Mathematics
Grade 10
15 min
Multiples of fractions
Multiples of fractions
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1
Introduction & Learning Objectives
Learning Objectives
Calculate the new dimensions of a 3D figure when its original dimensions are multiplied by a fractional scale factor.
Determine the fractional scale factor relating two similar three-dimensional figures.
Prove and apply the relationship between the linear scale factor (k), the surface area ratio (k^2), and the volume ratio (k^3) for similar solids.
Compute the new surface area of a solid after scaling its dimensions by a fractional multiple.
Compute the new volume of a solid after scaling its dimensions by a fractional multiple.
Solve for original dimensions, surface area, or volume given the properties of a fractionally scaled model.
Ever wonder how an architect's tiny model represents a massive skyscraper, or how a 3D printer creates a perfect half...
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Key Concepts & Vocabulary
TermDefinitionExample
Similar SolidsTwo three-dimensional figures that have the same shape, and all of their corresponding linear dimensions (like height, radius, or edge length) are proportional.A sphere with a radius of 2 cm is similar to a sphere with a radius of 6 cm. A cube with a side length of 5 inches is similar to a cube with a side length of 2.5 inches.
Scale Factor (k)The constant ratio of corresponding linear dimensions of two similar solids. If k > 1, it's an enlargement. If 0 < k < 1, it's a reduction.If a cone has a height of 10m and a similar cone has a height of 4m, the scale factor from the larger cone to the smaller is k = 4/10 = 2/5.
Fractional MultipleThe result of multiplying a quantity by a fraction. In this context, it refers to the new dimension...
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Core Formulas
Linear Scaling Rule
Dimension_{new} = k \cdot Dimension_{original}
To find any new linear dimension (length, width, height, radius) of a scaled figure, multiply the original dimension by the scale factor k.
Surface Area Scaling Rule
Area_{new} = k^2 \cdot Area_{original}
To find the new surface area of a scaled figure, multiply the original surface area by the square of the scale factor (k^2). This is because area is a two-dimensional measurement.
Volume Scaling Rule
Volume_{new} = k^3 \cdot Volume_{original}
To find the new volume of a scaled figure, multiply the original volume by the cube of the scale factor (k^3). This is because volume is a three-dimensional measurement.
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Challenging
A rectangular prism is scaled by a factor of 3/4. The resulting prism is then scaled again by a factor of 2/3. What is the final volume as a fraction of the original volume?
A.1/8
B.5/7
C.1/2
D.6/12
Challenging
The surface area of a solid is scaled by an unknown fractional multiple, x/y. The volume of the same solid is scaled by a fractional multiple of 8/27. What is the value of x/y?
A.2/3
B.4/9
C.4/6
D.16/81
Challenging
An architectural model of a skyscraper is made with a scale of 1/200. The volume of the model is 0.15 m³. The actual skyscraper will be constructed using concrete that costs $120 per cubic meter. What is the total cost of the concrete required for the actual skyscraper?
A.$1,200,000
B.$3,600,000
C.$144,000,000
D.$2,400,000
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