ACT Fractions and Decimals: Keep Exact Values Until the Last Step

To convert a fraction to a decimal, divide its numerator by its denominator. A terminating decimal can be written as a fraction using place value; a repeating decimal needs its repeating digits preserved. Keep an exact fraction or repeating form through your calculation, then round the final answer if the question requests it.

ACT’s mathematics standards include equivalent fractions, decimal place value, rounding, and ordering rational numbers. The examples here are original Kibo practice, not official ACT items. This guide focuses on converting and calculating; our number-types guide explains how to classify the results.

Read the repeating bar before converting

A bar tells you exactly which digits continue forever. Moving that bar changes the number. A finite decimal written as an exact value stops at its last digit.

OpenStax’s decimals-and-fractions lesson explains division and repeating-bar notation. A calculator’s finite display alone does not tell you whether the original decimal terminates. Use the fraction or stated repeating pattern to decide.

Convert a terminating value in either direction

For $\frac{9}{40}$, make the denominator a power of ten. Multiplying both parts by $25$ gives:

$$\frac{9}{40}=\frac{225}{1000}=0.225$$

The reverse conversion starts with place value: $0.225$ is $225$ thousandths. Reduce $\frac{225}{1000}$ by dividing numerator and denominator by $25$. You return to $\frac{9}{40}$.

The denominator counts the place occupied by the final decimal digit. OpenStax’s decimal lesson covers this place-value method and rounding.

Use remainders to recognize a repeating decimal

Divide $7$ by $12$. Since $7<12$, the whole-number part is zero. Continue into decimal places:

1. In the tenths place, $70=12\times5+10$. Write $5$ and keep remainder $10$.
2. In the hundredths place, $100=12\times8+4$. Write $8$ and keep remainder $4$.
3. Next, $40=12\times3+4$. Write $3$. The remainder is $4$ again, so every later step repeats this one.

$$\frac{7}{12}=0.58\overline{3}\approx0.58$$

The equality preserves the full repeating decimal. The approximation is rounded to the nearest hundredth: the thousandths digit is $3$, so the hundredths digit stays $8$.

Convert a decimal whose repetition starts later

Write $0.2\overline{7}$ as a fraction. Let $x=0.27777\ldots$. First move the nonrepeating digit past the decimal point, then shift one more place so the repeating tails line up:

$$10x=2.7777\ldots$$

$$100x=27.7777\ldots$$

Subtract the first equation from the second. The identical decimal tails cancel:

$$100x-10x=27-2$$

$$90x=25,\quad x=\frac{25}{90}=\frac{5}{18}$$

Check by division: $5\div18=0.27777\ldots$. Writing $\frac{27}{99}$ would instead give $0.\overline{27}$. When some digits do not repeat, line up the tails before subtracting.

The same check explains the table: for $x=0.\overline{58}$, subtraction gives $99x=58$. For $x=0.5\overline{8}$, subtract $10x$ from $100x$ to get $90x=58-5=53$.

Why rounding early can change the answer

Suppose a question asks for the exact value of $36\times\frac{7}{12}$. The fraction simplifies cleanly:

$$36\times\frac{7}{12}=3\times7=21$$

Replacing $\frac{7}{12}$ with its rounded value $0.58$ gives $36\times0.58=20.88$, which is $0.12$ too small. The multiplication is correct, but it uses a changed input. Retain the fraction until the requested final rounding step.

Rounding can also hide an ordering difference. Although $\frac{7}{12}$ rounds to $0.58$, it is larger than the exact decimal $0.58=\frac{29}{50}$. Their exact difference is:

$$\frac{7}{12}-\frac{29}{50}=\frac{175-174}{300}=\frac{1}{300}>0$$

If the problem gives a rounded measurement, use the precision stated in that problem. Converting its printed decimal to a fraction cannot recover unknown extra digits.

Try four original practice questions

Write the exact value first. For a rounding question, name the requested place and inspect the next digit.

1. Write the exact decimal $0.3125$ as a fraction in lowest terms.
2. Write $\frac{13}{24}$ as a repeating decimal, then round it to the nearest hundredth.
3. Convert $0.1\overline{6}$ to a fraction. Show which two multiples you subtract.
4. Find the exact value of $18\times\frac{5}{6}$. A student replaces the fraction with $0.83$ first. How far is that student's result from the exact answer?

Answers and mistake checks

1. $0.3125=\frac{3125}{10000}=\frac{5}{16}$. There are four decimal places, so start with ten-thousandths. Dividing both parts by $625$ reduces the fraction.
2. $\frac{13}{24}=0.541\overline{6}\approx0.54$. Only the six repeats. The thousandths digit is $1$, so the hundredths digit stays $4$; do not use a later six to decide the rounding.
3. Let $x=0.1\overline{6}$. Then $100x-10x=16-1=15$, giving $90x=15$ and $x=\frac{1}{6}$. Subtracting $x$ from $100x$ would leave unmatched decimal tails.
4. The exact result is $18\times\frac{5}{6}=15$. The rounded-input result is $18\times0.83=14.94$, which is $0.06$ too small. Keep the original fraction when the question asks for an exact answer.

Choose your next practice from the mistake

If the denominator was wrong, label the last decimal place before converting. If the repeating fraction was wrong, write both shifted equations and compare their tails. If an exact answer was slightly off, look for an intermediate rounding step. Record the specific slip in Kibo’s practice error log and try a fresh example later with the solution covered.

You can try a free ACT practice question with Kibo to apply your math skills.

Prepared with AI assistance. All original examples were verified with separate exact calculations; no human review is claimed. Sources checked October 17, 2026. This guide is not affiliated with or endorsed by ACT.