When there is a problem to solve, you work out or figure out the answer (find the solution), or sort out a mess (put it in order). Notice each one needs its preposition. You say "work out", not just "work".
Solving problems at work
work out /wɜːrk aʊt/ — Diego worked out a better timetable for the team.
figure out /ˈfɪɡjər aʊt/ — Mei figured out why the projector was not working.
sort out /sɔːrt aʊt/ — Sofia sorted out the files before the client meeting.
Try it: Lars needs time to ___ the new software before he can train the others. (figure out / carry out / break down) Answer: figure out — Figure out means understand or find the answer to something, which is what Lars is doing with the new software. "Carry out" means perform a task, and "break down" means stop working, neither of which fits "understanding" here.
Doing and accepting work
You carry out a task (do it, formal), take on work (accept new responsibility), or fill out / fill in a form (complete it). These are everywhere in workplace English.
carry out /ˈkæri aʊt/ — Yasmin carried out a short customer survey.
take on /teɪk ɑn/ — Lani took on extra work during the festival week.
fill out /fɪl aʊt/ — Wei filled out the application form online.
Try it: The team cannot ___ another project this month. They already have too much. (take on / fill out / work out) Answer: take on — Take on means accept new work or responsibility, and the team has no room for more. "Fill out" is for forms, and "work out" is for solving a problem, not for accepting a project.
When things stop working
Break down has two work uses: a machine breaks down (stops working), and talks or plans break down (fail, collapse). Same phrasal verb, both about something that stops.
break down /breɪk daʊn/ — The printer broke down before the meeting.
Not ""The negotiations broke." (missing the preposition)" — say ""The negotiations broke down." (they failed / collapsed)". The phrasal verb is break down. The preposition carries the "stop / fail" meaning.
Try it: Negotiations ___ after both sides rejected the plan. (broke down / carried out / took on) Answer: broke down — Break down means fail or collapse, and the negotiations ended without agreement. "Carry out" means perform a task and "take on" means accept work, neither of which describes talks that failed.
Postponing and cancelling
Put off means to move something to a later time, so it still happens. Call off means to cancel it completely, so it does not happen at all. The difference matters at work: something you put off is only delayed.
put off /ˌpʊt ˈɔːf/ — The team put off the product launch until March.
call off /ˌkɔːl ˈɔːf/ — Heavy snow forced the organizers to call off the match.
Not ""They put off the meeting, so it never happened."" — say ""They called off the meeting, so it never happened."". Put off = it happens later. Call off = it is cancelled and does not happen.
Try it: The supplier refused to lower the price, so Mei ___ the contract completely. (called off / put off / took on) Answer: called off — Call off means to cancel something so it does not happen at all, and "completely" shows the contract is finished. "Put off" would mean the contract still happens later, and "take on" means to accept new work.
Try it: Lars was not ready with the figures, so the manager ___ the review until next Monday. (put off / called off / carried out) Answer: put off — Put off means to move something to a later time, and "until next Monday" gives the new date. "Call off" would mean the review is cancelled and never happens, and "carry out" means to perform a task.
Producing ideas and cutting costs
Come up with means to produce an idea or a plan, and it needs all three parts. Cut down means to reduce the amount of something, such as costs or hours.
come up with /ˌkʌm ʌp ˈwɪð/ — Sofia came up with a new name for the campaign.
cut down /ˌkʌt ˈdaʊn/ — The office cut down its printing costs by half.
Not ""Sofia came up a new name for the campaign." (the preposition with is missing)" — say ""Sofia came up with a new name for the campaign."". Come up with takes with before the idea. Come up alone means a problem appears by itself.
Try it: The budget was too high, so the team looked for ways to ___ their travel spending. (cut down / take on / carry out) Answer: cut down — Cut down means to reduce the amount of something, which fits lowering a budget that is too high. "Take on" means to accept new work and "carry out" means to perform a task, so neither reduces spending.
Try it: The marketing team spent the afternoon trying to ___ a slogan for the new product. (come up with / carry out / put off) Answer: come up with — Come up with means to produce an idea or a plan, which is what inventing a slogan requires. "Carry out" means to perform a task that already exists, and "put off" means to move something to a later time.
Taking charge and keeping up
Take over means to take control of something from another person. Catch up on means to do work you should have done earlier, or to get up to date with information. Note that catch up on is used for work and news, while catch up with is used for people.
take over /ˌteɪk ˈoʊvər/ — Mei will take over the department in June.
catch up on /ˌkætʃ ʌp ˈɑn/ — Diego stayed late to catch up on his emails.
Not ""Mei will take the department in June." (the particle is missing)" — say ""Mei will take over the department in June."". Take over = take control from someone else. "Take" alone just means to carry or receive something.
Try it: Farid retires next month, and Ethan will ___ the department from him. (take over / take on / carry out) Answer: take over — Take over means to take control of something from another person, and "from him" points to the person handing it on. "Take on" means to accept new work but is not used with "from", and "carry out" means to perform a task.
Try it: Yasmin was away for a week, so on Monday she needed to ___ everything she had missed. (catch up on / take over / cut down) Answer: catch up on — Catch up on means to deal with work or news you missed earlier, which fits a week of absence. "Take over" means to take control from another person, and "cut down" means to reduce an amount.