The exact answer to "What is 33.33 in fractions?" depends on the context and how 33.33 is interpreted. While it can literally mean the decimal number 33.33, it is very commonly understood as an approximation for 33 and one-third percent, which has an exact fractional equivalent.
Understanding the Interpretations
When converting a number like 33.33 to a fraction, there are two primary ways to interpret it, each leading to a different "exact" fraction:
- 33.33 as a precise decimal number.
- 33.33 as an approximation of 33 and one-third percent, a common practice in mathematics to represent the repeating decimal 0.333...
Interpretation 1: 33.33 as a Percentage (Common Context)
In many mathematical and real-world scenarios, particularly when dealing with percentages, "33.33" is used as a rounded representation of $33 \frac{1}{3}\%$. This is because $33 \frac{1}{3}\%$ is equivalent to the repeating decimal $0.333...$, and 33.33 is a convenient truncation of this value.
To convert $33 \frac{1}{3}\%$ to a fraction:
- First, convert the mixed number to an improper fraction:
$33 \frac{1}{3} = \frac{(33 \times 3) + 1}{3} = \frac{99 + 1}{3} = \frac{100}{3}$ - Next, convert the percentage to a fraction by dividing by 100 (or multiplying by $\frac{1}{100}$):
$\frac{100}{3} \% = \frac{100}{3} \times \frac{1}{100} = \frac{100}{300} = \frac{1}{3}$
Therefore, in the context of percentages, 33.33% is exactly 1/3. This relationship is often noted in percentage-to-fraction conversions:
| Percent | Fraction |
|---|---|
| 30 % | 3/10 |
| 33.33 % | 1/3 |
| 37.5 % | 3/8 |
| 40 % | 2/5 |
Interpretation 2: 33.33 as an Exact Decimal
If "33.33" is treated purely as a finite decimal number without any implied percentage context, converting it to an exact fraction involves placing the number over a power of 10.
To convert 33.33 to a fraction:
- Identify the number of decimal places. In 33.33, there are two decimal places.
- Write the number without the decimal point (3333) as the numerator.
- For the denominator, use 1 followed by as many zeros as there are decimal places (100).
So, $33.33 = \frac{3333}{100}$.
This fraction is already in its simplest form because 3333 is not divisible by 2 or 5, which are the only prime factors of 100.
Why the Difference?
The difference stems from whether 33.33 is viewed as a precise, finite decimal or as a rounded approximation of a value whose exact fractional form involves a repeating decimal. When "33.33" is used to represent "one-third" in a percentage context, it's typically understood to refer to the exact fraction 1/3, which is the precise representation of the repeating decimal 0.333...
In conclusion, while 33.33 as a literal decimal is 3333/100, its common usage, especially when relating to percentages, implies the fraction 1/3.
