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Chapter 2: Problem 12

In \(8-12,\) write each rational number as a repeating decimal. $$ \frac{7}{8} $$

Short Answer

Expert verified
\( \frac{7}{8} = 0.875 \) (terminating decimal)

Step by step solution

01

Set Up the Division

To convert the fraction \( \frac{7}{8} \) into a decimal, we need to divide 7 by 8. Set up the long division with 7 as the dividend and 8 as the divisor.
02

Perform Long Division

Begin the division. Since 8 does not go into 7, we check how many times it goes into 70. Divide 70 by 8 to get 8, since 8 times 8 is 64, which is the nearest lower multiple of 8 to 70. Subtract 64 from 70 to get 6. Bring down 0 to make it 60.
03

Continue the Division

Repeat the process with 60. Divide 60 by 8, which gives 7, as 8 times 7 is 56. Subtract 56 from 60, leaving 4. Bring down another 0 to make it 40.
04

Finalize the Division

Divide 40 by 8, resulting in 5, since 8 times 5 is 40. Subtract 40 from 40 to finish at 0. This means the division is complete.
05

Write the Decimal

Since our division ends, we conclude the decimal representation of \( \frac{7}{8} \) is \( 0.875 \). There are no more remainders, so it is a terminating decimal as opposed to a repeating decimal.

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Key Concepts

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

Long Division
Long division is a technique used to divide large numbers by breaking down a division problem into a series of simpler steps. It's like a step-by-step process of finding out how many times a divisor fits into the dividend. Here鈥檚 how long division can be simplified:
  • Take the dividend (the number you're dividing, in this case, 7) and the divisor (the number you鈥檙e dividing by, here it's 8).
  • Since 8 doesn't go into 7, we look at 70 as a whole 鈥� multiply integers that bring us closest without exceeding.
  • 8 fits into 70 a total of 8 times since 8 times 8 is 64.
  • With this, we know our first digit in the quotient 鈥� the beginning of our answer.
Long division requires patience and careful subtraction to minimize any errors. Each step brings you closer to the complete answer, sometimes taking you beyond simple decimal points. It鈥檚 a foundational skill in math critical for understanding decimals.
Terminating Decimal
A terminating decimal is a decimal that comes to an end, as opposed to continuing infinitely. You know you've got a terminating decimal when you completely finish dividing, and no numbers or remainders are left. Here's why it鈥檚 significant:
  • When you divide a number and no remainers are left, the decimal "ends."
  • In this exercise, dividing 7 by 8 ends with 0.875, a clear example of a terminating decimal.
  • Terminating decimals are precise and are often easier to work with compared to repeating ones.
  • When the denominator, in this case 8, can be factored down into powers of 10 (such as 2 and 5), the decimal terminates since 10 is the base of our number system.
These decimals are important because they offer an exact value and help simplify complex calculations.
Fraction to Decimal Conversion
Converting a fraction into a decimal is a valuable mathematical process, and knowing how to do it can make numbers easier to work with. Here鈥檚 a straightforward approach:
  • Set up the fraction as a division problem 鈥� the numerator (top number) divided by the denominator (bottom number).
  • Perform long division. Keep dividing until you find that the division stops naturally, or it might become a pattern if repeating.
  • For the fraction \(\frac{7}{8}\), the result of long division equals 0.875, transforming 7 divided by 8 from a fraction into a decimal.
    • If you complete a division without any repeat, you have a terminating decimal.
    • If numbers begin to repeat, it becomes a repeating decimal.
  • This conversion is pivotal. It aids in understanding and comparing fractions in a simpler decimal form.
Mastering fraction to decimal conversion makes mathematical tasks more manageable and numbers more relatable in daily contexts.

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Most popular questions from this chapter

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{b}{b-1}-\frac{1}{2-2 b} $$

In \(13-24,\) divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{c^{2}-6 c+9}{5 c-15} \div \frac{c-3}{5} $$

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{3 x}{x-1} \cdot \frac{x^{2}-1}{x} \div \frac{x+1}{3} $$

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{2}{a^{2}-4}-\frac{1}{a^{2}+2 a} $$

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{2 x^{2}+1}{5 x}-\frac{7 x^{2}-1}{5 x} $$
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