Mathematics
Grade 10
15 min
Add fractions with unlike denominators
Add fractions with unlike denominators
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Introduction & Learning Objectives
Learning Objectives
Connect the arithmetic process of adding fractions to geometric transformations of congruent figures.
Determine the least common denominator (LCD) for two or more fractions representing scaled lengths or areas.
Convert fractions with unlike denominators into equivalent fractions with a common denominator.
Accurately add fractions with unlike denominators and express the sum in simplest form.
Solve geometric problems involving the sum of fractional lengths of scaled congruent figures.
Interpret the sum of fractional areas of congruent figures, including sums greater than one.
Justify the steps for adding fractions by relating them to the principle of combining like parts of a whole.
Imagine two identical blueprints for a house. 🏠 One is scaled down by 1...
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Key Concepts & Vocabulary
TermDefinitionExample
Congruent FiguresFigures that have the exact same size and shape. All corresponding sides and corresponding angles are equal in measure.Two triangles, ΔABC and ΔXYZ, are congruent if AB=XY, BC=YZ, AC=XZ, and ∠A=∠X, ∠B=∠Y, ∠C=∠Z.
Scale FactorA number which scales, or multiplies, some quantity. In geometry, it is the ratio of any two corresponding lengths in two similar geometric figures.If a side of length 5 cm in a figure is scaled by a factor of 2/3, its new length is 5 * (2/3) = 10/3 cm.
Unlike DenominatorsThe bottom numbers (denominators) of two or more fractions that are not equal. This indicates the wholes are divided into a different number of equal parts.In the fractions 1/3 and 1/4, the denominators 3 and 4 are unlike.
Least Common Denominator (LCD)The smalle...
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Core Formulas
Finding the Least Common Denominator (LCD)
LCD(b, d) = LCM(b, d)
To add fractions with unlike denominators, you must first find a common denominator. The most efficient one to use is the Least Common Denominator (LCD), which is the Least Common Multiple (LCM) of the original denominators.
Formula for Adding Fractions with Unlike Denominators
\frac{a}{b} + \frac{c}{d} = \frac{a \cdot (\frac{LCD(b,d)}{b})}{LCD(b,d)} + \frac{c \cdot (\frac{LCD(b,d)}{d})}{LCD(b,d)} = \frac{ad+bc}{bd}
This formula shows two methods. The first, more formal method involves converting each fraction to an equivalent fraction with the LCD. The second, quicker formula (ad+bc)/bd, works for any two fractions but may require simplification afterward.
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Challenging
Which statement provides the best geometric justification for why we must find a common denominator when adding fractions like 1/a and 1/b?
A.To combine fractional parts of congruent figures, the figures must be subdivided into an identical number of smaller, equal-sized units, which the LCD represents.
B.The formula (ad+bc)/bd is a proof that works for all numbers, so a geometric reason is not required.
C.Geometrically, adding denominators (a+b) would imply the whole figure has changed size, which is not true.
D.common denominator ensures the resulting shape is also a congruent figure.
Challenging
When adding scale factors a/b + c/d, the sum can be written as (ad+bc)/bd. In the geometric context of subdividing a congruent square to represent this sum, what does the denominator 'bd' represent?
A.The total number of shaded parts.
B.The perimeter of the newly subdivided square.
C.The simplest common number of subdivisions.
D.common, though not necessarily the least, number of equal-sized partitions of the original square.
Challenging
A large mosaic is made of 36 congruent square tiles. An artist decides to paint 1/4 of the total area blue and 2/9 of the total area red. Assuming no overlap, how many individual tiles are painted either blue or red?
A.3
B.26
C.17
D.1
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