Question

You investigate how long it would take you and a friend to fold 1000 origami cranes. You take 2 minutes to fold 1 crane. Let \(x\) represent the number of minutes it might take a friend to fold 1 crane. . Write an expression for the combined work rate per hour of you and your friend (the part of the job completed in 1 hour if you both work together).

Short answer

The total combined work rate per hour of you and your friend is \(30 + \frac{60}{x}\) cranes per hour.

Step-by-step solution

Convert the time to fold a crane from minutes to an hourly rate.

To start, consider that the rate for you is given as '2 minutes for 1 crane'. However, we need the rate as 'cranes per hour'. So to find out how many cranes you can fold in an hour, the conversion must be applied: 1 hour = 60 mins. So you can fold \(\frac{60}{2} = 30\) cranes in 1 hour.

Represent your friend's rate in terms of \(x\) and convert it into an hourly rate.

Now let's deal with your friend's rate. Per each crane, your friend takes \(x\) mins. Therefore, in 1 hour (60 mins), your friend will fold \(\frac{60}{x}\) origami cranes.

Slug the rates and find out the combined work rate.

Combining both rates, the combined work done by both per hour is going to be total cranes folded by you and your friend, which is \(30 + \frac{60}{x}\)

Key concepts

Expressions with Variables

Working with expressions that include variables allows you to model real-world situations mathematically. Variables, such as the letter 'x' in this case, serve as placeholders for values that may change or that we may not know yet. In the context of the problem, 'x' represents the unknown time in minutes it takes a friend to fold one crane. Using variables is essential as it allows us to express the relationship between different quantities, even when some of those quantities are not known beforehand.

Imagine a scenario where you'd like to calculate results under various conditions. If you have a formula that includes variables, you can easily swap out these variables with actual numbers when they become known, and this versatility is the power of using expressions with variables.

Rate Conversion

Rate conversion is a method used to change the measurement units of a given rate to another set of units for better comparison or integration in calculations. In the crane-folding problem, conversion from 'cranes per minute' to 'cranes per hour' is necessary to determine the combined work rate. Given that there are 60 minutes in an hour, you can determine your folding rate by dividing 60 by the time it takes to fold one crane.

This conversion grants a common basis for comparison or combination, which is key because the rates must be expressed in the same units to find the combined work rate accurately. Without rate conversion, it would be impossible to apply these calculations to real-life scenarios that often require different units of measurement.

Algebraic Representation

Algebraic representation involves writing the mathematical relationships between variables and numbers using algebraic expressions and equations. The step-by-step solution offers a clear example of this. Initially, the personal rate of folding is expressed as an algebraic expression, '30 cranes per hour'.

For your friend’s rate, the expression is \(\frac{60}{x}\) cranes per hour, which is a form of algebraic representation showing the rate in terms of the variable 'x'. This type of representation is extremely useful because it allows you to create an equation that would give you the ability to plug in different values of 'x' to find out how the combined rate of work changes with your friend's folding speed.

Rational Expressions

Rational expressions are fractions that involve polynomials in the numerator, denominator, or both and are used to represent quantities that can vary. In the exercise, your friend’s rate is given by the rational expression \(\frac{60}{x}\), which is the number of cranes folded in one hour per the time taken to fold one crane. This rational expression can be simplified or manipulated just like numerical fractions and is an algebraic representation of the rate.

Working with rational expressions is integral in algebra because it allows us to handle variable rates or ratios and understand how they change in relation to one another, such as when we’re combining work rates of two individuals to find out how much they can accomplish jointly.