Word Problems and Mathematical Modelling

Distance = Rate × Time

"Two cars leave at the same time, one at 60 mph and one at 45 mph. When are they 210 miles apart?"

$60t + 45t = 210$ → $105t = 210$ → $t = 2$ hours.

Age Problems

"Sarah is 4 years older than Tom. In 5 years, Sarah will be twice Tom's age."

Now: Sarah = $t + 4$, Tom = $t$. In 5 years: $t + 9 = 2(t + 5)$ → $t = -1$. Tom is −1? ← Re-check the problem setup. Actually: $t + 4 + 5 = 2(t + 5)$ → $t + 9 = 2t + 10$ → $t = -1$ means Tom is currently -1, which means the problem may state "In 5 years Sarah will be twice Tom's current age": $t + 9 = 2t$ → $t = 9$. Always read carefully!

Percent Problems

Original × (1 + rate) = New. For decrease: (1 − rate).

Work Problems

If A takes 6 hours and B takes 3 hours: combined rate = $\frac{1}{6} + \frac{1}{3} = \frac{1}{2}$. Together: 2 hours.

1. Define variables.
2. Write the equation.
3. Solve.
4. Check the answer makes sense.