Mathematics
Grade 10
15 min
Multiply fractions and mixed numbers in recipes
Multiply fractions and mixed numbers in recipes
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1
Introduction & Learning Objectives
Learning Objectives
Convert mixed numbers to improper fractions to facilitate multiplication.
Calculate the volume and surface area of standard three-dimensional figures (cylinders, spheres, cones).
Apply the multiplication of fractions and mixed numbers to solve problems involving scaling the volume of 3D figures.
Determine the required quantity of materials for a 3D object based on a 'recipe' that specifies an amount per unit of volume or surface area.
Interpret a word problem to identify the correct 3D figure, the relevant formula (volume or surface area), and the fractional quantities involved.
Prove the material requirements for a scaled 3D model by applying the principles of geometric scaling and fractional multiplication.
Ever wonder how a baker knows exactl...
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Key Concepts & Vocabulary
TermDefinitionExample
Mixed NumberA number consisting of a whole number and a proper fraction.2 3/4 cups of flour. This means 2 whole cups and 3/4 of another cup.
Improper FractionA fraction in which the numerator (top number) is greater than or equal to the denominator (bottom number).To convert 2 3/4 to an improper fraction: (2 * 4 + 3) / 4 = 11/4.
VolumeThe amount of three-dimensional space an object occupies, measured in cubic units.The volume of a cylindrical can of soup tells you how much soup it can hold, e.g., 473 cm³.
Surface AreaThe total area of the surface of a three-dimensional object, measured in square units.The surface area of a spherical ornament determines how much paint is needed to cover it, e.g., 201 in².
Scale FactorA number which scales, or multiplies, some quantity...
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Core Formulas
Multiplying Mixed Numbers
a \frac{b}{c} \times d \frac{e}{f} = \frac{ac+b}{c} \times \frac{df+e}{f}
To multiply mixed numbers, first convert each mixed number into an improper fraction. Then, multiply the numerators together and the denominators together. Simplify the resulting fraction if possible.
Volume of a Cylinder
V = \pi r^2 h
The volume (V) of a cylinder is found by multiplying pi (π ≈ 3.14159) by the square of the radius (r) of the base, and then multiplying by the height (h).
Volume Scaling of Similar Figures
V_{new} = V_{original} \times (k)^3
When a 3D figure is scaled by a linear scale factor (k), its volume increases by a factor of k cubed. This is critical for calculating materials for scaled-up models.
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Challenging
A student claims that if a cylinder's recipe calls for an ingredient amount proportional to its volume, and you scale the cylinder's radius by a factor 'k' but leave the height unchanged, the amount of ingredient needed will scale by k³. Which statement correctly analyzes this claim?
A.The claim is correct because volume always scales by k³.
B.The claim is incorrect; the new volume is π(kr)²h = k²(πr²h), so the ingredient amount scales by k².
C.The claim is incorrect; the new volume is πr²(kh) = k(πr²h), so the ingredient amount scales by k.
D.The claim is correct because both radius and height must be scaled for the principle to change.
Challenging
A recipe for a 3D-printed model of a cone is based on its volume. A new, larger model is printed that is geometrically similar to the original, but requires 3 3/8 times the amount of filament. What was the linear scale factor (k) applied to the original model's dimensions?
A.1 1/2
B.3 3/8
C.1 7/8
D.11 3/8
Challenging
A spherical ball is made using a recipe that requires 1 1/2 cups of polymer. A larger ball is created by scaling the linear dimensions of the original by a factor of 2 1/2. What is the difference in the amount of polymer required between the new ball and the original ball?
A.15 5/8 cups
B.21 15/16 cups
C.23 7/16 cups
D.1 1/2 cups
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