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 × hours = 1/1 tractors × 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 × hours = 1/x tractors × 1 hour = 1/x
It takes x tractors 4 hours to plow 1 field, so the rate for the tractor is 1/4x of a field per hour.
1/tractors × hours = 1/x tractors × 4 hours = 1/4x
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 × hours = 1/x tractors × 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 × 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 = 60x 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 × 15
45 = 3 × 15
2 × 3 × 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 × 15 + 1/3 × 15
= ( 3/3 ) ( 1/2 × 15 ) + ( 2/2 ) ( 1/3 × 15 )
= 3 + 2/3 × 2 × 15
= 5/3 × 2 × (3 × 5)
= 1/3 × 2 × 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 × 20
60 = 3 × 20
2 × 3 × 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 × 20 + 1/3 × 20
= ( 3/3 ) ( 1/2 × 20 ) + ( 2/2 ) ( 1/3 × 20 )
= 3 + 2/3 × 2 × 20
= 5/3 × 2 × (4 × 5)
= 1/3 × 2 × 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 + 20R
Multiply both sides by 20.
2.5 – 1 = 20R
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 × 60 minutes/1 hour = 20 minutes
Solution
The rate that one man works is 1/3 men × 8 hours = 1/3 × 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 ×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) t Multiply by the LCD.
24(7) = (24/8 + 24/6) t 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 × 36 = 35( 1/5 + 1/7 ) t
35 × 36 = ( 35/5 + 35/7 ) t
35 × 36 = (7 + 5) t
t = (35 × 36)/12 = 35 × 3
t = 105
Method 2
Factor and add fractions.
36 = ( 1/5 + 1/7 ) t
36 = ( 7/7 × 5 + 5/5 × 7 ) t
36 = ( 12/35 ) t
t = 35 × 36 /12= 35 × 3
t = 105
Video Quiz
Work
Best viewed in landscape mode
6 questions with video explanations
100 seconds per question
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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