Problem 23 - Prealgebra

Question

Express each percent as a fraction or mixed number in simplest form and as a decimal. $$350 \%$$

Step-by-Step Solution

Answer

350% is \(\frac{7}{2}\) or 3 \(\frac{1}{2}\), and in decimal form, it is 3.5.

Step 1: Understand the Percent

The number given is 350%. A percent means 'per hundred', so 350% is equivalent to 350 per 100.

Step 2: Convert Percent to Fraction

To express 350% as a fraction, write it as \( \frac{350}{100} \).

Step 3: Simplify the Fraction

Simplify \( \frac{350}{100} \) by finding the greatest common divisor of 350 and 100, which is 50. Divide both the numerator and the denominator by 50 to get \( \frac{7}{2} \).

Step 4: Convert to Mixed Number

Since \( \frac{7}{2} \) is an improper fraction, we can also express it as a mixed number. Divide 7 by 2 to get 3 as the whole number, with a remainder of 1, so \( \frac{7}{2} \) equals 3 \( \frac{1}{2} \).

Step 5: Convert to Decimal

To convert \( \frac{7}{2} \) to a decimal, divide 7 by 2 to obtain 3.5.

Key Concepts

Improper Fractions

An improper fraction is a fraction where the numerator is larger than or equal to the denominator. This makes the fraction represent more than one whole unit. For instance, in the exercise, we converted 350% to an improper fraction, which resulted in \( \frac{7}{2} \). Here is how you identify and work with improper fractions:

• Look at the numerator and the denominator in the fraction.
• If the numerator is greater than or equal to the denominator, you have an improper fraction.
• The value of an improper fraction can also be found by dividing the numerator by the denominator.
• Improper fractions can be simplified by finding a common factor, if possible.

By simplifying \( \frac{350}{100} \) to \( \frac{7}{2} \), we transformed it from a larger fraction to a simpler improper fraction.

Mixed Numbers

Mixed numbers combine whole numbers with fractions. They are especially helpful when you want to express improper fractions with whole parts. For instance, the improper fraction \( \frac{7}{2} \) can be rewritten as a mixed number, which is 3 \( \frac{1}{2} \). Converting improper fractions to mixed numbers involves:

• Dividing the numerator by the denominator to find the whole number part.
• The remainder becomes the numerator of the fractional part.
• The original denominator stays the same in the fractional part.

When 7 is divided by 2, the whole number is 3, and the remainder is 1, making the mixed number 3 \( \frac{1}{2} \).Expressing it in this way can make understanding and visualizing quantities easier.

Decimal Conversion

Decimal conversion is about switching a fraction into a number with a decimal point, which can make it easier to interpret for arithmetic operations or comparison. To convert a fraction to a decimal, simply divide the numerator by the denominator.Let's break it down:

• Perform direct division using the fraction's numerator and denominator.
• If necessary, add zeros after the decimal to complete the division.
• Be precise with division to avoid rounding errors.

In the exercise, \( \frac{7}{2} \) was converted to a decimal by dividing 7 by 2, resulting in 3.5. Decimals offer a different form to represent the same value, and they are crucial in calculations involving percentages, financial transactions, and measurements.