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Chapter 5: Problem 8

Convert the given fraction to a terminating decimal. \(\frac{7}{175}\)

Short Answer

Expert verified
The terminating decimal of \(\frac{7}{175}\) is 0.04.

Step by step solution

01

Simplify the Fraction

Before converting the fraction to a decimal, let's simplify it. We do this by finding the greatest common divisor (GCD) of the numerator 7 and the denominator 175. The GCD is 7. So, we divide both the numerator and the denominator by their GCD:\[ \frac{7}{175} = \frac{7 \div 7}{175 \div 7} = \frac{1}{25} \]
02

Convert to a Decimal

Now that the fraction is simplified to \(\frac{1}{25}\), convert it to a decimal. To do this, divide the numerator by the denominator:\( 1 \div 25 = 0.04 \) Hence, the decimal representation is 0.04.

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Key Concepts

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

Simplifying Fractions
Fractions are often not in their simplest form. Simplifying a fraction means reducing it so that the numerator and the denominator have no common factors other than 1.

To simplify a fraction:
  • Identify the greatest common divisor (GCD) of the numerator and the denominator.
  • Divide both the numerator and the denominator by the GCD.
This results in a fraction that is equivalent to the original, but in a simpler state.

For example, in the fraction \(\frac{7}{175}\), we find that the GCD is 7. Simplifying it involves dividing both 7 (numerator) and 175 (denominator) by 7, resulting in \(\frac{1}{25}\). This process makes calculations, like converting to decimals, much easier.
Greatest Common Divisor
The greatest common divisor (GCD) is the highest number that divides exactly into two or more numbers.

To find the GCD:
  • List the factors of each number.
  • Identify the largest factor that both numbers share.
Using the GCD helps in simplifying fractions, as it shows the largest fraction by which both the numerator and denominator can be divided evenly.

For instance, with the numbers 7 and 175, their GCD is 7 because 7 is the largest number that exactly divides both numbers. Thus, \(\frac{7}{175}\) simplifies to \(\frac{1}{25}\). Calculating the GCD can save time and ensures the fraction is simplified correctly.
Terminating Decimals
Decimals can be terminating or non-terminating. A terminating decimal ends after a finite number of digits.

When a fraction has a denominator that is a power of ten, converting it to a decimal will result in a terminating decimal.
To ensure a fraction converts to a terminating decimal:
  • Simplify the fraction first, as this may reduce the denominator to a form compatible with powers of ten.
  • Perform division of the numerator by the simplified denominator.
For example, the simplified form of \(\frac{7}{175}\) is \(\frac{1}{25}\). Dividing 1 by 25 gives us the terminating decimal 0.04.

Understanding that simplification and division lead to a neat ending decimal makes the conversion process straightforward.

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Most popular questions from this chapter

Your solutions should include a well-labeled sketch. The length of one leg of a right triangle is 4 meters, and the length of the hypotenuse is 8 meters. Find the length of the other leg. Round your answer to the nearest hundredth.

Your solutions should include a well-labeled sketch. The lengths of two legs of a right triangle are 9 meters and 12 meters. Find the exact length of the hypotenuse.

Compute the exact square root. If the square root is undefined, write "undefined". \(-\sqrt{196}\)

Your solutions should include a well-labeled sketch. The lengths of two legs of a right triangle are 6 meters and 22 meters. Find the exact length of the hypotenuse.

Your solutions should include a well-labeled sketch. The length of one leg of a right triangle is 3 meters, and the length of the hypotenuse is 15 meters. Find the length of the other leg. Round your answer to the nearest tenth.
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