Conversion between decimals and common fractions

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MODULE: Fractions, Proportions, and Percentages

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✖️ 1. Converting terminating decimals to fractions using place value denominators

🔢 Decimal to Fraction: Count the Digits

  • Write the decimal digits as the numerator without the decimal point.
  • The denominator is 1 followed by zeros matching the number of decimal places.
  • Simplify the fraction by dividing both parts by their greatest common factor.
  • One decimal place means denominator 10, two places means 100, three means 1000.

Example: 0.75 has 2 decimal places, so write 75/100, then simplify to 3/4.

💡 Count decimal places → that's how many zeros go under 1.

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1. Converting terminating decimals to fractions using place value denominators

Converting Terminating Decimals to Fractions Using Place Value Denominators

A terminating decimal is a decimal number with a finite number of digits after the decimal point. To convert it to a fraction, write the digits after the decimal point as the numerator and use the place value of the last digit as the denominator.

Intuition: The decimal 0.75 means "75 hundredths," which directly translates to the fraction 75100\frac{75}{100}.

Core Rules:

  • Identify the place value of the rightmost digit (tenths = 10, hundredths = 100, thousandths = 1000, etc.)
  • Write the decimal digits (without the decimal point) as the numerator
  • Use the place value as the denominator
  • Simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD)

This method works because our decimal system is base-10, so each position represents a power of 10.

Example: Convert 0.625 to a fraction. The rightmost digit is in the thousandths place, so 0.625=6251000=580.625 = \frac{625}{1000} = \frac{5}{8} after simplification.

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Convert the decimal 0.40.4 to a fully simplified fraction. Write your answer in the format a/b.

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✖️ 2. Understanding the prime factorization rule for why fractions produce terminating vs. repeating decimals

⚙️ Why Some Fractions Terminate

  • A fraction in lowest terms gives a terminating decimal only if the denominator has only 2s and 5s as prime factors.
  • If the denominator contains any other prime (like 3 or 7), the decimal repeats forever.
  • Check by factoring the denominator completely.
  • Denominators like 10, 20, 25, 40, 50, 80, 100, 125, 200 all work because they factor into 2s and 5s only.

Example: 7/8 terminates because 8 = 2×2×2, but 1/3 repeats because 3 is prime and not 2 or 5.

💡 Denominator = only 2s and 5s → decimal stops. Any other prime → it repeats.

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2. Understanding the prime factorization rule for why fractions produce terminating vs. repeating decimals

Prime Factorization Rule for Terminating vs. Repeating Decimals

A fraction in lowest terms produces a terminating decimal if and only if its denominator has no prime factors other than 2 and 5. If the denominator contains any other prime factors (3, 7, 11, etc.), the decimal representation repeats.

Intuition: Since 10=2×510 = 2 \times 5, only denominators that are products of 2s and 5s can be converted to exact powers of 10, which correspond to terminating decimals.

Core Rules:

  • Reduce the fraction to lowest terms first
  • Factor the denominator into primes
  • Terminating: denominator = 2a×5b2^a \times 5^b (where a,b≥0a, b \geq 0)
  • Repeating: denominator contains at least one prime other than 2 or 5

This explains why 18=123\frac{1}{8} = \frac{1}{2^3} terminates (0.125), but 13\frac{1}{3} repeats (0.333...).

Example: 740=723×5\frac{7}{40} = \frac{7}{2^3 \times 5} terminates as 0.175, while 712=722×3\frac{7}{12} = \frac{7}{2^2 \times 3} repeats as 0.58333...

TASK_1[0 / 3]
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STRC: TRANSFORM

Which of the following fractions will produce a terminating decimal?

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✖️ 3. Converting fractions to decimals via long division

➗ Fraction to Decimal: Divide Top by Bottom

  • Divide the numerator by the denominator using long division.
  • Add a decimal point and zeros after the numerator as needed.
  • If division ends with remainder zero, the decimal terminates.
  • If remainders start repeating, the decimal digits will repeat in a cycle.

Example: 3/8 means 3 ÷ 8 = 0.375 (terminates). 1/3 means 1 ÷ 3 = 0.333... (repeats).

💡 Top ÷ Bottom = decimal. Watch for repeating remainders.

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3. Converting fractions to decimals via long division

Converting Fractions to Decimals via Long Division

To convert any fraction ab\frac{a}{b} to its decimal form, perform long division of the numerator aa by the denominator bb. The quotient gives the decimal representation, which either terminates or eventually repeats.

Intuition: Division is the inverse of multiplication; finding how many times bb fits into aa (with remainders expressed as decimal places) reveals the decimal equivalent.

