FN10, HN10 - Fractions and terminating decimals

Decimal fractions

Is $0.125+\frac18$ less than, equal to or greater than one quarter? The two terms look different, but recognising equivalent forms will make the calculation simple.

A decimal tells you which fraction unit it counts. For example, $0.4$ is four tenths, so $0.4=\frac4{10}=\frac25$. Two decimal places count hundredths; three count thousandths. A value greater than $1$ works the same way: $1.4$ is fourteen tenths, giving $\frac{14}{10}=\frac75$.

To convert a terminating decimal to a fraction, count its decimal places, write the whole number of those units over $10$, $100$, $1000$ or the appropriate power of $10$, then simplify. If the number is negative, retain its sign. An exact conversion uses an equals sign because the value has not changed.

Add one whole to the same decimal. The denominator for thousandths remains $1000$, but the numerator must now include the whole.

For $-0.425$, the conversion is $-\frac{17}{40}$. The sign changes no place values. Trailing zeros also cause no problem: $0.4250=\frac{4250}{10\,000}$, which simplifies to the same $\frac{17}{40}$.

Include the whole-number part and handle the trailing zero carefully.

Division as conversion

A fraction is a division: $\frac78$ means $7\div8$. Written division finds the decimal digit in each place without changing the value.

Start with the whole-number digit. If a remainder remains, put a decimal point in the quotient and append a zero to the dividend. This expresses the remainder in the next smaller unit: $7$ ones become $70$ tenths, for example. Divide again to find the tenths digit, then continue through hundredths and thousandths. A zero remainder means the decimal has terminated.

Increase the numerator by one denominator, from $7$ to $15$. This adds one whole, so expect the same fractional digits with a different whole-number digit.

A negative fraction gives the corresponding negative decimal, so $-\frac78=-0.875$. Not every fraction gives a terminating decimal: if the written division never reaches zero remainder, stopping after a few digits would give an approximation. Keep the fraction when you need its exact value.

Continue a new division until its remainder is zero.

Choosing exact forms

Choose the form that makes a comparison or calculation easiest while preserving the exact value.

For a comparison, put the numbers into one common form. To compare $0.42$ and $\frac25$, write $\frac25=0.4=0.40$. Then $0.42>0.40$. You could instead use fractions: $\frac{42}{100}>\frac{40}{100}$. Both compare equal-sized place-value units.

For addition or subtraction, a fraction with a convenient common denominator can be quicker. For multiplication, simple fractions can allow cancellation. Exact terminating decimals are equally valid when their arithmetic is simpler. Convert the whole calculation consistently rather than treating a decimal numerator as if it were already a fraction.

Change only the numerator of the fraction being added. This time choose decimal arithmetic and check using fractions.

The same choice helps with a product. For example, $0.6\times\frac14=\frac35\times\frac14=\frac3{20}=0.15$. All these forms are exact, so choose a fraction or a terminating decimal unless a particular form is requested.

Return to the opening question: $0.125=\frac{125}{1000}=\frac18$. Therefore $0.125+\frac18=\frac28=\frac14$, exactly equal to one quarter.

Quick Recap

• Decimal places give fraction units: tenths, hundredths, thousandths and beyond.
• Simplify a decimal fraction without losing the whole-number part or its sign.
• Divide numerator by denominator to find a terminating decimal, keeping every quotient digit in its correct place.
• Convert to a convenient common exact form for a calculation or comparison. The opening sum is exactly one quarter.