Write 4.768 as a common fraction, proper or improper and then reduce. .
The correct answer and explanation is:
Sure! Let’s convert 4.768 to a common fraction and then reduce it.
Step 1: Write 4.768 as a fraction
4.768 means 4 whole units plus 0.768 as the decimal part.
We can write this as: 4.768=4+0.7684.768 = 4 + 0.768
Now, focus on converting 0.768 to a fraction.
Step 2: Convert 0.768 to a fraction
The decimal 0.768 has three digits after the decimal point, so we can write: 0.768=76810000.768 = \frac{768}{1000}
Step 3: Simplify 7681000\frac{768}{1000}
We find the greatest common divisor (GCD) of 768 and 1000.
- Factors of 768: 28×32^8 \times 3 (since 768=256×3768 = 256 \times 3)
- Factors of 1000: 23×532^3 \times 5^3
The common factors are 23=82^3 = 8.
So, GCD(768,1000)=8\text{GCD}(768, 1000) = 8
Divide numerator and denominator by 8: 768÷81000÷8=96125\frac{768 \div 8}{1000 \div 8} = \frac{96}{125}
So, 0.768=961250.768 = \frac{96}{125}
Step 4: Combine whole number and fraction
4.768=4+96125=4×125125+96125=500125+96125=5961254.768 = 4 + \frac{96}{125} = \frac{4 \times 125}{125} + \frac{96}{125} = \frac{500}{125} + \frac{96}{125} = \frac{596}{125}
Step 5: Check if the fraction 596125\frac{596}{125} can be simplified
- 125 is 535^3.
- 596 factors: 2 × 2 × 149 (149 is prime).
No common factors between 125 and 596 besides 1.
So, 596125\frac{596}{125} is in simplest form.
Final answer:
596125\boxed{\frac{596}{125}}
Explanation:
Converting a decimal to a fraction involves understanding place value and simplifying the fraction. Here, 4.768 is separated into its whole part (4) and decimal part (0.768). The decimal part is expressed as a fraction over a power of ten based on the number of decimal places—in this case, 3 decimal places means dividing by 1000. Then, simplifying the fraction by dividing numerator and denominator by their greatest common divisor ensures the fraction is in its simplest form. Finally, the whole number is converted to a fraction with the same denominator and added to the fractional decimal part to get the improper fraction form.
This process is fundamental in math because it helps convert between decimal and fractional representations, which are used differently in various contexts, such as measurements, probability, or algebraic calculations. It also reinforces understanding of prime factorization and simplifying fractions by GCD.