Fraction and Decimal Conversions

🔢 Fraction & Decimal Conversions

Part 1 of 5 — Two Names for the Same Number

Topics in This Part

🔑 Key Concept: A fraction and a decimal are just two ways of writing the same amount. $\dfrac{1}{2}$ and $0.5$ point to the exact same spot on the number line. This lesson teaches you to travel freely between the two forms.

Fractions and Decimals Mean the Same Thing

A fraction $\dfrac{a}{b}$ is really a division statement: it means "$a$ divided by $b$."

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

So $\dfrac{3}{4}$ literally means $3 \div 4 = 0.75$. The fraction bar is a division sign.

A decimal writes the same value using place value to the right of the decimal point:

🔑 Key Idea: The fraction bar means divide. That single fact is the engine behind every fraction-to-decimal conversion in this lesson.

Warm-Up 🧮

Use the table above (or divide top by bottom) to fill in the matching form.

1) $\dfrac{1}{2} = \,?$ (decimal) 2) $\dfrac{1}{4} = \,?$ (decimal) 3) $0.75 = \dfrac{?}{4}$ (numerator only)

Decimal Place Value

Each spot after the decimal point names a denominator that is a power of $10$:

So $0.7$ is 7 tenths $= \dfrac{7}{10}$, and $0.07$ is 7 hundredths $= \dfrac{7}{100}$.

Read the whole number, then attach the last place value. The decimal $0.349$ ends in the thousandths place, so it reads "three hundred forty-nine thousandths" $= \dfrac{349}{1000}$.

💡 The number of digits after the point tells you the denominator: 1 digit → $10$, 2 digits → $100$, 3 digits → $1000$. Each extra digit adds a zero.

Name That Place 🔽

Match each underlined idea to the correct place value.

Terminating vs. Repeating Decimals

When you turn a fraction into a decimal, exactly one of two things happens:

• A terminating decimal stops. The division comes out even. $\frac{3}{8} = 0.375 \qquad \frac{1}{4} = 0.25$

• A repeating decimal never stops — one or more digits repeat forever. We mark the repeat with a bar over the repeating block. $\frac{1}{3} = 0.333\ldots = 0.\overline{3} \qquad \frac{2}{11} = 0.1818\ldots = 0.\overline{18}$

💡 How to predict it: Put the fraction in lowest terms. If the denominator's only prime factors are $2$'s and $5$'s, the decimal terminates. Any other prime factor (like $3$ or $7$) makes it repeat. Example: $\dfrac{3}{8}$ has denominator $8 = 2^3$ → terminates; $\dfrac{1}{6}$ has denominator $6 = 2 \cdot 3$ → repeats.

Concept Check 🎯

🔢 Fraction & Decimal Conversions

Part 2 of 5 — Fraction → Decimal (Divide)

🔑 The Idea: To turn any fraction into a decimal, divide the top by the bottom: $\dfrac{a}{b} = a \div b$. The fraction bar told you to divide all along.

The Long-Division Method

To convert $\dfrac{3}{4}$, compute $3 \div 4$. Since $3$ is smaller than $4$, write $3$ as $3.000$ and divide:

$3.000 \div 4 = 0.75$

Step by step:

• $30 \div 4 = 7$ remainder $2$ → first decimal digit $7$
• bring down: $20 \div 4 = 5$ remainder $0$ → next digit $5$
• remainder is $0$, so we stop: $\dfrac{3}{4} = 0.75$ ✓

A few you should memorize

💡 Shortcut: If the denominator is already $10$, $100$, or $1000$, just write the digits in the matching place — no division needed. $\dfrac{7}{10} = 0.7$ and $\dfrac{23}{100} = 0.23$.

Concept Check 🎯

The "Build to a Power of 10" Trick

Sometimes you can rewrite a fraction with a denominator of $10$, $100$, or $1000$ instead of doing long division. Multiply the top and bottom by the same number:

$\frac{3}{25} = \frac{3 \times 4}{25 \times 4} = \frac{12}{100} = 0.12$

$\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100} = 0.35$

This works whenever the denominator divides evenly into a power of $10$ (that is, when it only has $2$'s and $5$'s as prime factors). For example, $\dfrac{9}{20} = \dfrac{45}{100} = 0.45$.

⚠️ You must multiply top and bottom by the same number, or you change the value. $\dfrac{3}{25}$ and $\dfrac{12}{100}$ are equal; $\dfrac{3}{25}$ and $\dfrac{12}{25}$ are not.

