Convert terminating decimals and equivalent fractions

Number • Fractions, decimals and percentages

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What is the decimal form of 3/8?

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3/8 = 0.375 by dividing 3 by 8 or by converting denominator to a power of ten equivalent.

Key concepts

What you'll likely be quizzed about

Definition of a terminating decimal

A terminating decimal contains a finite number of digits after the decimal point. If the decimal stops after a certain place value, then it is terminating and not repeating. Termination occurs because the number can be expressed exactly as an integer divided by a power of ten.

Convert a terminating decimal to a fraction

Count the digits after the decimal point and multiply the decimal by 10^n to remove the decimal point. The resulting numerator equals the multiplied value, and the denominator equals 10^n. Simplify the fraction by dividing numerator and denominator by their greatest common divisor. Example: 0.375 × 1000 = 375, so 0.375 = 375/1000 = 3/8 after simplification.

Decimals with whole-number part (mixed numbers)

Separate the whole-number part from the decimal fraction and convert the decimal part using the same power-of-ten method. Combine the whole number and simplified fractional part by converting the whole number into an equivalent fraction or by writing an improper fraction. Example: 3.5 = 3 + 0.5 = 3 + 1/2 = 7/2 after converting to an improper fraction and simplifying.

Deciding whether a fraction yields a terminating decimal

Simplify the fraction first. If the denominator has only prime factors 2 and/or 5, then the decimal expansion terminates because the denominator divides a power of ten. If the denominator contains any prime factor other than 2 or 5, then the decimal expansion repeats and does not terminate.

Limitations and common pitfalls

Failure to simplify before checking denominator factors can lead to incorrect conclusions about termination. Treat trailing zeros in decimal form as significant for place value when creating the denominator. Long division produces the same result as the power-of-ten method but introduces repeated steps; use simplification to reach the final reduced fraction efficiently.

Key notes

Important points to keep in mind

Count the number of decimal places and use 10^n as the denominator.

Multiply the decimal by 10^n to create an integer numerator before simplifying.

Always simplify the fraction by the greatest common divisor.

Simplify the original fraction first when checking for termination.

A simplified denominator with only factors 2 and/or 5 guarantees termination.

Denominators containing primes other than 2 or 5 produce repeating decimals.

Convert a mixed decimal by separating the whole number and fractional part.

Treat trailing zeros as significant when determining the power of ten.