Terminating Decimal
In short: Terminating decimal describes a number such as 0.525 whose digits run out after a few places, because dividing the fraction’s top by its bottom ends with nothing left over.
7th GradeNumbers and Operations7.NS.A.2d
What is Terminating Decimal?
The value \(0.525\) has three places and then simply ends, which makes it a terminating decimal. It equals \(\frac{525}{1000}\), a fraction whose bottom number is a power of ten.
A fraction in lowest terms ends this way exactly when its denominator contains no primes other than 2 and 5. Those two primes build every power of 10, so you can rescale the fraction to tenths, hundredths or thousandths with no leftover piece.
How to spot it
- The bottom of a fraction is 4, 8, 16, 20, 25 or 40, numbers made from nothing but twos and fives.
- Paper division lands on a remainder of 0 and the quotient quits growing.
- A calculator shows a short answer like 0.1875 with no looping tail.
- Prices, measurements to the nearest hundredth and most test scores already come in this form.
The method
- Cancel the fraction to lowest terms, then break the bottom number into prime factors.
- Only 2s and 5s present means the decimal ends; any other prime means it loops.
- Divide top by bottom, or scale the fraction until its bottom is 10, 100 or 1,000.
- Count the digits after the point to name the place value of the last digit.
Terminating Decimal: solved example
Problem
Does \(\frac{21}{40}\) become a terminating decimal? If so, write it.
Solution
The fraction is in lowest terms and \(40 = 2 \times 2 \times 2 \times 5\), so only 2s and 5s are present.
Multiply top and bottom by 25: \(\frac{21 \times 25}{40 \times 25} = \frac{525}{1{,}000}\). So \(\frac{21}{40} =\) \(0.525\), and the digits stop.
Try it
Question 1
Write \(\frac{9}{25}\) as a decimal.
Show the answer
Multiply top and bottom by 4: \(\frac{36}{100} = 0.36\).
Question 2
Will \(\frac{5}{18}\) terminate? Explain briefly.
Show the answer
No. \(18 = 2 \times 3 \times 3\) contains the prime 3, so the decimal repeats: \(0.2\overline{7}\).
Terminating Decimal vs. Repeating Decimal
A terminating decimal halts after finitely many digits, as if zeros followed forever. A repeating decimal keeps going with a cycling block. Oddly, one value can be written both ways, since 0.5 equals \(0.4\overline{9}\), but we always pick the shorter form.
Keep going
Lessons
Worksheets and quizzes
- Rational and Irrational Numbers – Worksheet
- Rational and Irrational Numbers – Quiz
- Multi-Digit and Decimal Operations – Worksheet
- Multi-Digit and Decimal Operations – Quiz
Related words
Whole dictionary: A–Z · 7th Grade · Numbers and Operations