Expert: Scott A Wilson - 9/24/2009

Question
im in eighth grade and im having trouble understanding unit rates and proportions. Can you help me???

Answer
Rates
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The speed of an item is the rate at which it travles.
Speed is in feet/sec, miles/hour, quarter miles/minute, and a variety of other ways.

In doing problems, it is best to take the units along.

For example, if a car travels at 10 mph and a runner travels a quarter mile in 1.5 minutes,
which is faster?

To do this, write out the units.  They are 10 mph = 10 miles/hour and
a quarter mile in 1.5 minutes = 0.25 miles / 1.5 minutes.

The units on the second one, 0.25/1.5, should have the top and bottom multiplied by 4.
This gives 1/6 miles/minute.

Next, since minutes are in the denominator, we need to multiply with minutes in in numerator
and hours in the denominator.  That makes minutes/hour, which is known to be 60.

So (1/6)(miles/minute)(60 minutes/hour) gives us (1/6)(60) miles/hour since the minutes cancel.

1/6 of 60 is 10, so as can be seen, the car and person are going at the same speed.

If they were racing, they would tie.  The question might be, 'Who will win on the next lap?'
The answer would most likely be they would tie again, but after awhile, the person would tire
and the car would win.  Of course, the car might run out of gas, and then it's a whole new game
as the runner would have no problem winning.

Proportions
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Proportions are the comparison of sizes.  If the bed is 7 feet long, but only 3 1/2 feet wide,
the proportion is 1:2 since (3 1/2) * 2 = 7.

If a house were 50 feet wide and 20 feet tall, the proportion would be that it could be said that the height of the house were 2:5 of the width since 20:50 is the same as 2:5.
Just divide both numbers by 10.

If one man measure his car 4 yards long and another man measure his car 13 feet long,
the units would have to be made the same first.  It is known that 1 yard = 3 feet,
so take the (4 yards)(3 feet/yard) = 3*4 feet = 12 feet since the yards cancel.
Now we can look at the proportion.  It can be said that the proportion of the car lengths
is 13:12.