4.3. Multiplying by a Constant

What would be your guess: can an American Corvette get away if chased by an Italian police Lamborghini?

corvette lamborgghini

The top speed of the Corvette is 186 mph ± 2 mph. The top speed of the Lamborghini Gallardo is 309 km/h ± 5 km/h. We know that 1 mile = 1.61 km. In order to convert the speed of the Corvette to km/h, we need to multiply it by the factor of 1.61. What should we do with the error?

Well, you've learned in the previous section that when you multiply two quantities, you add their relative errors. The relative error on the Corvette speed is 1%. However, the conversion factor from miles to kilometers can be regarded as an exact number.1 There is no error associated with it. Its relative error is 0%. Thus the relative error on the Corvette speed in km/h is the same as it was in mph, 1%. (adding relative errors: 1% + 0% = 1%.) It means that we can multiply the error in mph by the conversion constant just in the same way we multiply the speed. So our answer for the maximum speed of the Corvette in km/h is: 299 km/h ± 3 km/h.

Now we are ready to answer the question posed at the beginning in a scientific way. The highest possible top speed of the Corvette consistent with the errors is 302 km/h. The lowest possible top speed of the Lamborghini Gallardo consistent with the errors is 304 km/h. Bad news for would-be speedsters on Italian highways. No way can you get away from that police car.

The rule we discussed in this chase example is true in all cases involving multiplication or division by an exact number. You simply multiply or divide the absolute error by the exact number just as you multiply or divide the central value; that is, the relative error stays the same when you multiply or divide a measured value by an exact number.

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1 For this example, we are regarding the conversion 1 mile = 1.61 km as exact. Actually, the conversion factor has more significant digits.

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