Ratio: Comparison is a general phenomenon used in daily life to compare two similar quantities. It is used to compare how big or small one amount is compared to the other amount. Suppose for two numbers, \(a\) and \(b,\) we say \(a\) is one-fourth of \(b,\,a\) is triple of \(b,\,a\) is equal to \(b,\) etc. In our daily life, we too use the concept of ratio. For example, while baking a cake muffin or doughnut, we need a perfect flour, sugar, and butter ratio.

Ratio is a number that represents the relationship between two homogeneous quantities, which shows how many times one quantity is less or more than the other. In this article, we will learn about ratios, their definition, methods of calculation, different types of ratios and some solved examples.

What is a Ratio?

The relation of two quantities of the same kind and in the same unit obtained on dividing one quantity by the other is called their ratio. Thus, given any two similar quantities \(a\) and \(b,\) the ratio of \(a\) to \(b\) is denoted by \(a:b\) and is defined as \(a:b = \frac{a}{b},\) where \(b \ne 0.\)

The colon mark \(\left( : \right)\) denotes the ratio of two numbers or quantities. The first term of the ratio is called the antecedent, and the second term is called the consequent. Thus, for example, in the ratio \(3:7,\,3\) is the antecedent, and \(4\) is the consequent.

The value of the ratio remains unchanged if its antecedent and consequent are multiplied by the same non-zero number.

A ratio must always be expressed in its lowest terms. Whatever be the units of the terms of a ratio, the ratio has no unit. For example, the ratio of \(25\,{\rm{kg}}\) and \(30\,{\rm{kg}} = \frac{{{\rm{25}}\,{\rm{kg}}}}{{{\rm{30}}\,{\rm{kg}}}} = \frac{5}{6} = 5:6\) Here, the two quantities, \(25\,{\rm{kg}}\) and \(30\,{\rm{kg}},\) have the unit as \({\rm{kg,}}\) but their ratio \(5:6\) has no unit. Thus, on dividing, the unit cancels out.

Meaning of Two Quantities of the Same Kind and in the Same Unit

1. Both the quantities must be of the same kind, which means if one quantity is weight, then the other quantity must also be weight. If one quantity represents distance, then the other quantity must also represent the distance. The ratio between unlike quantities has no meaning. For example, the ratio of length to mass has no sense at all.

2. Both the quantities must be in the same unit means both quantities must have the same unit of measurements. For example, if the mass of both the objects is given in \(45\,{\rm{kg}}\) and \(2500\,{\rm{gm}},\) then before finding the ratio of one mass to the other, both the mass must be either be converted into \({\rm{kg}}\) or \({\rm{gm}}{\rm{.}}\)

Simplification of Ratios

A ratio has no unit and can be expressed as \(\frac{a}{b}.\) Since \(\frac{a}{b}\) is a fraction, it can have equivalent fractions like \(\frac{{\frac{a}{m}}}{{\frac{b}{m}}}\) and \(\frac{{ma}}{{mb}},\) for any integer \(m,\) where \(m \ne 0.\)

When we equate all three fractions, we get,
\(\frac{{\frac{a}{m}}}{{\frac{b}{m}}} = \frac{{ma}}{{mb}} = \frac{a}{b}\)
Rewriting the fractions using the ratio notation, we get,
\(\frac{a}{m}:\frac{b}{m} = ma:mb = a:b,\) for any integer \(m,\) where \(m \ne 0.\)

We call \(\frac{a}{m} = \frac{b}{m}\) and \(ma:mb\) as equivalent ratios of \(a:b,\) where \(a\) and \(b\) can be any rational numbers and \(b \ne 0.\)
The simplest form of a ratio \(a:b\) is where the terms \(a\) and \(b\) are integers and have no common factors other than \(1.\)

Let us understand this with the help of a couple of examples.

