Converting Terminating Decimals and Fractions

Fractions and decimals are two ways to write the same number

A fraction line means division:

$\frac{a}{b}=a\div b$

Therefore, converting between a fraction and a decimal does not change the value—it changes only its representation. For example, $\frac25$, $0.4$ and $\frac4{10}$ all mark the same point on a number line.

A terminating decimal ends after a finite number of decimal places. Examples include $0.4$, $0.45$ and $0.875$. This lesson concentrates on converting terminating decimals and fractions without a calculator. Division can sometimes produce a recurring decimal instead; recognising that outcome is useful, but converting recurring decimals back into fractions belongs to the next lesson.

Place value to the right of the decimal point

Our number system is based on powers of ten. Moving one column to the left multiplies a digit's value by 10; moving one column to the right divides it by 10.

In $6.789$:

• $6$ is in the units column;
• $7$ represents seven tenths, or $\frac7{10}$;
• $8$ represents eight hundredths, or $$;

Thus

$6.789=6+\frac7{10}+\frac8{100}+\frac9{1000}=\frac{6789}{1000}.$

The final occupied decimal column determines the power-of-ten denominator. One decimal place gives tenths, two give hundredths and three give thousandths. Trailing zeros do not change a decimal's value, so $0.4=0.40=0.400$.

Place value also prevents a common false comparison: more decimal digits do not necessarily mean a larger number. For example, $0.23>0.1237$ because $0.23=0.2300$, and $2300$ ten-thousandths is greater than $1237$ ten-thousandths.

Converting a terminating decimal to a fraction

Write the decimal digits as an integer over the appropriate power of 10, then simplify.

For $0.45$, there are two decimal places, so

$0.45=\frac{45}{100}=\frac9{20}.$

Both fractions are equivalent to the decimal, but is in .

The same method works when the number is greater than 1. Since $4.135$ has three decimal places,

$4.135=\frac{4135}{1000}=\frac{827}{200}=4\frac{27}{200}.$

An improper fraction or a mixed number may be suitable, depending on what the question asks for.

Converting a fraction by making a power-of-ten denominator

Sometimes an equivalent fraction with , or is easy to find. Whatever multiplication is applied to the denominator must also be applied to the .

For example,

$\frac25=\frac{2\times2}{5\times2}=\frac4{10}=0.4.$

Similarly,

$\frac78=\frac{7\times125}{8\times125}=\frac{875}{1000}=0.875.$

This method makes the decimal place value immediately visible.

Converting any simple fraction by division

Because the fraction line means divide, calculate the numerator divided by the denominator. To find $\frac78$, perform . Since 8 does not fit into 7 whole times, write and a decimal point, then regard as :

• 8 goes into 70 eight times, leaving 6;
• bring down the next zero: 8 goes into 60 seven times, leaving 4;
• bring down the next zero: 8 goes into 40 five times, with no remainder.

Therefore $\frac78=0.875$. The zero before the decimal point matters because the fraction is less than 1.

Division also reveals whether the decimal terminates. If the remainder becomes zero, the decimal terminates. For example, the division for $\frac78$ ends. By contrast,

$\frac16=0.1666\ldots$

because the division keeps producing the same non-zero remainder and the digit 6 repeats.

Predicting whether a fraction will terminate

First simplify the fraction fully. A fraction has a terminating decimal exactly when its simplified denominator contains no prime factors other than 2 and 5. This works because powers of 10 are made only from 2s and 5s:

$10^n=2^n\times5^n.$

For instance, $8=2^3$, so $\frac78$ can be rewritten with denominator $1000$. Also, first simplifies to ; since , its decimal terminates. Testing the unsimplified denominator alone could therefore give the wrong conclusion.