Numbers & Place Value

Depth · one concept from first principles · step 2 of 4

Spend downstream time first, bound upstream speed

A boat goes 100 km downstream at its own speed of 30 km/h on a river flowing at 2 km/h, then comes back upstream at a higher own speed. The whole round trip must take at most 5 hours. Find the smallest whole-number own speed for the upstream trip.
RatiosExpressions
Your answer
How to solve
Strategy Convert to Algebra — The downstream half is completely known, so finish it first: its time comes off the 5-hour budget. What is left is a time limit for the upstream half, which becomes one inequality in the unknown speed. Last, round up to a whole number and check the neighbors.
1STEP 1

Speed over the ground going downstream

Going with the current, the river carries the boat along, so its speed over the ground is its own speed plus the current.

30 + 2 = 32
2STEP 2

Time for the downstream trip

Time is distance divided by speed: hours = km ÷ (km per hour).

100/32 = 25/8
3STEP 3

Time left for the upstream trip

Subtract the downstream time from the 5-hour limit.

5 - 25/8 = 15/8
4STEP 4

Write the inequality

Let the boat's own speed upstream be x km/h. Against the current it covers x - 2 km/h, and must fit in the time left.

100/(x - 2) ≤ 15/8
5STEP 5

Clear the fractions

The boat must make progress, so x - 2 ＞ 0. Multiplying by the positive 8(x - 2) keeps the inequality's direction.

800 ≤ 15(x - 2), 830 ≤ 15x, x ≥ 166/3 = 55 1/3
6STEP 6

Round up and check

The speed is a whole number at least 55 1/3, so the smallest choice is 56. Check it and the number just below.

100/54 ≈ 1.852 ≤ 1.875, 100/53 ≈ 1.887 ＞ 1.875
Answer
at least 56 km/h
At 56 km/h the upstream ground speed is 54 km/h, so the round trip takes 25/8 + 100/54 ≈ 3.125 + 1.852 = 4.977 hours, just under 5. The upstream speed has to be much larger than 30 because the downstream leg already used more than 3 of the 5 hours.
Takeaway

When a trip has a known half, spend its time first -- the leftover time turns into a clean inequality for the unknown half.

  • Speed over the ground going downstream
  • Time for the downstream trip
  • Time left for the upstream trip
  • Write the inequality
  • Clear the fractions
  • Round up and check
Where next?
Level upsuggested
Two thresholds on one mixture give a range
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
One win count sets both players' scoresMoving-point triangle area by subtraction

A parent dashboard for the family lives at sensimlab.com.

▶ Practice — 12 problems