A desktop processor for a computer can calculate approximately 100 operations in $2.5 \times 10^{-7}$ seconds. At this rate, approximately how many operations can be calculated by this computer in one half hour?

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We can use a simple proportion, as long as we make sure the units correspond to each other. We can convert one half hour to 30 minutes, which is 1800 seconds.

$$\frac{100 \text{ operations}}{2.5 \times 10^{-7}\text{ seconds}} =\frac{O \text{ operations}}{1{,}800 \text{ seconds}}$$ $$100(1{,}800)= (2.5 \times 10^{-7})O$$ $$O = \frac{180{,}000}{2.5 \times 10^{-7}}$$ $$O = \boxed{7.2 \times 10^{11}} \text{ operations}$$

Using the standard unit conversion method:

Writing the rates and quantities so that they cancel out and leave us with operations:

$$\frac{100 \text{ operations }}{2.5 \times 10^{-7} \cancel{\text{ seconds }}} \cdot \frac{60 \cancel{\text{ seconds }}}{1 \cancel{\text{ minute }}} \cdot \frac{60 \cancel{\text{ minutes }}}{1 \cancel{\text{ hour }}} \cdot 0.5 \cancel{\text{ hours}}$$ $$\frac{100}{2.5 \times 10^{-7}} \cdot \frac{60}{1} \cdot \frac{60}{1} \cdot 0.5 \text{ operations}$$ $$= \boxed{7.2 \times 10^{11}} \text{ operations}$$