# Algebra Rate Problems (upstream/downstream)

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**Important tips for solving Algebra Rate problems.**

Algebra Rate problems are used to find the distance traveled or time required for traveling certain distance.

**Important note :**

For downstream ----> Rate of (boat /steamer) in still water + rate of stream

For upstream ----> Rate of (boat /steamer) in still water - rate of stream

With wind ----> Rate of the plane in still air + rate of the wind

against wind ----> Rate of the plane in still air - rate of the wind

For downstream ----> Rate of (boat /steamer) in still water + rate of stream

For upstream ----> Rate of (boat /steamer) in still water - rate of stream

With wind ----> Rate of the plane in still air + rate of the wind

against wind ----> Rate of the plane in still air - rate of the wind

1) A boat travels 30 km up a river in the same time it takes to travel 50 km down the same river. If the current of water is 5km/h . What is the speed of the boat in still water?

**Solution :**

Let speed of boat in still water = x km/h

Speed of the boat downstream = x + 5

Speed of the boat upstream = x - 5

Speed = Distance / time

So time = Distance / speed

Time taken by boat to travel 30 km upstream = 30/ (x – 5)

Time taken by boat to travel 50 km downstream = 50/ (x + 5)

Since time taken by the boat in both cases is the same ,

=> 30 (x + 5) = 50 (x -5)

=> 30 x +150 = 50x -250

=> 20x = 400

=> x = 20

Thus the speed of the boat in still water is 20 km/h.

2) A steamer goes downstream and covers the distance between two ports in 4 hrs., while it covers the same distance upstream in 5 hrs. If the speed of the stream is 2km/h , find the speed of the steamer in still water.

**Solution :**

Let speed of the steamer in still water = x km/h

Speed of the steamer in downstream = x +2

Speed of the steamer in upstream = x – 2

So time = Distance / speed

Distance = speed x time

D = (x +2) 4

D = (x – 2) 5

As the distance between the port is same.

4(x +2) = 5(x -2)

4x + 8 = 5x -10

-x = -18

X = 18

Speed of the the steamer in still water = 18 m/h

3) Joe traveled against the wind in a small plane for 3.2 hr. The return trip with the wind took 2.8 hr. Find the speed of the wind if the speed of the plane is still air is 170 mph.

**Solution :**

Let the speed of the plane in still air = x mph

Speed of the plane against the wind = 170 –x

Speed of the plane with the wind = 170 +x

Speed = Distance / time

Distance = speed x time

D = (170 –x) 3.2

D = ( 170 +x ) 2.8

As the distance is same, so

(170 –x) 3.2 = ( 170 +x ) 2.8

544 – 3.2 x = 476 + 2.8x

– 3.2 x - 2.8x = 476 – 544

-6 x = - 68

x = 68/6 = 11.33

So the speed of the wind = 11.33 mph

**Linear Equation in One Variable**

• Linear Equation in One Variable

• Algebraic Equations with Fractions

• Algebra Word Problems

• Algebra Rate Problems (upstream and downstream)

• Algebra Age Problems

• Algebra Digit Problems

• Linear Equation in One Variable

• Algebraic Equations with Fractions

• Algebra Word Problems

• Algebra Rate Problems (upstream and downstream)

• Algebra Age Problems

• Algebra Digit Problems

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