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Questions and Answers
What does the Invertendo property state about ratios?
What does the Invertendo property state about ratios?
- If $a/b = c/d$, then $a/c = b/d$.
- If $a/b = c/d$, then $b/a = d/c$. (correct)
- If $a/b = c/d$, then $b/a = c/d$.
- If $a/b = c/d$, then $a-b = c-d$.
Which operation is described by the Alternando property?
Which operation is described by the Alternando property?
- If $a/b = c/d$, then $a/c = b/d$. (correct)
- If $a/b = c/d$, then $a+c = b+d$.
- If $a/b = c/d$, then $b/d = a/c$.
- If $a/b = c/d$, then $c/d = a+b/b.
What is the result of applying the Dividendo property?
What is the result of applying the Dividendo property?
- If $a/b = c/d$, then $(a-b)/b = (c-d)/d$. (correct)
- If $a/b = c/d$, then $(a-b)/a = (c-d)/c$.
- If $a/b = c/d$, then $(a-b)/b = (c+d)/d$.
- If $a/b = c/d$, then $(a+b)/b = (c+d)/d$.
Which property provides an equation in the form $(a+b)/b = (c+d)/d$?
Which property provides an equation in the form $(a+b)/b = (c+d)/d$?
What utility do these properties of equal ratios provide?
What utility do these properties of equal ratios provide?
What is the result when applying the Componendo property to the ratio $ rac{a}{b} = rac{c}{d}$?
What is the result when applying the Componendo property to the ratio $ rac{a}{b} = rac{c}{d}$?
Which equation represents the Dividendo property when applied to the ratio $ rac{a}{b} = rac{c}{d}$?
Which equation represents the Dividendo property when applied to the ratio $ rac{a}{b} = rac{c}{d}$?
What is the result of combining the Componendo and Dividendo properties?
What is the result of combining the Componendo and Dividendo properties?
How is the Componendo property generalized according to the provided information?
How is the Componendo property generalized according to the provided information?
Which of the following describes the application of the Dividendo property multiple times?
Which of the following describes the application of the Dividendo property multiple times?
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Study Notes
Properties of Equal Ratios
- Invertendo: If two ratios are equal, then their reciprocals are also equal.
- Alternando: If two ratios are equal, then their alternate ratios are also equal.
- Dividendo: If two ratios are equal, then the ratio of their differences to their denominators is also equal.
- Componendo-Dividendo: If two ratios are equal, then the ratio of their sum to their difference is equal to the ratio of the sum of the corresponding terms of the other ratio to the difference of the corresponding terms.
Componendo
- Adding 1 to both sides of the equation $\frac{a}{b} = \frac{c}{d}$ yields $\frac{a+b}{b} = \frac{c+d}{d}$
Dividendo
- Subtracting 1 from both sides of the equation $\frac{a}{b} = \frac{c}{d}$ yields $\frac{a-b}{b} = \frac{c-d}{d}$
Componendo-Dividendo
- This property is a combination of Componendo and Dividendo.
General Forms
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Componendo: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a + mb}{b} = \frac{c + md}{d}$
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Dividendo: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a-mb}{b} = \frac{c-md}{d}$
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These general forms demonstrate the reversibility of Componendo and Dividendo operations.
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