9.2 Time, Timetables & Money

3. Time Calculations & Timetable Analysis

Time questions in competitive entrance tests fall into four main categories: 12-hour/24-hour conversions, analogue clock geometry, elapsed time durations, and multi-service timetable analysis.

12-Hour vs. 24-Hour Clock Conversions

• 24-Hour Format Rules: Always uses four digits, where the first two digits represent the hour ($00$ to $23$) and the last two digits represent minutes ($00$ to $59$). It never uses "am" or "pm".
• Converting am to 24-hour: Retain the hour, adding a leading zero if necessary (e.g., $7:25\text{ am} \to 07:25$, $11:40\text{ am} \to 11:40$). The exception is the midnight hour: $12:15\text{ am} \to 00:15$.
• Converting pm to 24-hour: Add $12$ to the hour figure (e.g., $3:15\text{ pm} \to 15:15$, $9:48\text{ pm} \to 21:48$). The exception is the noon hour: $12:30\text{ pm} \to 12:30$ (do not add 12).

┌────────────────────────────────────────────────────────┐
│             Key Clock Conversion Landmarks             │
├───────────────────────────┬────────────────────────────┤
│       12-Hour Clock       │       24-Hour Clock        │
├───────────────────────────┼────────────────────────────┤
│  12:00 am (Midnight)      │  00:00                     │
│  12:45 am                 │  00:45                     │
│   6:30 am                 │  06:30                     │
│  11:59 am                 │  11:59                     │
│  12:00 pm (Noon)          │  12:00                     │
│   1:15 pm                 │  13:15                     │
│   7:50 pm                 │  19:50                     │
│  11:59 pm                 │  23:59                     │
└───────────────────────────┴────────────────────────────┘

Analogue Clock Geometry

An analogue clock dial is a complete circle of $360^\circ$ partitioned into 12 hour divisions and 60 minute divisions:

• Minute Hand Speed: Travels $360^\circ$ in $60\text{ minutes} = 6^\circ\text{ per minute}$.
• Hour Hand Speed: Travels $360^\circ$ in $12\text{ hours} = 30^\circ\text{ per hour}$. In one minute, the hour hand advances $\frac{30^\circ}{60} = 0.5^\circ\text{ per minute}$.

Elapsed Time: The "Bridge to the Hour" Method

Because time uses base-60 (sexagesimal) rather than base-10 arithmetic, decimal-style subtraction without regrouping in 60s produces errors (e.g., subtracting $14:15 - 11:45$ using column subtraction yields $2.70$, which has no physical meaning). Instead, use the systematic three-step timeline method:

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1. Step 1 (Bridge to Next Whole Hour): From $11:45$ to $12:00$ is $15\text{ minutes}$.
2. Step 2 (Count Whole Hours): From $12:00$ to $14:00$ is $2\text{ hours}$.
3. Step 3 (Bridge to Final Time): From $14:00$ to $14:15$ is $15\text{ minutes}$.
4. Sum the Durations: $15\text{ mins} + 2\text{ hours} + 15\text{ mins} = 2\text{ hours } 30\text{ minutes}$.

Reading Timetables

Timetables arrange journeys in vertical columns or horizontal rows. In standard UK rail and bus timetables:

• Each column represents an individual train or bus service.
• Read down a column to trace the progress of a specific vehicle from departure to terminus.
• Read across a row to see when different services arrive at a particular station.
• If a cell contains a dash (—) or blank space, that service does not call at that station.

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4. Money Problems & Financial Reasoning

Financial arithmetic evaluates multi-step operational fluency, percentage application, and proportional comparison.

Correct Notation

• Never mix pounds and pence notation: write $\text{£}3.45$ or $345\text{p}$, never $\text{£}3.45\text{p}$.
• For consistent price presentation, include two decimal places: write £4.50. The value £4.5 is numerically the same, but two places make the pence clear.

Best-Buy Comparisons: Two Standard Strategies

When comparing different package sizes to determine the "best value for money", choose between two robust mathematical approaches:

Strategy A: The Unit Price Method (Price per $100\text{ g}$ or per unit)

Divide the total cost by the quantity to establish a standardized baseline:

$$\text{Unit Cost} = \frac{\text{Total Price}}{\text{Total Quantity}}$$

Strategy B: The Common Multiple (Scaling) Method

Scale both quantities up to their lowest common multiple (LCM). For example, if comparing $250\text{ g}$ for $\text{£}1.60$ against $400\text{ g}$ for $\text{£}2.40$, scale both to $2{,}000\text{ g}$:

• Option 1 ($250\text{ g}$): $2{,}000\text{ g} = 8 \times 250\text{ g} \implies 8 \times \text{£}1.60 = \text{£}12.80$
• Option 2 ($400\text{ g}$): $2{,}000\text{ g} = 5 \times 400\text{ g} \implies 5 \times \text{£}2.40 = \text{£}12.00$
• Comparing directly: Option 2 is cheaper by $\text{£}0.80$ for $2\text{ kg}$.

Multi-Buy Promotions Breakdown

• "Buy Two, Get One Free" (3 for the price of 2): Effective discount is $\frac{1}{3} = 33\frac{1}{3}\%$.
• "Buy One, Get Second Half Price": You pay $1 + 0.5 = 1.5$ times the single price for $2$ items. Effective discount is $\frac{0.5}{2} = 25\%$.
• "3 for $\text{£}5$ when singles are $\text{£}1.90$": Standard cost $= 3 \times \text{£}1.90 = \text{£}5.70$. Saving $= \text{£}0.70$.

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Timetable example

Worked Example 2: Train Timetable & Connecting Journey

Problem: Below is a rail timetable for services from Greenfield to City Centre:

Oliver must arrive at City Centre by 09:00 for an interview. It takes him 14 minutes to walk from his home to Greenfield station. What is the latest time Oliver can leave his home?

• Step 1 (Identify latest valid train):
  • Train A arrives at City Centre at 08:24 (before 09:00).
  • Train B arrives at City Centre at 08:51 (before 09:00).
  • Train C arrives at City Centre at 09:18 (too late; arrives after 09:00).
  • Therefore, Oliver must catch Train B.
• Step 2 (Check departure from Greenfield):
  • Train B departs Greenfield at 07:44.
• Step 3 (Subtract walking time):
  • Departure time: $07:44 - 14\text{ minutes} = 07:30$.
• Answer: 07:30 (or 7:30 am).