Understanding the Decimal Expression of a Fraction

The question asks about the decimal expression of the fraction \(\frac{3}{8}\) and how many digits are after the decimal point before it terminates. Converting a fraction to a decimal involves dividing the numerator by the denominator.

In this case, we need to divide 3 by 8.

Converting \(\frac{3}{8}\) to a Decimal

To convert \(\frac{3}{8}\) to a decimal, we perform the division \(3 \div 8\).

We can set up a long division:

We start by dividing 3 by 8. Since 3 is smaller than 8, we place a 0 in the quotient, add a decimal point, and add zeros to the dividend.

• \(3 \div 8\) is 0.
• Add a decimal point and a zero to 3, making it 3.0. Now divide 30 by 8.
• \(30 \div 8 = 3\) with a remainder of \(6\) (\(8 \times 3 = 24\), \(30 - 24 = 6\)). The first digit after the decimal is 3.
• Bring down another zero next to the remainder 6, making it 60. Now divide 60 by 8.
• \(60 \div 8 = 7\) with a remainder of \(4\) (\(8 \times 7 = 56\), \(60 - 56 = 4\)). The second digit after the decimal is 7.
• Bring down another zero next to the remainder 4, making it 40. Now divide 40 by 8.
• \(40 \div 8 = 5\) with a remainder of \(0\) (\(8 \times 5 = 40\), \(40 - 40 = 0\)). The third digit after the decimal is 5.

Since the remainder is 0, the decimal division terminates.

So, the decimal expression of \(\frac{3}{8}\) is \(0.375\).

Counting the Digits After the Decimal

The decimal expression is \(0.375\). The digits after the decimal point are 3, 7, and 5.

Let's count them:

1. Digit 1: 3
2. Digit 2: 7
3. Digit 3: 5

There are exactly 3 digits after the decimal point (3, 7, and 5) before the decimal expression comes to an end.

Why Does the Decimal Terminate?

A rational number (a fraction \(\frac{p}{q}\) where p and q are integers and \(q \neq 0\)) has a terminating decimal expansion if and only if the prime factors of the denominator (q), in its simplest form, contain only 2s and/or 5s.

Let's look at the denominator of \(\frac{3}{8}\), which is 8.

The prime factorization of 8 is \(2 \times 2 \times 2\), or \(2^3\).

Since the only prime factor is 2, the decimal expression of \(\frac{3}{8}\) must terminate.

The number of digits after the decimal point in a terminating decimal fraction \(\frac{p}{q}\) (in lowest terms) is determined by the highest power of 2 or 5 in the prime factorization of q. In \(8 = 2^3\), the highest power is 3 (for the factor 2). This corresponds to the number of digits after the decimal point.

The division confirms that the decimal expression of \(\frac{3}{8}\) is \(0.375\), which has 3 digits after the decimal point.

Revision Table: Terminating Decimals

Additional Information: Repeating Decimals

Not all fractions result in terminating decimals. If the prime factors of the denominator of a fraction (in lowest terms) include prime numbers other than 2 or 5 (like 3, 7, 11, etc.), the decimal expression will be a repeating decimal.

• For example, \(\frac{1}{3} = 0.333...\) This is a repeating decimal. The denominator is 3, which is a prime factor other than 2 or 5.
• Another example is \(\frac{1}{7} = 0.142857142857...\). The denominator is 7, a prime factor other than 2 or 5.

The fraction \(\frac{3}{8}\) results in a terminating decimal because its denominator, 8, only has 2 as a prime factor.

Therefore, the decimal expression of \(\frac{3}{8}\) is \(0.375\), which has 3 digits after the decimal point before it terminates.