Write the decimal as a proper fraction or as a mixed number and simplify. 3.14
The Correct Answer and Explanation is:
To convert the decimal 3.14 to a proper fraction or mixed number, let’s follow these steps:
Step 1: Express 3.14 as a fraction
First, express 3.14 as 3+0.143 + 0.143+0.14. Now, we’ll convert 0.140.140.14 into a fraction.
Step 2: Convert 0.140.140.14 to a fraction
To do this, recognize that 0.140.140.14 means 141414 hundredths. This can be written as:0.14=141000.14 = \frac{14}{100}0.14=10014
Next, simplify 14100\frac{14}{100}10014. The greatest common divisor (GCD) of 14 and 100 is 2, so we divide both the numerator and denominator by 2:14100=14÷2100÷2=750\frac{14}{100} = \frac{14 \div 2}{100 \div 2} = \frac{7}{50}10014=100÷214÷2=507
Step 3: Combine the whole number and the fraction
Now that we’ve simplified 0.140.140.14 to 750\frac{7}{50}507, we combine it with the whole number part (3):3.14=3+7503.14 = 3 + \frac{7}{50}3.14=3+507
This expression is a mixed number:3.14=37503.14 = 3 \frac{7}{50}3.14=3507
Step 4: Convert the mixed number to an improper fraction (if desired)
To convert 37503 \frac{7}{50}3507 into an improper fraction, we multiply the whole number (3) by the denominator (50) and then add the numerator (7):3×50=150so150+7=1573 \times 50 = 150 \quad \text{so} \quad 150 + 7 = 1573×50=150so150+7=157
Thus, the improper fraction is:3750=157503 \frac{7}{50} = \frac{157}{50}3507=50157
Final Answer
So, the decimal 3.143.143.14 can be written as the mixed number 37503 \frac{7}{50}3507, or the improper fraction 15750\frac{157}{50}50157. Both are equivalent and simplified forms of the original decimal.
Why simplify?
Simplifying fractions helps in understanding their simplest form, which makes calculations easier and provides clarity when working with different mathematical operations. The process of converting decimals into fractions is also essential for understanding relationships between numbers that are not readily obvious from the decimal form.
