Comparing Quantities: Definition, Formulas, Examples, NCERT Solutions

Comparing Quantities: In everyday life, we come across many situations, quantities, activities, etc. Many times, we need to compare and measure these quantities. The comparison is a daily activity among us. Sometimes we compare our weight, height, marks, speed, sometimes distance, quantity, etc. This list of quantities extends. We can only compare two exact quantities, i.e., a person’s weight cannot be compared with another person’s height. Hence, there should always be a common reference point for comparison.

Similarly, we need a standard measure or way for Comparing Quantities. The ratio and proportion are among them. So let us learn how to compare two quantities in terms of ratio and proportion. In this article, we will learn all about the comparison of different quantities.

Definition of Comparing Quantities

Comparing Quantities is the quantitative relation between two quantities which reflects the relative size of both the quantities. It is simply the means to compare the quantities.
Example:

In our daily life, there are many occasions when we compare two quantities. Suppose we are comparing the heights of Heena and Amir. We find that Heena is two times taller than Amir. Or Amir’s height is \(\frac{1}{2}\) of Heena’s height.

We see that the ratio for two different comparisons may be the same. However, remember that to compare two quantities, the units must be the same.

Ratio

A ratio is a comparison of two or more quantities with the same units. A ratio compares two quantities by division, with the dividend or number being divided termed the antecedent and the divisor or number that is dividing termed the consequent. The mathematical symbol used to denote ratio is “\(:\)” and read it as “is to”. For example: \(3:5 = \frac{3}{5}\)

Here, “\(3\)” is called the first term or antecedent, and “\(5\)” is called the second term or consequent. To compare the two quantities, their units must be the same. Two ratios can be compared by converting them into like fractions. If the two fractions are equal, we say that the two given ratios are equivalent or equivalent ratios.
Example: \(6\,{\text{cm}} : 3\,{\text{m}} = 6\,{\text{cm}} : 300\,{\text{cm}}\)
\( \Rightarrow \frac{{6\,{\text{cm}}}}{{6\,{\text{cm}}}} : \frac{{300\,{\text{cm}}}}{{6\,{\text{cm}}}}\)
\(\Rightarrow 1 : 50\)
So, \(6\,{\text{cm}} : 3\,{\text{m}} = 1 : 50\)

Therefore, \(6\,{\text{cm}} : 3\,{\text{m}}\) and \(1 : 50\) are equivalent ratios.

Example: Find the ratio of \(40\, {\text{cm}}\) and \(2\, {\text{m}}\).

Here, the given two quantities are not in the same unit. Therefore, we must convert them into the same units.
\({\text{2}}\,{\text{m=2}} \times {\text{100}}\,{\text{cm=200}}\,{\text{cm}}\)
Therefore, the required ratio is \(40\,{\text{cm}} {\text{:}} {\text{200}}\,{\text{cm}}\).
\(\frac{{40\,{\text{cm}}}}{{40\,{\text{cm}}}} {\text{:}} \frac{{{\text{200}}\,{\text{cm}}}}{{40\,{\text{cm}}}} = 1 : 5\)
Therefore, the required ratio is \(1 : 5\).

Proportion

Proportion is an equation that defines that the two given ratios are equivalent to each other. In other words, the proportion states the equality of the two fractions or the ratios. When two ratios are equal in value, then they are said to be in proportion. If two ratios are equivalent (or equal), then the involved four quantities are said to be in proportion.

A proportion is simply a statement that two ratios are equal. It can be written as
\(a:b = c:d\) or \(a:b :: c:d\)
Example: Are \(15,\,20,\,45\,\& \,60\) in proportion?
The given ratios can be written as \(15:20::45:60\)
The ratio of \(15\) to \(20\)\( = \frac{ {15}}{ {20}} = \frac{3}{4} = 3:4\)
The ratio of \(45\) to \(60\)\( = \frac{ {45}}{ {60}} = \frac{3}{4} = 3:4\)
Since, \(15:20=45:60\)

Therefore, the given numbers \(15\), \(20\), \(45\) and \(60\) are in proportion.

