3.2 Time, Speed & Money

Time, Speed & Money

Quick Summary: Time measurement, speed dynamics, and financial calculations are fundamental mathematical tools used daily. Mastery of 12-hour and 24-hour time conversions, elapsed time schedules, distance-speed-time relationships, and monetary transaction arithmetic forms a core component of the CPEA assessment.

Mathematical reasoning involving time, motion, and currency allows us to plan travel schedules, evaluate vehicle speeds, manage household budgets, and process commercial transactions accurately.

12-Hour vs. 24-Hour Clock Systems

Time can be expressed using either the standard 12-hour clock or the international 24-hour clock (often referred to as military or official transportation time).

Key Differences

• 12-Hour Clock: Uses numbers 1 through 12, accompanied by AM (Ante Meridiem – before noon) or PM (Post Meridiem – after noon).
  • Midnight is 12:00 AM; Noon is 12:00 PM.
• 24-Hour Clock: Uses four digits ranging from 00:00 (midnight) to 23:59 (one minute before midnight). It does NOT use AM or PM suffixes.

Conversion Rules

Step-by-Step Conversion Example

Problem: An inter-island ferry departs St. Vincent at 15:40 and arrives in Bequia at 16:25. Convert the departure and arrival times to 12-hour clock format.

• Departure (15:40): Since hour 15 is greater than 12, subtract 12: $15 - 12 = 3$. Format: 3:40 PM.
• Arrival (16:25): Subtract 12: $16 - 12 = 4$. Format: 4:25 PM.

Elapsed Time Calculations

Elapsed time is the total amount of time that passes from the start of an event to its conclusion.

Strategy: Counting Up (Timeline Method)

When calculating elapsed time across hour boundaries:

1. Count the minutes from the start time to the next nearest full hour.
2. Count the full hours to the final hour before the end time.
3. Count the remaining minutes to the final end time.
4. Add all intermediate hours and minutes together.

Worked Example: Inter-Island Flight Duration

Problem: A flight from Trinidad departs at 10:45 AM and lands in Saint Lucia at 1:20 PM. How long was the flight?

• Step 1 (Minutes to next hour): From 10:45 AM to 11:00 AM $= 15 \text{ minutes}$.
• Step 2 (Full hours to target hour): From 11:00 AM to 1:00 PM $= 2 \text{ hours}$.
• Step 3 (Remaining minutes): From 1:00 PM to 1:20 PM $= 20 \text{ minutes}$.
• Step 4 (Total time): $2 \text{ hours} + (15 + 20) \text{ minutes} = 2 \text{ hours } 35 \text{ minutes}$.
• Answer: The total elapsed flight time is $2 \text{ hours and } 35 \text{ minutes}$.

Average Speed, Distance & Time Formulas

Average speed is the rate at which an object covers distance over time. The primary relationships between Distance ($D$), Speed ($S$), and Time ($T$) are represented by the standard DST triangle:

$\text{Speed } (S) = \frac{\text{Distance } (D)}{\text{Time } (T)}$ $\text{Distance } (D) = \text{Speed } (S) \times \text{Time } (T)$ $\text{Time } (T) = \frac{\text{Distance } (D)}{\text{Speed } (S)}$

Units of Measurement

• Distance: kilometres (km) or metres (m)
• Time: hours (h) or seconds (s)
• Speed: kilometres per hour (km/h) or metres per second (m/s)

Critical Rule: Ensure time units match the speed units! If speed is in $\text{km/h}$, time MUST be expressed in hours or decimal hours (e.g., $30 \text{ minutes} = 0.5 \text{ hours}$, $15 \text{ minutes} = 0.25 \text{ hours}$, $45 \text{ minutes} = 0.75 \text{ hours}$).

Worked Example 1: Finding Average Speed

Problem: A express bus travels $180 \text{ kilometres}$ from Kingston to Montego Bay in $2.5 \text{ hours}$ ($2 \text{ hours } 30 \text{ minutes}$). What is its average speed?

• Calculation: $\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{180 \text{ km}}{2.5 \text{ h}} = 72 \text{ km/h}$
• Answer: The bus travelled at an average speed of $72 \text{ km/h}$.

Worked Example 2: Finding Distance

Problem: A cyclist travels at a constant speed of $18 \text{ km/h}$ for $3 \text{ hours and } 20 \text{ minutes}$. Calculate the total distance covered.

• Step 1: Convert $3 \text{ h } 20 \text{ min}$ into a fraction of an hour: $20 \text{ min} = \frac{20}{60} = \frac{1}{3} \text{ h}$. Total time $= 3\frac{1}{3} = \frac{10}{3} \text{ hours}$.
• Step 2: Calculate distance: $\text{Distance} = \text{Speed} \times \text{Time} = 18 \times \frac{10}{3} = 6 \times 10 = 60 \text{ km}$
• Answer: The cyclist covered $60 \text{ kilometres}$.

Money Transactions, Discounts & Making Change

Financial mathematics tests practical skills including calculating total costs, calculating percentage discounts, adding sales tax (VAT), and determining change when paying with currency notes.

Important Financial Formulas

1. Total Cost: $\text{Total Cost} = \sum (\text{Quantity of items} \times \text{Unit Price})$
2. Discount Amount: $\text{Discount Amount} = \text{Original Price} \times \frac{\text{Discount Percentage}}{100}$
3. Sale Price: $\text{Sale Price} = \text{Original Price} - \text{Discount Amount}$
4. Change Tendered: $\text{Change} = \text{Amount Paid} - \text{Total Cost}$

Worked Example: Back-to-School Shopping

Problem: Maya purchases 3 notebooks at $4.50 each, 2 pens at $1.50 each, and 1 school bag priced at $40.00. The store offers a 10% back-to-school discount on the total bill. Maya pays the cashier with a $100.00 bill. How much change should she receive?

• Step 1: Calculate subtotal before discount:
  • 3 Notebooks $= 3 \times $4.50 = $13.50$
  • 2 Pens $= 2 \times $1.50 = $3.00$
  • 1 School Bag $= $40.00$
  • Subtotal $= $13.50 + $3.00 + $40.00 = $56.50$
• Step 2: Calculate 10% discount: $\text{Discount} = 10\% \text{ of } \$56.50 = 0.10 \times \$56.50 = \$5.65$
• Step 3: Calculate final discounted price: $\text{Final Price} = \$56.50 - \$5.65 = \$50.85$
• Step 4: Calculate change from $100 bill: $\text{Change} = \$100.00 - \$50.85 = \$49.15$
• Answer: Maya should receive $49.15 in change.