Problem 10 - Prealgebra

Question

Convert the given fraction to a terminating decimal. \(\frac{15}{16}\)

Step-by-Step Solution

Answer

The fraction \( \frac{15}{16} \) as a terminating decimal is 0.9375.

Step 1: Understand the Fraction

We start with the fraction \( \frac{15}{16} \). This fraction means we will divide 15 by 16 to convert it to its decimal equivalent.

Step 2: Perform Long Division

Perform the division by dividing 15 by 16 using long division. Write 15.000 (add three zeros for accuracy) and divide by 16. It goes into 150 zero times, so: When 16 is divided into 150, it fits 9 times (16 x 9 = 144), leaving a remainder of 6. Bring down another 0 to make 60. 16 goes into 60 three times (16 x 3 = 48), remaining 12. Bring down a final 0 to make 120. 16 goes into 120 seven times (16 x 7 = 112), with a remainder of 8. Bring down another 0 to make 80. 16 goes into 80 five times exactly (16 x 5 = 80), leaving no remainder. The division terminates.

Step 3: Write the Decimal

The decimal representation of \( \frac{15}{16} \) is 0.9375 based on the repeated division steps resulting in no further remainder.

Key Concepts

Long Division for Fraction to Decimal Conversion

To convert a fraction into a decimal, you often use a process called long division. It's a straightforward method where the numerator, in our case 15, is divided by the denominator, which is 16. This kind of division helps us find out how many times 16 fits into 15, considering decimals. Here’s how we apply long division:

• Write the number 15, and give it a few extra zeros for accuracy (like 15.000).
• Then divide as if you're working with a whole number. Start by seeing how many times 16 fits into 150, which is 9 times, then subtract the result (144) from 150, resulting in a remainder.
• Continue this process by bringing down more zeros from the dividend, allowing you to see where 16 will fit into that new number.

Long division can seem confusing at first, but once you break it down into smaller steps and practice, it becomes much clearer and is a very useful tool for many mathematical operations.

Fraction to Decimal Conversion

Converting fractions to decimals involves dividing the numerator by the denominator. When dealing with fractions like \( \frac{15}{16} \), understanding the basic setup is key. Start by recognizing that the fraction signifies a division problem - 15 divided by 16.This type of conversion can result in two outcomes:

• A terminating decimal, where the decimal has a finite number of digits after the decimal point.
• A repeating decimal, where one or more numbers repeat infinitely after the decimal point.

For our fraction, \( \frac{15}{16} \), the resulting decimal is 0.9375, a terminating decimal. This means the division results in a remainder of zero at a certain point, ending the process.

Understanding the Remainder in Division

The remainder in division is what’s left after you’ve divided one number by another as completely as possible. When converting fractions to decimals, the remainder plays a crucial role. It helps us determine if the decimal will terminate or continue. During the division of 15 by 16:

• We had several remainders that eventually led us to a remainder of zero.
• A zero remainder indicates that the division has fully completed, resulting in a terminating decimal, 0.9375.
• If the remainder repeats or doesn’t resolve, the decimal representation could be repeating instead of terminating.

Understanding remainders helps determine the continuity of your decimal result, guiding you in mathematical conversions.