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Chapter 6: Problem 37

Write fraction notation. $$ 0.9375 $$

Short Answer

Expert verified
15/16

Step by step solution

01

Understand the decimal

The given decimal is 0.9375. We need to convert this decimal into a fraction.
02

Write the decimal as a fraction

Write the given decimal number 0.9375 as a fraction with 9375 as the numerator and 10000 as the denominator: dfrac{9375}{10000}. This is because there are four decimal places in the number.
03

Simplify the fraction

To simplify dfrac{9375}{10000}, we find the greatest common divisor (GCD) of 9375 and 10000, which is 125.dfrac{9375 div 125}{10000 div 125} = dfrac{75}{80}disce the fraction further by dividing both the numerator and the denominator: dfrac{75 div 5}{80 div 5} = dfrac{15}{16}. So the simplest form of the given fraction is dfrac{15}{16}

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Key Concepts

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

Converting Decimals to Fractions
Converting a decimal to a fraction is quite straightforward. Start by writing the decimal number without the decimal point. The decimal number 0.9375 becomes 9375. This number will be your numerator. The denominator will be a power of 10, depending on the number of decimal places. Because 0.9375 has four decimal places, you'll place it over 10000, thus creating the fraction:\( \frac{9375}{10000} \. \). This step is crucial for retaining the decimal's value.
Simplifying Fractions
When you have a fraction, it's often not in its simplest form. Simplifying fractions makes them easier to work with. Start by finding the greatest common divisor (GCD) of the numerator and the denominator. For \( \frac{9375}{10000} \. \), the GCD is 125. Now divide both the numerator and the denominator by the GCD:\( \frac{9375 \div 125}{10000 \div 125} = \frac{75}{80} \. \). To simplify further, repeat this process until you can no longer divide evenly. After one more round, dividing by 5, you get:\( \frac{75 \div 5}{80 \div 5} = \frac{15}{16} \. \).
Greatest Common Divisor (GCD)
The greatest common divisor (GCD) is the largest number that can divide two numbers without leaving a remainder. Finding the GCD is essential for fraction simplification. There are several methods to find the GCD, but a common one is using the Euclidean algorithm. For example, to find the GCD of 9375 and 10000, you would:
  • Divide 10000 by 9375 and get the remainder: 625
  • Divide 9375 by 625 and get the remainder: 0
  • Since the remainder is now 0, 625 is the GCD
With the GCD, you can simplify fractions by dividing both the numerator and the denominator by this value, as seen in the fraction \( \frac{9375}{10000} \. \).
Dividing both by 625 and continuing this process as needed will lead you to the simplest form of the fraction.

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