Rate, Distance & Time

Problem Setup

d = 60 × 2.5

Solve for

Examples

Time = D / (r₁ + r₂)

Distance Apart (D)

Speed 1 (r₁)

Speed 2 (r₂)

Two objects start D apart and travel toward each other. Combined rate = r₁ + r₂.

Examples

Time = H / (r₂ − r₁)

Head Start Distance (H)

Slower Speed (r₁)

Faster Speed (r₂)

Object 1 has a head start of H. Object 2 is faster and chases. Catch-up rate = r₂ − r₁.

Examples

b − c = d_up / t_up | b + c = d_dn / t_dn

Distance Downstream

Time Downstream (hr)

Distance Upstream

Time Upstream (hr)

Enter two trips. Downstream speed = b + c. Upstream speed = b − c. Solve for boat speed b and current speed c.

Examples

Solution

Enter values above and press Solve to see the answer, the d = rt equation used, and a distance table.

Answer

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Equation Used

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Distance Table (Rate × Time = Distance)

d = r × t Triangle

d = r × t | r = d / t | t = d / r

Cover the variable you want to find in the triangle — the remaining two show the operation.

Cover d: multiply r × t to get distance.

Cover r: divide d ÷ t to get rate.

Cover t: divide d ÷ r to get time.

Units must be consistent — if rate is mph, time must be hours and distance will be miles.

Always label a d = rt table with each object's rate and time before solving. This keeps multi-object problems organized.

Two Objects Strategy

Set distances equal (or sum to total)

Toward each other: Both travel for time t. Their distances add up to the total D apart.
r₁·t + r₂·t = D → t = D / (r₁ + r₂)

Same direction (catch-up): Slower has head start H. Faster gains at rate (r₂ − r₁).
r₁·t + H = r₂·t → t = H / (r₂ − r₁)

Upstream/Downstream: Let b = boat speed in still water, c = current speed.
Downstream speed = b + c | Upstream speed = b − c

Write two equations: d_dn = (b+c)·t_dn and d_up = (b−c)·t_up. Solve the system.