Multiply fractions by whole numbers: set 2

Learning Objectives Calculate the volume of prisms and pyramids when dimensions are fractional and multipliers are whole numbers. Determine the surface area of composite 3D figures involving fractional lengths and a whole number of components. Apply the multiplication of fractions by whole numbers to solve problems involving the density and mass of 3D objects. Analyze how scaling a 3D figure's dimensions by a whole number factor affects its volume when the original volume is a fraction. Solve multi-step word problems involving rates of change (e.g., filling a container) that require multiplying fractions by whole numbers within a 3D geometry context. Compute the total volume of a set of multiple, identical 3D objects with fractional dimensions. How do manufacturers calc...

TermDefinitionExample Volume of a PrismThe amount of space a prism occupies, calculated by multiplying the area of its base (B) by its height (h). The calculation often involves a fractional area or height multiplied by a whole number.A rectangular prism with a base area of 9/2 cm² and a height of 10 cm has a volume of 10 * (9/2) = 90/2 = 45 cm³. Volume of a Pyramid or ConeThe space occupied by a pyramid or cone, calculated as one-third of the product of its base area (B) and height (h). The formula itself introduces a fraction (1/3) that is often multiplied by whole numbers.A pyramid with a base area of 24 m² and a height of 7/2 m has a volume of (1/3) * 24 * (7/2) = 8 * (7/2) = 56/2 = 28 m³. Surface Area of Multiple ObjectsThe total exposed area of a set of objects. For 'n' id...

Multiplying a Whole Number by a Fraction n \cdot \frac{a}{b} = \frac{n \cdot a}{b} To multiply a whole number (n) by a fraction (a/b), multiply the whole number by the numerator (a) and keep the denominator (b) the same. This is the core operation for finding total volumes or scaling fractional dimensions. Volume of a Prism or Cylinder V = B \cdot h The volume (V) is the product of the base area (B) and the height (h). Use this when, for example, B is a fraction and h is a whole number, or vice versa. Volume Scaling Principle V_{new} = k^3 \cdot V_{original} If a 3D figure's linear dimensions are all multiplied by a whole number scaling factor (k), the new volume is k-cubed times the original volume. This is a key application of multiplying a whole number (k³) b...