Time and Work: AFCAT Numerical Ability

Formulas (4)

• Part of the work in some days · Whole job from a part
  $$\tfrac pq \text{ of the job in } d \text{ days} \;\Rightarrow\; T = d \times \dfrac qp$$
• Work done in stages, in percentages · A share of the remainder
  $$\text{done} = \tfrac{k}{100} \times (\text{work left})$$
• Efficiency and time · Time and efficiency
  $$\dfrac{T_A}{T_B} = \dfrac{E_B}{E_A}$$
• Output per hour: machines and shifts · Output
  $$\text{output} = \text{rate} \times \text{time}$$

Watch out for (6)

• Scale the part the right way
  → Part of the work in some days
• 'More days' is what is left
  → Part of the work in some days
• Of the remaining work
  → Work done in stages, in percentages
• More efficient is not the same percentage less time
  → Efficiency and time
• The ratio of times turns round
  → Efficiency and time
• Efficiency scales the rate
  → Output per hour: machines and shifts

Formulas (5)

• Adding rates · Two people together
  $$T = \dfrac{ab}{a + b}$$
• One person's time from the joint time · The other person's time
  $$b = \dfrac{aT}{a - T}$$
• Pairs working together · All three from the pairs
  $$A + B + C = \tfrac 12\left[(A + B) + (B + C) + (C + A)\right]$$
• Efficiency ratios with a joint time · One person from a ratio
  $$T_A = \dfrac{(a + b)\,T}{a}$$
• Work done in a given time, and each share · Work done together
  $$W = d\left(\dfrac 1a + \dfrac 1b\right)$$

Watch out for (6)

• Never add or average the days
  → Adding rates
• Subtract rates, not days
  → One person's time from the joint time
• Halve the total
  → Pairs working together
• Which way the ratio runs
  → Efficiency ratios with a joint time
• Done, or left?
  → Work done in a given time, and each share
• Pay follows the work, not the days
  → Work done in a given time, and each share

Formulas (2)

• Work in stages: one stops, another finishes · Stages
  $$\sum (\text{rate} \times \text{days}) = 1$$
• Leaving a few days before the end · Leaving k days early
  $$\dfrac{t - k}{a} + \dfrac{t}{b} = 1$$

Watch out for (3)

• More days, or days in all?
  → Work in stages: one stops, another finishes
• Use the rate of whoever is left
  → Work in stages: one stops, another finishes
• The one who left worked fewer days
  → Leaving a few days before the end

Formulas (2)

• Two people on alternate days · One cycle
  $$\text{work per cycle} = \dfrac 1a + \dfrac 1b$$
• Turns, helpers and rest days · Repeating blocks
  $$\text{blocks} = \dfrac{\text{work}}{\text{work per block}}$$

Watch out for (4)

• Do not run the cycles past the job
  → Two people on alternate days
• The starter works first in the last cycle
  → Two people on alternate days
• Rest after the end does not count
  → Turns, helpers and rest days
• A helper's day is one day per block
  → Turns, helpers and rest days

Formulas (5)

• Man-days: workers × days · Man-days
  $$M_1 D_1 = M_2 D_2$$
• Man-hours: when the working day changes · Man-hours
  $$M_1 H_1 D_1 = M_2 H_2 D_2$$
• When the amount of work changes · Work with output
  $$\dfrac{M_1 H_1 D_1}{W_1} = \dfrac{M_2 H_2 D_2}{W_2}$$
• Converting men, women and children · Equivalent workers
  $$p \text{ men} = q \text{ women} \;\Rightarrow\; 1 \text{ man} = \tfrac qp \text{ women}$$
• Two mixed teams: solve for one worker · Equal work
  $$(p\,m + q\,w)\, d_1 = (r\,m + s\,w)\, d_2$$

Watch out for (5)

• More workers, fewer days
  → Man-days: workers × days
• Hours a day are part of the work
  → Man-hours: when the working day changes
• Output goes under the labour
  → When the amount of work changes
• 'Or' is not 'and'
  → Converting men, women and children
• Equal work, not equal daily work
  → Two mixed teams: solve for one worker

Formulas (4)

• Work left after the team changes · Days after a change
  $$\text{more days} = \dfrac{M D - M d}{M'}$$
• Extra days against the plan · Extra days
  $$\text{extra} = \dfrac{L \times R}{M - L}$$
• Keeping to the deadline · Team needed
  $$\text{workers} = \dfrac{\text{work left}}{\text{days left}}$$
• Hours, efficiency and team size change together · Standard workers
  $$\text{effective team} = \text{regular workers} + \tfrac{p}{100} \times \text{others}$$

Watch out for (6)

• Take away the work already done
  → Work left after the team changes
• Extra is not the same as more
  → Extra days against the plan
• Joining saves days
  → Extra days against the plan
• Count from the workers still there
  → Keeping to the deadline
• Efficiency changes the team, not the job
  → Hours, efficiency and team size change together
• Each stage at its own hours
  → Hours, efficiency and team size change together

Formulas (4)

• Filling and emptying together · Net rate
  $$\dfrac 1T = \dfrac 1a + \dfrac 1b - \dfrac 1c$$
• Finding a leak from the delay · Time for the leak
  $$L = \dfrac{ab}{b - a}$$
• Pipes opened or closed partway · Stages
  $$\sum (\text{net rate} \times \text{time}) = 1$$
• Capacity, pipes in turn and mixtures · Capacity
  $$\text{capacity} = \dfrac{\text{litres per minute}}{\text{share of the tank per minute}}$$

Watch out for (5)

• Take the outlet away
  → Filling and emptying together
• A negative net rate never fills
  → Filling and emptying together
• The leak is slower than both times
  → Finding a leak from the delay
• Leave out the closed pipe
  → Pipes opened or closed partway
• The tank can fill partway through the cycles
  → Capacity, pipes in turn and mixtures