Problem 70 - Prealgebra

Question

Write as a percent. Round to the nearest tenth of a percent. $$1 \frac{2}{3}$$

Step-by-Step Solution

Answer

The percent is 166.7%.

Step 1: Convert the Mixed Number to an Improper Fraction

The mixed number is given as \(1 \frac{2}{3}\). First, convert this into an improper fraction by multiplying the whole number by the denominator and adding the numerator. So, \(1 \times 3 + 2 = 5\). Thus, \(1 \frac{2}{3} = \frac{5}{3}\).

Step 2: Convert the Fraction to a Decimal

To convert the fraction to a decimal, divide the numerator by the denominator: \(\frac{5}{3} = 1.6666\ldots\). This is a repeating decimal.

Step 3: Convert Decimal to Percent

To convert a decimal to a percent, multiply the decimal by 100 and add a percent sign. Here, \(1.6666\ldots \times 100 = 166.6666\ldots\%\).

Step 4: Round to the Nearest Tenth of a Percent

To round \(166.6666\ldots\%\) to the nearest tenth, consider the first two digits after the decimal point. Since the second digit is 6, round up, resulting in \(166.7\%\).

Key Concepts

Mixed Numbers

A mixed number combines a whole number with a proper fraction. For instance, in the mixed number \(1 \frac{2}{3}\), "1" is the whole number and “\(\frac{2}{3}\)” is the fraction.

Mixed numbers are often used in everyday situations because they can make numbers easier to visualize. For example, when measuring something that is more than one unit but less than two, a mixed number like \(1 \frac{2}{3}\) can be more intuitive than \(\frac{5}{3}\).

• To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction.
• Add the numerator to the result.
• Use the denominator of the original fraction for the improper fraction.

This conversion is especially useful when you need to perform operations like multiplication or division, where improper fractions are easier to handle.

Improper Fractions

Improper fractions may look a bit confusing, but they're actually simple once you get the hang of them. An improper fraction has a numerator that is larger than or equal to its denominator, like \(\frac{5}{3}\).

When dealing with improper fractions, remember:

• The larger numerator means that the fraction represents a value greater than or equal to one whole.
• They're often the first step in calculations, like converting to a decimal.

To convert back to a mixed number, divide the numerator by the denominator: the quotient becomes the whole number, and the remainder over the original denominator becomes the fraction part.

Improper fractions can be easier to work with in mathematical equations because they allow for more straightforward calculations compared to mixed numbers.

Decimal Conversion

Converting fractions to decimals can unlock their value in yet another way. For \(\frac{5}{3}\), you achieve this through division: the numerator (5) divided by the denominator (3). This transformation gives a repeating decimal \(1.6666\ldots\).

• Decimals provide an absolute value that is easy to work with in further calculations.
• They are particularly useful in percentage conversions, giving clear numerical results.

Decimals also come in handy when you require a concise representation of a number that can be used in further mathematical operations like adding or comparing quantities. In everyday uses, such as financial calculations, decimals provide clarity and simplicity.

Rounding Numbers

Rounding is the mathematical way to make numbers simpler while keeping their value close to the original. When rounding, you adjust a number to a specified approximation, based on certain rounding rules.

In the case of our decimal from \(\frac{5}{3}\) that turned into \(1.6666\ldots\), we applied these steps to round it to the nearest tenth of a percent.

• Identify the place you want to round to.
• Look at the digit immediately to its right.
• If it’s 5 or more, round up the number you’re keeping.
• If it’s less than 5, keep the digit the same.

After rounding based on these simple rules, \(166.6666\ldots\%\) becomes \(166.7\%\). Rounding not only simplifies the number but also makes it easier to read and communicate, especially in practical scenarios such as assigning grades, estimating costs, or measuring quantities.