List Five Rational Numbers Between -4/5 And -2/3 . Please Tell By Step By Step Explaintion

List Five Rational Numbers Between -4/5 And -2/3 . Please Tell By Step By Step Explanation

Finding rational numbers between two given fractions is a common problem in mathematics that helps develop understanding of number lines, fractions, and rational numbers. This article provides a comprehensive, step-by-step guide to identify five rational numbers lying between \(-\frac{4}{5}\) and \(-\frac{2}{3}\). We will systematically explain the process, including converting fractions to comparable forms, finding common denominators, and selecting suitable rational numbers. Whether you're a student preparing for exams or someone interested in mathematical reasoning, this guide aims to clarify the method thoroughly.

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Understanding Rational Numbers and the Given Fractions

What Are Rational Numbers?

• Rational numbers are numbers that can be expressed as the quotient or fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers, and \(q \neq 0\).
• Examples include \(\frac{1}{2}\), \(-\frac{3}{4}\), 0, and integers like 5 (which can be written as \(\frac{5}{1}\)).

Given Fractions: \(-\frac{4}{5}\) and \(-\frac{2}{3}\)

• Both fractions are negative, indicating they lie to the left of zero on the number line.
• Our goal is to find five rational numbers between these two fractions.

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Step 1: Convert Fractions to a Common Denominator

To compare fractions efficiently, especially when finding numbers lying between them, it's best to convert both to equivalent fractions with a common denominator.

Find the Least Common Denominator (LCD)

• Denominator of \(-\frac{4}{5}\): 5
• Denominator of \(-\frac{2}{3}\): 3
• LCD of 5 and 3: 15 (since 15 is the smallest number divisible by both 5 and 3)

Convert both fractions to have denominator 15

• \(-\frac{4}{5}\): multiply numerator and denominator by 3

\[ -\frac{4 \times 3}{5 \times 3} = -\frac{12}{15} \]

• \(-\frac{2}{3}\): multiply numerator and denominator by 5

\[ -\frac{2 \times 5}{3 \times 5} = -\frac{10}{15} \]

Now, the fractions are:

• \(-\frac{12}{15}\)


• \(-\frac{10}{15}\)

Since \(-\frac{12}{15}\) is less than \(-\frac{10}{15}\), the numbers lying between them are greater than \(-\frac{12}{15}\) and less than \(-\frac{10}{15}\).

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Step 2: Understand the Number Line and the Range

• The number line segment of interest is between \(-\frac{12}{15}\) and \(-\frac{10}{15}\).
• Recognize that these are negative numbers: \(-0.8\) and \(-0.666...\).
• Our task is to find rational numbers greater than \(-0.8\) and less than \(-0.666...\).

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Step 3: Find Rational Numbers Between \(-\frac{12}{15}\) and \(-\frac{10}{15}\)

To find rational numbers between these two fractions, consider their decimal equivalents:

• \(-\frac{12}{15} = -0.8\)


• \(-\frac{10}{15} \approx -0.6667\)

Any rational number greater than \(-0.8\) and less than \(-0.6667\) will satisfy the condition.

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Step 4: Selecting Rational Numbers Between the Two Fractions

There are multiple methods to find rational numbers in between. Here, we will explore two common approaches:

Method 1: Find fractions with a common denominator between 15 and a higher denominator for more options

• Increase the denominator to find finer fractions.
• For example, choose denominator 30 (double 15):
• Convert \(-\frac{12}{15}\) to denominator 30:

\[ -\frac{12 \times 2}{15 \times 2} = -\frac{24}{30} \]

• Convert \(-\frac{10}{15}\) to denominator 30:

\[ -\frac{10 \times 2}{15 \times 2} = -\frac{20}{30} \]

• Now, the fractions are:
• \(-\frac{24}{30}\) (which is \(-0.8\))
• \(-\frac{20}{30}\) (which is \(-0.6667\))
• Rational numbers between these can then be selected as fractions with denominator 30, with numerators between 20 and 24:
• \(-\frac{23}{30}\)
• \(-\frac{22}{30}\)
• \(-\frac{21}{30}\)
• Simplify where possible:
• \(-\frac{23}{30}\) (cannot be simplified)
• \(-\frac{22}{30} = -\frac{11}{15}\) (since 2 and 30 are divisible by 2)
• \(-\frac{21}{30} = -\frac{7}{10}\)
• So, three rational numbers between the original fractions are:
• \(-\frac{23}{30}\)
• \(-\frac{11}{15}\)
• \(-\frac{7}{10}\)

Method 2: Use decimal approximations and convert back to fractions

• Pick decimal numbers between -0.8 and -0.6667, such as:
• \(-0.75\)
• \(-0.72\)
• \(-0.69\)
• \(-0.68\)
• \(-0.67\)
• Convert these to fractions:
• \(-0.75 = -\frac{3}{4}\)
• \(-0.72 = -\frac{18}{25}\)
• \(-0.69 = -\frac{69}{100}\)
• \(-0.68 = -\frac{17}{25}\)
• \(-0.67 \approx -\frac{67}{100}\)
• Check if these are between \(-\frac{12}{15}\) and \(-\frac{10}{15}\):
• \(-\frac{3}{4} = -0.75\) (less than \(-0.8\), so not between)
• \(-\frac{18}{25} = -0.72\) (between \(-0.8\) and \(-0.6667\))
• \(-\frac{69}{100} = -0.69\) (between)
• \(-\frac{17}{25} = -0.68\) (between)
• \(-\frac{67}{100} = -0.67\) (just slightly greater than \(-0.6667\), so perhaps just outside the range)
• From this, select five suitable rational numbers:

1. \(-\frac{18}{25}\)
2. \(-\frac{69}{100}\)
3. \(-\frac{17}{25}\)
4. \(-\frac{33}{50}\) (which is \(-0.66\))
5. \(-\frac{13}{20}\) (which is \(-0.65\))

All these are between \(-0.8\) and \(-0.6667\).

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Step 5: Final List of Five Rational Numbers Between the Given Fractions

Based on the methods above, here are five rational numbers that lie between \(-\frac{4}{5}\) and \(-\frac{2}{3}\):

1. \(-\frac{23}{30}\) (approx. \(-0.7667\))


2. \(-\frac{11}{15}\) (approx. \(-0.7333\))


3. \(-\frac{7}{10}\) (exactly \(-0.7\))


4. \(-\frac{18}{25}\) (approx. \(-0.72\))


5. \(-\frac{69}{100}\) (approx. \(-0.69\))

All these fractions are greater than \(-0.8\) and less than \(-0.6667\), meaning they are strictly between \(-\frac{4}{5}\) and \(-\frac{2}{3}\).

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Additional Tips for Finding Rational Numbers Between Two Fractions