Question

Treat the given numbers in these problems as exact, and leave your answers in fractional form. Do not use your calculator. One crew can assemble \(M\) machines in \(p\) days. Another crew can assemble \(N\) of these machines in \(q\) days. Write an expression for the number of machines that both crews together can assemble in 1 day, combine into a single term, and simplify.

Short answer

\( \frac{qM + pN}{pq} \) machines per day

Step-by-step solution

Determine Individual Crew Production Rate

First, we find the rate at which each crew assembles machines. For the first crew, the rate is the number of machines, M, divided by the number of days, p. This is written as \( \frac{M}{p} \) machines per day. For the second crew, the rate is \( \frac{N}{q} \) machines per day.

Sum of Individual Production Rates

To find the combined rate, add the rate of the first crew to the rate of the second crew. The combined rate is \( \frac{M}{p} + \frac{N}{q} \) machines per day.

Combine Rates into Single Term

Now, to combine the rates into a single term, find a common denominator and combine the fractions. The common denominator here is pq. Thus, the expression becomes \( \frac{qM + pN}{pq} \) machines per day.

Simplify the Expression

There are no common factors between the numerator and the denominator, so the expression \( \frac{qM + pN}{pq} \) is already the simplest form for the number of machines assembled by both crews in one day.

Key concepts

Rates and Work

Figuring out how quickly work can be accomplished is a common problem in math, and it's particularly important in the context of jobs and productivity. In our exercise, we're looking at two crews assembling machines. The rate at which each crew works is the number of machines they can assemble per day.

Understanding rates in the context of work involves dividing the total quantity of work done (in this case, assembling machines) by the time taken to complete the work. Sometimes, this rate is called 'work rate' or 'production rate' and is often represented as units of work per time. To calculate the collective rate of two separate crews working together, we simply add their individual rates. This gives us an equation that allows us to predict how much work both crews can accomplish together in a day, which is an essential concept in optimizing resources and meeting production goals.

Adding Fractions

When combining the work rates of two crews, we are essentially adding fractions. Adding fractions is straightforward when they have the same denominator, but when they don’t, we must find a common denominator to combine them.

In our exercise, the common denominator is obtained by multiplying the denominators of each crew’s rate (that is, the days, 'p' and 'q'). Once this common denominator, 'pq', is found, we then adapt the numerators - which represent the machine production rates - to conform to the common base. This allows us to sum the fractions accurately and express the combined rate of both crews as a single fraction. It’s essential to remember the rule that the denominators must be the same to add fractions effectively, which is a crucial step in representing combined rates.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that helps make equations easier to work with. It involves reducing expressions to their most basic form without changing their value. We do this by combining like terms, factoring, and canceling out terms when possible.

In the context of the production rate problem, after finding a common denominator and combining the fractions into one, we look to simplify the resulting expression. If there are common factors between the numerator and the denominator, we can reduce them to simplify the fraction. However, in this scenario, the numerator (representing the combined work rate in terms of machines) and the denominator (the product of the days) do not share common factors, indicating that the expression derived for the rate at which both crews work together is already in its simplest form.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. In algebra, working with rational expressions is commonplace and understanding how to manipulate them is essential.

In our exercise, the final expression representing the collective production rate of both crews is a rational expression. The numerator consists of a linear combination of terms representing the total machines assembled, and the denominator is the product of the days taken by each crew. Rational expressions can be simplified, added, subtracted, multiplied, and divided, following specific rules for fractions. When dealing with rational expressions in the context of rates and work, it is crucial to interpret them correctly to make meaningful predictions about productivity and efficiency.