### 1. Work

The amount of work *W*, accomplished in time *T*, depends on the rate *R*, at which the work is done. This relationship is described by the equation:

*W* = *RT*

The amount of work is often one completed job, so usually *W* = 1.

The time *T* is how long it takes to complete the entire job.

The rate *R* is the job divided by time.

Rate = 1 job/time to do 1 job **or** jobs done/time

### 2. Rates

A work-rate sounds like a speed, but the unit is a job done, not a distance covered.

There are many work-rates in everyday life.

*She types 60 words per minute.**The repairman makes 6 service calls per day.**The hen lays 5 eggs per week.**In the formula, you need to use the unit rate.*

#### 2a. Examples of Unit Rates

It takes 1 tractor 10 hours to plow 1 field, so the rate for the tractor is 1/10 of a field per hour.

1/tractors Ã— hours = 1/1 tractors Ã— 10 hours = 1/10

It takes *x* tractors 1 hours to plow 1 field, so the rate for the tractor is 1/*x* of a field per hour.

1/tractors Ã— hours = 1/*x* tractors Ã— 1 hour = 1/*x*

It takes *x* tractors 4 hours to plow 1 field, so the rate for the tractor is 1/4*x* of a field per hour.

1/tractors Ã— hours = 1/*x* tractors Ã— 4 hours = 1/4*x*

It takes *x* tractors *y* hours to plow 1 field, so the rate for the tractor is 1/*xy* of a field per hour.

1/tractors Ã— hours = 1/*x* tractors Ã— *y* hours = 1/*xy*

## Example

A machine that folds and inserts letters into envelopes can do 3 envelopes in 8 seconds. How long will it take to do 240 envelopes?

### Solution

The work is 240 envelopes. The rate is 3 envelopes/8 seconds. The time *t* is what you need to find.

Using the formula:

240 = 3/8* t*

*t* = 240 Ã— 8/3 = 640

It would take 640 seconds, or 10 minutes 40 seconds.

*Review:*

640 seconds = 600 seconds + 40 seconds = 10 minutes 40 seconds

60 seconds/1 minute = 60*x* seconds/*x* minutes

## Example

It takes Jeff 30 minutes to rake the leaves in the yard, and it takes his brother Ken 45 minutes. How long would it take the two of them working together to rake the yard?

### Solution

The work is 1 yard. You want to find the time *t*.

Jeff’s rate is 1/30 and Ken’s rate is 1/45. So the combined rate is 1/30 + 1/45 .

Using the formula:

1 = ( 1/30 + 1/45 ) *t*

*t* = 18 It would take them 18 minutes to rake the yard working together.

*Review: *

There are multiple methods to deal with the fractions in work-rate equations.

*Method 1*

Multiply both sides of the equation by the LCD (least common denominator).

To find the LCD, first factor the denominators.

Then multiply each unique factor.

30 = 2 Ã— 15

45 = 3 Ã— 15

2 Ã— 3 Ã— 15 = 90

The LCD is 90.

90 = 90( 1/30 + 1/45 ) *t*

90 = ( 90/30 + 90/45 ) *t*

90 = (3 + 2) *t*

*t* = 18

*Method 2*

Factor and add fractions first.

1/30 + 1/45 = 1/2 Ã— 15 + 1/3 Ã— 15

= ( 3/3 ) ( 1/2 Ã— 15 ) + ( 2/2 ) ( 1/3 Ã— 15 )

= 3 + 2/3 Ã— 2 Ã— 15

= 5/3 Ã— 2 Ã— (3 Ã— 5)

= 1/3 Ã— 2 Ã— 3

= 1/18

1 = 1/18 *t*

*t* = 18

## Example

Michelle can input a day’s invoices into the computer system in 40 minutes, and John can input the same invoices in 60 minutes. How long will it take both of them, working simultaneously, to input the invoices?

### Solution

The work is 1 day’s invoices. You want to find the time* t*.

Michelle’s rate is 1/40 and John’s rate is 1/60 . So the combined rate is 1/40 + 1/60 .

Using the formula:

1 = ( 1/40 + 1/60 ) *t*

t = 24 It would take them 24 minutes to do the job working together.

*Review: *

*Method 1*

Multiply by the LCD

40 = 2 Ã— 20

60 = 3 Ã— 20

2 Ã— 3 Ã— 20 = 120

The LCD is 120.

