing the first term and the second term by the same non-zero number, we can get the equivalent fraction. At last, we represent it in the ratio form.

Example: Find the equivalent ratio of \(1:5.\)

The fraction form of the ratio is \(\frac{1}{5}\)

\( \Rightarrow \frac{{1 \times 2}}{{5 \times 2}} = \frac{2}{{10}}\)

The equivalent ratio for \(\frac{1}{5}\) is \(\frac{2}{{10}}.\)

Let us take another example, \(8:24.\)

The fraction form of \(8:24\) is \(\frac{8}{{24}}.\)

We will divide the numerator and the denominator by the same non-zero number.

\( \Rightarrow \frac{{8 \div 4}}{{24 \div 4}} = \frac{2}{6}\) is the equivalent fraction of \(\frac{8}{{24}}.\)

Hence, the equivalent ratio is \(2:6\).

Let us see some examples of equivalent ratios. For example, when the first and the second term of the ratio \(2:5\) are multiplied by \(2,\) we get \((2×2):(5×2)\) or \(4:10.\) Here, \(2:5\) and \(4:10\) are equivalent ratios. Similarly, when both the terms of the ratio \(4:10,\) are divided by \(2,\) it gives the ratio as \(2:5.\)

If we multiply both the terms of \(1:6\) by \(100,\) we will get,

\(1×100 :6×100=100 :600.\)

Now, again dividing \(100:600\) by \(5\) we get,

\(100÷5 :600÷5=20 :120.\)

Similarly, dividing \(20:120\) by \(4\)

\(20÷4:120÷4=5 :30.\)

Therefore, \(100:600, 20:120, 5:30\) are the equivalent ratios of \(1:6.\)

To find the equivalent fractions, first, we should represent the given ratios in fraction form and then simplify them to see whether they are equivalent ratios or not. Simplification of the ratios can be done till both the antecedent and the consequent are still be whole numbers.

There are some different methods to check if the given ratios are equivalent or not.

1. Making the consequents the same

2. Finding the decimal form of both the ratios

3. Cross multiplication method

4. Visual method

The consequents of the ratios \(3:5\) and \(6:10\) are \(5\) and \(10.\) To make the process simple, we will represent it in fraction form that is \(\frac{3}{5}\) and \(\frac{6}{10}.\) The least common multiple (LCM) of the denominators \(5\) and \(10\) is \(10\). Now make the denominators of both fractions \(10,\) by multiplying them with suitable numbers.

\( \Rightarrow \frac{3}{5} = \frac{{3 \times 2}}{{5 \times 2}} = \frac{6}{{10}}\)

\( \Rightarrow \frac{6}{{10}} = \frac{{6 \times 1}}{{10 \times 1}} = \frac{6}{{10}}\)

Note that both the fractions are equivalent to the same fraction \(\frac{6}{10}\) or the ratio \(6:10.\) Thus, the given ratios are equivalent.

In this method, we find the decimal form of both the ratios after converting it to fraction form by actually dividing them.

We have to check whether \(\frac{3}{5}\) and \(\frac{6}{10}\) have the same value.

So, first, find the decimal value of each ratio.

\( \Rightarrow \frac{3}{5} = 0.6\)

\( \Rightarrow \frac{6}{{10}} = 0.6\)

The decimal values of both the fractions are the same, i.e., \(0.6.\)

Therefore, \(3:5\) and \(6:10\) are equivalent ratios.

To test whether two given ratios \(a:b, c:d\) are equivalent or not, first, write them in fraction form that is \(\frac{a}{b},\frac{c}{d}\) respectively.

Cross multiply the given fractions \(\frac{a}{b}\) & \(\frac{c}{d}.\)

If \(ad=bc,\) we say that* *\(a:b, c:d\)* *are equivalent ratios, otherwise not.

To check whether \(3:5\) and \(6:10\) are equivalent, we cross multiply the fraction forms of the ratios. If both the products are the same, then the fractions are equivalent ratios.

The fraction forms of the ratios are \(\frac{3}{5}\) and \(\frac{6}{10}.\)

\(⇒3×10=30, 5×6=30\)

\(⇒30=30\)

Note that both the obtained products are \(30,\) so the given ratios are equivalent.

Let us represent each of the ratios \(1:2, 2:4, 4:8\) pictorially on identical shapes and can identify if the shaded portions of both are equal or not. We can consider the portion of pizza as a shaded part.

We can see that the shaded portions of the circular shape are equal. Hence, the given ratios are equivalent.

** Q.1. Are the ratios** \(2:7\)

Then, we will cross multiply and get,

\(2 \times 12\,{\rm{\& }}\,7 \times 4\)

\( \Rightarrow 24 \ne 28\)

Therefore, \(2:7\) and \(4:12\) are not equivalent ratios.

