Apply rational equations to work and rate problems. Math 20-1 Alberta.
Lesson 6.6 of Rational Expressions and Equations in Math 20-1 — Alberta curriculum lessons.
The v = (d)/(t) Relationship. The fundamental relationship for any travel problem is:
speed = distance ÷ time, or v = (d)/(t)
This formula can be rearranged to solve for any of the three quantities:
Distance: d = v · t
Time: t = (d)/(v)
When a problem gives you distance and speed, the form t = (d)/(v) is what produces the rational expressions in these problems — because times from two legs often add up to a known total or are set equal to each other.
Setting Up a Chart. For most distance-speed-time problems, organize the information into a chart with two rows (one for each leg of the trip or each vehicle). The strategy is:
1. Fill in the distance and speed for each row using the problem statement.
2. Use only one variable (call it x) for the unknown speed.
3. Fill in time using t = (d)/(v) for each row.
4. Set up an equation using the relationship the problem describes — times add to a total, or times are equal.
Russell rows his boat 24 km downstream and back to where he began. When the average speed of the current is 2 km/h, he can complete the journey in 9 hours. What is his average rowing speed in still water?
Example 1 — Steps 1 to 4: Define, Set Up, Identify NPVs. Step 1 — Define the variable.
Let x = Russell's rowing speed in still water (km/h).
Step 2 — Determine speeds.
Going downstream, the current aids him: speed = x + 2.
Coming upstream, the current opposes him: speed = x - 2.
Step 3 — Fill in the chart.
Each row has d = 24 km. Times are t = (d)/(v):
Downstream time = (24)/(x + 2)
Upstream time = (24)/(x - 2)
Step 4 — Identify NPVs.
x + 2 = 0 → x = -2 and x - 2 = 0 → x = 2, so x ≠ -2 and x ≠ 2.
In context, x is a speed and Russell must actually row against the current, so we also need x > 2.
Example 1 — Steps 5 to 10: Solve and Verify. Step 5 — Set up the equation.
The total time for both legs is 9 hours:
(24)/(x + 2) + (24)/(x - 2) = 9
Step 6 — Multiply through by LCD = (x + 2)(x - 2).
(x+2)(x-2) · (24)/(x+2) + (x+2)(x-2) · (24)/(x-2) = (x+2)(x-2) · 9
24(x-2) + 24(x+2) = 9(x^2 - 4)
Step 7 — Expand the left side.
24x - 48 + 24x + 48 = 9x^2 - 36
48x = 9x^2 - 36
Step 8 — Move everything to one side.
0 = 9x^2 - 48x - 36
Divide every term by 3: 0 = 3x^2 - 16x - 12
Step 9 — Quadratic formula (a = 3, b = -16, c = -12).
x = (16 ± √(256 + 144))/(6) = (16 ± √(400))/(6) = (16 ± 20)/(6)
x = (36)/(6) = 6 or x = (-4)/(6) = -(2)/(3)
Step 10 — Check each solution against context.
x = 6: positive and greater than 2 — valid.
x = -(2)/(3): negative, not a valid rowing speed — rejected.
Answer: Russell's rowing speed in still water is 6 km/h.
Sanity check: downstream speed = 8 km/h, time = (24)/(8) = 3 h; upstream speed = 4 km/h, time = (24)/(4) = 6 h; total = 9 h. ✓
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