Example: Express each ratio in the simplest form:
(a) \(1:\frac{1}{3}\)
(b) \(2\frac{1}{4}:3\frac{3}{5}\)
Ans:
(a) \(1:\frac{1}{3}\)
\(1:\frac{1}{3} = 1 \times 3:\frac{1}{3} \times 3\,\left( {a:b = ma:mb} \right)\)
\( \Rightarrow 3:1\)
(b) \(2\frac{1}{4}:3\frac{3}{5}\)
\(2\frac{1}{4}:3\frac{3}{5} = \frac{9}{4}:\frac{{18}}{5}\)
\( \Rightarrow \frac{9}{4} \times 20:\frac{{18}}{5} \times 20\)
\( \Rightarrow 45:72 = \frac{{45}}{9}:\frac{{72}}{9}\)
\( \Rightarrow 5:8\)

Ratio of Three Quantities \(x:y:z\)

A ratio can also be used to represent the relationship of more than two quantities. Three quantities \(x, y\) and \(z,\) all of the same kind and with the same unit, are said to be in ratio \(x:y:z,\) the quantities can be taken as \(xk:yk\) and \(zk\) respectively, where \(k\) is a positive integer, i.e., for ratio \(x:y:z.\)

A ratio involving three quantities cannot be written as a fraction. However, it can be simplified by multiplying or dividing each term by the same constant.

The first quantity \( = xk,\) the second quantity \( = yk,\) and the third quantity \(=zk.\)

For example,
If \(X:Y:Z = 4:5:6\) and \(k = 3\)
Then, \(A = 4\,k = 4 \times 3 = 12,\)
\(B = 5\,k = 5 \times 3 = 15\) and
\(C = 6\,k = 6 \times 3 = 18\)

In the same way,
In the ages of Messi, Ronaldo and Neymar are in the ratio \(11:12:13\) and \(k=3,\) then,
The age of Mess \( = 11\,k = 11 \times 3 = 33\) years,
The age of Ronaldo \( = 12\,k = 12 \times 3 = 36\) years,
And the age of Neymar \( = 10\,k = 10 \times 3 = 30\) years

Let us solve an example.
Simplify the ratio \(\frac{1}{6}:\frac{1}{3}:\frac{1}{8}\)
LCM of consequents (denominators) \(6, 3\) and \(8=24\)
Therefore, \(\frac{1}{6}:\frac{1}{3}:\frac{1}{8} = \frac{1}{6} \times 24:\frac{1}{3} \times 24:\frac{1}{8} \times 24\)
\( = 4:8:3\)

Comparing the Ratios

For any ratios \(\frac{x}{y}\) and \(\frac{p}{q},\) if
1. \(x \times q = y \times p \Rightarrow \frac{x}{y} = \frac{p}{q}\) i.e., both the ratios are equal.
2. \(x \times q > y \times p \Rightarrow \frac{x}{y} > \frac{p}{q}\) i.e., \(\frac{x}{y}\) is greater than \(\frac{p}{q}.\)
3. \(x \times q < y \times p \Rightarrow \frac{x}{y} < \frac{p}{q}\) i.e., \(\frac{x}{y}\) is smaller than \(\frac{p}{q}.\)

Let us understand this with the help of examples.

Example 1: Which ratio is greater \(\frac{{12}}{{17}}\) or \(\frac{{15}}{{19}}?\)
Solution: \(\frac{{12}}{{17}}\) or \(\frac{{15}}{{19}} \Rightarrow 12 \times 19\) or \(17 \times 15\)
\( \Rightarrow 228\) or \(225\)
Since \(225 > 228 \Rightarrow \frac{{15}}{{19}}\) is greater.

Example 2: Which ratio is smaller \(\frac{5}{8}\) or \(\frac{8}{{11}}?\)
Solution: \(\frac{5}{8}\) or \(\frac{8}{{11}} \Rightarrow 5 \times 11\) or \(8 \times 8\)
\( \Rightarrow 55\) or \(64\)
Since \(55 < 64 \Rightarrow \frac{5}{8}\) is smaller.

Increase or Decrease in a Given Ratio

1. If the quantity is increased in the ratio \(x:y\) (where \(y > x\)), then, the new (resulting) quantity \( = \frac{y}{x} \times \)the given quantity.
2. If the quantity is decreased in the ratio \(x:y\) (where\(y < x\)), then, the new (resulting) quantity \( = \frac{y}{x} \times \)the given quantity.

Let us understand the above points of increase and decrease with the help of examples.

Example 1: Increase \(345\) in the ratio \(3:4.\)
Solution: The increased quantity \( = \frac{4}{3} \times 345 = 4 \times 115 = 460\)

Example 2: Decrease \(570\) in the ratio \(5:2.\)
Solution: The decreased quantity \( = \frac{2}{5} \times 570 = 2 \times 114 = 228\)

Word Problems

Ratios can be applied to solve problems such as those involving a part to the whole, mixture, and sharing by ratios.

