### 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*

### 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

### 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

### 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

### 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 GMAT 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

### 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.

### 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.

### 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

#### Video Quiz

#### Work

Best viewed in landscape mode

6 questions with video explanations

100 seconds per question

# Are you sure you want to refresh the question?

https://www.youtube.com/watch?v=BcdaMhbYqOY

https://www.youtube.com/watch?v=FtZS4iJGkL4

https://www.youtube.com/watch?v=DphkFeJ9sus

https://www.youtube.com/watch?v=qoC-V187JsM

https://www.youtube.com/watch?v=YYD6ACGoirs

https://www.youtube.com/watch?v=dYyd6vhfHPU

**Before attempting these problems, be sure to review this section on data sufficiency questions.**

https://www.youtube.com/watch?v=pbUSV1v1yTE

https://www.youtube.com/watch?v=bNUr_VJoE4s