Problem 9 - Prealgebra

Question

Convert the given fraction to a terminating decimal. \(\frac{3}{2}\)

Step-by-Step Solution

Answer

The fraction \( \frac{3}{2} \) converts to the terminating decimal 1.5.

Step 1: Identify the Division

The fraction \( \frac{3}{2} \) can be converted to a decimal by performing the division \(3 \div 2\). Analyze the dividend and divisor in this fraction.

Step 2: Perform the Division

Divide 3 by 2. Since 3 divided by 2 goes 1 time, write down 1 and subtract 2 \( (1 \times 2 = 2) \) from 3 to leave a remainder of 1.

Step 3: Introduce the Decimal Point

After obtaining a remainder of 1, introduce a decimal point in the quotient, and bring down a 0 to make the new dividend 10. The quotient now starts at 1. and continues.

Step 4: Continue the Division Process

Divide 10 by 2, which goes 5 times exactly. Write down 5 after the decimal point. The new part of the quotient is 1.5. Since there is no remainder left, the division stops here.

Key Concepts

Fraction to Decimal Conversion

Converting fractions to decimals helps us understand numbers in a different form. When you see a fraction like \( \frac{3}{2} \), it means we want to divide 3 by 2. This is called the numerator over the denominator.
A decimal representation shows how many whole parts and partial parts make up a number. For fractions with denominators that are powers of 10 (like 10, 100, 1000), conversion is straightforward as each digit in the decimal represents a power of 10.
For other fractions, like \( \frac{3}{2} \), we directly perform division to convert them into decimals. Understanding this conversion helps in comparing sizes of different quantities and in practical applications like measuring and currency.

Division Process

The division process begins with identifying the fraction's numerator and denominator. For \( \frac{3}{2} \), 3 is the dividend, and 2 is the divisor. Performing the actual division \( 3 \div 2 \) is our way to find out how many times 2 fits into 3.
- **Start the division**: Check how many times 2 goes into 3, which is 1 time.- **Write down the quotient part**: This gives us the whole number part of our decimal.When there's still some leftover, as with our remainder of 1, we add a decimal point to continue dividing. This step-by-step breakdown ensures you get an accurate decimal representation. Remember, the goal is to keep dividing until no remainder is left or until you reach a satisfactory level of precision.

Remainder in Division

A remainder appears when a number cannot be entirely divided by the divisor without any leftovers. In the example \( 3 \div 2 \), after taking \( 2 \times 1 = 2 \), a remainder of 1 is left.
The remainder guides the continuation of the division in decimal conversion. We can bring down extra zeros to represent fractional parts, allowing the division to proceed further. Here, bringing down a zero after the remainder allows us to proceed. Now, the new dividend becomes 10.- **Divide further**: With a new dividend of 10, dividing by 2 results in 5, with no remainder left.- **Complete the process**: No remainder signals the end of our division process, giving a terminating decimal.Understanding remainders is crucial because it dictates whether we stop the division or keep going for greater precision.