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Rate Problems

[6000] You have 2 candles both 10 cm tall and you light both at the same time. Candle A finishes burning in 3 hours, candle B in 4 hours. When was candle B twice the height of candle A and how tall were the candles at the time?

Solution: The first thing we have to do is work out the rate per hour of the 2 candles.
Candle A: 10 cm per 3 hours = 10/3 cm per hour
Candle B: 10 cm per 4 hours = 10/4 cm per hour

Next, we will let t represent the amount of time that has passed in hours. For candle A, t will range between 0 and 3, for candle B, between 0 and 4.

The next thing we have to do is work out the height of each candle after t hours. For each candle we have to subtract from 10 the amount of the candle that has burned. For candle A, the height after t hours is

10 - 10t/3

For candle B, the height after t hours is

10 - 10t/4

We want the height of candle B to be twice the height of candle A:

10 - 10t/4 = 2(10 - 10t/3)
10 - 10t/4 = 20 - 20t/3
(40 - 10t)/4 = (60 - 20t)/3

Cross-multiplying, we get:

120 - 30t = 240 - 80t
50t = 120
t = 2.4

So, the height of candle B is twice that of candle A after 2.4 hours or 2 hours and 24 minutes. To find the height of each candle after 2.4 hours we plug it into both height expressions. For candle A, the height is 10 - 10(2.4)/3 cm = 2 cm. The height of candle B is 10 - 10(2.4)/4 cm = 4 cm.

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[6001] Jim and Jack are running the Boston Marathon. Let's assume the distance is 26 miles. Jim runs 12 miles per hour, Jack 9 miles per hour.
a) How long will it take Jim to finish the race to the nearest minute?
b) When Jim finishes the race, how much farther does Jack have to run?

Solution part a): 26 miles divided by 12 miles/hour equals 2 and 1/6 hours. Since an hour has 60 minutes, 1/6 hour = 60/6 minutes = 10 minutes. So, Jim finishes the race in 2 hours and 10 minutes.

Solution part b): The first thing to do is convert Jack's rate per hour to a rate per minute:

9 miles per hour
= 9 miles per 60 minutes
= 0.15 miles per minute

by dividing 9 by 60. Next, we have to figure out how far Jack has run. Jim finished the race in 2 hours and 10 minutes. We need to change that to just minutes since we have Jack's rate per minute. That works out to 2(60) + 10 minutes = 130 minutes. So, after 130 minutes Jack has run 130(0.15) miles = 19.5 miles. Since the race is 26 miles, Jack still has to run 26 - 19.5 miles = 6.5 miles.

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[6002] Mike is packing cans in boxes for shipping. He has 500 boxes to pack and he figures he can do it in 5 hours.
a) How many boxes per hour does he have to pack to finish in 5 hours?
b) After 2 hours, he had packed 200 boxes. But then his girlfriend called him on his cell and tied him up for half an hour. If he still wants to finish in 5 hours total, how many boxes per minute does he have to pack now?

Solution part a): This is pretty basic.

500 boxes per 5 hours
= 100 boxes per hour

which we get by dividing 500 by 5. So Mike has to pack 100 boxes per hour to finish in 5 hours.

Solution part b): If Mike has packed 200 boxes, he still has 300 left to pack. Since he spent 0.5 hours not packing, he now has only 2.5 hours left to pack the 300 boxes. Since the question asks the rate per minute, we have to convert:

300 boxes per 2.5 hours
= 300 boxes per 150 minutes
= 2 boxes per minute

So, if Mike wants to finish in 5 hours, he now has to pack 2 boxes per minute.

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Proportion Problems

[2000] Y is directly proportional to x. When x = 3, y = 10. What does y equal when x = 6?

First we set up the general equation:

y = cx

Then we plug in our values for x and y to solve for c:

10 = (c)(3)

We get c = 10/3. Our equation is now:

eq001

So, when x = 6,we have y = 60/3 = 20

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[2001] Y is directly proportional to the cube of x. When x = 2, y = 24. What does y equal when x = 4?

First we set up the general equation:

y = cx3

Then we plug in our values for x and y to solve for c:

24 = (c)(23)

24 = (c)(8)

We get c = 24/8 = 3. Our equation is now:

y = 3x3

So, when x = 4,we have y = (3)(43) = (3)(64) = 192

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[2002] Y is directly proportional to the square root of x. When x = 16, y = 2. What does x equal when y = 5/4?

First we set up the general equation:

eq001

Since the square root of 16 is 4, there is no problem solving for c:

2 = (c)(4)

So c = 1/2. Our new equation is now:

eq002

To solve for x, we plug y = 5/4 into the equation:

eq003

Multiplying both sides by 2, we get:

eq004

eq005

Squaring both sides, the final solution is x = 25/4

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[2003] Sales is inversely proportional to price. Sales are $200/day when the price is $5. According to this formula, what would be the daily sales if the price were $4?

The first thing we do is make sales the y variable and price the x variable. Now we can set up our general equation:

xy = c

Now we can plug in our values to solve for c:

(5)(200) = c

So c = 1000. Our equation is:

xy = 1000

Solving for y when x = 4 we get y = 1000/4 = 250. So, according to this formula, the daily sales would be $250 if the price is $4.

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[2004] Y is inversely proportional to the cube of x. When x = 2, y = 3. What would y equal when x = 5?

First we set up our general equation:

x3y = c

Then we plug in our values to solve for c:

(23)(3) = c

(8)(3) = c

So c = 24. Our equation is now:

x3y = 24

Solving for y when x = 5 we get y = 24/125 because 53 = 125

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[2005] Y is inversely proportional to the square root of x. When x = 9, y = 4. What would x equal when y = 6?

First we set up our general equation:

eq001

Since the square root of 9 is 3, we get:

(3)(4) = c

We have c = 12. Our equation is now:

eq002

To solve for x we plug y = 6 into the equation:

eq003

eq004

Squaring both sides, we get x = 4

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