Question

Treat the given numbers in these problems as exact, and leave your answers in fractional form. Do not use your calculator. A certain work crew can grade 7 mi of roadbed in 3 days, and another crew can do 9 mi in 4 days. How much can both crews together grade in 1 day? Hint: First find the amount that each crew can do in one day (e.g., the first crew can grade \(\frac{7}{3}\) mi per day). Then add the separate amounts to get the daily total. You can use a similar approach to the other work problems in this group.

Short answer

\frac{55}{12} \text{ mi/day}

Step-by-step solution

Determine the daily rate for the first crew

Calculate the amount of work the first crew can do per day by dividing their total work by the number of days it takes them to complete that work. For the first crew: \(\frac{7 \text{ mi}}{3 \text{ days}}\) = \(\frac{7}{3} \text{ mi/day}\).

Determine the daily rate for the second crew

Perform the same calculation for the second crew to determine their daily work rate. For the second crew: \(\frac{9 \text{ mi}}{4 \text{ days}}\) = \(\frac{9}{4} \text{ mi/day}\).

Combine the daily rates of both crews

Add the daily rates of the two crews together to find out how much roadbed they can grade working together in one day. \(\frac{7}{3} \text{ mi/day} + \frac{9}{4} \text{ mi/day} = \frac{28}{12} \text{ mi/day} + \frac{27}{12} \text{ mi/day} = \frac{55}{12} \text{ mi/day}\).

Key concepts

Fractions in Mathematics

Fractions are foundational in math and are used to represent parts of a whole. They consist of two numbers: a numerator and a denominator. The numerator indicates how many parts are being accounted for, while the denominator shows the total number of equal parts the whole is divided into. For instance, the fraction 1/2 �, means we are considering one part out of two equal parts.

Rate Calculation

Rate calculation in mathematics is a method used to measure one quantity in relation to another. When we talk about work rate, we focus on the amount of work done per unit of time. Using fractions, we can express the work done by individuals or machines, such as how many miles a crew can grade in a day. This value is essential for solving work rate problems effectively. It is represented as a fraction of work done divided by the time taken, for example, miles per day in our case. When you encounter a rate calculation, always remember that it signifies a ratio – it is work (the numerator) over time (the denominator).

Combined Work Rates

When we have more than one person or group working together on a task, we calculate their combined work rate by adding their individual rates together. This requires a good understanding of fractions, as individual work rates are often expressed as such. Using rate calculation, we can determine how much two or more combined forces can accomplish in a given time frame.

To accurately forecast the combined output, ensure you add the individual work rates taking into account their least common multiple (LCM) to find the equivalent fractions with the same denominator. Doing so allows for a straightforward addition of the numerators to calculate the combined work rate.

In our road grading example, by combining the fractions representing the rates of two crews, we determine the total miles they can grade jointly in one day. The concept of combined work rates is critical in project management and planning, as it helps allocate resources more efficiently and predict project timelines.