3.2 Financial & Distance Math: Interest, Profit/Loss, Speed & Time

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3 Chapter 3: Arithmetic Reasoning (AR)

3.1 Word Problems: Rates, Ratios, Proportions & Percentages 3.2 Financial & Distance Math: Interest, Profit/Loss, Speed & Time 3.3 Work & Mixture Problems: Joint Rates & Percentage Concentrations 3.4 Multi-Step Problem Solving & Algebraic Word Problems

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3.2 Financial & Distance Math: Interest, Profit/Loss, Speed & Time

Key Takeaways Explain This Topic AI

Simple interest formula I = P * r * t requires interest rate r in decimal form and time t expressed strictly in years.

Profit and loss percentages are calculated relative to the Cost Price (CP): Profit % = (Selling Price - Cost Price) / Cost Price * 100%.

The uniform motion formula d = r * t connects distance, rate, and time; ensure all measurement units are uniform before solving.

For round-trip journeys with different speeds over equal distance, average speed is the harmonic mean: Average Speed = 2 * v1 * v2 / (v1 + v2).

Relative speed rules: objects moving towards each other add speeds (r1 + r2); objects moving in the same direction subtract speeds (|r1 - r2|).

Last updated: July 2026

3.2 Financial & Distance Math: Interest, Profit/Loss, Speed & Time

Financial mathematics and distance-rate-time motion problems are useful Arithmetic Reasoning practice types. Mastering these topics requires a strong command of specific formulas, unit consistency, and algebraic modeling.

1. Simple Interest & Compound Interest Basics

Simple interest is calculated solely on the initial principal amount. The formula is: I = P ⋅ r ⋅ t I = P \cdot r \cdot t I = P ⋅ r ⋅ t

• I = I = I = Total interest earned or owed (in dollars)
• P = P = P = Principal starting amount (in dollars)
• r = r = r = Annual interest rate (expressed as a decimal, e.g., 6 % = 0.06 6\% = 0.06 6% = 0.06 )
• t = t = t = Time duration ( strictly in years )
• Total Accumulated Balance

The total balance ( A A A ) accumulated after time t t t is the principal plus interest: A = P + I = P ( 1 + r ⋅ t ) A = P + I = P(1 + r \cdot t) A = P + I = P ( 1 + r ⋅ t )

Time Conversions for t t t

If time is given in months or days, it must be converted to years before substituting into the formula:

Months to Years: t = Number of Months 12 t = \frac{\text{Number of Months}}{12} t = 12 Number of Months ​

Days to Years (Standard Bank Year): t = Number of Days 360 t = \frac{\text{Number of Days}}{360} t = 360 Number of Days ​ or Number of Days 365 \frac{\text{Number of Days}}{365} 365 Number of Days ​

Step-by-Step Worked Example 1: A service member deposits $6,000 into an account earning 5% simple annual interest. What is the total balance after 18 months?

Step 1: Convert time into years: t = 18 12 = 1.5 years t = \frac{18}{12} = 1.5\text{ years} t = 12 18 ​ = 1.5 years .

Step 2: Convert rate to decimal: r = 0.05 r = 0.05 r = 0.05 .

Step 3: Calculate interest: I = $ 6 , 000 × 0.05 × 1.5 = $ 450 I = \text{\textdollar}6,000 \times 0.05 \times 1.5 = \text{\textdollar}450 I = $ 6 , 000 × 0.05 × 1.5 = $ 450 .

Step 4: Calculate total balance: A = $ 6 , 000 + $ 450 = $ 6 , 450 A = \text{\textdollar}6,000 + \text{\textdollar}450 = \text{\textdollar}6,450 A = $ 6 , 000 + $ 450 = $ 6 , 450 .

2. Cost Price, Selling Price, Profit & Loss

In retail and procurement math, transactions revolve around Cost Price ( C P CP CP ) and Selling Price ( S P SP SP ).

• Concept Formula Condition
• Profit (Gain) Profit = S P − C P \text{Profit} = SP - CP Profit = SP − CP When S P > C P SP > CP SP > CP
• Loss Loss = C P − S P \text{Loss} = CP - SP Loss = CP − SP When C P > S P CP > SP CP > SP

Percentage Profit Profit % = S P − C P C P × 100 % \text{Profit \%} = \frac{SP - CP}{CP} \times 100\% Profit % = CP SP − CP ​ × 100% Always based on C P CP CP

Percentage Loss Loss % = C P − S P C P × 100 % \text{Loss \%} = \frac{CP - SP}{CP} \times 100\% Loss % = CP CP − SP ​ × 100% Always based on C P CP CP

Finding Cost Price from Selling Price and Profit %

To determine the original Cost Price when Selling Price and Profit percentage are given: S P = C P ⋅ ( 1 + Profit % in decimal ) ⟹ C P = S P 1 + Profit % SP = CP \cdot (1 + \text{Profit \% in decimal}) \implies CP = \frac{SP}{1 + \text{Profit \%}} SP = CP ⋅ ( 1 + Profit % in decimal ) ⟹ CP = 1 + Profit % SP ​

Step-by-Step Worked Example 2: A surplus vendor sells a piece of reconditioned equipment for $1,320, realizing a 10% profit over the original reconditioning cost. What was the cost price?

Step 1: Express relationship: S P = C P × ( 1 + 0.10 ) = 1.10 × C P SP = CP \times (1 + 0.10) = 1.10 \times CP SP = CP × ( 1 + 0.10 ) = 1.10 × CP .

Step 2: Substitute known value: $ 1 , 320 = 1.10 × C P \text{\textdollar}1,320 = 1.10 \times CP $ 1 , 320 = 1.10 × CP .

