Fractions to Terminating Decimals

Only factors of two and five allow termination

A terminating decimal ends after a finite number of digits. Once a fraction is simplified, its decimal terminates exactly when its denominator has no prime factors other than 2 and 5. This is the structure to keep in mind before choosing a calculation or rewriting the number.

Simplify first, then factor the denominator. Multiply by enough 2s or 5s to make a power of ten, or use division and look for a remainder of zero. Make a quick estimate or identify the relevant unit first; that gives you a check independent of the written steps.

For 7/20, when the denominator can become 10, 100, or 1000, scale the numerator by that same multiplier. For example, 3/8 becomes 375/1000 because both parts are multiplied by 125. The equivalent fraction makes each decimal digit a visible base-ten unit.

When terminating decimals is useful

Terminating decimal forms are convenient when a context requires a finite written decimal, such as many measurements and prices. The fraction remains the exact form if rounding would otherwise be needed.

Treat the fraction as a division and track the remainder after each decimal digit. For a terminating result, multiplying 0.35 by the original denominator should return the numerator; for a repeating result, keep the repeat exact rather than using a truncated display.

Estimate 7/20 to tell whether division will produce a whole-number part before decimal digits. For an improper fraction, divide first to find that whole part, then convert the remainder over the original denominator. This keeps the size of the whole separate from the decimal expansion of what remains.

The mathematics behind terminating decimals

In 7/20, the fraction bar means numerator divided by denominator. After reducing, check the denominator’s prime factors: only 2s and 5s combine into a power of ten, so a decimal ends exactly in that case. Any other prime factor leaves a repeating remainder pattern.

For 7/20, watch the remainder rather than guessing how many digits to write. A zero remainder ends the decimal; a repeated remainder starts the same sequence of digits again. Multiplying 0.35 by the denominator checks the conversion when the decimal terminates.

A calculator display can hide whether the exact decimal ends. If the display stops after a set number of digits, compare it with long division or track remainders before calling the result terminating. For 0.35, use an ellipsis, repeating bar, or approximation sign when the digits continue.

What happens when terminating decimals changes

Starting with 7/20, try a nearby fraction whose reduced denominator has a different factor pattern. Only twos and fives allow a terminating decimal; another prime factor means a remainder must eventually repeat.

For 7/20, record each remainder beside the next decimal place as you divide. That sequence explains why the digits end or cycle, and multiplying the exact decimal by the denominator checks that the numerator is recovered.

The fraction bar means numerator divided by denominator, so preserve that order in 7/20. A quick size estimate can catch reversal: a proper fraction must be less than one, while a numerator larger than its denominator gives a value above one. Check the fraction’s size before trusting the quotient.

Prompts for reviewing terminating decimals

Use these questions after finishing a problem about fractions to terminating decimals:

• Can the denominator in 7/20 become a power of ten?
• If you divide, what remainder is carried to the next place?
• Does the decimal multiplied by the denominator recover the numerator?

Verify the reasoning in terminating decimals

Before you accept a result, pause and ask:

• Keep numerator and denominator in their correct roles.
• Track each remainder in the next decimal place.
• Distinguish a terminating result from a repeating one.
• Multiply 0.35 by the denominator to check the conversion.