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Chapter 6: Problem 39

Find fraction notation. Simplify. $$ 62.5 \% $$

Short Answer

Expert verified
\( \frac{5}{8} \)

Step by step solution

01

Convert Percentage to Fraction

To convert the percentage to a fraction, write it over 100. For 62.5%, we get \ \[ \frac{62.5}{100} \ \frac{62.5}{100} \]
02

Simplify the Fraction

To simplify the fraction \( \frac{62.5}{100} \), we need to get rid of the decimal point. Multiply the numerator and the denominator by 10: \ \[\frac{62.5 \times 10}{100 \times 10} \ = \frac{625}{1000} \]
03

Further Simplify the Fraction

Continue simplifying the fraction \( \frac{625}{1000} \) by finding the greatest common divisor (GCD) of 625 and 1000, which is 125. Divide both the numerator and the denominator by 125: \ \[ \frac{625 \div \ 125}{1000 \div \ 125} \ = \frac{5}{8} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Percentage to Fraction Conversion
The first step in converting a percentage to a fraction is to understand that 'percentage' means 'per hundred.'
To convert a percentage to a fraction, you simply place the percentage value over 100.
For example, to convert 62.5% to a fraction, write it as \(\frac{62.5}{100}\).

Next, we must address the decimal point in the numerator, 62.5.
To eliminate the decimal, multiply both the numerator (62.5) and the denominator (100) by 10.
This is because 62.5 is the same as 62.5 * 10 divided by 10. After performing this multiplication, you get \(\frac{625}{1000}\).

Now we have a more manageable fraction representation of 62.5%.
Simplifying Fractions
Once you have the fraction \(\frac{625}{1000}\), the next step is to simplify it.
Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator cannot be divided by any number other than 1.

However, finding that number requires understanding the concept of the greatest common divisor (GCD).
You'll use the GCD to reduce both the numerator and denominator by this common value.
Greatest Common Divisor
The Greatest Common Divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder.
To simplify the fraction \(\frac{625}{1000}\), we need the GCD of 625 and 1000.

Here鈥檚 how you can find the GCD:
  • List the factors of both numbers.
  • The factors of 625 are: 1, 5, 25, 125, 625.
  • The factors of 1000 are: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000.

The common factors are 1, 5, 25, and 125.
The greatest common factor is 125.

Now divide both the numerator and the denominator of \(\frac{625}{1000}\) by 125: \(\frac{625 \div 125}{1000 \div 125} = \frac{5}{8}\).
The fraction \(\frac{5}{8}\) is the simplified form of 62.5%.

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