Sure, let's convert the given decimal [tex]\( 5.625 \)[/tex] into an improper fraction step-by-step.
1. Separate the decimal into its whole number and fractional parts: The given decimal [tex]\( 5.625 \)[/tex] can be separated into the whole number part [tex]\( 5 \)[/tex] and the decimal part [tex]\( 0.625 \)[/tex].
2. Convert the decimal part to a fraction:
- To convert [tex]\( 0.625 \)[/tex] to a fraction, we consider it as [tex]\( \frac{625}{1000} \)[/tex].
- Simplify this fraction by finding the greatest common divisor (GCD) of 625 and 1000. [tex]\( \text{GCD}(625, 1000) = 125 \)[/tex].
- Divide both numerator and denominator by 125:
[tex]\[
\frac{625 \div 125}{1000 \div 125} = \frac{5}{8}
\][/tex]
3. Convert the whole number part to a fraction:
- The whole number [tex]\( 5 \)[/tex] can be written as [tex]\( \frac{5}{1} \)[/tex].
4. Combine the whole number and the fractional part:
- Express [tex]\( 5 \)[/tex] as a fraction with the same denominator as the fractional part [tex]\( \frac{5}{8} \)[/tex].
- Multiply the numerator and the denominator of [tex]\( \frac{5}{1} \)[/tex] by 8 to get a common denominator:
[tex]\[
5 = \frac{5 \times 8}{1 \times 8} = \frac{40}{8}
\][/tex]
- Add the two fractions:
[tex]\[
\frac{40}{8} + \frac{5}{8} = \frac{40 + 5}{8} = \frac{45}{8}
\][/tex]
5. Result:
- The improper fraction representation of the decimal [tex]\( 5.625 \)[/tex] is [tex]\( \frac{45}{8} \)[/tex].
So, the answer is [tex]\( \frac{45}{8} \)[/tex].
