Other Rates

Links and useful resources

• gr7's Prealgebra Course Outline START HERE<<<
• AoPS Online Textbook
• IXL Grade 7 index
• IXL Grade 8 index
• AoPS Alcumus
• IXL prealgebra practice index
• OpenSTAX Prealgebra-1 textbook

Lesson-specific resource links

• AOPS Rates 1
• AoPS Rates 2

Concept summary and lesson

• rates of change
• related rates

Combined rates

Sometimes you have a problem in which the speeds of two different things are given (how quickly two people can paint a fence, usually, for some reason. How many people in the modern world have ever actually done that?). The question then asks how quickly they can do it working together.

The trick with that kind of problem is to realize that you have to add their rates to get a combined rate, then use that to solve the problem. They'll often give you the time it takes to do a job, or the amount of job they can finish in a given time. You can't add the time or the jobs done directly, but you can add rates. So the trick is always to get to where you have the rate for each thing, and add those. Example:

Bob can bail three gallons of water per minute out of a canoe, and Jane can bail four gallons per minute out. How long will it take them to bail out their canoe if it's leaking water at 5 gallons per minute, and it started out with 10 gallons in it?

First things first, we identify our unknown: $t$ = time it takes to get the canoe empty. That suggests that we should be using the amount of water in the canoe at a given time, and find the moment when that is zero.

1. What are our rates? $B=3\text{ gal}:1\text{ min}$, $J=4\text{ gal}:1\text{ min}$.
2. How much water is in the canoe after $t$ minutes?
   1. $W=10\text{ gal}+5t\text{ gal}$
3. Working together, their rate is $7\text{ gal}:1\text{ min}$, so their contribution to the canoe's water looks like
   1. $W=-7t\text{ gal}$
4. The total water in the canoe is then
   1. $W=10+5t-7t=10-2t$
5. So, after five minutes, they will have the canoe emptied and be able to keep up.

That one was pretty easy, but what if the problem was changed:

The canoe can hold 30 gallons of water before it sinks. If it takes Bob 10 minutes to bail 30 gallons of water, and it takes Jill 7.5 minutes, how long would it take them together to bail 10 gallons out if the canoe is leaking 5 gallons per minute while they work on it?

1. This time, we aren't given their rates, we're given their overal time and amounts. This problem is identical to the last one, except that we have to start by figuring out their rate of bailing!
2. Once we have that (it's the same as last time, just divide the amount they can bail by the time it takes), the problem works exactly the same way.

Media resources

• Youtube search for "rates of change"
• Youtube search for "related rates"

Guided practice

• Tom can paint Mr. Thatcher’s fence in 6 hours, while Huck can paint Mr. Thatcher’s fence in 5 hours. If they work together, then how long will it take them to paint the fence?
• If 5 woodchucks could chuck 50 cords of wood in 4 days, then how many cords of wood could 7 woodchucks chuck in 6 days?
• Julie wants to give a 45-minute speech, and she speaks 120 words per minute. Her written notes contain 500 words per page. How many pages should she prepare?

Homework

• #hw (math) (gr7) 7.6 Other Rates (p. 310) 2025-02-10