Mathematics
Grade 10
15 min
Convert decimals to fractions
Convert decimals to fractions
What you'll learn
- Identify the place value (tenths, hundredths, thousandths) of a decimal number to correctly write it as a fraction.
- Write decimals (up to the thousandths place) as fractions with denominators of 10, 100, or 1000 with 80% accuracy on a worksheet.
- Simplify fractions resulting from decimal conversions to their simplest form in at least 3 out of 4 examples.
- Explain the relationship between decimals and fractions using place value, providing at least two examples.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Differentiate between terminating, repeating, and non-repeating decimals based on their fractional representations.
Formulate a logical, step-by-step algorithm for converting any terminating decimal into its simplest fractional form.
Construct an algebraic proof to convert any purely repeating decimal (e.g., 0.333...) into a rational number.
Develop and apply an algebraic method to convert mixed repeating decimals (e.g., 0.12333...) into fractions.
Justify why non-repeating, non-terminating decimals (irrational numbers) cannot be expressed as a fraction of two integers.
Apply the concept of infinite geometric series to validate the conversion of repeating decimals to fractions.
Is 0.999... really and truly equal to 1? 🤔 Let's use the power of algebr...
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Key Concepts & Vocabulary
TermDefinitionExample
Rational NumberAny number that can be expressed as a quotient or fraction p/q of two integers, where p is the numerator and q is a non-zero denominator. All terminating and repeating decimals are rational.7/3, -5 (as -5/1), and 0.25 (as 1/4) are all rational numbers.
Terminating DecimalA decimal number that has a finite number of digits after the decimal point. It can be written as a fraction with a denominator that is a power of 10.0.875 is a terminating decimal.
Repeating DecimalA decimal number that has a digit or a block of digits that repeat infinitely. The repeating block is denoted with a bar (vinculum) over the digits.0.555... is written as 0.overline{5}. 2.1343434... is written as 2.1overline{34}.
Period (of a repeating decimal)The digit or block of digits t...
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Core Formulas
Terminating Decimal Conversion Algorithm
d = \frac{n}{10^k}
For a terminating decimal `d`, write the digits of the decimal `n` as the numerator. The denominator is 10 raised to the power of `k`, where `k` is the number of digits after the decimal point. Always simplify the resulting fraction.
Purely Repeating Decimal Conversion (Algebraic Method)
Given x = 0.\overline{p}, then 10^k x - x = p. Solve for x.
Let `x` be the decimal. Let `k` be the length of the period (the repeating block). Multiply `x` by `10^k`. Subtract the original equation (`x`) from the new one to eliminate the repeating part, then solve for `x`.
Mixed Repeating Decimal Conversion (Algebraic Method)
Given x = 0.n\overline{p}, then 10^{k+j} x - 10^k x = \text{integer}. Solve for x.
Let `k` be the nu...
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Challenging
Convert the complex mixed repeating decimal 1.23overline{45} into a single, simplified fraction (improper fraction).
A.12222/9900
B.12333/9990
C.679/550
D.679/55
Challenging
Which statement provides the most rigorous logical justification for why a non-repeating, non-terminating decimal (an irrational number) cannot be expressed as a fraction of two integers?
A.The number of digits is infinite, so the denominator would have to be infinitely large.
B.Calculators cannot display the full number, so a fraction cannot be found.
C.The division of any two integers (p/q) will always result in a decimal that either terminates or eventually enters a repeating cycle of digits.
D.Irrational numbers were defined by the ancient Greeks as numbers that are not fractions, so they cannot be fractions.
Challenging
For the decimal x = 0.overline{7}, demonstrate that the algebraic method (10x - x) and the infinite geometric series formula (S = a / (1-r)) produce the same fractional result.
A.Both methods correctly result in 7/9.
B.The algebraic method gives 7/9, but the series method gives 7/10.
C.Both methods are approximations and result in slightly different fractions.
D.The algebraic method gives 7/10, and the series method gives 7/9.
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Frequently asked questions
What grade level is "Convert decimals to fractions"?
Convert decimals to fractions is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Convert decimals to fractions?
You'll be able to: Identify the place value (tenths, hundredths, thousandths) of a decimal number to correctly write it as a fraction; Write decimals (up to the thousandths place) as fractions with denominators of 10, 100, or 1000 with 80%….
Is "Convert decimals to fractions" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Convert decimals to fractions?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.