To express 2114% as a fraction, you start by converting the percentage to a fraction.
2114% can be expressed as:
\[ \frac{2114}{100} \]
Next, we simplify this fraction. To simplify, we find the greatest common divisor (GCD) of 2114 and 100.
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The prime factorization of 100 is \(2^2 \times 5^2\).
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To find the GCD, we first factor 2114:
- 2114 is an even number, so it can be divided by 2: \[ 2114 \div 2 = 1057 \]
- Next, we check if 1057 is prime. Upon checking for divisibility by primes (2, 3, 5, 7, 11, 13, etc.), we find that 1057 has no factors apart from itself, confirming it is prime.
Thus, the factorization of 2114 is: \[ 2114 = 2 \times 1057 \]
The GCD of 2114 and 100 is 2 (the only common prime factor).
Now we can simplify the fraction: \[ \frac{2114 \div 2}{100 \div 2} = \frac{1057}{50} \]
Now, we look at the responses to see if any match \(\frac{1057}{50}\). However, none of the options match this value.
Since none of the provided options accurately represent 2114% as a fraction when simplified, let's double-check which option is the closest valid representation.
The original mathematical expression for percentages is often derived from the following relationships between fractions and percentages:
- Percent can usually be represented as \(\frac{n}{100}\). For example, 50% = \(\frac{50}{100} = \frac{1}{2}\), etc.
Checking the provided options:
- 13200: This is just a whole number, not a fraction relevant to our percentage conversion.
- 13 over 200: This decimal approximately equals 0.065, not near 21.14.
- 17 over 40: This decimal is approximately 0.425, again not close.
- 1780: Another whole number, not a fraction.
- 17 over 80: This decimal is 0.2125, which is not approximate to our conversion.
None of the given responses accurately represent the conversion of 2114%. The correct fraction is \(\frac{1057}{50}\).