Problem 80 - Prealgebra

Question

Write as a percent. Write the remainder in fractional form. $$1 \frac{5}{9}$$

Step-by-Step Solution

Answer

The number is 155% with a remainder of \(\frac{5}{9}\).

Step 1: Convert Mixed Number to Improper Fraction

First, we need to convert the mixed number \(1 \frac{5}{9}\) into an improper fraction. To do this, multiply the whole number (1) by the denominator of the fractional part (9), then add the numerator of the fractional part (5). This gives \((1 \times 9) + 5 = 14\). So, \(1 \frac{5}{9} = \frac{14}{9}\).

Step 2: Perform Division for Decimals

Next, divide the numerator (14) by the denominator (9) to get a decimal. Performing the division, \(14 \div 9 = 1.5555...\). This is a repeating decimal where the digit 5 repeats indefinitely.

Step 3: Convert Decimal to Percentage

To write the decimal as a percent, move the decimal point two places to the right. Thus, \(1.5555...\) becomes \(155.555...\%\).

Step 4: Express the Remainder as a Fraction

The repeating decimal \(0.555...\) corresponds to the fraction \(\frac{5}{9}\) since 5 is the digit repeating after the decimal point, and 9 is the original denominator. So, the remaining part of \(0.555...\) is \(\frac{5}{9}\). Therefore, the remainder in fractional form is \(\frac{5}{9}\).

Key Concepts

Improper Fractions

Improper fractions might sound complex, but they're quite straightforward! An improper fraction is simply a fraction in which the numerator is greater than or equal to the denominator. This means the fraction represents a number equal to or greater than one.

• For example, if we have \(\frac{14}{9}\), 14 (the numerator) is larger than 9 (the denominator), making it an improper fraction.
• In our exercise, the mixed number \(1 \frac{5}{9}\) was converted into \(\frac{14}{9}\), illustrating how we can express mixed numbers as improper fractions for easier computation.
• Always remember the key formula: \((\text{whole number} \times \text{denominator}) + \text{numerator} = \text{new numerator}\).

Understanding this concept is crucial because it allows for easier calculations with fractions and is a stepping stone to converting fractions to decimals and percentages.

Repeating Decimals

Repeating decimals occur when a division between two numbers results in a decimal wherein one or more digits repeat indefinitely. In our exercise, when we divided 14 by 9, we got \(1.5555...\), where 5 is the repeating digit.

• Repeating decimals can often be represented more neatly with a bar, so \(1.5555...\) can be written as \(1.\overline{5}\).
• Each repeating decimal has a fractional equivalent. For \(0.\overline{5}\), this is \(\frac{5}{9}\).
• This conversion is essential for exact calculations where decimal may not suffice, or when needing to revert a repeating decimal back to a fraction.

By converting decimals to fractions, we maintain precision, especially when dealing with repeating sequences.

Mixed Numbers

Mixed numbers are a way to express amounts that are greater than one using both a whole number and a fraction. The challenge is often understanding how they relate to fractions and decimals.

• In the exercise, \(1 \frac{5}{9}\) represented a whole quantity plus a fractional part.
• Converting mixed numbers into improper fractions is often necessary for ease of calculation, as it aligns the number into a single fraction.
• We must understand mixed numbers to fully utilize them in context, such as in recipes or measurements, where mixing whole and fractional quantities occur often.

Learning to work with mixed numbers means understanding their breakdown into simple components, allowing easier conversions and carry forward in mathematical operations.