Convert to a Decimal 🧮

Divide top by bottom (or build to a power of $10$). Each of these terminates.

1) $\dfrac{1}{4} = \,?$ 2) $\dfrac{7}{8} = \,?$ 3) $\dfrac{3}{5} = \,?$ 4) $\dfrac{11}{20} = \,?$

When the Division Repeats

If the remainder never reaches $0$, the decimal repeats. Convert $\dfrac{2}{3}$:

$2 \div 3 = 0.6666\ldots = 0.\overline{6}$

Every step gives a remainder of $2$ again, so the $6$ repeats forever.

Convert $\dfrac{5}{6}$: dividing gives $0.8333\ldots$, where only the $3$ repeats: $\frac{5}{6} = 0.8\overline{3}$

💡 Watch the remainders. Once a remainder repeats, the digits start repeating too — that's your signal to stop and draw the bar.

Repeating Decimals 🧮

Convert each fraction to a decimal, rounding to exactly $3$ decimal places — for example, enter $0.333$ for $0.\overline{3}$.

1) $\dfrac{1}{3} = \,?$ (type $0.333$) 2) $\dfrac{1}{9} = \,?$ (type $0.111$)

🔢 Fraction & Decimal Conversions

Part 3 of 5 — Terminating Decimal → Fraction

🔑 The Idea: Read the decimal out loud. The last place value gives you the denominator; the digits give you the numerator. Then simplify.

Place Value → Fraction → Simplify

Three steps to turn a terminating decimal into a fraction:

1. Write the digits (after the point) as the numerator.
2. Use the last place value as the denominator ($10$, $100$, $1000$, …).
3. Simplify by dividing top and bottom by their greatest common factor (GCF).

Worked Example: $0.6$

The last digit is in the tenths place, so: $0.6 = \frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5}$

Worked Example: $0.48$

The last digit is in the hundredths place, so: $0.48 = \frac{48}{100} = \frac{48 \div 4}{100 \div 4} = \frac{12}{25}$

⚠️ Always simplify. $\dfrac{6}{10}$ is correct but not finished — the fully reduced answer is $\dfrac{3}{5}$. Divide top and bottom by their GCF.

Pick the Starting Fraction 🔽

Before simplifying, choose the place-value fraction for each decimal.

Watch the Zeros

The place-value step trips people up with zeros. Two rules keep you safe:

• A leading zero inside the decimal still counts as a place. $0.07$ has two decimal digits, so the denominator is $100$: $0.07 = \dfrac{7}{100}$, not $\dfrac{7}{10}$.
• A trailing zero doesn't change the value. $0.40 = \dfrac{40}{100} = \dfrac{2}{5}$, the same as $0.4 = \dfrac{4}{10} = \dfrac{2}{5}$.

⚠️ Count every digit after the point — including zeros — to choose the right denominator. Then simplify.

Concept Check 🎯

Finishing the Job: Simplifying

After writing the place-value fraction, reduce it.

Worked Example: $0.125$

$0.125 = \frac{125}{1000}$

The GCF of $125$ and $1000$ is $125$: $\frac{125 \div 125}{1000 \div 125} = \frac{1}{8}$

So $0.125 = \dfrac{1}{8}$. ✓ (This matches the table you saw in Part 2.)

Mixed numbers

If there's a whole-number part, keep it out front. $2.75 = 2\dfrac{75}{100} = 2\dfrac{3}{4}$, or as an improper fraction $\dfrac{11}{4}$.

💡 If you can't spot the GCF at once, reduce in stages: $\dfrac{48}{100} \to \dfrac{24}{50} \to \dfrac{12}{25}$. Halving twice gets you there.

Decimal → Fraction 🧮

Convert each terminating decimal to a fraction in lowest terms. Type fractions like $3/5$.

1) $0.4 = \,?$ 2) $0.75 = \,?$ 3) $0.05 = \,?$ 4) $0.16 = \,?$

🔢 Fraction & Decimal Conversions

Part 4 of 5 — Repeating Decimal → Fraction

🔑 The Idea: A repeating decimal hides a clean fraction. We catch it with a little algebra: set the decimal equal to $x$, multiply to shift the repeat, then subtract to make the repeating tails cancel.

The Subtraction Trick (one repeating digit)

Convert $0.\overline{4} = 0.4444\ldots$ to a fraction.

