Mathematics
Grade 10
15 min
Compare decimals and fractions
Compare decimals and fractions
What you'll learn
- Identify whether a decimal or fraction is greater than, less than, or equal to another decimal or fraction with at least 80% accuracy.
- Solve at least 3 out of 4 word problems that require comparing decimals and fractions in real-world scenarios.
- Explain, using mathematical vocabulary, the steps taken to compare a decimal and a fraction, providing a clear and logical justification for their answer in at least 2 out of 3 attempts.
- Apply knowledge of equivalent fractions and decimals to convert fractions to decimals and decimals to fractions in order to compare them, achieving 75% accuracy on a conversion worksheet.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Construct a logical argument to justify the ordering of a fraction and a decimal.
Apply the Trichotomy and Density properties of rational numbers to compare and order fractions and decimals.
Prove the relationship between two rational numbers using the cross-multiplication property of inequality.
Analyze the structure of terminating and repeating decimals to determine their exact fractional equivalents for precise comparison.
Evaluate the validity of mathematical statements involving inequalities between decimals and fractions.
Formulate a counterexample to disprove a false claim about the comparison of two numbers.
Which is larger: 0.999... or 1? 🤔 The answer is a classic logical puzzle that reveals the true nature of infinite series and number represen...
2
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. As a decimal, a rational number will either terminate or repeat.The number 3/8 is a rational number. It can be expressed as the terminating decimal 0.375.
Trichotomy PropertyA fundamental axiom of ordering real numbers. For any two real numbers, 'a' and 'b', exactly one of the following three statements is true: a < b, a = b, or a > b.When comparing 1/2 and 0.51, the Trichotomy Property guarantees that only one relationship is possible. Since 1/2 = 0.5, we can conclude that 1/2 < 0.51 is the only true statement.
Density Property of Rational NumbersThis property states that between any two...
3
Core Formulas
Cross-Multiplication Property of Inequality
For fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), with \( b > 0 \) and \( d > 0 \): \( \frac{a}{b} > \frac{c}{d} \) if and only if \( ad > bc \).
This is a powerful logical tool for comparing two fractions without finding a common denominator or converting to decimals. It transforms the comparison of fractions into a comparison of integers.
Conversion of a Purely Repeating Decimal to a Fraction
For a repeating decimal \( x = 0.\overline{d_1d_2...d_n} \) with \( n \) repeating digits, the fractional equivalent is \( x = \frac{d_1d_2...d_n}{10^n - 1} \).
This formula provides a direct method for converting an infinite repeating decimal into its exact rational form, which is essential for precise comparisons.
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Sign Up Free to ContinueSample Practice Questions
Easy
Which property states that for any two rational numbers 'a' and 'b', exactly one of the following is true: a < b, a = b, or a > b?
A.Density Property
B.Trichotomy Property
C.Commutative Property
D.Cross-Multiplication Property
Easy
Which of the following fractions will convert to a terminating decimal?
A.7/12
B.4/15
C.9/40
D.5/18
Easy
A rational number p/q (in simplest form) will result in a repeating decimal if the prime factorization of its denominator 'q' contains...
A.only the prime factor 2
B.only the prime factor 5
C.only the prime factors 2 and 5
D.a prime factor other than 2 or 5
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Frequently asked questions
What grade level is "Compare decimals and fractions"?
Compare decimals and fractions is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Compare decimals and fractions?
You'll be able to: Identify whether a decimal or fraction is greater than, less than, or equal to another decimal or fraction with at least 80% accuracy; Solve at least 3 out of 4 word problems that require comparing decimals and fractions in….
Is "Compare decimals and 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 Compare decimals and fractions?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.