Answer :
Certainly! Let's convert the decimal number 0.75 into a rational number, which is a fraction in its simplest form. Here is a step-by-step approach:
### Step 1: Understand the Decimal
The given decimal is 0.75.
### Step 2: Express the Decimal as a Fraction
To convert a decimal to a fraction, we need to express it in the form [tex]\( \frac{a}{b} \)[/tex].
0.75 can be read as "seventy-five hundredths." This means it can be written as:
[tex]\[ \frac{75}{100} \][/tex]
### Step 3: Simplify the Fraction
To simplify the fraction [tex]\(\frac{75}{100}\)[/tex], we need to find the greatest common divisor (GCD) of the numerator (75) and the denominator (100) and then divide both by this GCD.
1. Find the GCD of 75 and 100. The factors of 75 are [tex]\(1, 3, 5, 15, 25, 75\)[/tex] and the factors of 100 are [tex]\(1, 2, 4, 5, 10, 20, 25, 50, 100\)[/tex]. The largest common factor is 25.
2. Divide both the numerator and the denominator by their GCD (25):
[tex]\[ \frac{75 \div 25}{100 \div 25} = \frac{3}{4} \][/tex]
### Step 4: Write the Simplified Fraction
The fraction [tex]\(\frac{3}{4}\)[/tex] is already in its simplest form, as the GCD of 3 and 4 is 1 (meaning they have no other common factors other than 1).
### Conclusion
Therefore, the decimal 0.75 can be expressed as the rational number [tex]\(\frac{3}{4}\)[/tex].
### Step 1: Understand the Decimal
The given decimal is 0.75.
### Step 2: Express the Decimal as a Fraction
To convert a decimal to a fraction, we need to express it in the form [tex]\( \frac{a}{b} \)[/tex].
0.75 can be read as "seventy-five hundredths." This means it can be written as:
[tex]\[ \frac{75}{100} \][/tex]
### Step 3: Simplify the Fraction
To simplify the fraction [tex]\(\frac{75}{100}\)[/tex], we need to find the greatest common divisor (GCD) of the numerator (75) and the denominator (100) and then divide both by this GCD.
1. Find the GCD of 75 and 100. The factors of 75 are [tex]\(1, 3, 5, 15, 25, 75\)[/tex] and the factors of 100 are [tex]\(1, 2, 4, 5, 10, 20, 25, 50, 100\)[/tex]. The largest common factor is 25.
2. Divide both the numerator and the denominator by their GCD (25):
[tex]\[ \frac{75 \div 25}{100 \div 25} = \frac{3}{4} \][/tex]
### Step 4: Write the Simplified Fraction
The fraction [tex]\(\frac{3}{4}\)[/tex] is already in its simplest form, as the GCD of 3 and 4 is 1 (meaning they have no other common factors other than 1).
### Conclusion
Therefore, the decimal 0.75 can be expressed as the rational number [tex]\(\frac{3}{4}\)[/tex].
