Terminating Decimals, Repeating Decimals, and Rounding

A reduced fraction has a terminating decimal exactly when its denominator has no prime factors other than 2 and 5. For example, $3/8 = 0.375$ terminates because $8 = 2^3$ . By contrast, $1/3 = 0.333\ldots$ repeats because the reduced denominator includes a factor other than 2 or 5.

Long division shows the reason: each decimal digit leaves a remainder. If a remainder reaches zero, the decimal ends. If a nonzero remainder appears again, the same digits repeat from that point. For $5/6$ , the digits are $0.8\overline{3}$ : the 8 is non-repeating and the 3 repeats.

An exact fraction and a displayed decimal can have different precision. For example, to three places, $1/3 \approx 0.333$ ; the approximation sign matters because $0.333$ is not equal to $1/3$ . Rounding early can also change a later result: multiplying the rounded $0.333$ by 3 gives $0.999$ , while the exact fraction gives 1.

Predict whether a reduced fraction terminates

First reduce the fraction. Then factor its denominator. If every prime factor is 2 or 5, the decimal terminates; if any other prime factor remains, the decimal repeats. For example, $7/20$ is already reduced and $20 = 2^2 × 5$ , so it terminates as 0.35. But $7/12$ has a denominator of $2^2 × 3$ , so its decimal repeats: 0.58\overline3.

The reason is that a terminating decimal can be written with a denominator that is a power of 10. Since $10^n = 2^n × 5^n$ , a reduced denominator must use only 2s and 5s to divide some power of 10. The number of places needed depends on how many of those factors are required. For $3/40$ , the denominator $40 = 2^3 × 5$ becomes 1000 after multiplying by 25, so $3/40 = 75/1000 = 0.075$ .

Use remainders to locate the repeating block

In long division, each step multiplies the current remainder by 10 and records the next digit. If the remainder becomes zero, there are no more nonzero digits. If a remainder repeats, the division is back in the same state, so the following digits repeat from the same point.

For $5/6$ , the first step gives digit 8 and remainder 2. The next step gives digit 3 and remainder 2 again. Since the remainder 2 has returned, the 3 repeats: $5/6 = 0.8\overline{3}$ . For $1/7$ , long division cycles through six nonzero remainders before returning to its start, producing the six-digit cycle 142857.

This method distinguishes a repeating block that starts immediately, as in $1/3 = 0.\overline{3}$ , from one with a non-repeating prefix, as in $5/6 = 0.8\overline{3}$ . A rounded decimal may hide the cycle, so keep the fraction when exactness matters.

Round only when a display precision is needed

The exact value of $1/3$ is the fraction $1/3$ . Its three-place decimal display is approximately 0.333. Treating that rounded display as exact changes subsequent arithmetic: $0.333 × 3 = 0.999$ , while $(1/3) × 3 = 1$ . Carry the exact fraction, or enough unrounded precision, through intermediate steps and round at the point specified by the task.

Rounding and truncation are different. Rounding examines the digit after the last place to keep; truncation simply cuts off later digits. For a positive value rounded to three decimal places, 2/3 becomes 0.667 because the next digit after 0.666 is 6. A tool's selected display precision controls what it shows; it does not change the exact rational value represented by the input fraction. OpenStax's section on decimals explains decimal notation and fraction conversion.

Choose a useful precision

For a homework check, follow the requested number of places. For measurement, use the tolerance or resolution in the drawing or instrument specification. More digits do not automatically mean a measurement is more accurate, and rounding a displayed value does not add information that was absent from the original measurement.

The fraction to decimal calculator accepts non-negative fractions and mixed numbers, labels repeating results, and offers display precision controls. For a practical time example with a fractional hour, see why 1:30 is 1.5 hours. The calculator does not decide a grading policy or engineering tolerance; use the requirements for your task.

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Predict whether a reduced fraction terminates

First reduce the fraction. Then factor its denominator. If every prime factor is 2 or 5, the decimal terminates; if any other prime factor remains, the decimal repeats. For example,

7/20

is already reduced and

20 = 2^2 × 5

, so it terminates as 0.35. But

7/12

has a denominator of

2^2 × 3

, so its decimal repeats: 0.58\overline3.

The reason is that a terminating decimal can be written with a denominator that is a power of 10. Since

10^n = 2^n × 5^n

, a reduced denominator must use only 2s and 5s to divide some power of 10. The number of places needed depends on how many of those factors are required. For

3/40

, the denominator

40 = 2^3 × 5

becomes 1000 after multiplying by 25, so

3/40 = 75/1000 = 0.075

Use remainders to locate the repeating block

For

5/6

, the first step gives digit 8 and remainder 2. The next step gives digit 3 and remainder 2 again. Since the remainder 2 has returned, the 3 repeats:

5/6 = 0.8\overline{3}

. For

1/7

, long division cycles through six nonzero remainders before returning to its start, producing the six-digit cycle 142857.

This method distinguishes a repeating block that starts immediately, as in

1/3 = 0.\overline{3}

, from one with a non-repeating prefix, as in

5/6 = 0.8\overline{3}

. A rounded decimal may hide the cycle, so keep the fraction when exactness matters.

Round only when a display precision is needed

The exact value of

1/3

is the fraction

1/3

. Its three-place decimal display is approximately 0.333. Treating that rounded display as exact changes subsequent arithmetic:

0.333 × 3 = 0.999

, while

(1/3) × 3 = 1

. Carry the exact fraction, or enough unrounded precision, through intermediate steps and round at the point specified by the task.

Choose a useful precision