Compound Ratio: A ratio is a comparison of two or more numbers that indicates their sizes in relation to each other. Candidates might have some idea about a ratio but the concept of Compound Ratio might be entirely new to them. Let us understand what is Compound Ratio. When two or more ratios are multiplied termwise, the ratio obtained thus is called Compound Ratio. The first quantity of the ratio is called antecedent and the second quantity is called consequent. 

For instance, for two ratios, a:b and m:n, the resultant compound ratio is am: bn. Compound Ratio examples include Duplicate Ratio and Triplicate Ratio. A Compound ratio with two equal ratios is called a Duplicate Ratio. A Compound ratio with three equal ratios is called a Triplicate Ratio. Let us explore this article to learn the compound ratio formula in Maths, how to solve ratio sums, and more.

Ratio Formula in Maths

A ratio is a method of comparing quantities or numbers by division. We use ratios to compare two things and the sign used to denote a ratio is \(‘ : ‘\)

A ratio is a way of comparing quantities or numbers by division. As a ratio, we need to read \(\frac{1}{3}\) or \(1: 3\) as \(1\) is to \(3\), and not as one-third. The two numbers in the ratio are called the terms of a ratio. In \(1: 3,1\) and \(3\) are terms of a ratio. The first term is called an antecedent, and the second term is called a consequent.

The ratio should exist between the amounts of the same kind; while comparing two things, the units should be the same. In a ratio, the order of the terms is important. Suppose we are asked for the student to teacher ratio in a school. The number of students must be the antecedent, and the number of teachers should be the consequent. If it is represented in the other way, it will be meaningless.

A ratio gives us a good way to compare the size of two quantities. For example, we are making any dish. Then the recipe sometimes says to mix any ingredient and water in the ratio \(3\) parts to \(1\) part. So that means if you are using \(3\) cups of ingredient, then mix it with \(1\) cup of water. Both the terms should be non-zero to have a meaningful comparison. The ratio is either represented as \(p: q\) or \(\frac{p}{a}\).

Ratio in Simplest Form

A ratio \(m: n\) is said to be in the simplest form if its antecedent \(m\) and consequent \(n\) have no common factors other than \(1\) .

A ratio in the simplest form is also said as the ratio in the lowest form. The ratio \(85: 51\) is not in the simplest form because \(17\) is the common factor of its antecedent and consequent. The simplest form of the ratio \(85: 51\) is \(5: 3\), dividing the first and the second term by \(17\).

Properties of Ratio

1. To find the ratio of two quantities, they must be expressed in the same units.
   In a ratio, we compare two quantities. The comparison becomes meaningless if the quantities being compared are not of the same kind. That is, if not measured in the same units. For example, we cannot compare \(20\) hens with \(200\) cows.
   
2. The ratio has no unit, or it is independent of the units used in the quantities compared.
   Since the ratio of two quantities of the same kind determines how many times one quantity is contained by the other, the ratio of any two amounts of the same type is theoretical.
   
3. The order of the terms in a ratio is significant.
   For example, the ratio \(5:3\) is different from the ratio of \(3:5\).

What Is Compound Ratio?

For a compound ration, there should be at least two quantities. When two or more ratios multiplied term-wise, the ratio thus obtained is called compound ratio. In other words, for two or more ratios, if we take antecedent as the product of antecedents of the ratios and consequent as the product of consequents of the ratios, then the ratio thus formed is called mixed or compound ratio.

Thus, the product of two or more simple ratios like \(a:c\) and \(b:d\), i.e., \(ab:cd\) is a ratio compounded ratio.

For example, the compounded ratio of \(m:n, o:p\) and \(q:r\) is \(moq:npr\).

Compound Ratio Facts and Examples

Suppose the ratio is said to be a compound ratio. In that case, we should take antecedent as the product of antecedents of the ratios and consequent as the consequent product. For ratio \(a: b\) and \(c: d\), the compound ratio is \((a \times c):(b \times d)\). For ratio \(m: n, p: q\) and \(r: s ;\) the compound ratio is \((m \times p \times r):(n \times q \times s)\). Let us see some examples.

The compound ratio of \(5: 4\) and \(8: 5\) is \((5 \times 8):(4 \times 5)=40: 20=2: 1\). The compound ratio of \(2: 3,9: 4\) and \(1: 7\) is \((2 \times 9 \times 1):(3 \times 4 \times 7)=18: 84\)

Duplicate Ratio

A duplicate ratio is the compounded ratio of two equal ratios.

