Lesson Notes By Weeks and Term v4 - SHS 3
PROPORTIONAL REASONING
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PROPORTIONAL REASONING
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Subject: Mathematics
Class: SHS 3
Term: 1st Term
Week: 6
Grade code: 3.1.2.LI.2
Strand code: 1
Sub-strand code: 2
Content standard code: 3.1.2.CS.2
Indicator code: 3.1.2.LI.2
Theme: NUMBERS FOR EVERYDAY LIFE
Subtheme: PROPORTIONAL REASONING
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In our previous studies, we learned how two quantities can be related through direct and inverse variation. However, real-world situations are often more complex. For instance, the cost of organising an event isn't just based on the number of guests; it might involve a fixed hall rental fee as well. The yield of a maize farm depends on both the amount of rainfall and the quantity of fertiliser used. This lesson extends our understanding of variation to these more realistic scenarios. We will explore Joint Variation, where a quantity depends on two or more other quantities, and Partial Variation, where a quantity is made up of a fixed part and a variable part.
A. Quick Recap: Direct and Inverse Variation
Before we build our new house, let's check the foundation. Direct Variation: Two quantities, say `y` and `x`, vary directly if their ratio is constant. As one increases, the other increases at the same rate. Symbolically: `y ∝ x` Equation: `y = kx`, where `k` is the constant of variation. *Example:* The more hours you work (`x`), the more money you earn (`y`). Inverse Variation: Two quantities, `y` and `x`, vary inversely if their product is constant. As one increases, the other decreases proportionally. Symbolically: `y ∝ 1/x` Equation: `y = k/x`, where `k` is the constant. *Example:* The more workers (`x`) you put on a job, the less time (`y`) it takes to complete. B. Joint Variation
Definition: Joint variation describes a situation where one variable depends on two or more other variables. It is an extension of direct variation. A quantity varies "jointly" as a set of other variables if it varies directly as their product.
Mathematical Form: If `z` varies jointly as `x` and `y`, it means: `z ∝ x * y` Which we write as the equation: `z = kxy`, where `k` is the constant of variation.