Problem 57 - Prealgebra

Question

The average times it takes Miguel to cut his lawn and his neighbor's lawn are given in the table. Last summer, he cut his lawn 10 times and his neighbor's 5 times. About how many hours did he spend cutting both lawns? $$\begin{array}{|c|c|}\hline \text { Lawn } & \begin{array}{c}\text { Time of Cut } \\\\\text { (hours) }\end{array} \\\\\hline \text { Miguel's } & \frac{3}{4} \\\\\hline \text { Neighbor's } & \frac{2}{4} \\\\\hline\end{array}$$ A. 8 \frac{1}{2} h B. 9 h C. 9 \frac{1}{2} \mathrm{l} D. 10 h

Step-by-Step Solution

Answer

Miguel spent 10 hours cutting both lawns.

Step 1: Understand the Tasks

Miguel cuts two lawns - his own, and his neighbor's. We need to find out how many hours he spent cutting both lawns over the summer.

Step 2: Identify the Time per Cutting

Identify the average time for one cut of each lawn as given in the table. Miguel's lawn takes \(\frac{3}{4}\) hour per cut, and his neighbor's lawn takes \(\frac{2}{4}\) hour per cut.

Step 3: Calculate Time Spent on Miguel's Lawn

Since Miguel cut his lawn 10 times, calculate the total time spent: \(10 \times \frac{3}{4} = \frac{30}{4} = 7.5\) hours.

Step 4: Calculate Time Spent on Neighbor's Lawn

Miguel cut his neighbor's lawn 5 times. Total time spent: \(5 \times \frac{2}{4} = \frac{10}{4} = 2.5\) hours.

Step 5: Add the Total Times

Add the time spent on both lawns to find the total time: \(7.5 + 2.5 = 10\) hours.

Step 6: Confirm the Answer

Compare the calculated total time with the given options in the question. The total time spent is 10 hours, which matches option D.

Key Concepts

Time calculation

Time calculation is an essential skill that helps us understand how to manage and allocate time for different activities. In this exercise, we are tasked with finding out how much time Miguel spent mowing lawns over the summer.

By calculating the total time spent on repetitive tasks, such as lawn mowing, we gain insights into effective time management. This practice involves multiplying the tasks by the time spent on each individual task and then summing the totals.

Time calculations help us to make informed decisions and optimize our schedules to be more efficient.

Fractions

Fractions are a mathematical way of representing parts of a whole. In this exercise, the average time it takes Miguel to mow each lawn is given as a fraction of an hour: \(\frac{3}{4}\) hours for his lawn and \(\frac{2}{4}\) hours for his neighbor's lawn.

Understanding fractions is crucial when we need to perform calculations involving parts of a whole unit, like hours in a day. By multiplying fractions, Miguel can find the total time spent on each lawn across multiple mowings.

Recognizing how to work with fractions allows us to handle everyday situations involving division, measurement, and timekeeping with ease.

Problem solving

Problem solving is the process of identifying a challenge and determining the steps needed to overcome it. Miguel's problem involves calculating the time spent on lawn mowing over a specific period.

To solve this, we break down the problem into smaller steps: figuring out the average time for each lawn mowing, multiplying by the number of mowings, and then adding totals. These steps structure our approach, allowing us to find a solution methodically.

Problem solving hones critical thinking and analytical skills, empowering us to tackle diverse problems efficiently and confidently.

Mathematics skills

Mathematics skills include knowledge and understanding of number operations, calculations, and other mathematical processes. This exercise highlights several mathematics skills, such as working with fractions, multiplication, and sum calculation.

Specifically, Miguel needs to multiply fractions representing time, sum those results, and check against possible answers. These actions demonstrate a range of skills crucial in mathematics, such as attention to detail and logical reasoning.

Strengthening mathematical skills equips individuals to approach everyday tasks, like time calculation, with greater ease and accuracy, and problem solving becomes a more intuitive process.