AFCAT Numerical Ability · Clock

Clocks That Gain or Lose Time

A faulty clock drifts by a fixed amount each hour: multiply to find what it shows, divide to find when it is right, and use its rate to turn shown time into real time.

All Clock notes

Why this matters

Most questions on this page are EASY and need one multiplication or one division. Marks go when an hour or noon is crossed carelessly, when a fast clock is treated as slow, and when an hour of gain is taken to set a clock right, which needs 12 hours.

Concept 1 of 4: What a gaining or losing clock shows

A clock that gains a few minutes every hour gets that much further ahead after each real hour. Count the real hours since it was right, multiply by the rate, and add the gain to the real time or take the loss away.

Definition

Drift $=$ rate $\times$ real hours since the clock was right.
A rate per 2 or 3 hours: count the spells. 6 hours at 5 minutes per 2 hours is 3 spells, so 15 minutes.
Time shown $=$ real time $+$ gain, or real time $-$ loss.
Adding minutes to a clock time carries into the hour: 9:50 plus 25 minutes is 10:15. Passing noon turns AM into PM.

Time shown

\text{shown} = \text{real} \pm \text{rate} \times \text{hours}

Worked example

A clock gains 4 minutes every hour. It was set right at 7:00 AM. What does it show at 3:00 PM?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Clock · Faulty Clocks

A faulty clock gains 12 minutes every hour. If it was set correctly at 1:00 AM, what time will it show at 1:00 PM on the same day?

Carry into the hour

A clock 75 minutes ahead at 4:00 PM shows 5:15 PM, not 4:75.

Count the real hours across noon

From 9:00 AM to 2:00 PM is 5 hours. Taking

9 - 2 = 7

hours counts the wrong way and overstates the drift.

Drill 15 more on what a gaining or losing clock shows

Concept 2 of 4: When a fast or slow clock shows the right time

A clock that is ahead and loses time closes its error at its rate per hour. It is right when the error reaches zero, so divide the error by the rate.

Definition

Hours until right $= \dfrac{\text{minutes fast or slow}}{\text{minutes corrected per hour}}$ .
A fast clock must lose time to come right, and a slow clock must gain.
Turn a fraction of an hour into minutes: 2.5 hours is 2 hours 30 minutes.

Hours until right

t = \dfrac{\text{error}}{\text{rate}}

Worked example

A clock is 15 minutes fast at 9:00 AM and loses 3 minutes every hour. When does it show the right time?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Clock · Faulty Clocks

A clock is 40 minutes fast at 8:00 AM and loses 6 minutes every hour. When will it show the correct time?

A third of an hour is 20 minutes

6\tfrac{1}{3}

hours is 6 hours 20 minutes, not 6 hours 33 minutes. Multiply the fraction by 60.

A fast clock that gains never catches up

A clock 10 minutes fast that gains 2 minutes an hour only gets further ahead. It looks right again only after gaining a whole 12 hours, the last concept on this page.

Drill 6 more on when a fast or slow clock shows the right time

Concept 3 of 4: Turning shown time into real time

A clock that loses 5 minutes an hour shows 55 minutes for every 60 real ones. Shown time and real time are in the ratio

55 : 60

, so convert with that ratio rather than by adding the loss.

Definition

Gaining $g$ minutes an hour: shown $:$ real $= (60 + g) : 60$ . Losing $g$ minutes: $(60 - g) : 60$ .
Real time passed $=$ shown time passed $\times \dfrac{60}{60 \pm g}$ .
'Loses 1 minute in every 10' means it shows 9 minutes for every 10, a ratio of $9 : 10$ . '30 seconds slow every minute' means it runs at half speed.

Shown and real time

\text{real} = \text{shown} \times \dfrac{60}{60 \pm g}

Worked example

A watch loses 5 minutes every hour. It was set right at noon. What is the real time when it shows 5:30 PM?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Clock · Faulty Clocks

A watch gains 6 minutes every hour. If it was set correctly at 10:00 AM, what will be the actual time when the watch shows 2:00 PM?

The rate is per real hour

A watch gaining 12 minutes an hour shows 6 hours gone. Taking off

6 \times 12 = 72

minutes gives 4 hours 48 minutes, but those were the watch's hours. The ratio gives

360 \times \tfrac{60}{72} = 300

minutes, exactly 5 hours.

Drill 4 more on turning shown time into real time

Concept 4 of 4: When a faulty clock is right again

A 12-hour dial looks right again once the clock has gained or lost exactly 12 hours, which is 720 minutes. Divide 720 by the drift per hour.

Definition

Hours until right again $= \dfrac{720}{\text{minutes gained or lost per hour}}$ .
Two faulty clocks set together show the same time again when the gap between them reaches 720 minutes. One gains and one loses: add the rates. Both gain, or both lose: subtract them.
Turn large answers into days if the options ask: 360 hours is 15 days.

Right again after

t = \dfrac{720}{\text{drift per hour}} \text{ hours}

Worked example

A clock gains 8 minutes every hour. After how many hours does it show the right time again?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Clock · Faulty Clocks

A clock loses

10

minutes every hour. After how many hours will it show the correct time again?

An hour of gain is not enough

A clock 60 minutes ahead shows the wrong hour. Only a gain of 12 hours, 720 minutes, brings both hands back to the right place.

Both gain: subtract the rates

Clocks gaining 5 and 3 minutes an hour pull apart by only 2 minutes an hour, so they match again after 360 hours, not

720 \div 8 = 90

Drill 8 more on when a faulty clock is right again

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 (4)

What a gaining or losing clock shows
Time shown
$\text{shown} = \text{real} \pm \text{rate} \times \text{hours}$
When a fast or slow clock shows the right time
Hours until right
$t = \dfrac{\text{error}}{\text{rate}}$
Turning shown time into real time
Shown and real time
$\text{real} = \text{shown} \times \dfrac{60}{60 \pm g}$
When a faulty clock is right again
Right again after
$t = \dfrac{720}{\text{drift per hour}} \text{ hours}$

Watch out for (7)

Carry into the hour
→ What a gaining or losing clock shows
Count the real hours across noon
→ What a gaining or losing clock shows
A third of an hour is 20 minutes
→ When a fast or slow clock shows the right time
A fast clock that gains never catches up
→ When a fast or slow clock shows the right time
The rate is per real hour
→ Turning shown time into real time
An hour of gain is not enough
→ When a faulty clock is right again
Both gain: subtract the rates
→ When a faulty clock is right again

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