Comparison of Quantities Using Ratio: When comparing two quantities of the same kind, we use the ratio. According to the ratio formula for two numbers, \(a\) and \(b\) are equal to \(a:b\) or \(\frac{a}{b}.\) Two or more such ratios are said to be in proportion when they are equal. The fundamental concepts of ratio and proportion are ratios and fractions.

How to Compare Quantities with Ratios?

The units of two quantities must be the same to compare them. By transforming two ratios into like fractions, they may be compared. We say the two supplied ratios are equivalent if their equivalent fractions are equal. The four quantities involved are in proportion if two ratios are equivalent (or equal).
We frequently encounter circumstances in which we must compare quantities regarding their magnitudes/measures in our daily lives. For example, at the time of admission to a school, we compare students’ grades, whereas we compare candidates’ weights/heights at the time of police recruiting. In most cases, the comparison is made in one of two ways:
(i) By calculating the difference between two quantities’ magnitudes. This is referred to as a difference comparison.
(ii) By determining the magnitudes of two quantities and dividing them. This is referred to as a divisional comparison.

Difference Between Quantities and Ratios

A quantity is a measurement, a number, or an amount. Quantities are compared using ratios. Ratios assist us in comparing and determining the relationship between numbers. A ratio is a result of dividing one quantity by the other to compare two similar quantities. A ratio is an abstract number since it is simply a comparison or relationship between two quantities.

We frequently encounter circumstances in which we must compare quantities regarding their magnitudes/measurements in our daily lives. In general, we compare two quantities by calculating the difference between their magnitudes or dividing their magnitudes. We find the difference in their magnitudes when we wish to see how much more or less one quantity is than the other, and this is known as a comparison by difference. We find the ratio or division of their magnitudes to see how many times more or less one quantity is than the other, and this is known as a comparison by division. When we divide two amounts of the same kind, we establish a ratio of the two quantities.
Example: Assume Heena is \(150\,{\rm{cm}}\) tall, and her brother Amir is \(100\,{\rm{cm}}\) tall. If we compare Heena and Amir’s heights by difference, we may claim Heena is \(50\,{\rm{cm}}\) taller than her brother Amir.

Examples of Comparison of Quantities Using Ratio

A ratio is a comparison of two numbers by division. The value of a ratio is the quotient that results from dividing the two numbers.

For example, if Nishitha and Naira are two sisters with their weights as \(35\,{\rm{kg}}\) and \(25\,{\rm{kg}},\) respectively, we can say that Nishitha’s weight is greater than Naira’s weight \(\left( {35 – 25} \right)\,{\rm{kg = 10}}\,{\rm{kg}}.\) The weight of Nishitha and Naira can also be compared by finding their division.
We have, \(\frac{{{\rm{weight}}\,{\rm{of}}\,{\rm{Nishitha}}}}{{{\rm{weight}}\,{\rm{of}}\,{\rm{Naira}}}} = \frac{{35}}{{25}} = \frac{7}{5}.\)
So, we can say Nishita’s weight is \(\frac{7}{5}\) times the weight of Naira.

How to Compare Two Quantities Using Ratio?

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.

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 \(a\) and \(b\) are called the terms of the ratio. The former \(‘a’\) is called the first term or antecedent, and the later \(‘b’\) is known as the second term or consequent.
Thus, in the ratio \(a:b,\) the term \(a\) is called an antecedent, and the term \(b\) is called a consequent.

Solved Examples – Comparison of Quantities Using Ratio

Q.1. Express the following in the language of ratios: The length of the rectangle is double its breadth.
Ans: The ratio of length to breadth of the rectangle is \(2:1,\) or the length and breadth of the rectangle are in the ratio \(2:1.\)

Q.2. Express the ratio \(150:400\) in its simplest form.
Ans: The given ratio is \(150:400 = \frac{{150}}{{400}}.\)
To express this ratio in the simplest form, we will have to find the \({\rm{HCF}}\) of \(150\) and \(400.\) It is \(50.\)
Dividing each term of the ratio by the \({\rm{HCF}}\) of its terms, i.e. \(50,\) we get
\(\frac{{150}}{{400}} = \frac{{150 \div 50}}{{400 \div 50}} = \frac{3}{8}\) or \(3:8.\)
Hence, the simplest form of the ratio \(150:400\) is \(3:8.\)

