To determine whether each given fraction is a decimal fraction, we need to understand what makes a fraction a decimal fraction. A fraction is a decimal fraction if it can be expressed as a finite decimal number. This is true if and only if the denominator, in its simplest form, is a product of the primes 2 and/or 5 only. Let's evaluate each fraction individually.
### Part a: [tex]\(\frac{3}{100}\)[/tex]
The denominator is 100.
- [tex]\(100 = 2^2 \times 5^2\)[/tex]
Since the denominator is a product of the primes 2 and 5 only, [tex]\(\frac{3}{100}\)[/tex] is a decimal fraction.
Result for a: [tex]\(\checkmark\)[/tex]
### Part b: [tex]\(\frac{7}{10}\)[/tex]
The denominator is 10.
- [tex]\(10 = 2 \times 5\)[/tex]
Since the denominator is a product of the primes 2 and 5 only, [tex]\(\frac{7}{10}\)[/tex] is a decimal fraction.
Result for b: [tex]\(\checkmark\)[/tex]
### Part d: [tex]\(\frac{72}{1000}\)[/tex]
The denominator is 1000.
- [tex]\(1000 = 2^3 \times 5^3\)[/tex]
Since the denominator is a product of the primes 2 and 5 only, [tex]\(\frac{72}{1000}\)[/tex] is a decimal fraction.
Result for d: [tex]\(\checkmark\)[/tex]
### Part e: [tex]\(\frac{25}{42}\)[/tex]
The denominator is 42.
- [tex]\(42 = 2 \times 3 \times 7\)[/tex]
Since the denominator includes prime factors other than 2 and 5 (specifically 3 and 7), [tex]\(\frac{25}{42}\)[/tex] is not a decimal fraction.
Result for e: [tex]\(x\)[/tex]
### Part C: [tex]\(\frac{7}{9}\)[/tex]
The denominator is 9.
- [tex]\(9 = 3^2\)[/tex]
Since the denominator includes a prime factor other than 2 and 5 (specifically 3), [tex]\(\frac{7}{9}\)[/tex] is not a decimal fraction.
Result for C: [tex]\(x\)[/tex]
### Part f: [tex]\(\frac{19}{25}\)[/tex]
The denominator is 25.
- [tex]\(25 = 5^2\)[/tex]
Since the denominator is a product of the prime 5 only, [tex]\(\frac{19}{25}\)[/tex] is a decimal fraction.
Result for f: [tex]\(\checkmark\)[/tex]
So, the final results are:
- a. [tex]\(\frac{3}{100} \)[/tex] : [tex]\(\checkmark\)[/tex]
- b. [tex]\(\frac{7}{10} \)[/tex] : [tex]\(\checkmark\)[/tex]
- d. [tex]\(\frac{72}{1000} \)[/tex] : [tex]\(\checkmark\)[/tex]
- e. [tex]\(\frac{25}{42} \)[/tex] : [tex]\(x\)[/tex]
- C. [tex]\(\frac{7}{9} \)[/tex] : [tex]\(x\)[/tex]
- f. [tex]\(\frac{19}{25} \)[/tex] : [tex]\(x\)[/tex]
The results align with our analysis and understanding of decimal fractions.