Example 1: The strength of a class is \(50\) with \(30\) girls and the remaining boys. Find the ratio of the number of boys to the number of girls in the class.
Solution: The strength of the class \(=50\)
And, the number of girls in the \(=30\)
The number of boys in the class \(=50-30=20\)
Thus, the required ratio \( = \frac{{{\rm{Number}}\,{\rm{of}}\,{\rm{boys}}\,{\rm{in}}\,{\rm{the}}\,{\rm{class}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{girls}}\,{\rm{in}}\,{\rm{the}}\,{\rm{class}}}} = \frac{{20}}{{30}} = \frac{2}{3}\)
\( = 2:3\)

Example 2: The ages of \(X\) and \(Y\) are in the ratio 5:4. If \(Y\)’s age is \(16\) years, find the age of \(X.\)
Solution: \(X\)’s age\(:Y\)’s age \(=5:4\)
Therefore, if \(Y\)’s age \( = 4\) years, \(X\)’s age is \(5\) years
And, if \(Y\)’s age \( = 1\) year, \(X\)’s age is \(\frac{5}{4}\) years
Thus, if \(Y\)’s age\( = 16\) year, \(X\)’s age is \(\frac{5}{4} \times 16 = 20\) years.
Hence, the age of \(X\) is \(20\) years.

Example 3: The monthly salaries of Priya and Jyoti are in the ratio \(5:6.\) When Priya’s salary increases by \(₹2000\) and Jyoti’s salary increases by \(₹3300,\) their salaries ratio become \(4:5.\) Find their original monthly salaries.
Solution: Let the monthly salaries of Priya and Jyoti be \(₹5x\) and \(₹5x,\) respectively.
Increased salary of Priya \( = ₹\left( {5x + 2000} \right)\) and increased salary of Jyoti \( = ₹\left( {6x + 3300} \right).\)
The ratio of their new salary\( = 4:5\)
Therefore, \(\frac{{5x + 2000}}{{6x + 3300}} = \frac{4}{5}\)
After cross-multiplication, we get,
\(5\left( {5x + 2000} \right) = 4\left( {6x + 3300} \right)\)
\( \Rightarrow 25x + 10000 = 24x + 13200\)
\( \Rightarrow 25x – 24x = 13200 – 10000\)
\( \Rightarrow x = 3200\)
Hence, the salary of Priya \( = ₹\left( {5 \times 3200} \right) = ₹16000\) and salary of Jyoti \( = ₹\left( {6 \times 3200} \right) = ₹19200\)

Types of Ratios

There are 4 types of ratios. We have listed the types below:

1. Simple Ratio: Such a ratio in which both the terms are prime among themselves, then it is called a simple ratio. For example: x : y or 4 : 5
2. Inverse Ratio: The ratio in which the two terms are interchanged is called inverse proportion. For example: The inverse ratio of x : y will be y : x.
3. Compound Ratio: The new ratio formed by the products of the previous terms and the last terms of two or more ratios is called mixed ratio. For example: The mixed ratio of two ratios (a : b) and (c : d) will be (ac : bd).
4. Square Ratio: If a ratio is mixed with the same to make a new ratio, then it is called square ratio. For example: Square Ratio of X : Y X × X : Y × Y i.e. X² : Y²

Solved Examples – Ratio

Q.1. Find the ratio of \(5\,{\rm{km}}\) to \(600\,{\rm{m}}.\)
Ans: Let us first convert the distance to the same unit.
Therefore, \(5\,{\rm{km}} = 5 \times 1000\,{\rm{m}} = 5000\,{\rm{m}}.\)
Now, the required ratio \( = 5\,{\rm{km}}:600\,{\rm{m}}\)
\( = 5000\,{\rm{m}}:600\,{\rm{m}}\)
\( = 5000:600\)
\( = \frac{{5000}}{{200}}:\frac{{600}}{{200}}\)
\( = 25:3\)
Hence, the required ratio is \(25:3.\)

Q.2. Express as simplest ratio \(3\frac{1}{2}:2\frac{1}{3}.\)
Ans: Divide the first term of the ratio by its second term and then simplify.
\(3\frac{1}{2}:2\frac{1}{3} = \frac{7}{2}:\frac{7}{3}\)
\( = \frac{{\frac{7}{2}}}{{\frac{7}{3}}}\)
\( = \frac{7}{2} \times \frac{7}{3} = \frac{3}{2}\)
\( = 3:2\)
Hence, the required ratio is \(3:2.\)

