9 - RTW

...

2 Objects begin together, but 1 stops before Completion Key to identify who works for the shorter time Billy leaves after working 2 hours leaving Sam to finish the job. Billy's time is 2 hours, Sam's time is 2+t hours, with t representing the additional time Sam works

...

Rate × Time = Work Work is usually in units

...

Relative Rate Work Problems - rate of 1 worker is expressed as a multiple of the rate of another worker - rate of 1 worker is slower/faster than the rate of another worker

...

% or Fraction of a Job done: i.e. looking for [W₁ / W total] Can allow you to calculate the % of work done without knowing the time "t"

if object 1 is x times fast as object 2 and we let object 2's rate be the variable r, object 1's rate would be xr

- rate of 1 worker is expressed as a multiple of the rate of another worker

If object 1 takes x minutes longer than object 2 to complete a job & we let object 2's time be t minutes, object 1's time would be (t + x). Then, object 1's rate would be 1/(t+x) job/min, object 2's rate would be 1/t job/min

- rate of 1 worker is slower/faster than the rate of another worker

if object₁ takes x % fewer minutes than object₂ to complete a job and we let object 2's work be t minutes, object 1's work time would be t(100-x/100) minutes. Then, object 1's rate would be 1/t(100-x/100) = 100/t(100-x) job/min, and object 2's rate would be 1/t job/min

1 worker can complete a job in some %/fraction greater or less than the time it takes another worker to do the same job (see back)

2 hours Rate of 1 man: 10(r)t = 1 → r = ¹/₂₀ Rate of 2 men: ²/₂₀ = ¹/₁₀ Time it will take to dig ¹/₅ of the ditch → ¹/₅ ÷ ¹/₁₀ → 2 hrs

10 men, each working at a uniform rate, can dig a ditch in 2 hours. How long will it take 2 workers to dig ¹/₅ of the ditch

5 pistons

10 pistons, working at a constant rate, can spin an engine 30 times in 10 seconds. How many fewer pistons would have to be used to spin the engine the same # of times in 20 seconds?

...

2 Objects begin together, but 1 has an unknown time We can represent the rate at which the object completes the job as 1/t, where 1 represents the 1 job. We can then solve for t to determine the object's rate or the time it takes the object to complete a job by itself If the job being done is more or less than the 1 job, we need to account for it in the numerator, eg A can drink 100 sodas in t hours, rate = 100/t

...

3 Ways that Work is presented to us: i) A specific quantity of work completed in a specific amount of time - machine can produce 15 cars in 1 month ∴ rate = 15/1 cars per month - Jack can paint 20 fences in 4 weeks ∴ rate = 20/4 → 5 fences per wk ii) Work as a Single Job or Task Completed in a Specific amount of time -object is completing 1 job, when this occurs we need to remember that it can be expressed as rate = 1 job/ x minutes - a hose can fill pool in 4 hours, rate is 1 pool/4 hours i.e. ¼ of pool pr hour iii) Object completes a Fractional Amount of Work in a specific amt of time - mow ¼ of lawn, fill ½ of pool, build ¾ of house

Combined rate: ½ + ¼ = ¾ To complete the 1 job together "t hours" = 1 ÷ ¾ → ⁴/₃ hours ⁴/₃ hrs = ³/₃ + ¹/₃ → 1hr + ¹/₃ hr → 1+ (¹/₃ × ⁶⁰/₁) = 1 hr 20 mins

A can paint a fence in 2 hours, B can paint it in 2 hrs. How long will it take if they both work together.

x(60p+q) / 3600y

A can produce x light bulbs in y hours. How many can it produce in p minutes & q seconds?

Rate A: 1/2 Rate B: 1/3 Work₁ + Work₂ = Work Total Time A: (1 + t) Time B: (1) ∴ ½ × (1 + t) + ¹/₃ ×(1) → 5+3t = 6 t = 1/3 = 20 mins

A takes 2 hours to build a car. B takes 3 hours. If they work together for 1 hour & then B breaks down, how much additional minutes will it take A to finish the car by itself?

ty/60 hours

A takes t hours to assemble 40 widgets. B is 50% faster, how many hows does B take to make y widgets?

