## Algebra 1 Tutorial

*Defining Units*

#### Intro

**Labor**

Imagine doing some labor. All things being equal, you would expect to

do twice as much work in 2 hours as you would in only 1 hour. Using

much the same reasoning, you would also expect, if you had another

person helping you, to perform twice as much labor in a given 1 hour

period as the amount of work you could do by yourself in the same 1

hour period.

By using the above reasoning, humankind has devised a definition for

labor which allows us to use our understanding of counting and

grouping to make predictions (or at least estimates) concerning the

amount of time and number of people required to perform a given

quantity of work.

definition: person hour

personhour;

The amount of labor performed by 1 person in a period of 1 hour is,

by definition, 1 personhour of labor.

From the definition of personhour and the definition for counting

multiplication, it is obvious that, if you have people performing

labor for hours, personhours of labor will be

performed.

Suppose it has been found that 20 carpenters can build a shopping

plaza in 56 days. From this information, estimate how long it will

take 70 carpenters to build the same shopping plaza.

From past experience, we know that it takes

(1)

persondays of labor to build the given structure.

Therefore, we might estimate that it takes 70 carpenters

(2)

days to complete this job.

If it is desired that the shopping plaza in the above illustration

be completed in 10 days, then we would estimate that it will require

(3)

carpenters on the project.

Suppose that at a factory that manufactures widgets, it requires 2.7

personhours of labor to manufacture 3000 widgets. How many

people are required if it is desired that widgets are

produced in a total production time of 30 hours?

The amount of labor required per widget is

(4)

personhours. Since we wish to manufacture

widgets, we require

(5)

personhours of labor. Since we also wish

to complete this work in a period of 30 hours, we will require

(6)

workers at this factory.

**Re-purposing the Concept of Labor**

We can make definitions similar in spirit to the definition of personhour, in situations where the underlying reasoning is

the same. In this section, we will provide an illustration where it

would be advantageous to have the following definition;

Definition: oxweek

oxweek of grass;

The amount of grass that 1 ox eats in a 1 week period of time is

said to be 1 oxweek of grass.

If 12 oxen eat the grass growing on

acres\footnote{The acre is an archaic unit of rectilinear area that

still remains in common use in English speaking countries for the

purpose of measuring land area.} in 4 weeks and 21 oxen eat the

grass growing on 10 acres in 9 weeks, how many oxen will eat up 24

acres in 18 weeks, assuming that the grass is equal on every acre,

grows uniformly, and, that the each oxen eats the same amount of

grass each week?

Let the amount of initial grass per acre be , and, let be

the amount of grass that grows on one acre in one week. If we also

let the area of the pasture be , then the amount of grass that

grows in a period of time may be given by , and the

amount of initial grass may be given by , therefore the total

amount of grass may be given by;

(7)

From the definition of oxweek, Equation ?? and the problem statement, we have;

(8)

(9)

From Equations 8 and 9, we have;

(10)

From Equations 8 and 10, we have;

(11)

From Equations 7, 10, and

11 the amount of grass on 24 acres after 18 weeks

will be;

(12)

Therefore, a total of oxweeks of grass will be

found on 24 acres after 18 weeks. The number of oxen that will eat

this much grass in 18 weeks will then be given by;

(13)

Therefore, 36 oxen will eat the grass growing on 24 acres in 18 weeks.

#### Sample Problem

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