Q.3. If the ratio between \(x + 3\) and \(2x – 3\) is \(5:7;\) find \(x.\)?
Ans: Given, \(x + 3\) and \(2x – 3\) is \(5:7\)
Thus, \(\frac{{x + 3}}{{2x – 3}} = \frac{5}{7}\)
After cross-multiplication, we get,
\( = 7\left( {x + 3} \right) = 5\left( {2x – 3} \right)\)
\( = 7x + 21 = 10x – 15\)
\( = 21 + 15 = 10x – 7x\)
\( = 36 = 3x\)
And, \(x = \frac{{36}}{3} = 12\)
Hence, the value of \(x\) is \(12.\)

Q.4. Decrease \(800\) in the ratio \(5:3\) and then increase the result in the ratio \(2:5.\)
Ans: The given quantity \( = 800\) and decrease in the ratio is \(5:3.\)
Therefore, the decreased quantity \( = \frac{3}{5} \times 800 = 480\)
Now, the quantity is increased in the ratio of \(2:5.\)
Therefore, the resulting quantit \( = \frac{5}{2} \times 480 = 1200\)

Q.5. Coffee costs \(₹80\) per 50 g and tea costs |(₹440\) per \({\rm{kg}}{\rm{.}}\) Find the ratio of cost of \({\rm{1}}\,{\rm{g}}\) coffee to \({\rm{1}}\,{\rm{g}}\) tea.
Ans: Cost of \({\rm{50}}\,{\rm{g}}\) coffee \({\rm{ = ₹80}}\)
Therefore, the cost of 1 g coffee\( = ₹\frac{{80}}{{50}} = ₹\frac{8}{5}\)
Again, the cost of \(1000\,{\rm{g}}\) tea \(=₹440\)
Therefore, the cost of \(1\,{\rm{g}}\) tea \( = ₹\frac{{440}}{{1000}} =₹ \frac{{11}}{{25}}\)
Required ratio\( = \frac{{{\rm{Cost}}\,{\rm{of}}\,{\rm{1}}\,{\rm{g}}\,{\rm{coffee}}}}{{{\rm{Cost}}\,{\rm{of}}\,{\rm{1}}\,{\rm{g}}\,{\rm{tea}}}}\)
\( \Rightarrow \frac{{\frac{8}{5}}}{{\frac{{11}}{{25}}}} = \frac{8}{5} \times \frac{{25}}{{11}} = 40:11\)

Summary

In this article, we learned about ratios. We studied simplifying the ratios and also finding the ratio between the quantities. We now know that a ratio does not have any unit. In addition to this, we learned to compare the ratios and increase and decrease in a ratio. To make our grip strong on ratio concepts, we learned to solve examples based on ratios.

FAQs

Q.1. How do you calculate ratios?
Ans: To calculate the ratio between two quantities, say \(a\) and \(b,\) convert them to the same units and then divide them. The ratio will be \(\frac{a}{b}\) or can also be written as \(a:b.\) We can also reduce the terms to the lowest form by dividing them by their HCF.

Q.2. Define ratio.
Ans: The relation of two quantities (both of the same kind and in the same unit) obtained on dividing one quantity by the other is called their ratio. Thus, given any two similar quantities \(a\) and \(b,\) the ratio of \(a\) to \(b\) (denoted by \(a:b\)) is defined as \(a:b = \frac{a}{b},\) where \(b \ne 0 .\) The first term of the ratio is called the antecedent, and the second term is called the consequent.

Q.3. What are the different ways to write a ratio?
Ans: The ratio of two quantities \(a,\) and \(b,\) both of the same kind in the same unit, is \(\frac{a}{b},\) and is often written as \(a:b\) (read as \(a\) to \(b\) or \(a\) is to \(b\)).

Q.4. How do you simplify a ratio?
Ans: Ratios can be simplified in the same way as fractions.

Q.5. How do you convert ratios?
Ans: A ratio can be expressed as \(\frac{a}{b}.\) Since \(\frac{a}{b}\) is a fraction, it can also be represented as,
\(\frac{{\frac{a}{m}}}{{\frac{b}{m}}}\) and \(\frac{{ma}}{{mb}},\) for any integer \(m,\) where \(m \ne 0\)
After equating all three fractions, we get,
\(\frac{{\frac{a}{m}}}{{\frac{b}{m}}} = \frac{{ma}}{{mb}} = \frac{a}{b},\)
\(\frac{a}{m} = \frac{b}{m} = ma:mb = a:b,\) for any integer \(m,\) where \(m \ne 0.\)

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