Rate C: 1/42 Rate C+D: 1/28 ∴1/28 = 1/42 + rD rD = 1/28 - 1/42 → 1/84 i.e 84 minutes Book Method: W₁ + W₂ = W total (r₁ × t) + (r₂ × t) = W total ←|seems like weighted avg formula Work C does: ¹ / ₄₂ × 28 → ²/₃ Work D does: ¹ / t × 28 → ²⁸/t ²/₃ + ²⁸/t = 1

C can wash a load of dishes in 42 minutes. If she works with D, it takes 28 mins. How long would it take D to complete the job alone?

Both seem to say the same thing i.e. y is 2x as fast if you are 2 times as fast you do the work in half the time 1) (D)

Can Y produce 100 light bulbs in less than 60% of the time it would take X to produce 100 bulbs? 1) Y takes 50% less time than X to produce 4000 bulbs 2) Y works twice as fast as X

...

Change in Workers Problems -involve 2 or more different size groups of workers performing a certain job or jobs - these problems assume that all workers in a group work at the same rate -whenever there is a change in the # of workers performing the job, there is a change in the rate at which the group as a whole works on that task Eg: if you double the # of workers working on a project, assuming they all work at the same pace, you double the rate at which they can perform the work

...

Combined Worker Problems W₀₁ + W₀₂ = W total Work done by object 1 + object 2 must equal the total work done by the 2 objects The job completed will also be faster than if each object is working alone

You can either assume the # of gallons in the sink, I supposed 4 gallons, or: W₁ - W₂ = W total t/20 - t/36 = 1

Drain closed, faucet can fill sink in 20 mins Drain open, faucet off, water drains in 36 mins How long will it take for the sink to fill if the faucet is on and drain open

40 mins

F & S work together to shovel snow off the driveway. They complete the task in 24 mins. If S works alone, it takes him 20 more minutes than it takes F if he works alone. How long does it take F to shovel the snow alone?

Rate A + B = 1/6 +1/8 = 7/24 Time to do 100% or 1 = 1 × 24/7 → ²⁴/₇ hrs 75% or .75 or 3/4 will take ²⁴/₇ ×¾ → ¹⁸/₇ → ₂ + ⁴/₇ hours

Hose A can fill pool in 6 hours, B can fill pool in 8 hours. How long will it take if they both work together to fill 75% of pool

C

How long will A take to produce 20 units? 1) A and B together can produce 40 units in 2 mins 2) B working alone can produce 20 units in 4 mins

...

If a pool is being filled at a rate of 100 gallons per hr but the pool water is evaporating at 5 gallons per hour, the pool will be filled at 95 gallons per hour. Here the evaporation reduces the rate at which the pool could be filled by 5 gallons per hour, thus increasing the time it will take it to fill the entire pool

24 hrs

J can paint a fence in 50% less time than T. Together they can paint fence in 8 hours. How long does it take T to paint the fence alone

192 fish

J catches fish at a rate of 4 times as fast as M. Together they catch 60 fish per hour. If M starts catching fish as fast as J, how many fish will they catch in 2 hours?

20%

Machine A: can build engine in 8 hrs Machine B: can build engine in 4 hrs Machine C: can build engine in 4 hrs If all 3 work together to build 1 engine, what % of work is done by A

important thing to remember in opposing worker problems is to subtract the work done by 1 object from the work done by the other object The total task equals the "difference" b/w their work values

Opposing Worker Problems - faucet filling a sink & a drain emptying it - one worker digging hole and another putting dirt back into the hole - water filling pool & water evaporating from pool - investment that makes money & an investment that loses money see back

3 days

Single Worker Problems A can build 6 cars in 12 hours, how many days to build 36 cars?

60xz/y sodas

Single Worker Problems with Variable in the ACs S can drink x sodas in y minutes. # of sodas S can drink in z hours?

...

We must always be careful to express the work rate as "work per unit of time" and not "time per unit of work"