Q.3. Find the ratio of \({\rm{200}}\,{\rm{gms}}\) to \({\rm{4}}\,{\rm{kg}}.\)
Ans: Given, \({\rm{200}}\,{\rm{gms}}\) and \({\rm{4}}\,{\rm{kg}}.\)
We need to find the ratio of given quantities.
That is, \(200\,{\rm{gms}}:4\,{\rm{kg}}\)
\( = 200\,{\rm{gms}}:4000\,{\rm{gms}}\left( {∵ \,1\,{\rm{kg}} = 1000\,{\rm{gms}}} \right)\)
\( = \frac{{200}}{{4000}} = \frac{{200 \div 200}}{{4000 \div 200}}\) (∵ \({\rm{HCF}}\) of \(200\) and \(4000\) is \(200\))
\(\frac{1}{{20}}\) or \(1:20.\)
Hence, the ratio of the given quantities is \(1:20.\)

Q.4. Divide 108 in two parts in the ratio \(4:5.\)
Ans: We have, sum of the terms of the ratio \( = \left( {4 + 5} \right) = 9\)
Thus, the first part \( = \frac{4}{9} \times 108 = 4 \times 12 = 48\) and the second part \( = \frac{5}{9} \times 108 = 5 \times 12 = 60.\)
Hence, the two parts of \(108\) are \(48\) and \(60.\)

Q.5. If \(\left( {4x + 5} \right):\left( {3x + 11} \right) = 13:17,\) find the value of \(x.\)
Ans: We have, \(\left( {4x + 5} \right):\left( {3x + 11} \right) = 13:17.\)
\( \Rightarrow \frac{{4x + 5}}{{3x + 11}} = \frac{{13}}{{17}}\)
\( \Rightarrow 17\left( {4x + 5} \right) = 13\left( {3x + 11} \right)\)
\( \Rightarrow 68x + 85 = 39x + 143\)
\( \Rightarrow 68x – 39x = 143 – 85\)
\( \Rightarrow 29x = 58\)
\( \Rightarrow x = \frac{{58}}{{29}} = 2\)
Hence, the value of \(x\) is \(2.\)

Summary

In this article, we learnt about the definition of quantities, how to compare quantities with ratios, the difference between quantities and ratio, examples of comparison of quantities using ratio, ratio the comparison of two quantities, solved examples on comparison of quantities using ratio, FAQs on comparison of quantities using ratio.
This article’s learning outcome is that we learnt how to compare two or more quantities using ratios.

Frequently Asked Questions – Comparison of Quantities Using Ratio

Q.1. What is the comparison of quantities using ratio?
Ans: The term comparison of quantities using ratio refers to examining the relationship between two or more ratios. A ratio is a quantitative relationship between two amounts or numbers, and when two or more quantities are involved, a comparison of ratios is required.

Q.2. How do you compare quantities with ratios?
Ans: The units of two quantities must be the same to compare them. By transforming two ratios into like fractions, they may be compared. We say the two supplied ratios are equivalent if the two fractions are equal. The four quantities involved are in proportion if two ratios are equivalent (or equal).

Q.3. What are the three ratio comparisons?
Ans: The terms of the proportion are \(A,\,B,\,C\) and \(D.\) Its extremes are \(A\) and \(D,\) while its means are \(B\) and \(C.\,A\) continuous proportion is defined as the equality of two or more ratios, such as \(AD = BC.\)

Q.4. Is a ratio the comparison of two quantities?
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.
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 \(a\) and \(b\) are called the terms of the ratio. The former \(‘a’\) is called the first term or antecedent, and the later \(‘b’\) is known as the second term or consequent.

Q.5. How do you compare two quantities?
Ans: A ratio results from dividing one quantity by the other to compare two similar quantities.
Two quantities must have the same units to be compared. By transforming two ratios into like fractions, you can compare them. The two given ratios are said to be equivalent if their fractions are equal. The relevant four quantities are said to be in proportion if two ratios are equivalent (or equal).

Q.6. What are some examples of comparison of quantities using ratio?
Ans: For example, if Nishitha and Naira are two sisters with their heights as \(165\,{\rm{cm}}\) and \(155\,{\rm{cm}},\) respectively, we can say that Nishitha’s weight is greater than Naira’s weight \(\left( {165 – 155} \right)\,{\rm{cm}} = 10\,{\rm{cm}}.\)
The weight of Nishitha and Naira can also be compared by finding their division.
We have, \(\frac{{{\rm{Height}}\,{\rm{of}}\,{\rm{Nishitha}}}}{{{\rm{Height}}\,{\rm{of}}\,{\rm{Naira}}}} = \frac{{165}}{{155}} = \frac{{33}}{{31}}.\)
So, we can say Nishita’s weight is \(\frac{{33}}{{31}}\) times the weight of Naira.