For example, the duplicate ratio of \(x: y\) is \(x^{2}: y^{2} \). In other words, the duplicate ratio of the ratio \(x: y=\) Compound ratio of \(x: y\) and \(x: y\).

Example: The duplicate ratio of \(4: 7=4^{2}: 7^{2}=16: 49\).

Triplicate Ratio

The triplicate ratio is the compound ratio of three equal ratios.

The triplicate ratio of the ratio \(x: y\) is \(x^{3}: y^{3}\). The triplicate ratio of the ratio \(x: y=\) Compound ratio of \(x: y, x: y\) and \(x: y\).

Example: The triplicate ratio of \(4: 7=4^{3}: 7^{3}=64: 343\)

Solved Compound Ratio Examples

Q1. Find the compound ratio of the following ratios.
a. \(3: 5\) and \(8: 15\)
b. \(a: b\) and \(b: c\)
c. \(\left(x^{2}-y^{2}\right):\left(x^{2}+y^{2}\right)\) and \(\left(x^{4}-y^{4}\right):(x+y)^{4}\)
A1:
a. Given \(3: 5\) and \(8: 15\)
For ratio \(a: b\) and \(c: d\), the compound ratio is \((a \times c):(b \times d)\)
So, the compound ratio of \(3: 5\) and \(8: 15\) is \((3 \times 8):(5 \times 15)=24: 75=8: 25.\)
b. Given: \(a: b\) and \(b: c\)
The compound ratio of \(a: b\) and \(b: c\) is \((a \times b):(b \times c)=a b: b c=a: c\)
c. Given: \(\left(x^{2}-y^{2}\right):\left(x^{2}+y^{2}\right)\) and \(\left(x^{4}-y^{4}\right):(x+y)^{4}\)
The compound ratio of \(\left(x^{2}-y^{2}\right):\left(x^{2}+y^{2}\right)\) and \(\left(x^{4}-y^{4}\right):(x+y)^{4}\) is
\(\left(x^{2}-y^{2}\right) \times\left(x^{4}-y^{4}\right):\left(x^{2}+y^{2}\right)(x+y)^{4}\)
\(\Rightarrow \frac{\left(x^{2}-y^{2}\right)}{\left(x^{2}+y^{2}\right)} \times \frac{\left(x^{4}-y^{4}\right)}{(x+y)^{4}}\)
\(\Rightarrow \frac{\left(x^{2}-y^{2}\right)}{\left(x^{2}+y^{2}\right)} \times \frac{\left(x^{2}-y^{2}\right) \times\left(x^{2}+y^{2}\right)}{(x+y)^{4}}\)
\(\Rightarrow \frac{\left(x^{2}-y^{2}\right)^{2}}{(x+y)^{4}}\)
\(\Rightarrow \frac{[(x-y)(x+y)]^{2}}{(x+y)^{4}}\)
\(\Rightarrow \frac{(x-y)^{2}}{(x+y)^{2}}\)
Therefore, the compound ratio of \(\left(x^{2}-y^{2}\right):\left(x^{2}+y^{2}\right)\) and \(\left(x^{4}-y^{4}\right):(x+y)^{4}\) is \((x-y)^{2}:(x+y)^{2}\)

Q2. Find the compound ratio of each of the following three ratios.
a. \(2: 3,9: 14\) and \(14: 27\)
b. \(x: y, y: z\) and \(z: w\)
c. \(2 a: 3 b, m n: x^{2}\) and \(x: n\)
A2:
a. Given: \(2: 3,9: 14\) and \(14: 27\)
For ratio \(m: n, p: q\) and \(r: s ;\) the compound ratio is \((m \times p \times r):(n \times q \times s)\).
So, the compound ratio of \(2: 3,9: 14\) and \(14: 27\) is \((2 \times 9 \times 14):(3 \times 14 \times 27)\)
\(\Rightarrow 252: 1134\)
Dividing the antecedent and consequent by \(14\), we get
\(18: 81\)
Dividing the antecedent and consequent by \(9\), we get
\(2: 9\)
Therefore, the compound ratio of \(2: 3,9: 14\) and \(14: 27\) is \(2: 9\)
b. Given: \(x: y, y: z\) and \(z: w\)
The compound ratio of \(x: y, y: z\) and \(z: w\) is \((x \times y \times z):(y \times z \times w)=x y z: y z w=x: w\)
Therefore, the compound ratio of \(x: y, y: z\) and \(z:w\) is \(x:w.\)
c. Given: \(2 a: 3 b, m n: x^{2}\) and \(x: n\)
The compound ratio of \(2 a: 3 b, m n: x^{2}\) and \(x: n\) is \(\left( {2\,a \times mn \times x} \right):\left( {3\,b \times {x^2} \times n} \right)\)
\( = \left( {2\,amn\,x} \right):\left( {3\,b{x^2}n} \right)\)
\(=2am:3bx\)

