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Chapter 4: Problem 29

Find decimal notation. \(\frac{4}{3}\)

Short Answer

Expert verified
1.3333... or \( 1.\overline{3} \)

Step by step solution

01

- Understand the fraction

The fraction given is \( \frac{4}{3} \), which means 4 divided by 3.
02

- Perform the division

Divide 4 by 3 by performing the long division. You will find that 4 divided by 3 equals 1.3333..., which is a repeating decimal.
03

- Write the decimal notation

Since the division results in a repeating decimal, the decimal notation for \( \frac{4}{3} \) is written as 1.3333... or \( 1.\overline{3} \), where the bar indicates that 3 repeats.

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Key Concepts

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

Fractions
Fractions represent parts of a whole. In the fraction \(\frac{4}{3}\), the number 4 is called the numerator and the number 3 is called the denominator. The numerator shows how many parts we have, and the denominator shows how many parts make up a whole. So, \(\frac{4}{3}\) means we have 4 parts out of a total of 3 parts.

Fractions can be converted to decimals by division. By dividing the numerator by the denominator, we can see how the parts relate to the whole in a decimal form. Sometimes, this division gives a result that is an exact decimal (like 0.5), whereas other times, it might give a repeating or terminating decimal.
Long Division
Long division is a method used to divide larger numbers. The steps can be broken down as follows:
  • Set up the division problem with the numerator (4) inside the division bracket and the denominator (3) outside.
  • Determine how many times 3 goes into 4. The answer is 1, since 3 times 1 is 3.
  • Write 1 above the division bracket.
  • Multiply 1 by 3 to get 3 and subtract this from 4, which leaves us with a remainder of 1.
  • Bring down a 0 (since we are moving to decimal places now), making the number 10.
  • Determine how many times 3 goes into 10, which is 3 times (since 3 times 3 is 9).
  • Subtract 9 from 10 to get a remainder of 1.
  • Repeat the process to notice that the remainder of 1 keeps occurring, making the decimal 1.3333...
Long division helps show clearly why certain fractions become repeating decimals.
Repeating Decimals
A repeating decimal occurs when after a certain point in the division, the digits start repeating themselves indefinitely. In our example of dividing 4 by 3, the remainder after the first unit of division is 1, and bringing down more zeros keeps giving us that remainder. This makes the decimal 1.3333..., where the 3 repeats forever.

Mathematically, we can write repeating decimals using a bar notation, like this: \[ 1.\overline{3} \] . The bar over the 3 indicates that it repeats continuously. Understanding repeating decimals is key in many areas of math because they often appear in real-world applications, whether through measurements, finances, or scientific calculations.

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Most popular questions from this chapter

The Brunners own a house in Indiana with an assessed value of \(\$ 234,500 .\) For every \(\$ 1000\) of assessed value, they pay \(\$ 8.70\) in property taxes each year. How much do they pay in property taxes each year?

A board \(\frac{7}{10}\) in. thick is glued to a board \(\frac{3}{5}\) in. thick. The glue is \(\frac{3}{100}\) in. thick. How thick is the result?

Add. $$ 65.987+9.4703+6744.02+1.0003+200.895 $$

Subtract. $$ 100.16-0.118 $$

Rita's gross pay (before deductions) is \$495.72. Her deductions are \(\$ 59.60\) for federal income tax, \(\$ 29.00\) for FICA, and \(\$ 29.00\) for medical insurance. What is her take-home pay?
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