1. Let $x = 0.4444\ldots$
2. Multiply by $10$ (one repeating digit → shift one place): $10x = 4.4444\ldots$
3. Subtract the first equation from the second — the endless tails cancel: $\begin{aligned} 10x &= 4.4444\ldots \\ -\;\;x &= 0.4444\ldots \\ \hline 9x &= 4 \end{aligned}$
4. Solve: $x = \dfrac{4}{9}$.

So $0.\overline{4} = \dfrac{4}{9}$. ✓

🔑 Pattern for a single repeating digit: the fraction is just that digit over $9$. $0.\overline{1} = \frac{1}{9}$, $0.\overline{2} = \frac{2}{9}$, … , $0.\overline{8} = \frac{8}{9}$.

Concept Check 🎯

Two Repeating Digits

When two digits repeat, multiply by $100$ instead. Convert $0.\overline{27} = 0.272727\ldots$:

1. Let $x = 0.272727\ldots$
2. Multiply by $100$ (two repeating digits → shift two places): $100x = 27.272727\ldots$
3. Subtract: $\begin{aligned} 100x &= 27.2727\ldots \\ -\;\;\;x &= \;\,0.2727\ldots \\ \hline 99x &= 27 \end{aligned}$
4. Solve and simplify: $x = \dfrac{27}{99} = \dfrac{27 \div 9}{99 \div 9} = \dfrac{3}{11}$.

🔑 Pattern: one repeating digit → denominator $9$; two repeating digits → denominator $99$; three → $999$. Then simplify. So $0.\overline{27} = \frac{27}{99} = \frac{3}{11}$.

Build the Equation 🔽

You're converting $0.\overline{6}$ to a fraction. Choose each step.

The Fast Shortcut (use after you understand the why)

Once you trust the algebra, you can skip straight to the answer for a decimal where the whole part after the point repeats:

$\text{repeating block over a string of 9's, one 9 per repeating digit, then simplify}$

⚠️ Count the 9's carefully: it's one 9 per repeating digit, not one 9 total. $0.\overline{45}$ has two repeating digits, so it's $\frac{45}{99}$, not $\frac{45}{9}$.

Repeating → Fraction 🧮

Convert each repeating decimal to a fraction in lowest terms. Type fractions like $1/3$.

1) $0.\overline{8} = \,?$ (one repeating digit) 2) $0.\overline{3} = \,?$ 3) $0.\overline{18} = \,?$ (two repeating digits)

🔢 Fraction & Decimal Conversions

Part 5 of 5 — Comparing, Ordering & Mastery Check

You can now go both directions: fraction → decimal (divide) and decimal → fraction (place value or the subtraction trick). Let's use that skill to compare numbers and then prove mastery.

Comparing Fractions and Decimals

It's hard to compare $\dfrac{5}{8}$ and $0.6$ in different forms. Convert to one form — usually decimals — and compare digit by digit.

$\frac{5}{8} = 0.625 \qquad\text{vs.}\qquad 0.6 = 0.600$

Line up the places: $0.625 > 0.600$, so $\dfrac{5}{8} > 0.6$.

Worked Example: order $\dfrac{3}{4},\; 0.7,\; \dfrac{2}{3}$

Convert each to a decimal: $\frac{3}{4} = 0.75 \qquad 0.7 = 0.700 \qquad \frac{2}{3} = 0.6\overline{6} \approx 0.667$

Least to greatest: $\dfrac{2}{3} \,(0.667) < 0.7 < \dfrac{3}{4}\,(0.75)$.

💡 Tip: Give every number the same number of decimal places (pad with zeros) before you compare. $0.6$ becomes $0.600$ — now it's easy to see $0.600 < 0.625$.

Compare Them 🔽

Convert to decimals in your head (or on paper), then pick the correct symbol.

Quick Reference

⚠️ Two reminders: a fraction-to-decimal answer must use division (top ÷ bottom, not bottom ÷ top), and a decimal-to-fraction answer is not done until it's in lowest terms.

Mixed Practice 🎯

Almost There — Final Reminders

Before the exit quiz, run through the three skills one last time:

• Fraction → decimal: divide top ÷ bottom (not the other way around).
• Terminating decimal → fraction: digits over the last place value, then reduce to lowest terms.
• Repeating decimal → fraction: repeating block over a string of $9$'s (one $9$ per repeating digit), then reduce.

🔑 If you can do those three moves, you can convert any number between the two forms. Take the quiz below to lock it in.

Exit Quiz ✅

Answer all three to finish the lesson.