Problem 4 - Algebra A Combined Function

Question

Write each decimal as a fraction. $$ 7.23 $$

Step-by-Step Solution

Answer

7.23 as a fraction is \( \frac{723}{100} \).

Step 1: Identify the Decimal Place Values

Recognize the place values of the decimal 7.23. The number 7 is in the units place, 2 is in the tenths place, and 3 is in the hundredths place.

Step 2: Express the Decimal as a Fraction

Write the decimal as a fraction. Since 2 is in the tenths place and 3 is in the hundredths place, the decimal 7.23 can be expressed as \( \frac{723}{100} \) since there are two digits after the decimal point.

Step 3: Simplify the Fraction if Possible

Check if \( \frac{723}{100} \) can be simplified. Since 723 and 100 do not have any common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Identifying Decimal Place Values

When converting a decimal to a fraction, the first step is to identify the place values in the decimal number. This is essential because it tells us the denominator of our fraction. Let's consider the decimal 7.23.

• **7** is in the units place. This tells us it is a whole number and contributes to the whole part of our number.
• **2** is in the tenths place. The tenths place is the first digit to the right of the decimal point. This means that this value is 2 tenths or \(\frac{2}{10}\).
• **3** is in the hundredths place. The hundredths place is the second digit to the right of the decimal point. This represents 3 hundredths or \(\frac{3}{100}\).

Understanding these place values is the foundation for expressing the decimal as a fraction.

Fraction Simplification

Once you convert a decimal to a fraction, the next step is to check if the fraction can be simplified. Simplification involves reducing the fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. For 7.23, once expressed as the fraction \(\frac{723}{100}\), we need to determine if we can simplify it:

• Check for common factors. Look for any number besides 1 that divides both 723 and 100 equally.
• In this case, the greatest common divisor (GCD) is 1. Hence, the fraction \(\frac{723}{100}\) is already in its simplest form.

By simplifying fractions, we ensure that the fraction is easy to interpret and handle in mathematical calculations.

Place Value Understanding

Grasping place values is not just about identifying which position a digit holds in a number; it also involves understanding how these positions impact the overall number. Each position in a decimal number has a different weight:

• The first position right of the decimal point is the tenths place, contributing \(\frac{1}{10}\) of the whole.
• The next position is the hundredths place, adding \(\frac{1}{100}\) of the whole.

For example, knowing that 3 in 0.03 is in the hundredths place means it contributes 3 parts out of 100 to the number. This understanding aids in accurately converting decimals to fractions and vice versa, providing clarity in mathematical operations.This principle of place value applies not just to decimals but to whole numbers as well, although in reverse (i.e., units, tens, hundreds). Recognizing the significance of each digit based on its position is fundamental to mastering various mathematical concepts.