Alan rewrites a fraction less than 1 as a decimal. The numerator is a whole the number greater than 0 . For which denominator will the fraction always convert to the terminating decimal?
The correct answer and explanation is:
To determine for which denominator a fraction will always convert to a terminating decimal, we must examine the relationship between the numerator and denominator of a fraction.
Definition of Terminating Decimal:
A terminating decimal is a decimal number that has a finite number of digits after the decimal point. For example, 12=0.5\frac{1}{2} = 0.5, 34=0.75\frac{3}{4} = 0.75, and 78=0.875\frac{7}{8} = 0.875 are terminating decimals.
Fraction Conversion to Decimal:
To convert a fraction to a decimal, we divide the numerator by the denominator. Some fractions, when divided, result in decimals that either repeat (like 13=0.3333…\frac{1}{3} = 0.3333\ldots) or terminate after a finite number of decimal places (like 12=0.5\frac{1}{2} = 0.5).
Criteria for Terminating Decimal:
A fraction ab\frac{a}{b} will always convert to a terminating decimal if and only if the denominator bb (after simplifying the fraction) has no prime factors other than 2 or 5. In other words, the denominator must be of the form 2m×5n2^m \times 5^n, where mm and nn are non-negative integers.
Why does this work?
- The long division process works by repeatedly dividing the numerator by the denominator. For a fraction to have a terminating decimal, the division must end after a finite number of steps.
- If the denominator has prime factors other than 2 or 5, the division will continue indefinitely, leading to a repeating decimal. For example, 13\frac{1}{3} gives a repeating decimal because 3 is not divisible by 2 or 5.
- If the denominator is a power of 2, a power of 5, or a combination of both (like 2, 4, 5, 8, 10, 25, 50, etc.), the division process will terminate.
Examples of Denominators that Work:
- Denominators like 2, 4, 5, 8, 10, 20, 25, 50, 100, etc., will always convert to terminating decimals.
- For example, 18=0.125\frac{1}{8} = 0.125 and 110=0.1\frac{1}{10} = 0.1.
Conclusion:
The denominator of a fraction must be a product of powers of 2 and 5 (i.e., 2m×5n2^m \times 5^n) for the fraction to convert to a terminating decimal. Thus, denominators like 2, 4, 5, 8, 10, 25, 50, 100, etc., will always result in a terminating decimal.