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Simplifying fractions is a way to make them easier to work with by reducing their size without changing their value. When a fraction is expressed in its simplest form, the numerator and the denominator do not have any common factors other than 1. Take the fraction \( \frac{86}{100} \). It can be simplified by finding a number that divides both the top number (numerator) and the bottom number (denominator) exactly without leaving a remainder.
To do this, we use the greatest common divisor (GCD). The GCD is the largest number that can evenly divide both parts of the fraction. For \( 86 \) and \( 100 \), it's \( 2 \). Dividing both the numerator and the denominator by \( 2 \) gives us \( \frac{43}{50} \). This form is simpler because the numerator and the denominator can't be reduced further.

Understanding decimal place value helps in reading and writing decimals correctly. Each number in a decimal represents fractions of ten, reducing in size as you move to the right of the decimal point.
For the number \( 0.86 \), the \( 8 \) is in the tenths place, indicating \( 8 \times \frac{1}{10} = 0.8 \). The \( 6 \) is in the hundredths place, meaning \( 6 \times \frac{1}{100} = 0.06 \). Together, they total to \( 0.86 \). Each position after the decimal point represents a power of ten.
This understanding is crucial when converting decimals to fractions, because the decimal's place value helps set up the correct fraction format. In \( 0.86 \), the tenths and hundredths places tell us to set up our fraction as \( \frac{86}{100} \), since it encompasses up to the hundredths place.

The greatest common divisor (GCD) is a key concept when simplifying fractions. It is the highest number that can exactly divide two other numbers.
When simplifying a fraction like \( \frac{86}{100} \), calculating the GCD of \( 86 \) and \( 100 \) is essential. Both numbers are divisible by \( 1 \), \( 2 \), and themselves, but the largest mutual factor is \( 2 \). Determining the GCD can be tackled using a few methods:

• Listing divisors: Write out all factors of both numbers, and identify the largest one they share.
• Prime factorization: Break down both numbers into their prime factors and multiply the common factors.
• Euclidean algorithm: Use a series of divisions to find the GCD by dividing the larger number by the smaller and then using remainders.

Once the GCD is found, you divide the numerator and the denominator of the fraction by it to simplify the fraction. This makes the fraction easier to read and work with.