These questions ask you to move exactly between decimals and fractions, simplify the result, and decide whether a fraction ends or repeats in decimal form. Place value provides the conversion; the denominator's factors provide the termination check.
Use powers of ten as the bridge
For decimal to fraction, read the final decimal place. The decimal 0.42 means 42 hundredths, so write 42/100 and simplify to 21/50. Keep the numerator and denominator together while simplifying: divide both by the same non-zero whole number.
For fraction to decimal, either divide numerator by denominator or build a denominator of 10, 100, 1000, and so on. For example, 7/50 becomes 14/100, so it is 0.14. If no convenient power-of-ten denominator appears, simplify first and inspect the denominator. A simplified denominator made only from factors 2 and 5 terminates; a remaining factor such as 3 or 7 makes the decimal repeat. Thus 3/14 does not terminate because 14 includes the factor 7.
| Signal | Method |
|---|---|
| One decimal place | Write the visible digits over 10, then simplify. |
| Two decimal places | Write the visible digits over 100, then simplify. |
| Three decimal places | Write the visible digits over 1000, then simplify. |
| Fraction to decimal | Make an equivalent power-of-ten denominator when convenient; otherwise divide. |
| Will it terminate? | Simplify first, then factor the denominator and look only for 2s and 5s. |
Common wrong turns
| Tempting route | How to reject it |
|---|---|
| Copy decimal digits into a unit-fraction denominator | Visible digits do not name the denominator. The final decimal place determines whether the starting denominator is 10, 100, or 1000. |
| Round an exact fraction conversion | A terminating conversion has an exact final digit. Continue the division or build a power-of-ten denominator instead of stopping at a nearby rounded value. |
| Shift every decimal digit one place too far | Check by multiplying the decimal by the denominator. The product must return the numerator exactly. |
| Treat hundredths as tenths | Two digits after the point represent hundredths, so the unsimplified denominator is 100, not 10. |
| Assume a familiar-looking fraction must terminate | Simplify first and factor the denominator. Any remaining prime factor other than 2 or 5 makes the decimal repeat. |
Try it
Enter the decimal equal to 9/20.
Which fraction in simplest form is equal to 0.125?
- 1/125
- 1/8
- 1/12
- 8/10
Which fraction has a nonterminating decimal form?
- 7/40
- 11/20
- 5/8
- 2/9