Tag: Numerator

Introduction

There are numerous quantities and measures that cannot be stated just in terms of integers. Rational numbers were crucial in expressing how such quantities were measured. These quantities included time, money, length, and weight. Some of the quantities for which the rational numbers are most frequently employed include those ones. Rational numbers are also required in trigonometry in addition to counting and measuring. The trigonometric ratios are expressed as rational numbers. Calculations based on the Pythagorean Theorem employ a specific kind of rational integer. 

Rational Number 

The rational number can be described as the ratios expressed in numbers. The term “rational” contains the word “ratio” as well. Therefore, any ratio is represented by rational numbers. These ratios may be lower than one to one or higher than one. Let’s comprehend how rational numbers should be explained. Rational numbers are defined as any number which can be expressed in the form of where a and b are coprime integers and b ≠ 0. The denominator is not equal to zero and both the numerator “a” and denominator “b” have integer values. The outcome of the division method used to simplify the rational number is in decimal form. The decimal representation of a rational number can either be non-terminating repeating decimals or terminating decimals.

How to find Rational Numbers

Verify that each given number meets the following requirements.

• The amount must be expressed as a fraction with a denominator greater than or equal to 0.
• To get the decimals, the fraction can be further decomposed.
• Positive, negative, and 0 are all included in the set of rational numbers, which is represented as a fraction. Because they may be written as a fraction, each whole number and integer is a rational number.

Types of Rational Numbers

Positive Rational Numbers

The positive rational numbers are signified as the rational numbers having positive numerators and denominators. The rational numbers  and are positive rational numbers.

Negative Rational Numbers

The negative rational numbers are signified as the rational numbers having any one of the numerators and denominators less than 0. The rational numbers  and are negative rational numbers.

Integers

The integers can be expressed as fractions having a denominator of one. Therefore, all integers are a class of rational numbers. Integers can have the forms of 0, -8, 56 etc.

You can also read our detailed article on Positive and Negative Rational Numbers.

Terminating Decimals

The decimals are the outcome of simplifying rational numbers. Some values following the decimal point may be where these decimals end. Terminating decimals are the name given to these rational numbers. For example: 0.235, 0.056, etc.

Non-Terminating Repeating Decimals are one Type of Rational Number.

Any rational integer is a non-terminating repeating decimal if, after simplification, the outcome is a decimal with repeating digits after the decimal point. A single digit or a group of digits can be one of the recurring values. For example: 0.5533, 0,222, 0.659659, etc.

Summary

Rational numbers are the numbers that can be written in the form of a fraction, where numerator and denominator are integers. The rational numbers are represented in the form of p/q where,q the denominator is not equal to 0.  Five separate categories of rational numbers exist. Both the numerator and the denominator are bigger than zero with positive rational numbers. Any numerator or denominator of a negative rational number is less than zero. Rational numbers that have a denominator of 1 are known as integers. The rational numbers also include recurring decimals that do not terminate.

Practice Solved Example

Example: The decimal expansions of some real numbers are given below. In each case, decide whether they are rational or not. If they are rational, write in the form of p/q. 

a. 0.140140014000140000…    

We have, 0.140140014000140000… It is a non-terminating and non-repeating. So, it is irrational. It cannot be written in the form of P/q.

b.

We have,    a non-terminating but repeating decimal expansion. So, it is Rational.

Let x =

Then, x = 0.1616 ——–1

100x = 16.1616 —-2

On subtracting 1 from 2 we get,

100x – x = 16.1616-0.1616

99x = 16

x =

Frequently Asked Questions

1.The number of Rational numbers between 25 and 26 is Finite. State the give statement is True or False.

Ans: False, any two rational numbers can be integrated by an infinite number of other rational numbers. Therefore, there are infinite rational numbers between 21 and 26.

2. Why does the Rational Number not have a 0 as its Denominator?

Ans: The outcome is not a defined value if the denominator of the rational number is 0. As a result, the rational number’s denominator never equals 0.

3. Can a Rational Number have a Numerator and Denominator of Zero?

Ans: No, the numerator may equal 0. However, for every rational number, the denominator can never be 0.

4. Which Technique is used to Transform a Rational number’s Standard form to Decimals?

Ans: The standard form of a rational number is converted to decimals using the division method.

Introduction

Ratios are said to be equivalent if they can be made simpler or reduced to the same number. In other words, a ratio is said to be equivalent if it can be expressed as a multiple of another ratio.