Step 3: Solve for C P CP CP : C P = 1 , 320 1.10 = $ 1 , 200 CP = \frac{1,320}{1.10} = \text{\textdollar}1,200 CP = 1.10 1 , 320 ​ = $ 1 , 200 .

3. Distance, Rate & Time Core Principles

The fundamental uniform motion formula connecting Distance ( d d d ), Rate/Speed ( r r r ), and Time ( t t t ) is: d = r ⋅ t ⟺ r = d t ⟺ t = d r d = r \cdot t \quad \Longleftrightarrow \quad r = \frac{d}{t} \quad \Longleftrightarrow \quad t = \frac{d}{r} d = r ⋅ t ⟺ r = t d ​ ⟺ t = r d ​

Unit Consistency Rule

Before plugging values into d = r ⋅ t d = r \cdot t d = r ⋅ t , ensure all units align. If rate is in miles per hour (mph), distance must be in miles, and time must be in hours. If time is given in minutes, convert to hours by dividing by 60.

4. Relative Speed and Pursuit Word Problems

When two objects move simultaneously, solve using relative speed concepts.

Case 1: Moving in Opposite Directions (Towards or Away)

When two objects move towards each other (or away from each other in opposite directions), their rates add together because the distance between them changes at a faster combined rate: r relative = r 1 + r 2 ⟹ Time until Meeting = Initial Distance r 1 + r 2 r_{\text{relative}} = r_1 + r_2 \implies \text{Time until Meeting} = \frac{\text{Initial Distance}}{r_1 + r_2} r relative ​ = r 1 ​ + r 2 ​ ⟹ Time until Meeting = r 1 ​ + r 2 ​ Initial Distance ​

Case 2: Moving in the Same Direction (Pursuit / Catch-Up)

When a faster object pursues a slower object that started earlier or ahead, their rates subtract : r relative = r fast − r slow ⟹ Time to Catch Up = Head-Start Distance r fast − r slow r_{\text{relative}} = r_{\text{fast}} - r_{\text{slow}} \implies \text{Time to Catch Up} = \frac{\text{Head-Start Distance}}{r_{\text{fast}} - r_{\text{slow}}} r relative ​ = r fast ​ − r slow ​ ⟹ Time to Catch Up = r fast ​ − r slow ​ Head-Start Distance ​

Step-by-Step Worked Example 3 (Catch-Up Pursuit): Truck A leaves a logistics base traveling north at 40 mph. 2 hours later, Truck B leaves the same base traveling north along the same highway at 60 mph. How many hours after Truck B departs will it catch up to Truck A?

Step 1: Calculate Truck A's head-start distance: d head-start = 40 mph × 2 hrs = 80 miles d_{\text{head-start}} = 40\text{ mph} \times 2\text{ hrs} = 80\text{ miles} d head-start ​ = 40 mph × 2 hrs = 80 miles .

Step 2: Calculate relative catch-up rate: r relative = 60 mph − 40 mph = 20 mph r_{\text{relative}} = 60\text{ mph} - 40\text{ mph} = 20\text{ mph} r relative ​ = 60 mph − 40 mph = 20 mph .

Step 3: Calculate time for Truck B to close the gap: t = 80 miles 20 mph = 4 hours t = \frac{80\text{ miles}}{20\text{ mph}} = 4\text{ hours} t = 20 mph 80 miles ​ = 4 hours .

5. Round-Trip Average Speed & The Harmonic Mean Trap

classification practice Critical Warning: If a vehicle travels to a destination at speed v 1 v_1 v 1 ​ and returns along the exact same route at speed v 2 v_2 v 2 ​ , the average speed for the round trip is NOT the simple average v 1 + v 2 2 \frac{v_1 + v_2}{2} 2 v 1 ​ + v 2 ​ ​ .

Because the vehicle spends more time traveling at the slower speed, average speed is weighted by time. When distance is equal in both directions, average speed is calculated using the harmonic mean formula : Average Speed = 2 ⋅ v 1 ⋅ v 2 v 1 + v 2 \text{Average Speed} = \frac{2 \cdot v_1 \cdot v_2}{v_1 + v_2} Average Speed = v 1 ​ + v 2 ​ 2 ⋅ v 1 ​ ⋅ v 2 ​ ​

Step-by-Step Worked Example 4: An aircraft flies from Base X to Base Y (distance 300 miles) at 150 mph. On the return flight, headwinds reduce its speed to 100 mph. What is the aircraft's average speed for the entire round trip?

Step 1: Apply harmonic mean formula: Average Speed = 2 × 150 × 100 150 + 100 \text{Average Speed} = \frac{2 \times 150 \times 100}{150 + 100} Average Speed = 150 + 100 2 × 150 × 100 ​ .

Step 2: Simplify numerator and denominator: 30 , 000 250 = 120 mph \frac{30,000}{250} = 120\text{ mph} 250 30 , 000 ​ = 120 mph .

Verification via Total Distance / Total Time:

Time out: 300 150 = 2 hours \frac{300}{150} = 2\text{ hours} 150 300 ​ = 2 hours . Time back: 300 100 = 3 hours \frac{300}{100} = 3\text{ hours} 100 300 ​ = 3 hours . Total time = 5 hours = 5\text{ hours} = 5 hours .

Total distance = 300 + 300 = 600 miles = 300 + 300 = 600\text{ miles} = 300 + 300 = 600 miles .

Average Speed = 600 miles 5 hours = 120 mph \text{Average Speed} = \frac{600\text{ miles}}{5\text{ hours}} = 120\text{ mph} Average Speed = 5 hours 600 miles ​ = 120 mph .