Mathematics
Grade 10
15 min
Add 3 or more fractions with unlike denominators
Add 3 or more fractions with unlike denominators
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Introduction & Learning Objectives
Learning Objectives
Identify the Least Common Multiple (LCM) for three or more denominators to establish a common unit of measure.
Convert three or more fractions with unlike denominators into equivalent fractions with a common denominator.
Accurately sum three or more fractions and express the result in simplest form.
Apply the Segment Addition Postulate to segments whose lengths are fractional parts of a congruent unit.
Calculate the total area of a composite figure formed by combining fractional parts of congruent shapes.
Justify geometric steps in a proof that require the summation of fractional lengths or areas.
Imagine a designer creating a new pattern using identical (congruent) tiles, but using only 1/2 of the first tile, 1/3 of the second, and 3/4 of the third. How...
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Key Concepts & Vocabulary
TermDefinitionExample
Congruent FiguresFigures that have the exact same size and shape. All corresponding sides and angles are equal. In this context, they serve as a standard unit of measure.Two squares with side lengths of 5 cm are congruent. A line segment AB of length 10 is congruent to a line segment CD of length 10.
Unlike DenominatorsThe bottom numbers (denominators) of two or more fractions are different, indicating the whole has been divided into a different number of equal parts.In the fractions 1/2, 2/3, and 3/5, the denominators 2, 3, and 5 are unlike.
Least Common Multiple (LCM)The smallest positive integer that is a multiple of two or more numbers. For fractions, we find the LCM of the denominators to create a Least Common Denominator (LCD).For denominators 4, 6, and 8, the...
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Core Formulas
Finding the Least Common Denominator (LCD)
LCD(d_1, d_2, ..., d_n) = LCM(d_1, d_2, ..., d_n)
To add fractions with unlike denominators (d_1, d_2, etc.), you must first find a common unit of division. The most efficient common unit is the Least Common Multiple (LCM) of all the denominators, which becomes the Least Common Denominator (LCD).
Principle of Adding Fractions
\frac{a}{d_1} + \frac{b}{d_2} + \frac{c}{d_3} = \frac{a \cdot k_1}{LCD} + \frac{b \cdot k_2}{LCD} + \frac{c \cdot k_3}{LCD} = \frac{ak_1 + bk_2 + ck_3}{LCD}
Once you have the LCD, convert each fraction to an equivalent fraction with that denominator. The factor 'k' for each fraction is found by dividing the LCD by the original denominator (e.g., k_1 = LCD / d_1). Then, add the new numerators and place...
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Challenging
A composite shape is formed from fractional areas of three congruent regular pentagons, each with area P. The total area is (1/2 + 3/5 + 4/7)P. A step in a proof aims to show this area is greater than two full pentagons (2P). Which calculation and conclusion is correct?
A.The sum is (8/14)P, which is less than P.
B.The sum is (35+42+40)/70 P = (117/70)P, which is less than 2P because 117 < 140.
C.The sum is (35+42+40)/70 P = (117/70)P, which is greater than 2P because 117 > 70.
D.The sum is (8/70)P, which is less than P.
Challenging
A proof contains the statement: 'Given segments AB=(2/3)U, BC=(1/4)U, and CD=(3/5)U are collinear, the total length AD is (2+1+3)/(3+4+5) U = 6/12 U = 1/2 U.' Which statement best identifies and explains the flaw in this reasoning?
A.The error is in the final simplification; 6/12 simplifies to 1/3, not 1/2.
B.The error is adding the denominators; this violates the principle that fractions must have a common unit (denominator) before their counts (numerators) can be added.
C.The error is forgetting the geometric context; the unit 'U' should not be in the final answer.
D.The error is in applying the Segment Addition Postulate; it does not apply to three segments.
Challenging
An artist creates a sculpture using parts of three congruent cubes, each with volume V. The total volume of the sculpture is (1/3 + 3/8 + 5/6)V. If this total volume is used to form as many complete cubes of volume V as possible, what fractional part of a single cube V remains?
A.1/24
B.11/24
C.1 and 11/24
D.13/24
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