Problem 13 - Prealgebra

Question

Convert the given fraction to a terminating decimal. \(\frac{9}{8}\)

Step-by-Step Solution

Answer

The fraction \( \frac{9}{8} \) as a decimal is 1.125.

Step 1: Understand the Problem

We need to convert the fraction \( \frac{9}{8} \) into a decimal. To do this, we'll divide the numerator by the denominator.

Step 2: Perform Division

Divide 9 by 8 using long division. Start by seeing how many times 8 fits into 9. It fits 1 time, as 8 times 1 is 8. This leaves a remainder of 1.

Step 3: Add Zeros to the Dividend

Since there's a remainder, add a decimal point and a zero to the dividend, making it 10. Now, divide 10 by 8.

Step 4: Continue Division

Determine how many times 8 fits into 10. It fits once, and 8 times 1 is 8, leaving a remainder of 2. Add another zero, making it 20.

Step 5: Complete Division

Now, divide 20 by 8. Eight goes into 20 two times (with 8 times 2 being 16), leaving a remainder of 4. Add another zero, making it 40.

Step 6: Finish the Division

Divide 40 by 8. Eight goes into 40 exactly five times with no remainder. The process ends here as there is no remainder left.

Step 7: Write the Decimal Result

The decimal result of dividing 9 by 8 is 1.125.

Key Concepts

Long Division

Long division is a great method to convert fractions to decimals by dividing the numerator by the denominator. It's similar to regular division but provides a systematic approach, especially when dealing with larger numbers. The process begins by calculating how many times the divisor fits into the dividend for the first set of digits. This is done step-by-step.

• Write down the dividend inside and the divisor outside of the division symbol.
• Determine how many times the divisor fits into the first part that is large enough in the dividend.
• Write this number above the division symbol. Multiply it by the divisor and subtract the result from the dividend.
• Bring down the next digit of the dividend and repeat until you either reach zero or complete the long division process.

Long division not only helps in finding decimal representations of fractions but is also essential for grasping division algorithm insights easily.

Fraction to Decimal Conversion

The process of converting a fraction like \(\frac{9}{8}\) into a decimal involves dividing the numerator by the denominator. We're looking for a decimal that either terminates (ends) or repeats. Here's how the conversion works:

• First, perform the division of the numerator by the denominator.
• If the division results in a remainder of zero, like in our exercise, the decimal is terminating.
• If you find a repeating pattern that continues forever, the decimal is recurring.

When you perform the actual long division for \(\frac{9}{8}\), you see that it results in 1.125, a terminating decimal. This means the decimal representation comes to an end after three digits following the decimal point.

Remainders in Division

Understanding remainders is crucial for mastering division. A remainder is what is left over in a division problem when the numbers do not divide evenly. During long division, when each step is executed, any remaining value signifies a remainder.

• Initially, with 9 divided by 8, we get a remainder of 1.
• The division process continues by bringing a zero down (after inserting a decimal), making it 10, which again results in a remainder of 2 after division by 8.
• This process repeats, producing new remainders until a remainder of zero is achieved, signaling the end of the division.

Remainders guide us through the division method until completion, indicating when more digits are needed, or if the division is complete. Understanding them ensures accurate fraction-to-decimal conversion.