A ratio can be expressed using a fraction. The concept of an equivalent ratio is comparable to the concept of equivalent fractions. The antecedent and consequent of a ratio can be multiplied or divided by the same number, other than zero, to create an equivalent ratio.

To get a ratio that is equal to the given ratio, we must first represent the ratio in fraction form. Then, by multiplying or dividing the first term and second term by the same non-zero value, the equivalent fraction can be found. We finally convert it to a ratio.

What are Equivalent Ratios

We must first comprehend what equivalence is to understand the equivalence of ratios. Equivalence is very similar to the well-known mathematical relation equal to, as well as to the same mathematical relationship between different objects. In mathematics, equivalence refers to the concept that two objects are equal but distinct because they have the same overall value. When two ratios share the same simplest form, they are said to be equivalent.

Examples of Equivalent Ratios

We can simply create equivalent ratios by multiplying the antecedent and consequent of a ratio by any real number other than the number zero. Thus, creating some examples of equivalent ratios is a very simple task.

For example, we need to find 5 ratios equivalent to 6:10

We can multiply the given ratio by any real number, let’s multiply it by ½

6: 10 = = 3: 5

Thus 3:5 is a ratio equivalent to 6:10

Other such ratios are, 9:15, 12:20, 15:25, 18:30, etc. these all are ratios equivalent to 6:10.

Methods of Finding Equivalent Ratios

There are two methods to find the equivalence of ratios, these methods are

• Cross Multiplication Method

In this method, we multiply the antecedent of the 1st ratio with the consequent of the 2nd ratio and the antecedent of the 2nd ratio with the consequent of the first. If the two products are equal then we can say that the two ratios are equivalent, otherwise, the ratios are not equivalent.

For example: Let’s say we need to use the cross-multiplication method to determine whether the ratios 3:4 and 6:8 are equivalent.

Therefore, we will multiply each ratio’s antecedent by the other ratio’s consequent.

We can say that the ratios are equivalent if the two products are equal.

In this example: 1st Product

3 × 8 = 24

2nd Product

6 × 4 = 24

Since,

Product 1 = Product 2

The ratios 3:4 and 6:8 are equivalent.

• HCF Method

In this method, we first need to represent the ratios as fractions, then we will reduce that fraction into standard form by finding the HCF of the numerator and the denominator, and then dividing the numerator and denominator by that HCF. After both fractions are reduced to the standard form, if they are equal, then the original fraction, i.e., the ratios were equivalent.

For example: Let’s say that we need to use the HCF method to determine whether the ratios 4:14 and 6:21 are equivalent.

To divide the antecedent and consequent of both ratios with their respective HCFs, we will first check the HCF of the antecedent and consequent for each ratio. If the two ratios are equal after the division, the original ratios were equivalent.

In this example: 1st ratio

4:14

HCF (4, 14) = 2

The ratio in the simplest form,

4: 14 == 2: 7

2nd ratio

6: 21

HCF (6, 21) = 3

The ratio in the simplest form

6: 21 = = 2: 7

Since both ratios in their simplest forms are 2:7, thus the original ratios were equivalent.

Use of Equivalent Ratios

In mathematics and other sciences, equivalent ratios have many applications. Some examples of uses are:

• To make the ratios provided simpler.
• We use equivalent ratios to solve any ratio-related problem.
• To calculate ratios between various fractions.
• Various direct proportionality-related scientific issues.

There are also a lot more uses like this.

Read: Applications of Percentage

Summary

We learned about the circumstances under which ratios or proportions are equivalent in this article. We discussed a few instances of equivalent ratios. The following ideas we learned were how to find equivalent ratios. We also discovered how ratios are equivalent using these techniques. Last but not least, we solved several cases that illustrated the concept of equivalent ratios.

Frequently Asked Questions

1. What are Ratios? What are the Components of Ratios?

Ans: Ratios are defined as a comparison between two quantities of the same type. A ratio has 3 parts, 2 parts are the numbers representing the compared quantities antecedent and consequent, and the third part is a symbol, specifically the ‘:’ (colon) symbol, that is put between the two to represent the comparison.

2. Why are Equivalent Ratios Important?

Ans: The equivalent ratios can be used to explain certain relationships between objects in daily life. For instance, if two pens cost Rs. 10, we can use equivalent ratios to determine the price of any other number of pens or the number of pens that can be purchased with a given sum of money. Many other real-world issues can be resolved using equivalent ratios.

3.What are Proportions? What is the Symbol of Proportions?

Ans: Proportions are a comparison between two or more ratios. If two ratios are in proportion, then they are also equivalent.