Problem 19 - Prealgebra

Question

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$7 \frac{3}{10}$$

Step-by-Step Solution

Answer

The mixed number \(7 \frac{3}{10}\) as a decimal is 7.3.

Step 1: Understand the Mixed Number

The mixed number is given as \(7 \frac{3}{10}\). This consists of the whole number 7 and the fraction \(\frac{3}{10}\).

Step 2: Convert the Fraction to a Decimal

The fraction \(\frac{3}{10}\) means 3 divided by 10. When you divide 3 by 10, you get 0.3. So, the fraction \(\frac{3}{10}\) as a decimal is 0.3.

Step 3: Combine Whole Number and Decimal

Add the decimal form of the fraction (0.3) to the whole number part of the mixed number (7). This gives you: \(7 + 0.3 = 7.3\). So, the mixed number \(7 \frac{3}{10}\) as a decimal is 7.3.

Key Concepts

Fractions

Fractions represent parts of a whole. They consist of two numbers: the numerator, which is the top part, and the denominator, which is the bottom part. For example, in the fraction \(\frac{3}{10}\), the 3 is the numerator and the 10 is the denominator.

• The numerator indicates how many parts you have.
• The denominator shows the total number of equal parts the whole is divided into.

To convert a fraction to a decimal, you simply divide the numerator by the denominator. This is because fractions and division are closely linked. By dividing 3 by 10, you perform the operation 3 ÷ 10 to get 0.3.
This conversion helps in utilizing fractions in decimal-based calculations and comparisons.

Mixed Numbers

A mixed number combines a whole number with a fraction. It represents a number that is between integers, showing that there is a portion of a whole added to a complete one. For example, the mixed number \(7 \frac{3}{10}\) contains the whole number 7, plus the fraction \(\frac{3}{10}\).

• This means you have 7 whole units and an additional part equal to three-tenths of a unit.
• The fraction part makes the number more precise than just a whole number alone.

To convert a mixed number to a decimal, first change the fractional part to a decimal. Then, sum this decimal with the whole number. In the case of \(7 \frac{3}{10}\), you convert \(\frac{3}{10}\) to 0.3 and add it to 7, leading to the decimal 7.3.

Repeating Decimals

Repeating decimals are decimals where one or more digits repeat indefinitely. They occur when the division of the numerator by the denominator in a fraction does not end, and instead results in a repeating pattern.
For instance, the fraction \(\frac{1}{3}\) becomes the repeating decimal 0.333..., which is often written as 0.\overline{3}.

• Repeating decimals can be identified by a bar placed over the repeating digit(s).
• This bar notation indicates that the digit(s) underneath the bar repeat endlessly.

Not all fractions result in repeating decimals; some convert to terminating decimals, like \(\frac{3}{10}\), which gives the finite decimal of 0.3. Recognizing patterns in division helps students identify these repeating sequences and apply the appropriate notations.