Core Rules:

  • Divide the numerator by the denominator using standard long division
  • Add a decimal point and zeros to the dividend as needed
  • Continue until the remainder is zero (terminating) or a remainder repeats (repeating)
  • Mark repeating digits with a bar notation (e.g., 0.3‾0.\overline{3})

The process always terminates or repeats because there are only finitely many possible remainders (0 through b−1b-1).

Example: Convert 58\frac{5}{8} to decimal. Dividing 5.000... by 8 yields 0.625 (terminates). Converting 23\frac{2}{3} yields 0.666... = 0.6‾0.\overline{6} (repeats).

TASK_1[0 / 3]
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STRC: TRANSFORM

Convert the fraction 3/43/4 to its decimal form using long division.

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✖️ 4. Converting repeating decimals to fractions using algebraic methods

🔁 Repeating Decimal to Fraction: The 10x Trick

  • Let xx equal the repeating decimal.
  • Multiply xx by 10 (or 100 or 1000) to shift the repeating part one full cycle to the left.
  • Subtract the original xx from this new equation to cancel the repeating digits.
  • Solve for xx and simplify the resulting fraction.

Example: Let x=0.666...x = 0.666..., then 10x=6.666...10x = 6.666..., so 10x−x=610x - x = 6, giving 9x=69x = 6, thus x=6/9=2/3x = 6/9 = 2/3.

💡 Multiply to shift repeats, subtract to cancel, then solve.

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4. Converting repeating decimals to fractions using algebraic methods

Converting Repeating Decimals to Fractions Using Algebraic Methods

A repeating decimal can be converted to a fraction by setting the decimal equal to a variable xx, multiplying by an appropriate power of 10 to shift the repeating block, then subtracting to eliminate the repetition.

Intuition: Multiplying by 10, 100, or 1000 aligns the repeating parts so subtraction cancels them, leaving a solvable equation.

Core Rules:

  • Let xx equal the repeating decimal
  • Multiply xx by 10n10^n where nn is the number of repeating digits
  • Subtract the original equation from the multiplied equation
  • Solve for xx and simplify the resulting fraction

This method exploits the periodic nature of repeating decimals to create a finite algebraic expression.

Example: Convert 0.27‾0.\overline{27} to a fraction. Let x=0.272727...x = 0.272727..., then 100x=27.272727...100x = 27.272727.... Subtracting: 100x−x=27100x - x = 27, so 99x=2799x = 27 and x=2799=311x = \frac{27}{99} = \frac{3}{11}.

TASK_1[0 / 3]
LVL_2
STRC: TRANSFORM

Convert the repeating decimal 0.444...0.444... to a fraction.

Write your answer in the form a/b.

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✖️ 5. Applications: Converting imperial fractional inches to decimal inches in machining

🔧 Machining: Fractions to Decimals for Precision

  • Machinists measure in decimal inches but blueprints often show fractional inches like 3/16 or 5/32.
  • Convert the fraction to decimal by dividing numerator by denominator.
  • Digital calipers and CNC machines require decimal input for accuracy.
  • Common conversions: 1/8 = 0.125, 1/4 = 0.25, 3/8 = 0.375, 1/2 = 0.5, 3/4 = 0.75.

Example: A drill bit size of 7/32 inch converts to 7 ÷ 32 = 0.21875 inches for the machine setting.

💡 Blueprint fractions → divide → decimal for the machine.

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5. Applications: Converting imperial fractional inches to decimal inches in machining

Applications: Converting Imperial Fractional Inches to Decimal Inches in Machining

In precision machining and engineering, measurements are often given in fractional inches (e.g., 316\frac{3}{16} inch) but must be converted to decimal inches for CNC programming, digital calipers, and technical drawings.

Intuition: Machines and digital tools require decimal input for accuracy; converting fractions ensures precise fabrication and measurement consistency.

Core Rules:

  • Divide the numerator by the denominator to obtain the decimal equivalent
  • Common conversions: 12=0.5\frac{1}{2} = 0.5, 14=0.25\frac{1}{4} = 0.25, 18=0.125\frac{1}{8} = 0.125, 116=0.0625\frac{1}{16} = 0.0625
  • Round appropriately based on machine tolerance (typically 3–4 decimal places)
  • Memorizing common fractional-to-decimal conversions speeds workflow

This conversion is critical for avoiding costly errors in manufacturing where tolerances are measured in thousandths of an inch.

Example: A blueprint specifies a hole diameter of 716\frac{7}{16} inch. Converting: 716=0.4375\frac{7}{16} = 0.4375 inches, which is entered into the CNC machine.

TASK_1[0 / 3]
LVL_2
STRC: TRANSFORM

A digital caliper reads a measurement for a part that needs to be exactly 3/83/8 inch thick. Convert this fractional inch measurement to a decimal inch.

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