Proportion Switches 767316

1. **State the problem:** We are given four numbers $0.5$, $4$, $1.6$, and $0.2$ and asked to write proportions using these numbers and perform all possible switches and changes. 2. **Recall the definition of proportion:** A proportion is an equation stating that two ratios are equal, i.e., $$\frac{a}{b} = \frac{c}{d}$$ where $a,b,c,d$ are numbers and $b,d \neq 0$. 3. **Write the initial proportion:** Using the numbers in order, one proportion is: $$\frac{0.5}{4} = \frac{1.6}{0.2}$$ 4. **Check if the proportion is true:** Calculate each ratio: $$\frac{0.5}{4} = 0.125$$ $$\frac{1.6}{0.2} = 8$$ Since $0.125 \neq 8$, this proportion is false. 5. **Try switching terms to find true proportions:** - Switch means swapping terms either across the equal sign or within a ratio. - For example, switch means: $$\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{b}{a} = \frac{d}{c}$$ (invert both ratios) - Or swap means: $$\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a}{c} = \frac{b}{d}$$ (cross swap) 6. **Try switching the first ratio:** $$\frac{4}{0.5} = \frac{1.6}{0.2}$$ Calculate: $$\frac{4}{0.5} = 8$$ $$\frac{1.6}{0.2} = 8$$ This is true. 7. **Try switching the second ratio:** $$\frac{0.5}{4} = \frac{0.2}{1.6}$$ Calculate: $$\frac{0.5}{4} = 0.125$$ $$\frac{0.2}{1.6} = 0.125$$ This is also true. 8. **Try cross switching:** $$\frac{0.5}{1.6} = \frac{4}{0.2}$$ Calculate: $$\frac{0.5}{1.6} = 0.3125$$ $$\frac{4}{0.2} = 20$$ False. 9. **Try inverting both sides:** $$\frac{4}{0.5} = \frac{0.2}{1.6}$$ Calculate: $$\frac{4}{0.5} = 8$$ $$\frac{0.2}{1.6} = 0.125$$ False. 10. **Summary of true proportions found:** - $$\frac{4}{0.5} = \frac{1.6}{0.2}$$ - $$\frac{0.5}{4} = \frac{0.2}{1.6}$$ These are valid proportions formed by switching numerator and denominator pairs. **Final answer:** The true proportions using the numbers $0.5$, $4$, $1.6$, and $0.2$ are: $$\frac{4}{0.5} = \frac{1.6}{0.2}$$ and $$\frac{0.5}{4} = \frac{0.2}{1.6}$$