Q3. What is the triplicate ratio of \(3:5\) ?
A3: We know that the triplicate ratio of the ratio \(x: y\) is \(x^{3}: y^{3}\).
Therefore, the triplicate ratio of \(3: 5=3^{3}: 5^{3}=27: 125\)

Q4. The total number of students in a class is \(65.\) If the total number of girls in the class is \(35\), then the ratio of the total number of boys to the total number of girls is
A4: Given: Total number of students including boys and girls \(=65\)
Number of girls \(=30\)
Number of boys \(=(65-30)=35\)
The ratio of the total number of boys to the total number of girls is \(30: 35\).
Dividing the antecedent and consequent by \(5\), we get \(6: 7\).
So, the ratio of the total number of boys to the total number of girls is \(6: 7\)

Q5. What is the duplicate ratio of \(2: 7 ?\)
A5: We know that the duplicate ratio of \(x: y\) is \(x^{2}: y^{2}\).
So, the duplicate ratio of \(2: 7\) is \(2^{2}: 7^{2}=4: 49\)

Summary of Compound Ratio Formula

In this article, we have learned the definition of ratio, examples of ratio and the notation of the ratio. Also, we have known the meaning of compound ratio, examples of compound ratio, facts of compound ratio. Also, we got to know about the importance of duplicate ratio and triplicate ratio and solved some problems based on the ratio, compound ratio, duplicate ratio and triplicate ratio.

FAQs on Compound Ratio Sums

Q1. How do you find the compound ratio?
A1: Suppose the ratio is said to be a compound ratio. In that case, we should take antecedent as the product of antecedents of the ratios and consequent as the consequent’s product.

Q2. Explain compound ratio with example?
A2: For two or more ratios, if we take antecedent as the product of antecedents of the ratios and consequent as the product of the ratios consequents, then the ratio thus formed is called mixed ratio or compound ratio. For example, the compound ratio of \(a:b\) and \(c:d\) is \(a×c:b×d\).

Q3. What is the ratio of \(2\) by \(3\)?
A3: A ratio is a way of comparing quantities or numbers by division. The two numbers in a ratio can only be compared when they have the same unit. We make use of ratios to compare two things. The sign used to denote a ratio is \(‘:’\)
Therefore, the ratio of \(\frac{2}{3}\) is \(2: 3\).

Q4. What is the duplicate ratio of \(5\) is to \(7\) ?
A4: We know that the duplicate ratio of \(x: y\) is \(x^{2}: y^{2}\).
So, the duplicate ratio of \(5: 7\) is \(5^{2}: 7^{2}=25: 49\).

Q5. What is the compound ratio of \(5\) is to \(4\), and \(8\) is to \(5\)?
A5: Given \(5: 4\) and \(8: 5\).
For ratio \(a: b\) and \(c: d\), the compound ratio is \((a \times c):(b \times d)\).
So, the compound ratio of \(5: 4\) and \(8: 5\) is \((5 \times 8):(4 \times 5)=40: 20=2: 1\)

Q6. Where can I solve mock questions on Compound Ratio?
A6: Students can solve mock questions on Compound Ratio, for on Embibe.

Some other helpful articles by Embibe are provided below:

Foundation Concepts	Class-wise Mathematical Formulas
HCF And LCM	Maths Formulas For Class 6
Algebra Formulas	Maths Formulas For Class 7
BODMAS Rule	Maths Formulas For Class 8
Properties Of Triangles	Maths Formulas For Class 9
Trigonometry Formulas	Maths Formulas For Class 10
Mensuration Formulas	Maths Formulas For Class 11
Differentiation Formulas	Maths Formulas For Class 12

We hope this article on the compound ratio 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! Stay tuned to Embibe for the latest news and updates on Compound Ratio