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. The expression \(\frac{60}{1000 x}\) is your friend's work rate per hour. Explain why.

Short answer

The expression \(\frac{60}{1000x}\) refers to the friend's work rate per hour because \(x\) is the number of minutes your friend takes to fold one crane, \(\frac{x}{60}\) then converts this time in minutes to hours. \(\frac{60}{x}\) is the number of cranes your friend can fold in an hour. And since we want to find out the rate of folding 1000 cranes, we multiply \(\frac{60}{x}\) by 1000 resulting in the expression \(\frac{60}{1000x}\).

Step-by-step solution

Understand the Context

In this context, \(x\) represents the time (in minutes) it takes a friend to fold one origami crane. A work rate is defined as the amount of work done in a certain amount of time. In this case, it is the number of origami cranes folded per hour.

Convert minutes to hours

The time recorded, \(x\), is in minutes. However, the work rate that we want to find is in terms of hours. As we know, there are 60 minutes in an hour. Therefore, to convert the friend's time from minutes to hours, we divide \(x\) by 60.

Explain the Expression

After converting to hours, the expression then states that she folds 1 crane in \(\frac{x}{60}\) hours. When we want to find out how many cranes are folded in 1 hour, we take the reciprocal of \(\frac{x}{60}\) which gives us \(\frac{60}{x}\) as the rate. Now, the exercise asks for the rate for folding 1000 cranes. So we multiply \(\frac{60}{x}\) by 1000, leading to the expression \(\frac{60}{1000x}\), which is more accurately written as \(\frac{60}{x} * 1000\). That's why \(\frac{60}{1000x}\) represents the friend's work rate per hour in terms of folding 1000 cranes.

Key concepts

Rate of Work

Understanding the rate of work is essential when calculating how efficiently a task is performed. In the given problem, the task is folding origami cranes, and the efficiency is measured by the number of cranes that can be folded in a period, specifically, per hour.

Imagine you are part of a crane-folding team, and you need to keep track of how quickly each member folds. The rate of work for this activity tells you just that. If you fold 1 crane in 2 minutes, your friend might vary, taking 'x' minutes for 1 crane. To find the overall rate of work, you combine the individual rates, keeping in mind that the unit of time should be consistent—here, we focus on an hourly rate.

Formally, if a person's rate of work is 'r' tasks per unit time, then in 't' units of time, the total work done will be 'rt' tasks. When dealing with rates, always be attentive to the units; this will guide you in performing the correct calculations for consistent results.

Algebraic Expressions

An algebraic expression represents a mathematical phrase that includes numbers, variables, and operation symbols. In the context of our crane-folding scenario, we introduced an expression to define the work rate algebraically. Let's dissect the expression \( \frac{60}{x} \) further.

This expression is derived from the inverse relationship between time taken and the rate of work; as 'x' (time to fold one crane) increases, the work rate decreases, and vice versa. Algebra helps to generalize the problem, enabling us to compute the work rate for any given time 'x' without having to plug in specific numbers until needed.

In algebra, always ensure that you establish what each variable represents and how it interacts with the constants and coefficients within your expression. Remembering the proper order of operations is also vital when solving or simplifying algebraic expressions.

Units Conversion

When working with rates of work, we often encounter the need to convert units to align with the context of the problem—this is where units conversion enters the picture.

In our example, the initial time measurement is in minutes, but we want the rate per hour. To do this, we convert 'x' minutes into hours by dividing by 60, since there are 60 minutes in an hour (\( \frac{x}{60} \) hours). It's essential to understand the conversion factors between units—like 60 minutes in an hour or 100 centimeters in a meter—because using the wrong factor can lead to errors in your calculations.

Quick Conversion Reference

• Minutes to Hours: Divide by 60.
• Hours to Minutes: Multiply by 60.
• Centimeters to Meters: Divide by 100.
• Meters to Centimeters: Multiply by 100.

Being comfortable with units conversion helps ensure that your rate of work and other measurements are accurately represented and comparable.

Reciprocal of a Number

The reciprocal of a number is simply 1 divided by that number. It represents an inverse relationship and is especially useful in rate of work problems, where the time taken to complete a task and the number of tasks completed in a time frame are inversely related.

For example, in our crane problem, we've got the time it takes to fold a single crane in hours (\(\frac{x}{60}\)). To find out how many cranes can be folded in one hour—essentially the rate—we take the reciprocal of this time which flips the numerator and denominator, yielding \(\frac{60}{x}\) cranes per hour.

To understand reciprocals better, think of common fractions and their reciprocals:

• The reciprocal of \(\frac{1}{2}\) is 2. (Double the rate means half the time!)
• The reciprocal of 3 (which can be seen as \(\frac{3}{1}\)) is \(\frac{1}{3}\).

Whenever you switch from 'per task' to 'tasks per', you are making use of the concept of reciprocals to redefine your rate of work.