Time and Work: SSC CGL Quantitative Aptitude

Formulas (4)

• Workers together: add the one-day rates · Working together
  $$\text{Rate} = \frac{1}{n} \qquad T = \frac{ab}{a + b} \qquad T = \frac{1}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}$$
• Part of the job done in a given time · Work done and work left
  $$W = \frac{t}{n} \qquad \text{Left} = 1 - W \qquad \text{Whole time} = \frac{d}{f}$$
• One worker's time from the team's time · Finding one worker
  $$x = \frac{ab}{b - a} \qquad A + B + C = \frac{(A + B) + (B + C) + (C + A)}{2}$$
• Sharing the pay by work done · Pay shared by work
  $$\text{Share} = \frac{\text{work done}}{\text{whole job}} \times \text{total pay}$$

Watch out for (8)

• Adding the days instead of the rates
  → Workers together: add the one-day rates
• Using product over sum for three workers
  → Workers together: add the one-day rates
• Marking the part done when the part left is asked
  → Part of the job done in a given time
• Scaling the days the wrong way
  → Part of the job done in a given time
• Forgetting to halve the pair total
  → One worker's time from the team's time
• Subtracting the times
  → One worker's time from the team's time
• Sharing the pay in the ratio of days
  → Sharing the pay by work done
• Giving a helper an equal share
  → Sharing the pay by work done

Formulas (2)

• Efficiency ratio and time ratio · Efficiency and time
  $$\frac{T_A}{T_B} = \frac{E_B}{E_A} \qquad \text{Work} = E \times T$$
• A faster worker who takes fewer days · Faster worker, fewer days
  $$t = \frac{D}{k - 1} \qquad T_{\text{together}} = \frac{kt}{k + 1}$$

Watch out for (3)

• Not flipping the ratio
  → Efficiency ratio and time ratio
• Reading 60% more efficient as 60% less time
  → Efficiency ratio and time ratio
• Dividing the gap by k
  → A faster worker who takes fewer days

Formulas (3)

• One worker starts and another finishes · Work in stretches
  $$\sum (\text{rate} \times \text{days}) = 1 \qquad pA + qB = p(A + B) + (q - p)B$$
• A team that changes partway · Team changes partway
  $$x(A + B) + dB = 1$$
• Leaving a few days before the end · Leaving k days before the end
  $$\sum r_i (T - k_i) = 1$$

Watch out for (4)

• Taking the days for the rest as the whole time
  → One worker starts and another finishes
• Giving the days after the change when the total is asked
  → A team that changes partway
• Forgetting the old members in a joiner's rate
  → A team that changes partway
• Taking the leaving days off the leaver's own time
  → Leaving a few days before the end

Watch out for (5)

• Rounding a part day up to a whole day
  → Taking turns on alternate days
• Ignoring who starts
  → Taking turns on alternate days
• Taking one cycle too many
  → Taking turns on alternate days
• Putting the helpers' day first
  → Helpers who join every few days
• Reading a day off after every 4 days as a 4-day rhythm
  → Helpers who join every few days

Formulas (3)

• Workers times days is fixed for a job · Man-days rule
  $$M_1 D_1 = M_2 D_2 \qquad \frac{M_1 D_1}{W_1} = \frac{M_2 D_2}{W_2}$$
• Hours a day and part of a job · Man-hours rule
  $$\frac{M_1 D_1 H_1}{W_1} = \frac{M_2 D_2 H_2}{W_2}$$
• Workers added or leaving partway · Changing workforce
  $$\text{Days left} = \frac{n(D - t)}{n'} \qquad dn - \frac{kd(d - 1)}{2} = Dn$$

Watch out for (5)

• Working the proportion the wrong way
  → Workers times days is fixed for a job
• Giving the total when "how many more" is asked
  → Workers times days is fixed for a job
• Ignoring a change in hours a day
  → Hours a day and part of a job
• Putting the part of the job on the wrong side
  → Hours a day and part of a job
• Starting the leaving sequence one day early
  → Workers added or leaving partway

Formulas (2)

• Converting men into women · Equivalent workers
  $$a \, d_1 \text{ man-days} = b \, d_2 \text{ woman-days} \qquad 1 \text{ man} = \frac{b \, d_2}{a \, d_1} \text{ women}$$
• Two mixed teams doing the same job · Two mixed teams
  $$d_1(x_1 m + y_1 w) = d_2(x_2 m + y_2 w)$$

Watch out for (5)

• Averaging the two teams' times
  → Whole teams working together
• Setting men equal to women without the days
  → Converting men into women
• Converting the wrong way
  → Converting men into women
• Setting the teams equal when the days differ
  → Two mixed teams doing the same job
• Not checking with the second team
  → Two mixed teams doing the same job

Formulas (3)

• Inlets and outlets open together · Net rate of pipes
  $$\text{Net} = \sum \frac{1}{a_i} - \sum \frac{1}{b_j} \qquad T = \frac{ab}{b - a}$$
• Finding a leak from the slower fill · Time for the leak to empty the tank
  $$T_{\text{leak}} = \frac{ab}{b - a}$$
• One pipe so many times as fast as another · Pipes in a speed ratio
  $$T_{\text{slow}} = (k + 1)T \qquad T_{\text{fast}} = \frac{(k + 1)T}{k}$$

Watch out for (6)

• Adding an outlet's rate
  → Inlets and outlets open together
• Answering in the wrong unit
  → Inlets and outlets open together
• Taking the extra hours as the leak's time
  → Finding a leak from the slower fill
• Multiplying the joint time by k instead of k + 1
  → One pipe so many times as fast as another
• Keeping the wrong root
  → One pipe so many times as fast as another
• Running whole cycles with an outlet in them
  → Pipes opened or closed partway