Question

You check the oil level in your car, and it is down a quart. You dutifully add an estimated \(\frac{1}{3}\) quart because that is all you have on hand. a. Later, you buy another quart of oil and add about \(\frac{1}{2}\) quart before you stop and ask yourself this question: How much more oil will it take without overfilling? b. Estimate the amount wasted if you pour in the remaining half quart, assuming that anything over the fill line spills out.

Short answer

Answer: It will take 1/6 quart more to fill the car without overfilling, and if the remaining half quart is poured in, 1/3 quart of oil will be wasted.

Step-by-step solution

Determine the total amount of oil added so far

Add the initial \(\frac{1}{3}\) quart that was added and the later \(\frac{1}{2}\) quart: \(\frac{1}{3}+\frac{1}{2}\). To add the fractions, we need a common denominator, which is the least common multiple (LCM) of 3 and 2: LCM(3,2) = 6.

Find the common denominator

Convert the fractions to having a common denominator of 6 by multiplying the numerator and denominator by the correct factor: \(\frac{1}{3} \times \frac{2}{2} = \frac{2}{6}\) and \(\frac{1}{2} \times \frac{3}{3} = \frac{3}{6}\).

Add the fractions with the common denominator

Now, add the two fractions with the common denominator of 6: \(\frac{2}{6}+\frac{3}{6}=\frac{2+3}{6}=\frac{5}{6}\). So, so far a total of \(\frac{5}{6}\) quarts of oil were added.

Calculate the amount needed to fill up the quart

Since the car was short by one quart initially, and we have added \(\frac{5}{6}\) quarts until now, to find how much more oil is needed to fill the quart without overfilling, subtract the added oil amount from 1: \(1-\frac{5}{6}=\frac{6}{6}-\frac{5}{6}=\frac{6-5}{6}=\frac{1}{6}\). Therefore, it will take \(\frac{1}{6}\) quarts more to reach the full quart capacity without overfilling. b. Estimate the amount wasted if you pour in the remaining half quart

Calculate the oil wasted if the remaining half quart is added

In this case, you would be adding \(\frac{1}{2}\) quart to the car that only needs \(\frac{1}{6}\) quart more to be full. To find the quantity of oil wasted, subtract the needed amount from the added amount: \(\frac{1}{2}-\frac{1}{6}\).

Find the common denominator for the fractions to be subtracted

We find the LCM of 2 and 6 to have a common denominator: LCM(2,6) = 6. Convert the fractions to have the common denominator of 6: \(\frac{1}{2} \times \frac{3}{3} = \frac{3}{6}\) and \(\frac{1}{6} = \frac{1}{6}\).

Subtract the fractions with the common denominator

Now, subtract the two fractions with the common denominator of 6: \(\frac{3}{6}-\frac{1}{6}=\frac{3-1}{6}=\frac{2}{6}\).

Simplify the fraction

The fraction \(\frac{2}{6}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD): GCD(2,6) = 2. Then, \(\frac{2}{6} \div \frac{2}{2} = \frac{1}{3}\). Therefore, if the remaining half quart is poured into the car, an estimated \(\frac{1}{3}\) quart of oil will be wasted.