Write the following decimal number in its equivalent fraction form. Show all work for full credit 0.225 WRITER
The Correct Answer and Explanation is:
Decimal Number: 0.225
Step-by-Step Conversion to Fraction:
Step 1: Understand the place value of the decimal
The number 0.225 has three decimal places. That means it goes to the thousandths place:0.225=22510000.225 = \frac{225}{1000}0.225=1000225
Step 2: Simplify the fraction
We now simplify the fraction 2251000\frac{225}{1000}1000225 by finding the greatest common divisor (GCD) of the numerator and the denominator.
- Prime factorization of 225:
225=15×15=3×5×3×5=32×52225 = 15 \times 15 = 3 \times 5 \times 3 \times 5 = 3^2 \times 5^2225=15×15=3×5×3×5=32×52
- Prime factorization of 1000:
1000=10×10×10=(2×5)3=23×531000 = 10 \times 10 \times 10 = (2 \times 5)^3 = 2^3 \times 5^31000=10×10×10=(2×5)3=23×53
The common prime factors are 525^252.
So, the GCD of 225 and 1000 is 25.
Now divide both numerator and denominator by 25:225÷251000÷25=940\frac{225 \div 25}{1000 \div 25} = \frac{9}{40}1000÷25225÷25=409
Final Answer:
0.225=940\boxed{0.225 = \frac{9}{40}}0.225=409
Explanation
When converting a decimal to a fraction, the key idea is to express the decimal as a ratio of two integers. A decimal like 0.225 can be read as “two hundred twenty-five thousandths,” which directly gives the fraction 2251000\frac{225}{1000}1000225. This is called writing the decimal in fractional notation based on place value. The number 0.225 has three digits after the decimal point, so we place 225 over 1000.
The next step is simplifying the fraction. This means we divide both the numerator (top) and denominator (bottom) by their greatest common divisor (GCD) — the largest number that evenly divides both numbers. The GCD of 225 and 1000 is 25. This is determined by finding the prime factors of each number. For 225, the prime factors are 32×523^2 \times 5^232×52, and for 1000, they are 23×532^3 \times 5^323×53. The shared factor here is 52=255^2 = 2552=25. Dividing the numerator and denominator by 25 simplifies 2251000\frac{225}{1000}1000225 to 940\frac{9}{40}409.
Thus, the simplest form of the fraction equivalent to 0.225 is 940\frac{9}{40}409. This form is preferred because it’s the most reduced and clearly shows the ratio between two whole numbers without any unnecessary complexity. Converting decimals to fractions is a fundamental skill in mathematics, especially useful in algebra, measurements, and real-world applications involving ratios and proportions.
