Converting Terminating Decimals and Fractions

Converting Decimals to Fractions

Translating a decimal back into a fraction is as easy as looking at its final place value column.

• A terminating decimal is one that has a finite number of digits and eventually ends.
• To convert one to a fraction, the number of decimal places dictates the power of $10$ used as the denominator. One decimal place means tenths ($10$), two means hundredths ($100$), and three means thousandths ($1000$).
• After writing the fraction, you must reduce it to its simplest form, meaning the numerator and denominator share no common factors other than $1$.

Worked Example: Convert $0.325$ into a fraction in its simplest form.

Step 1: Identify the final place value column.

• The last digit ($5$) is in the third decimal place, which represents thousandths.

Step 2: Write the decimal as a fraction over the correct power of $10$.

• $\frac{325}{1000}$

Step 3: Divide both the numerator and denominator by common factors to simplify.

• Divide by $25$: $\frac{325 \div 25}{1000 \div 25} = \frac{13}{40}$
• The fraction is now in its simplest form.

Converting Fractions to Decimals by Scaling

Why divide a fraction manually when you can simply scale it up to a neat power of ten?

• If a fraction's denominator is a factor of $10$, $100$, or $1000$, we can convert it into an equivalent fraction by scaling.
• This involves multiplying both the top and bottom by the same number.
• A fully simplified fraction will only result in a terminating decimal if its denominator's prime factors consist strictly of $2$ and/or $5$. If it contains any other prime factors (like $3$, $7$, or $11$), it will not terminate and will instead be a recurring decimal.

Worked Example: Convert $\frac{11}{25}$ into a terminating decimal.

Step 1: Identify the multiplier needed to turn the denominator into $10$, $100$, or $1000$.

• $100 \div 25 = 4$

Step 2: Scale both the numerator and denominator by this multiplier.

• $\frac{11 \times 4}{25 \times 4} = \frac{44}{100}$

Step 3: Use place value to write the fraction as a decimal.

• $44$ hundredths is written as $0.44$.

Converting Fractions to Decimals using Short Division

What happens when a fraction refuses to scale neatly into a power of ten?

• A fraction line is essentially a division symbol ($a \div b$). When the denominator cannot easily scale to $100$ or $1000$, we use short division (the "bus stop" method).
• The top number (numerator) becomes the dividend and goes inside the bus stop. The bottom number (denominator) becomes the divisor and stays on the outside.
• We add a decimal point and trailing zeros to the dividend, carrying remainders to calculate the final quotient.

Worked Example: Convert $\frac{7}{8}$ into a decimal using short division.

Step 1: Set up the short division.

• The dividend ($7$) goes inside, and the divisor ($8$) goes outside. Add a decimal point and trailing zeros to the dividend.

Step 2: Divide $8$ into $7$. It goes $0$ times.

• Write $0$ and a decimal point in the quotient directly above the dividend's decimal point.
• Carry the $7$ to the first zero to make $70$.

Step 3: Divide $8$ into $70$. It goes $8$ times ($64$) with a remainder of $6$.

• Write $8$ in the tenths column of the quotient. Carry the $6$ to the next zero to make $60$.

Step 4: Divide $8$ into $60$. It goes $7$ times ($56$) with a remainder of $4$.

• Write $7$ in the hundredths column. Carry the $4$ to the next zero to make $40$.

Step 5: Divide $8$ into $40$. It goes exactly $5$ times ($40$).

• Write $5$ in the thousandths column.
• Final Answer: $0.875$