** Q.2. Are the ratios** \(1:6\)

Cross multiplying the fraction forms of the ratios, we get

\( \Rightarrow 1 \times 12\,{\rm{\& }}\,2 \times 6\)

\(⇒12=12\)

Therefore, \(1:6\) and \(2:12\) are equivalent ratios.

** Q.3. Write ratio equivalent to** \(3:4\)

Here, we need to multiply both the numerator and the denominator with \(5\) as \(3×5=15.\)

\( \Rightarrow \frac{{3 \times 5}}{{4 \times 5}} = \frac{{15}}{{20}} = 15:20\)

Therefore, the equivalent ratio with the antecedent \(915\) is \(15:20.\)

** Q.4. Write three equivalent ratios for** \(5:6.\)

\( \Rightarrow \frac{{5 \times 2}}{{6 \times 2}} = \frac{{10}}{{12}} = 10:12\)

\( \Rightarrow \frac{{5 \times 3}}{{6 \times 3}} = \frac{{15}}{{18}} = 15:18\)

\( \Rightarrow \frac{{5 \times 4}}{{6 \times 4}} = \frac{{20}}{{24}} = 20:24\)

Therefore, \(10:12, 15:18, 20:24\) are three equivalent ratios of \(5:6.\)

** Q.5. Write the ratio equivalent to** \(2:5\)

Here, we need to multiply both the numerator and the denominator with \(5\) as \(5×5=25.\)

\( \Rightarrow \frac{{2 \times 5}}{{5 \times 5}} = \frac{{10}}{{25}} = 10:25\)

Therefore, the equivalent ratio with the consequent \(25\) is \(10:25.\)

In this article, we learnt in detail about ratios, equivalent ratios, and how to check the equivalent ratios. We have learned that to find the equivalent ratios of a given ratio, we need to write the fraction form of it. Then, we will multiply the numerator and the denominator of a fraction by the same non-zero number. The equivalent ratio of a given ratio does not change the value of the ratio.

The most frequently asked queries about equivalent ratios are answered below:

Q.1. How do you simplify equivalent ratios? To simplify an equivalent ratio, first, we write it in fraction form and then divide the numerator and the denominator of it by the same non-zero number.Ans:Example: Find the simplest form of equivalent ratio for \(3:6.\) \(\Rightarrow \frac{{3 \div 3}}{{6 \div 3}} = \frac{1}{2} = 1:2\) is the equivalent ratio for \(3:6.\) Therefore, the simplest equivalent ratio for \(3:6\) is \(1:2.\) |

Q.2. Explain the equivalent ratio with examples. A ratio that we get either by multiplying or dividing by the same non-zero number to the antecedent and the consequent of a ratio is called an equivalent ratio. For example, when the terms of the ratio \(2:5\) are multiplied by \(2,\) we get, \((2×2):(5×2)\) or \(4:10.\) Here, \(2:5\) and \(4:10\) are equivalent ratios.Ans: |

\(2:3\)Q.3. Write three examples of equivalent ratio? The ratio is \(2:3.\)Ans:Three examples of the equivalent ratios of \(2:3\) are \(4:6, 6:9, 8:12.\) \(2:3=4:6=6:9=8:12\) these ratios are called equivalent ratios. |

Q.4. How do you find the equivalent ratios? To find the equivalent ratios of a given ratio, we need to write the fraction form of it. Then, we will multiply the numerator and the denominator of a fraction by the same non-zero number. The equivalent ratio of a given ratio does not change the value of the ratio.Ans:\(1:2 = \frac{1}{2} = \frac{{1 \times 2}}{{2 \times 2}} = \frac{{1 \times 3}}{{2 \times 3}} = \frac{{1 \times 4}}{{2 \times 4}} = \frac{{1 \times 5}}{{2 \times 5}}\) Similarly, \(\frac{{2 \div 2}}{{4 \div 2}} = \frac{{3 \div 3}}{{6 \div 3}} = \frac{{4 \div 4}}{{8 \div 4}} = \frac{{5 \div 5}}{{10 \div 5}} = \frac{1}{2} = 1:2\) |

Q.5. How to test whether two given ratios are equivalent or not? let us say \(a:b, c:d\) are the ratios and the fraction form of them are \(\frac{a}{b}\,\& \,\frac{c}{d}\) respectively.Ans:Cross multiply the given fractions \(\frac{a}{b}\,\& \,\frac{c}{d}.\) If \(ad=bc,\) we say that \(a:b, c:d\) are the equivalent ratios, otherwise not. |

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