Question

Jim rented a lawnmower from 9:00 a.m. until 5:30 p.m. He paid a total of \(\$ 53.04\). What was the hourly rental fee? (Note: You may not use a calculator for this problem.) (A) \(\$ 5.58\) (B) \(\$ 6.24\) (C) \(\$ 6.63\) (D) \(\$ 7.07\)

Short answer

The hourly rental fee was approximately (C) $6.63.

Step-by-step solution

Calculate the Rental Duration

Jim rented the lawnmower from 9:00 a.m. until 5:30 p.m. To find the total duration of the rental, we can calculate the difference between these times. From 9:00 a.m. to 5:00 p.m., there are 8 hours. We then need to add the additional 30 minutes from 5:00 p.m. until 5:30 p.m. We convert 30 minutes to 0.5 hours (30 minutes / 60 minutes), so the total duration is 8.5 hours.

Find the Hourly Rental Fee

Now that we found the duration in hours, we can calculate the hourly rental fee. To do this, we'll divide the total cost Jim paid, $53.04, by the total duration in hours, 8.5. So, the hourly rental fee is: \(\frac{53.04}{8.5}= \frac{5304}{850}\)

Simplify the Fraction and Compare it with the Given Options

For this step, we need to simplify the fraction \(\frac{5304}{850}\) without using a calculator. Let's divide both numerator and denominator by their greatest common divisor. The greatest common divisor of 5304 and 850 is 2, so we divide both numbers by 2. \(\frac{5304}{850} = \frac{5304 \div 2}{850 \div 2} = \frac{2652}{425}\) Now compare this simplified fraction with the given options: (A) \(5.58 = \frac{558}{100}\) (B) \(6.24 = \frac{624}{100}\) (C) \(6.63 = \frac{663}{100}\) (D) \(7.07 = \frac{707}{100}\) Notice that (C) \(\frac{663}{100}\) is close to our simplified fraction \(\frac{2652}{425}\). As we cannot use a calculator, we can safely approximate that the hourly rental fee was (C) \( \$6.63\).

Key concepts

Fraction Simplification

Fraction simplification is an essential skill in mathematics, especially useful for problems where you can't use a calculator. The core idea is to reduce a fraction to its simplest form by finding the greatest common divisor (GCD) of the numerator and denominator.
To simplify the fraction \(\frac{5304}{850}\), we need to follow a step-by-step approach:

• Identify the GCD of 5304 and 850.
• Divide both numbers by the GCD. In this case, the GCD is 2.
• Simplifying gives us \(\frac{2652}{425}\).

Fraction simplification helps to make calculations easier and assists in comparing fractions, as in our problem where we needed to identify the best match from the options.

Hourly Rate Calculation

Calculating the hourly rate involves dividing the total cost by the total hours worked or rented. In our exercise, Jim rented a lawnmower and spent a total of \(\\(53.04\) over 8.5 hours.
To find the hourly rate, we create a division problem:

• The total cost is \(\\)53.04\).
• The total hours are 8.5.

Thus, the problem becomes simplified to determining the result of \(\frac{53.04}{8.5}\). Breaking this fraction down further involves cutting it into smaller, manageable parts like simplifying to \(\frac{2652}{425}\), which eventually leads us close to option (C) in the problem. Understanding hourly rate calculations is crucial for budgeting and managing time resources efficiently.

No Calculator Math Problems

Facing math problems without a calculator enhances your arithmetic skills and helps improve mental math abilities. It involves a series of techniques like improving estimation skills, breaking down complex calculations into smaller steps, and understanding the properties of numbers such as divisibility, common factors, and multiples.
In no calculator math problems such as our exercise, it's essential to:

• Break down calculations into simple steps.
• Use estimations to check the accuracy of your answer.
• Simplify fractions manually as we did.

Utilizing these strategies helps to solve problems efficiently and boosts confidence in handling everyday math scenarios without technological aid.