Problem 52 - Algebra A Combined Function

Question

Write each fraction as a decimal. If the decimal is a repeating decimal, write using the bar notation and then round to the nearest hundredth. $$ \frac{7}{9} $$

Step-by-Step Solution

Answer

0.777... equals \(0.\overline{7}\); rounded to the nearest hundredth, it is 0.78.

Step 1: Convert Fraction to Decimal

To convert the fraction \( \frac{7}{9} \) into a decimal, divide the numerator (7) by the denominator (9). This will give us the decimal representation of the fraction.

Step 2: Perform the Division

Divide 7 by 9 using long division. You will find that 9 goes into 7, zero times initially. Append a zero to 7 to make it 70. 9 goes into 70, 7 times (since 9 x 7 = 63) . The remainder is 7, and we append another 0, getting 70 again, leading to another 7 in the decimal place. Hence, the division gives 0.777... which is a repeating decimal.

Step 3: Identify the Repeating Pattern

Observe that the decimal 0.777... has the digit 7 repeating indefinitely. This forms a repeating decimal and can be expressed using bar notation as \( 0.\overline{7} \).

Step 4: Round to the Nearest Hundredth

Since we have the repeating decimal \( 0.\overline{7} \), and we need to round to the nearest hundredth, we look at the first two digits after the decimal. The hundredths digit is 7, and the thousandths is also 7, indicating 0.777... rounds to 0.78.

Key Concepts

Understanding Long Division

To convert fractions into decimals, long division is a reliable strategy. Long division involves dividing the numerator of your fraction by the denominator. This process helps us find the decimal representation of a fraction.

Here's a simple step-by-step breakdown:

• Start by seeing how many times the denominator fits entirely into the numerator. If it doesn't fit as a whole number, you'll need to extend the division.
• Add a decimal point and include zeros after the numerator for ease of division. These zeros are vital when the numerator is smaller than the denominator.
• Continue dividing, carrying down any remainder each time, until the division stops or shows a repeating pattern.

For example, when converting \( \frac{7}{9} \) into a decimal, you divide 7 by 9. You will notice that 9 fits into 70 (after adding a zero) exactly 7 times \(9 \times 7 = 63\), leaving a remainder of 7. This cycle informs the repeating decimal.

Decoding Repeating Decimals

Repeating decimals occur when a sequence of digits starts to repeat indefinitely after the decimal point. This pattern signals that your division has reached a limit where it repeats the same remainder every cycle.

Recognizing repeating decimals is simple once you understand the pattern:

• Watch the numbers carefully. If your long division process is cycling the same remainder over and over, it's a repeating decimal.
• Typically, repeating decimals are represented with a bar over the repeating digits. For example, \(0.777...\) is written as \(0.\overline{7}\).

In our case with \( \frac{7}{9} \), the repeating decimal is clear since each cycle of division after appending a zero results in adding another 7 in the decimal sequence.

Rounding Decimals Effectively

Rounding decimals is an essential skill, especially when working with non-terminating or repeating decimals. It's about bringing the number to a specific decimal place by following certain rounding rules.

Here's how you round to the nearest hundredth:

• Identify the place value you need to round to, which in this case is the hundredths—the second decimal place.
• Look at the digit right after it, the thousandths place, to decide how to round. If it's 5 or more, round up the hundredths digit. If less, you leave it as it is.

For \( \frac{7}{9} \), when you have \(0.\overline{7}\), and you need to round to the nearest hundredth, the decimal becomes 0.78. This is because the repeating 7 suggests that rounding requires moving up from 7 to 8 in the hundredths place.