What is the correct way of setting up questions like these?
Working together, Amy, Menelda, and Rosa can set the table in 4 minutes. If Amy can set the table in 7 minutes on her own, and Menelda can set the table in 28 minutes on her own, how long would it take Rosa to set the table by herself?
Working alone, Anita can cook 20-minute rice in 4 hours, and Roger can cook the same type of rice in 15 hours. After working together for 2 hours, what fraction of the inedible rice still remains uncooked?
Rishi can rake a lawn in 45 minutes by himself, and Adam can rake the same lawn in 90 minutes by himself. How long would it take the two of them working together to rake a lawn?
Where did these questions come from? I’m pretty sure these aren’t SAT Math questions…(that is, not from anything written by College Board and, thus - luckily! - not to be expected on the actual exam).
That said, this is a work question (like a variation of a distance = rate x time question). Further it’s an advanced work problem because it involves combining rates.
Basic work question set up
Work = work rate x time (work rate = work completed/time it took to complete - always express work rates with work completed over time it took to complete)
Advanced/combined work question set up
Version 1: Are the rates all the same? Work = (number of machines/people * individual rate) x time
Version 2: Are the rates different? Work = (machine/person 1 rate + machine/person 2 rate…) x time
Applying this to the first practice question:
A = 1 table/7 mins, M = 1 table/28 mins, R = 1 table/x mins
All the rates are different, so this is version 2 of the advanced/combined work rates set up
Work = (person 1 rate + person 2 rate + person 3 rate) x time
1 table = (1 table/7 mins + 1 table/28 mins + 1 table/x mins) x 4 mins
1 table/4 mins = (1 table/7 mins + 1 table/28 mins + 1 table/x mins) [divide both sides by 4 mins]
1 table/4 mins = (5 tables/28 mins + 1 table/x mins) [convert 1 table/7 mins to 4 tables/28 mins to add to 1 table/28 mins]
1 table/4 mins - 5 tables/28 mins = 1 table/ x mins [subtract 5 tables/28 mins from each side]
2 tables/28 mins = 1 table/x mins [convert 1 table/4 mins to 7 tables/28 mins to subtract 5 tables/28 mins]
At this point you can either cross multiply to solve for x mins or reduce 2 tables/28 mins to 1 table/14 mins. Answer is 14 mins.
This is how I usually think about these, in case it’s helpful.
In one minute, Amy sets 1/7 of the table, Menelda sets 1/28 of the table, and Rosa sets 1/x of the table. In four minutes, Amy sets 4/7 of the table, Menelda sets 4/28 = 1/7 of the table, and Rosa sets 4/x of the table. So we have
(4/7) + (4/28) + (4/x) = 1
=> 16x + 4x + 112 = 28x (multiplying by 28x on both sides)
The second question is a little weird, but the third question is a classic example.
Basically you want to start off by asking how much work they can do per (insert unit of time here). Since they’re using minutes, lets go with that.
Rishi can do 1/45 of the job per minute. Adam can do 1/90 of the job per minute. By adding those two values, you’ll find out how much work BOTH of them get done per minute. That is, 1/30 of the job per minute. Now, to see how long it’ll take both of them to get the whole job done. Let “j” be the total time it takes them to get the job done. Using the same logic as before, they get 1/j of the job done per minute. Now you just solve the equation
1/30 = 1/j . Flip both of them and you have your answer. 30 minutes.
In one hour, Anita cooks 1/4 of the Rice and Roger cooks 1/15 of the rice. After two hours, they have cooked 2*((1/4) + (1/15)) = 19/30 of the rice. This means 11/30 of the rice remains uncooked.
OK, first things first: please don’t “bump” a question after 50 minutes. Realize that, while you’re sitting at your computer awaiting an answer, most of the people who have the answers you want are off the computer-- running errands, doing laundry, being with our family or friends.
When I teach them, I do so with a chart: Rate of work x time = part of job done.
Rate of work is 1/( length of time to do the job alone).
Time = time spent doing the job together.
The parts of job done, for the various people working together, should add up to one whole job.