Comparing Quantities Examples

Ratio Formula

Let us consider that we have two quantities (or two numbers or two entities), and we have to find the ratio of these two. Then, the formula for ratio is defined as.
\(m:n = \frac{m}{n}\)
Where \(m\) and \(n\) could be any two quantities.

Here, \(“m”\) is called the first term or antecedent, and \(“n”\) is called the second term or consequent.
Example: The ratio \(6 : 11\) is represented by \(\frac{6}{ {11}}\) where \(6\) is antecedent and \(11\) is consequent.
If we multiply and divide each term of the ratio by the same number (non-zero), it does not affect the ratio.
Example: \(6:11 = 12:22 = 18:33\)

Proportion Formula

Now, let us assume that, in proportion, the two ratios are \(p:q\) and \(r:s\). The two terms ‘\(q\)’ and ‘\(r\)’ are called ‘means or mean term,’ whereas the terms ‘\(p\)’ and ‘\(s\)’ are known as ‘extremes or extreme terms.’
\(\frac{p}{q} = \frac{r}{s}\) or \(p:q::r:s\)
\( \Rightarrow {\text{Product of extreames}} = {\text{Product of means}}\)
\( \Rightarrow {\text{first term}} \times {\text{fourth term}} = {\text{second term}} \times {\text{third term}}\)
\( \Rightarrow pr = qs\)

Example: Are \(60\), \(80\), \(90\) & \(120\) in proportion?
The given can be written as \(60:80::90:120\)
Ratio of \(30\) to \(40\)\( = \frac{ {30}}{ {40}} = \frac{3}{4} = 3:4\)
Ratio of \(45\) to \(60\)\( = \frac{ {45}}{ {60}} = \frac{3}{4} = 3:4\)
Since, \(30:40 = 45:60\)

Therefore, the given numbers \(60\), \(80\), \(90\) and \(120\) are in proportion.

Alternative method:
\( {\text{Product of extreames}} = {\text{Product of means}}\)
\( \Rightarrow {\text{first term}} \times {\text{fourth term}} = {\text{second term}} \times {\text{third term}}\)
\(\Rightarrow 60 \times 120 = 80 \times 90\)
\(\Rightarrow 7200 = 7200\)
\( \Rightarrow {\text{L}}.{\text{H}}.{\text{S}} = {\text{R}}.{\text{H}}.{\text{S}}\)
Therefore, the given numbers \(60\), \(80\), \(90\) and \(120\) are in proportion.

Solved Examples – Comparing Quantities

Q.1. Find the ratio of \(90\, {\text{cm}}\) & \(1.5\, {\text{m}}\).
Ans:The given two quantities are not in the same unit. Therefore, we must convert them into the same units.
\(1.5\,{\text{m}} = 1.5 \times 100\,{\text{cm}} = 150\,{\text{cm}}\)
Therefore, the required ratio is \(90 : 150\).
\( \Rightarrow \frac{ {90}}{ {150}} = \frac{3}{5} = 3:5\)
Therefore, the required ratio is \(3 : 5\)

Q.2. Are \(30\), \(40\), \(45\) & \(60\) in proportion?
Ans: Ratio of \(30\) to \(40\)\( = \frac{ {30}}{ {40}} = \frac{3}{4} = 3:4\)
The ratio of \(45\) to \(60\)\( = \frac{ {45}}{ {60}} = \frac{3}{4} = 3:4\)
Since, \(30:40 = 45:60\)
Therefore, the given numbers \(30\), \(40\), \(45\) and \(60\) are in proportion.

Q.3. There are \(50\) boys and \(20\) girls in Supriya’s class. Find the ratio of the number of boys to the number of girls in Supriya’s class.
Ans: From the given information we get,
Number of boys \(=50\), Number of girls \(=20\)
The ratio of boys to girls \( = \frac{ { {\text{Number of boys}}}}{ { {\text{Number of girls}}}}\)
\( \Rightarrow \frac{ {50}}{ {20}} = \frac{5}{2} = 5:2\)
Therefore, the ratio of the number of boys to the number of girls in Supriya’s class is \(5:2\)

