7.2 Applied Age Word Problems, Uniform Motion, and Relative Speed

7.2 Applied Age Word Problems, Uniform Motion, and Relative Speed

Civil service examinations consistently feature applied word problems because they evaluate an applicant's capacity to translate narrative real-world conditions into rigorous quantitative models. Diplomatic attachés and foreign service officers must schedule multi-vehicle diplomatic convoys, track repatriation flights across varying jet-stream winds, coordinate regional consular outreach itineraries, and evaluate long-term demographic and career seniority distributions. Understanding the underlying algebraic invariances across time and motion eliminates confusion and accelerates problem-solving speed.

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Chronological Age Word Problems: The Invariance Framework

Age word problems describe mathematical relationships connecting the ages of two or more individuals across different chronological epochs (past, present, and future). Solving them systematically requires two principles:

1. The Fundamental Age Invariance

The chronological age difference between any two living persons remains strictly constant across all time:

$$(A + t) - (B + t) = A - B$$

If Senior Diplomat $A$ is $15$ years older than Attaché $B$ today, $A$ was $15$ years older than $B$ ten years ago, and will remain exactly $15$ years older than $B$ thirty years from now. Ratios change over time, but the absolute difference is invariant.

2. The Epoch Matrix Setup

Never attempt to solve multi-epoch age problems in your head. Immediately construct a $2 \times 3$ grid in the test booklet margin:

Algebraic Formulation Steps:

1. Choose a single reference variable for the present age of one individual (usually the younger person, so coefficients remain positive).
2. Express the other person's present age in terms of that variable using the first relational premise.
3. Advance or regress both individuals into the second epoch by adding or subtracting the elapsed years.
4. Set up an equation connecting their ages in that second epoch and solve for the unknown variable.
5. Re-read the final question carefully: verify whether the question asks for the present age, an age in the past, an age in the future, or the sum of their ages.

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Uniform Motion Fundamentals: The $d = r \times t$ Model

The examples here assume uniform motion unless the stem says otherwise: $d=r\times t$. Real acceleration, traffic, and stops require additional information. Do not assume every official word problem uses constant speed.

Unit Consistency and Dimensional Analysis

Before writing an equation, ensure all units match:

• If distance is in kilometers ($\text{km}$) and time is in hours ($\text{h}$), speed must be in $\text{km/h}$.
• To convert speed from $\text{km/h}$ to meters per second ($\text{m/s}$), multiply by $\frac{5}{18}$:

$$72\text{ km/h} = 72 \times \frac{5}{18} = 4 \times 5 = 20\text{ m/s}$$

• To convert minutes to hours, divide by $60$ (e.g., $45\text{ minutes} = \frac{45}{60} = \frac{3}{4}\text{ hour} = 0.75\text{ hour}$).

The Three Core Uniform Motion Archetypes:

1. Equal Distance Travelled ($d_1 = d_2$): When two vehicles travel the same distance (e.g., an outbound trip and return trip, or two vehicles traveling between the same two cities), equate their distance expressions:

$$r_1 t_1 = r_2 t_2$$

1. Total Combined Distance ($d_1 + d_2 = D_{\text{total}}$): When two vehicles travel toward each other or travel in opposite directions starting from the same point, the sum of their individual distances equals the total separation:

$$r_1 t + r_2 t = D_{\text{total}} \implies (r_1 + r_2)t = D_{\text{total}}$$

1. Distance Separation / Difference ($d_2 - d_1 = \Delta d$): In pursuit problems where a faster vehicle overtakes a slower vehicle, the faster vehicle must cover the initial head-start distance plus whatever distance the slower vehicle travels.

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Relative Speed: Convergence and Pursuit Dynamics

Relative speed is the rate at which the distance between two moving bodies decreases or increases:

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1. Objects Moving in Opposite Directions (Convergence / Separation)

When two vehicles travel toward each other from two points separated by distance $D$, they close the gap simultaneously:

$$\text{Relative Closing Speed} = r_1 + r_2$$ $$\text{Time until Collision / Meeting} = t_{\text{meet}} = \frac{D}{r_1 + r_2}$$ $$\text{Distance from Point 1 where they meet} = d_1 = r_1 \times t_{\text{meet}}$$

2. Objects Moving in the Same Direction (Pursuit and Overtake)

When Vehicle 2 pursues Vehicle 1, Vehicle 1 has already established a head start:

1. Compute the head-start distance gap: $d_{\text{lead}} = r_1 \times t_{\text{head-start}}$.
2. Compute the net rate at which Vehicle 2 gains on Vehicle 1: $r_{\text{rel}} = r_2 - r_1$ (where $r_2 > r_1$).
3. Calculate the time required for Vehicle 2 to eliminate the gap:

$$t_{\text{overtake}} = \frac{d_{\text{lead}}}{r_2 - r_1}$$

1. The total distance from the starting point at the moment of overtake is:

$$d_{\text{overtake}} = r_2 \times t_{\text{overtake}}$$

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Water Current and Wind Resistance Problems

When a vessel moves through water or an aircraft flies through the atmosphere, the motion of the surrounding fluid medium alters the vehicle's speed relative to the fixed ground:

• Let $r_v$ = speed of vehicle in still water or calm air (cruising speed).
• Let $r_m$ = speed of the water current or wind medium.

Resolving Vessel and Medium Speeds from Observed Travel Speeds:

Adding the downstream and upstream rates eliminates the medium speed:

$$r_{\text{down}} + r_{\text{up}} = (r_v + r_m) + (r_v - r_m) = 2r_v \implies r_v = \frac{r_{\text{down}} + r_{\text{up}}}{2}$$

Subtracting the upstream rate from the downstream rate isolates the current:

$$r_{\text{down}} - r_{\text{up}} = (r_v + r_m) - (r_v - r_m) = 2r_m \implies r_m = \frac{r_{\text{down}} - r_{\text{up}}}{2}$$

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Average Speed and the Harmonic Mean Formula

A classic trap on the CSE-FSO is calculating the average speed of a round trip by taking the simple arithmetic mean of the two speeds: $\frac{v_1 + v_2}{2}$. For unequal speeds over equal distances with no stops, use the harmonic mean instead; with equal times, an arithmetic mean is appropriate.

The Fundamental Definition of Average Speed

Average speed is defined strictly as total distance divided by total elapsed time:

$$v_{\text{avg}} = \frac{d_{\text{total}}}{t_{\text{total}}} = \frac{d_1 + d_2}{t_1 + t_2}$$

The Harmonic Mean for Equal Distances

When an object travels an equal distance $d$ at speed $v_1$ and returns over the same distance $d$ at speed $v_2$:

$$t_1 = \frac{d}{v_1} \quad \text{and} \quad t_2 = \frac{d}{v_2}$$ $$v_{\text{avg}} = \frac{2d}{\frac{d}{v_1} + \frac{d}{v_2}} = \frac{2d}{d\left(\frac{1}{v_1} + \frac{1}{v_2}\right)} = \frac{2}{\frac{v_1 + v_2}{v_1 v_2}} = \frac{2 v_1 v_2}{v_1 + v_2}$$