AFCAT Numerical Ability · Speed, Time and Distance

Late or Early: Finding the Usual Time and the Distance

A slower or faster speed turns the usual time into a known number of minutes late or early, and two different speeds with their delays fix the distance.

All Speed, Time and Distance notes

Why this matters

No AFCAT paper has set one of these here, but the bank leans on them and most are MODERATE: each is one equation. The minutes late or early are a fraction of the usual time. Marks go on taking the fraction of the wrong time, or on comparing two speeds without adding the minutes late to the minutes early.

Concept 1 of 3: Slower than usual: the delay gives the usual time

\tfrac pq

of the usual speed the trip takes

\tfrac qp

of the usual time. The part over and above the usual time is the delay, so the delay is a known fraction of the usual time.

Definition

Usual time $t$ , speed times $\tfrac pq$ with $p < q$ : the new time is $\tfrac qp\,t$ and the delay is $\tfrac{q - p}{p}\,t$ .
'Reduces his speed by 20%' means $\tfrac45$ of the speed, so the delay is $\tfrac t4$ .
If only part of the journey is slow, only that part's time stretches.
The usual time comes out in the unit of the delay: minutes in, minutes out.

The delay at a slower speed

\text{delay} = \left(\dfrac qp - 1\right) t

Worked example

Walking at

\tfrac45

of his usual speed, a man is 12 minutes late. Find his usual time.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Speed, Time and Distance · Late, Early and Changed Speed

Walking at 3/4 of his usual speed, a man reaches his office 20 minutes late. What is his usual time to reach the office?

The fraction is of the usual time

\tfrac45

of the speed the delay is a quarter of the usual time, not a fifth. The fifth is the drop in speed; the time grows by

\tfrac54

, which is a quarter more.

Answer what is asked

Some questions want the slower trip's time, not the usual time. With a usual time of 48 minutes and a 12 minute delay, the slow trip takes 60 minutes.

Drill 26 more on slower than usual: the delay gives the usual time

Concept 2 of 3: Faster than usual: the time saved, and a late start

\tfrac ab

of the usual speed with

a > b

, the trip takes

\tfrac ba

of the usual time, so a fraction

\tfrac{a - b}{a}

of it is saved. Leaving late but arriving on time means the saving equals the late start.

Definition

Speed times $\tfrac ab$ , $a > b$ : saving $= \left(1 - \tfrac ba\right) t$ .
'Started 10 minutes late and still arrived on time': the saving is 10 minutes.
Late at one speed, asked how early at another: find $t$ from the first, then use it in the second.
Doubling the speed saves half the usual time.

The saving at a faster speed

\text{saving} = \left(1 - \dfrac ba\right) t

Worked example

Walking at

\tfrac65

of his usual speed, a man saves 9 minutes. Find his usual time.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 2 · Speed, Time and Distance · Late, Early and Changed Speed

Anil walks to his office daily. Today he started

15

minutes late but by increasing his speed by

25\%

, he reached exactly on time. What is his usual time to walk to office?

The saving is a fraction of the usual time

\tfrac65

of the speed the saving is a sixth of the usual time, not a fifth. The time becomes

\tfrac56

of usual.

A late start is a saving

'Started 20 minutes late but arrived on time' tells you the faster speed saved 20 minutes. It is not a delay to add.

Drill 21 more on faster than usual: the time saved, and a late start

Concept 3 of 3: Two speeds and the gap between their times

The distance is fixed and so is the right time. At the slower speed he is late and at the faster he is early, so the two travel times differ by the minutes late plus the minutes early. That gap fixes the distance.

Definition

Late by $L$ at speed $a$ , early by $E$ at speed $b$ ( $a < b$ ): $\dfrac da - \dfrac db = L + E$ , in hours.
Late both times, by $L_1$ and $L_2$ : the gap is $L_1 - L_2$ .
$\dfrac da - \dfrac db = \dfrac{d(b - a)}{ab}$ . Then the right time is $\tfrac da - L$ .
Speed unknown, with 'k km/h faster would take m hours less': $\dfrac dv - \dfrac{d}{v + k} = m$ , which leads to $v(v + k) = \dfrac{dk}{m}$ .

The gap between two travel times

\dfrac da - \dfrac db = \text{gap in hours}

Worked example

At 3 km/h a boy reaches school 8 minutes late; at 5 km/h he is 8 minutes early. Find the distance.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Speed, Time and Distance · Late, Early and Changed Speed

A man walks at 4 km/hr and reaches his office 20 minutes late. If he walks at 6 km/hr, he reaches 10 minutes early. Find the distance to the office.

Late and early add

8 minutes late at one speed and 8 minutes early at the other is a gap of 16 minutes, not 0. Subtract only when both are late or both early.

Minutes into hours before solving

With speeds in km/h, the gap must be in hours. Writing

\tfrac d3 - \tfrac d5 = 16

gives 120 km instead of 2 km.

Drill 15 more on two speeds and the gap between their times

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Slower than usual: the delay gives the usual time
The delay at a slower speed
$\text{delay} = \left(\dfrac qp - 1\right) t$
Faster than usual: the time saved, and a late start
The saving at a faster speed
$\text{saving} = \left(1 - \dfrac ba\right) t$
Two speeds and the gap between their times
The gap between two travel times
$\dfrac da - \dfrac db = \text{gap in hours}$

Watch out for (6)

The fraction is of the usual time
→ Slower than usual: the delay gives the usual time
Answer what is asked
→ Slower than usual: the delay gives the usual time
The saving is a fraction of the usual time
→ Faster than usual: the time saved, and a late start
A late start is a saving
→ Faster than usual: the time saved, and a late start
Late and early add
→ Two speeds and the gap between their times
Minutes into hours before solving
→ Two speeds and the gap between their times

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