120 = 120( 1/40 + 1/60 ) *t*

120 = ( 120/40 + 120/60 ) *t*

120 = (3 + 2)* t*

*t* = 24

*Method 2*

Factor and add fractions first

1/40 + 1/60 = 1/2 Ã— 20 + 1/3 Ã— 20

= ( 3/3 ) ( 1/2 Ã— 20 ) + ( 2/2 ) ( 1/3 Ã— 20 )

= 3 + 2/3 Ã— 2 Ã— 20

= 5/3 Ã— 2 Ã— (4 Ã— 5)

= 1/3 Ã— 2 Ã— 4

= 1/24

1 = 1/24 *t*

*t* = 24

## Example

Kelly and Shelley can together type a manuscript in 8 hours. Kelly can type the manuscript alone in 20 hours. How long would it take Shelley to type the manuscript?

### Solution

The work is 1 manuscript. Kelly’s rate is 1/20. Let Shelley’s rate be *R.* Time working together is 8 hours.

Using the formula:

1 = ( 1/20 + *R*) 8

1/8 = 1/20 + *R*

Divide both sides by 8.

20/8 = 1 + 20*R*

Multiply both sides by 20.

2.5 â€“ 1 = 20*R*

*R* = 1.5/20 = 3/40

So Shelley’s rate is 3/40, or she does 3/40 of the job in an hour.

Don’t fall for the GRE trick by stopping here; it doesn’t answer the question. Use Shelley’s rate to find the time for her to type the entire manuscript alone:

1 = 3/40* t*

*t* = 40/3 = 13/1/3

So Shelley would take 13/1/3 hours, or 13 hours 20 minutes to type the manuscript alone.

*Review:* 1/3 hour = 1/3 Ã— 60 minutes/1 hour = 20 minutes

## Example

It takes 3 men 8 hours to paint a house. How long will it take 5 men to paint the same house?

### Solution

The rate that one man works is 1/3 men Ã— 8 hours = 1/3 Ã— 8 = 1/24 house per hour.

The rate that 5 men work is 5/24

Using the equation:

1 = 5/24/*t*

*t* = 24/5 = 4.8

It would take 5 men 4.8 hours or 4 hours and 48 minutes.

*Review*: 0.8 hour = 0.8 Ã—60 minutes/1 hour = 48 minutes

Some jobs involve completing more than just one unit of work, so it’s not always the case that *W* = 1.

## Example

It takes Monique 8 minutes to frost a cake and it takes Cheri 6 minutes to frost a cake. How long will it take them working together to frost 7 cakes?

### Solution

The work is 7 cakes. You want to find the time* t*. Monique’s rate is 1/8 and Cheri’s rate is 1/6.

So the combined rate is 1/8 + 1/6.

Using the formula:

7 = (1/8 + 1/6) *t*

24(7) = 24(1/8 + 1/6) *tÂ Â * Multiply by the LCD.

24(7) = (24/8 + 24/6) *tÂ Â * Keep numbers in factored form.

24(7) = (3 + 4) *t*

24 = *t*

It will take them 24 minutes to frost 7 cakes.

## Example

Jiro makes 3 sushi rolls in 15 minutes. Michiko makes 4 sushi rolls in 28 minutes. How long will it take them to make 36 rolls?

### Solution

The work is 36 rolls. You want to find the time *t*.

Jiro’s rate is 3/15 = 1/5 and Michiko’s rate is 4/28 = 1/7. So the combined rate is 1/5 + 1/7.

Using the formula:

36 = ( 1/5 + 1/7 ) *t*

*t* = 105

It would take them 105 minutes, or 1 hour 45 minutes.

*Review: *105 minutes = 60 + 45 minutes = 1 hour 45 minutes

*Method 1*

Multiply by LCD

35 Ã— 36 = 35( 1/5 + 1/7 ) *t*

35 Ã— 36 = ( 35/5 + 35/7 ) *t*

35 Ã— 36 = (7 + 5) *t*

*t* = (35 Ã— 36)/12 = 35 Ã— 3

*t* = 105

*Method 2*

Factor and add fractions.

36 = ( 1/5 + 1/7 )* t*

36 = ( 7/7 Ã— 5 + 5/5 Ã— 7 )* t*

36 = ( 12/35 ) *t*

*t* = 35 Ã— 36 /12= 35 Ã— 3

*t* = 105