Consider the formula 𝑝 is equal to four 𝑘 over 𝐿. 𝑘 is 24 to two significant figures, and 𝐿 is 1.2 to two significant figures. Work out the upper and lower bounds for 𝑝. Give your answer to three significant figures.
Before we do anything, let’s find the upper and lower bounds for 𝑘 and 𝐿. 𝑘 is 24 to two significant figures. To work out the upper and lower bounds, we need to consider what it could have rounded to had it rounded up to the next number or had it rounded down to the number before.
If it had rounded to the number before, it would have been 23. And if it had rounded to the next number up, it would have been 25. The upper and lower bounds are found by finding the midpoint of these two numbers. The midpoint of 23 and 24 is 23.5. This is the absolute lowest value it could have been to still round to two significant figures. The midpoint of 24 and 25 is 24.5.
Now you might have noticed that 24.5 would actually round to 25. That’s true. But 24.4, whilst it does round to 24, isn’t the absolute largest number that does. 24.49 rounds to 24. 24.499 rounds to 24. And in fact, we can keep on going and keep adding nines to show that 24.49 recurring also rounds to 24. That gets so close to 24.5 that we say the upper bound is 24.5. So the lower bound for 24 is 23.5, and the upper bound for 24 is 24.5.
Let’s repeat this process for 𝐿. It’s 1.2. The number immediately below that is 1.1, and the number immediately after it is 1.3. The lower bound is the halfway point between 1.1 and 1.2. It’s 1.15. And the upper bound is the halfway point between 1.2 and 1.3. That’s 1.25. So the lower bound for 𝐿 is 1.15, and the upper bound for 𝐿 is 1.25.
We now need to use this information in our formula for 𝑝. We are dividing. To get the smallest possible value of 𝑝, we need to choose the smallest possible value of 𝑘 and then divide by the largest possible value of 𝐿. Essentially, what we’re doing is we’re dividing a smaller number into larger pieces. So each piece will be much smaller.
If we substitute the lower bound of 𝑘 and the upper bound of 𝐿 into our formula, we get four multiplied by 23.5 all over 1.25. That’s 75.2. We were told to give our answer to three significant figures, but that doesn’t affect this number at all.
Next, we’re going to find the upper bound for 𝑝. We need to find the largest possible value of 𝑘. And then we’re going to divide by the smallest possible value of 𝐿. This is essentially saying let’s take the biggest possible number and divide it into the fewest pieces. That’s going to give us ultimately larger pieces.
The upper bound for 𝑘 is 24.5, and the lower bound for 𝐿 is 1.15. So the upper bound for 𝑝 is four multiplied by 24.5 all over 1.15. That’s 85.217. The third significant figure is two. And the digit immediately to its right is the deciding digit. Since the deciding digit is less than five, we round our number down. And the upper bound is 85.2. The lower bound for 𝑝 is 75.2, and the upper bound is 85.2.