Time & Work

Efficiency is rate with a comparison

"Efficient" here does not mean clever — it means works faster. Efficiency is another name for rate. The one relation to remember:

$\frac{\text{efficiency of A}}{\text{efficiency of B}} = \frac{\text{time of B}}{\text{time of A}}$

Efficiency and time are inversely proportional: twice as efficient → half the days. Three times as efficient → one-third of the days. Write every statement as a rate ratio before doing anything else.

The unit trick for "k times as good"

If A is k times as efficient as B, let B do 1 unit a day and A do k units a day. Together $(k+1)$ units a day. If together they take T days, the whole job is $(k+1)T$ units, so: $\text{B alone} = (k+1)T \text{ days}, \qquad \text{A alone} = \frac{(k+1)T}{k} \text{ days}$ A is twice as efficient as B; together 18 days → job $= 3 \times 18 = 54$ units → B alone 54 days, A alone 27. Notice B (the weaker worker) takes the longer time — if you assign 54 to A, you inverted the ratio.

Efficiency given as a ratio

"A and B work in the ratio 3 : 2" — same idea. Rates $3u$ and $2u$; together $5u$. If the together time is 15 days, the job is $75u$, so B alone takes $\frac{75u}{2u} = 37.5$ days. If B's own time is given instead, first get u from it.

"50% more efficient" and friends

Percentages become ratios at sight: 50% more efficient → ratio $3 : 2$; 25% more efficient → $5 : 4$; 20% less efficient → $4 : 5$. Then continue exactly as above. A is 50% more efficient than B and B takes 27 days → A's rate $= \frac{3}{2} \times \frac{1}{27} = \frac{1}{18}$; together $\frac{1}{18} + \frac{1}{27} = \frac{5}{54}$ → $10.8$ days.

Days-difference questions

"A is twice as fast as B and finishes 12 days earlier." Times are $x$ and $2x$, and $2x - x = 12$, so A = 12, B = 24; together $\frac{12 \times 24}{36} = 8$ days. With ratio $a : b$ for A : B efficiency, times are in $b : a$ — set the difference equal to the given gap and everything falls out.

Why the inverse rule never fails

Rate × time = 1 whole job (constant). If the rate is multiplied by k, the time must be divided by k for the product to stay 1. That is all "inversely proportional" means, and it is why scaling days in the wrong direction gives an impossible answer — the strongest worker always takes the fewest days.

Common traps

• Reading "twice as efficient" as "twice the days" — efficiency and time move opposite ways.
• Giving the k-times worker $(k+1)T$ instead of $\frac{(k+1)T}{k}$ — that belongs to the weaker worker.
• Treating "25% more efficient" as a ratio of 25 : 100; it is 125 : 100 = 5 : 4.
• Mixing up whose time is longer in days-difference questions.

Quick revision

• Efficiency ratio = rate ratio = inverse time ratio.
• k-times worker, together T days: weaker $(k+1)T$, stronger $\frac{(k+1)T}{k}$.
• 50% / 25% / 20% more efficient → 3:2 / 5:4 / 6:5.
• Days gap with times $bx$ and $ax$: $(b-a)x =$ gap.
• Sanity check: faster worker → fewer days, always.