Problem 32 - Prealgebra

Question

Express each decimal or fraction as a percent. Round to the nearest tenth,if necessary. $$\frac{10}{3}$$

Step-by-Step Solution

Answer

The fraction \( \frac{10}{3} \) is 333.3% when expressed as a percent rounded to the nearest tenth.

Step 1: Convert Fraction to Decimal

First, convert the given fraction \( \frac{10}{3} \) into a decimal. This is done by dividing 10 by 3. When you carry out the division, you get 3.3333... which is a repeating decimal.

Step 2: Convert Decimal to Percent

To convert a decimal to a percent, multiply the decimal by 100. Take 3.3333... and multiply by 100 to get 333.3333...

Step 3: Round to the Nearest Tenth

Since we are asked to round to the nearest tenth, look at the number in the hundredths place, which is 3. Since it is less than 5, we round down. This gives us 333.3.

Key Concepts

Understanding Fractions

Fractions represent a part of a whole and are made up of two main components: the numerator and the denominator. The numerator is the top number and shows how many parts we have. The denominator is the bottom number and indicates the total number of equal parts the whole is divided into. For example, in the fraction \( \frac{10}{3} \), 10 is the numerator, and 3 is the denominator.

Fractions can be converted to decimals through division. This process involves dividing the numerator by the denominator. When you perform the division for \( \frac{10}{3} \), you carry out the operation 10 ÷ 3, which results in a repeating decimal: 3.3333... Understanding how to convert fractions to decimals is essential for making further calculations, like converting them to percent.

Fractions can also be greater than one, like in our example, and represent an improper fraction. Improper fractions can always be converted to mixed numbers or directly into decimals to better represent their value.

The Decimal System

Decimals are another way of representing numbers that are not whole, similar to fractions. They use a point, known as the decimal point, to separate the whole number part from the fractional part.

When you convert a fraction to a decimal, like \( \frac{10}{3} \), you often end up with a repeating decimal. A repeating decimal is a decimal number that has digits that continue infinitely in a repeating pattern, such as 3.3333... in our case.

To further work with decimals, especially when converting to a percent, it often involves multiplying the decimal number by 100. This is because percent values are parts per hundred, and multiplying by 100 shifts the decimal point two places to the right, effectively converting the decimal to its percent equivalent.

• Example: 3.3333... as a decimal becomes 333.3333... as a percent.

Mastering Rounding Techniques

Rounding is a useful tool to simplify numbers and make them easier to work with. It means altering a number to one that is less precise but simpler and more manageable.

In mathematics, we often round to a specific place value, either up or down. When rounding to the nearest tenth, like in our example, you look at the number in the hundredths place.

• If the digit is 5 or more, you round up.
• If the digit is less than 5, you round down.

In the decimal 333.3333..., the digit in the hundredths place is 3, which is less than 5, leading us to round down to 333.3.

Rounding is especially handy when dealing with long decimals or repeating numbers. It provides a reasonable estimate without being overly detailed, ensuring more efficient calculations in our daily lives.