Mathematics
Grade 10
15 min
Decompose fractions into unit fractions
Decompose fractions into unit fractions
What you'll learn
- Identify and explain the commutative property of multiplication (e.g., 3 x 4 = 4 x 3) with at least 80% accuracy on a worksheet.
- Solve multiplication problems using the associative property (e.g., (2 x 3) x 4 = 2 x (3 x 4)) and demonstrate the steps involved in solving a problem on a quiz.
- Apply the distributive property of multiplication to solve problems where a number is multiplied by a sum (e.g., 3 x (2 + 4) = (3 x 2) + (3 x 4)) with 75% accuracy in a problem-solving activity.
- Explain how the identity property of multiplication (multiplying by 1) impacts the product, using examples, during class discussion.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define a unit fraction and an Egyptian fraction.
Apply the Greedy Algorithm (Fibonacci-Sylvester method) to decompose any proper fraction into a sum of distinct unit fractions.
Prove that any positive rational number can be expressed as a finite sum of distinct unit fractions.
Decompose a given fraction into unit fractions using algebraic manipulation and factorization.
Analyze why Egyptian fraction representations are not unique.
Solve basic Diophantine equations related to unit fraction decomposition.
Ever wonder how ancient Egyptians performed complex calculations without decimals or modern notation? 📜 They used a unique system of 'unit fractions' to represent any rational number!
This tutorial explores the fascinating world of Egyptian fra...
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Key Concepts & Vocabulary
TermDefinitionExample
Unit FractionA rational number written as a fraction where the numerator is 1 and the denominator is a positive integer.1/2, 1/5, and 1/148 are all unit fractions.
Egyptian FractionA finite sum of distinct unit fractions. The denominators must all be different.2/3 can be written as the Egyptian fraction 1/2 + 1/6. The denominators 2 and 6 are distinct.
Proper FractionA fraction where the absolute value of the numerator is less than the absolute value of the denominator.4/7 is a proper fraction because 4 < 7. The decomposition methods we study apply to proper fractions.
Greedy Algorithm (Fibonacci-Sylvester Method)An iterative method for finding an Egyptian fraction representation. In each step, it chooses the largest possible unit fraction that is smaller than the...
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Core Formulas
Greedy Algorithm Formula
Given a fraction \( \frac{x}{y} \), the first unit fraction in the decomposition is \( \frac{1}{n_1} \) where \( n_1 = \lceil \frac{y}{x} \rceil \). The next step is to decompose the remainder: \( \frac{x}{y} - \frac{1}{n_1} \).
This is the core iterative step of the Fibonacci-Sylvester method. You repeatedly apply this rule to the remainder until the remainder itself is a unit fraction.
Splitting Identity
\( \frac{1}{n} = \frac{1}{n+1} + \frac{1}{n(n+1)} \)
This identity is useful for creating alternative decompositions or for breaking down a unit fraction into two smaller, distinct unit fractions. It proves that decompositions are not unique.
General Decomposition Form
\( \frac{a}{b} = \sum_{i=1}^{k} \frac{1}{n_i} = \frac{1}{n_1} + \frac{1}{n...
4 more steps in this tutorial
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Challenging
Find all positive integer pairs (x, y) with x < y that solve the Diophantine equation 1/x + 1/y = 1/6.
A.(7, 42) and (8, 24) and (9, 18)
B.(7, 42), (8, 24), (9, 18), and (10, 15)
C.(7, 42), (8, 20), and (10, 15)
D.(1, 5), (2, 4), (3, 3)
Challenging
For any odd integer n > 1, the fraction 2/n can be decomposed into a two-term Egyptian fraction. Which of the following is a general formula for this decomposition?
A.2/n = 1/n + 1/n
B.2/n = 1/((n+1)/2) + 1/(n(n+1)/2)
C.2/n = 1/n + 1/(2n)
D.2/n = 1/((n-1)/2) + 1/(n(n-1)/2)
Challenging
The Greedy Algorithm for x/y produces a sequence of denominators n₁, n₂, n₃, ... . A known (but not provided in the tutorial) property is that nₖ₊₁ > nₖ(nₖ - 1). Given the decomposition of 4/5 is 1/2 + 1/4 + 1/20, which part of this sequence demonstrates this property?
A.n₂ > n₁(n₁ - 1) because 4 > 2(2 - 1) = 2
B.n₃ > n₂(n₂ - 1) because 20 > 4(4 - 1) = 12
C.Both A and B are correct demonstrations.
D.Neither A nor B are correct as the property does not hold.
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What grade level is "Decompose fractions into unit fractions"?
Decompose fractions into unit fractions is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Decompose fractions into unit fractions?
You'll be able to: Identify and explain the commutative property of multiplication (e.g., 3 x 4 = 4 x 3) with at least 80% accuracy on a worksheet; Solve multiplication problems using the associative property (e.g., (2 x 3) x 4 = 2 x (3 x 4)) and….
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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 Decompose fractions into unit fractions?
This lesson includes 48 practice questions across multiple difficulty levels, each with instant feedback and explanations.