Q.4. The length of a lizard is \(25\,\rm{cm}\), and the length of a crocodile is \(4\,\rm{m}\). Find the ratio of their lengths.
Ans: From the given information we get,
Length of a lizard \( = 25\, {\text{cm}}\), length of a crocodile \( = 4\, {\text{m}} = 400\, {\text{cm}}\).
The ratio of the length of lizard to the length of crocodile \( = \frac{{{\text{length of lizard}}}}{{{\text{length of crocodile}}}}\)
\( = \frac{ {25}}{ {400}} = \frac{1}{ {16}} = 1:16\)
Therefore, the ratio is \(= 1:16\)

Q.5. Find if the ratios given \(7:3\) and \(5:2\) are equal and find if four members are in proportion.
Ans: The given information can be written as \(7:3=5:2\)
\( \Rightarrow \frac{7}{3} = \frac{5}{2}\)
\( \Rightarrow 14 \ne 15\)
Therefore, the four numbers are not in proportion.

Q.6. Are the ratios \(1:2\) and \(2:3\) equivalent?
Ans: To check this, we need to know whether \(\frac{1}{2} = \frac{2}{3}\)
We have,
\(\frac{1}{2} = \frac{ {1 \times 3}}{ {2 \times 3}} = \frac{3}{6}\) ; \(\frac{2}{3} = \frac{ {2 \times 2}}{ {3 \times 2}} = \frac{4}{6}\)
We find that, \(\frac{3}{6} < \frac{4}{6}\), which means that \(\frac{1}{2} < \frac{2}{3}\).
Therefore, the ratio \(1:2\) is not equivalent to the ratio \(2:3\).

Summary

To compare the two quantities, their units must be the same. Two ratios can be compared by converting them into like fractions. If the two fractions are equal, we say that the two given ratios are equivalent. If two ratios are equivalent (or equal), then the involved four quantities will be in proportion. This article helps to learn about Comparing Quantities, ratio, proportion, and its formulas. Furthermore, it helps to solve the problems of ratio and proportion.

Frequently Asked Questions (FAQ) – Comparing Quantities

Q.1. What are the formulas for Comparing Quantities?
Ans: The formula for ratio is defined as,
\(m:n = \frac{m}{n}\), where \(m\) and \(n\) could be any two quantities.
Here, \(“m”\) is called the first term or antecedent, and \(“n”\) is called the second term or consequent.
In proportion, the two ratios are \(p:q\) & \(r:s\). The two terms ‘\(q\)’ and ‘\(r\)’ is called ‘means or mean term, whereas the terms ‘\(p\)’ and ‘\(s\)’ are known as ‘extremes or extreme terms’.
Then, the formula for proportion is given by,
\(\frac{p}{q} = \frac{r}{s}\) or \(p:q::r:s\)

Q.2. What are Comparing Quantities?
Ans: Comparing Quantities is the quantitative relation between two quantities, reflecting the relative size of both quantities. It is simply the means to compare the quantities.

Q.3. How do you teach Comparing Quantities?
Ans: When teaching the comparison of two or more quantities, it is important for a child to understand the order of number, and this can be taught by using a number line. Once the order of numbers can be visualized in a sequence, we can practice counting objects.

Q.4. How do you understand Comparing Quantities?
Ans: We know, Comparing Quantities is the quantitative relation between two quantities that reflect both quantities’ relative size. It is simply the means to compare the quantities. To compare means to examine the differences between numbers, quantities, or values to decide if it is greater than, smaller than or equal to another quantity.

Q.5. Define ratio.
Ans: A ratio is a comparison of two or more quantities with the same units. A ratio compares two quantities by division, with the dividend or number being divided termed the antecedent and the divisor or number that is dividing termed the consequent. The mathematical symbol used to denote ratio is “\(:\)” and read it as “is to”.

Q.6. What are equivalent ratios?
Ans: To compare two quantities, their units must be the same. Two ratios can be compared by converting them into like fractions. If the two fractions are equal, we say that the two given ratios are equivalent or equivalent ratios. Equivalent ratios are obtained by dividing or multiplying both the antecedent and consequent of the given ratio by the same number.

Some other helpful articles by Embibe are provided below:

We hope this article on Comparing Quantities has provided significant value to your knowledge. If you have any queries or suggestions, feel to write them down in the comment section below. We will love to hear from you. Embibe wishes you all the best of luck!