Problem 17 - Prealgebra

Question

Write each fraction as a decimal correct to the hundredths column. $$\frac{2}{23}$$

Step-by-Step Solution

Answer

The fraction \( \frac{2}{23} \) is approximately 0.09 when rounded to the hundredths place.

Step 1: Set up the division

To convert the fraction \( \frac{2}{23} \) to a decimal, we need to perform long division of 2 divided by 23. We start by setting up the division, placing 2 inside the division bracket and 23 outside to divide into 2.

Step 2: Perform long division

Since 23 does not go into 2, we add a decimal point and a zero, turning the 2 into 20. Now, determine how many times 23 can go into 20: it doesn't fit, so we add another zero making it 200. 23 fits into 200 about 8 times as 23 times 8 is 184.

Step 3: Continue division to find the hundredths

Subtract 184 from 200, which gives us 16. Bring down another 0 to make it 160. 23 goes into 160 approximately 6 times (23 times 6 is 138). Subtract 138 from 160 to get 22. Bring down another zero to make it 220. 23 fits into 220 nine times (23 times 9 is 207). Subtract 207 from 220, resulting in 13.

Step 4: Round to the hundredths

With the decimal obtained from the long division as approximately 0.0869565, we round to the hundredths place. Since the third decimal digit is 6, which is 5 or more, we round up the second decimal digit from 8 to 9.

Key Concepts

Fractions to Decimals

When you start converting fractions into decimals, you’re essentially diving into dividing the numerator by the denominator. Consider the fraction \( \frac{2}{23} \). Here, 2 is the numerator and 23 is the denominator. Converting this fraction into a decimal involves seeing how many times 23 fits into 2, which signifies performing a division. Here’s what you do:

• Set up the division by placing 2 inside the division bracket and 23 outside.
• Since 23 is larger than 2, you need to add a decimal point and bring down zeros to the right of 2 to make it divisible.
• Continue dividing as you add zeros for precision beyond the decimal point.

You keep adding zeros until you can find a clear quotient. This process helps in converting any fraction to decimals, making it more accessible for further arithmetic computations.

Long Division

Long division is the method you use to solve the division problem when converting fractions to decimals. It involves a series of steps that help break down the process:First, understand that 23 (divisor) doesn’t fit into 2 (dividend). Add a decimal and a zero to transform 2 into 20. How many times does 23 go into 20? It doesn’t, hence add another zero making it 200. Now:

• Determine that 23 fits into 200 about 8 times, as \(23 \times 8 = 184\).
• Subtract 184 from 200, leaving a remainder of 16.
• Bring down another 0, turning 16 into 160, where 23 fits approximately 6 times (\(23 \times 6 = 138\)).
• Continue this process until you achieve a remainder close to 0 or decide upon a decimal place to stop.

This meticulous breakdown ensures the conversion from fraction to decimal is accurate. It's like following a recipe that gets complicated calculations simplified into manageable bites.

Rounding Decimals

Once you have your decimal, which in this example is roughly 0.0869565, you need to round it to the hundredths place for clarity and precision. Rounding decimals involves:

• Identifying your target decimal place, in this case, the hundredths, which is the second digit after the decimal point.
• Checking the digit right after your target; here, it’s 6 (from 0.0869...).
• Since 6 is greater than 5, you round the target place value up, changing 8 to 9, and thus, 0.0869565 becomes approximately 0.09.

Rounding helps in simplifying numbers for easier interpretation and use, especially where approximations are acceptable. It’s crucial in mathematic computations to ensure results are concise and understandable.