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Monday, October 15, 2012

Diogo's Proportion Post


The relationship between two measures, 
expressed as the number of times one is bigger or smaller than the other.

Consider the two bars on the right. One has a height of 10 meters the other 40 meters. The taller one is obviously four times taller than the other, so we say thier heights are in the ratio of four to one. What this means is that for every one unit of height on the left bar, the tall bar has four height units. Hence "four to one".

Using a colon

Ratios are usually written as two numbers separated by a colon:
the ratio of their heights is 4:1
The colon between the 4 and 1 is read as "to". So the sentence would be read as
"The ratio of the heights is four to one".
Obviously the taller one is first here. If you considered the smaller bar first then the ratio would be 1:4.

As a fraction

Using the bar example again, we ca also write the ratio as a fraction, where the top (numerator) is one height, and the bottom (denominator) is the other. We can pick either one to be the top, so using the smaller one, we can write:
the ratio of their heights is 
which reduces to 1/4.
This fraction can then also be written as a decimal. In this case 0.25. This would be read as "the smaller bar is 0.25 times the height of the larger one"

Ratios have no units

A ratio is how many times bigger one thing is than another. It's a number you multiply by to get one thing from another. But remember, when you find the ratio of two quantities, they must be in the same units.

It's easy to get ratios backwards

If you know that John and Jim have weights in the ratio 1.2 : 1, there can be confusion about whom is the heavier of the two. By convention we assume that since John is mentioned first then his weight is the 1.2 in the ratio since that is also first. But you should be as explicit as you can. It is better to say John's weight is 1.2 times Jim's.
When the ratio is expressed as a fraction, the convention is that the first mentioned item is on top (numerator).

Don't confuse ratio with difference

The bars above have heights in the ratio 4:1, but the difference in their height is 30 meters.
  • A ratio is obtained by division, and has no units
  • Difference is obtained by subtraction and does have units

Definition of Rate
  • Rate is a ratio that compares two quantities of different units.
Examples of Rate
  • 20 oz of juice for $4, miles per hour, cost per pound etc. are examples of rate.
More about Rate
  • Unit rate: Unit rate is a rate in which the second term is 1. 
    For example, Jake types 10 words in 5 seconds. 
    Jake’s unit rate is the number of words he can type in a second. 
    His unit rate is 2 words per second.
Solved Example on Rate
"4 lb of meat costs $5." Identify the two rates given by the statement.
A. and 
B.  and 
C.  and 
D. none of these
Correct Answer: A
Step 1: Rate is a ratio that compares two quantities of different units.
Step 2: The two rates given by the statement are  and .
Related Terms for Rate
A proportion is a name we give to a statement that two ratios are equal. It can be written in two ways:
  • two equal fractions, 
  • or,
  • using a colon,    a:b = c:d
When two ratios are equal, then the crossproduct of the ratios are equal.
That is, for the proportion, a:b = c:d ,  a x d = b x c       
   Part 2/3   

a. six pairs of shorts cost $96. How much will nine pairs of shorts cost?
b. Carmen used a 5ft of string to make 4 mobiles. how much string will she need to make 18 mobiles?
c. five art magazines cost $14.25. how much will eight art magazines cost?
d. four baseball caps cost $25. how manny baseball caps can you buy with $43.75?
e. Two helicopters were used for rescue missions from 8 am to 11 am. how many helicopters will be used for rescue missions in a twelve hour period?

  a) 6 pairs cost $96. Therefore 9 pairs cost 9/6 x $96 = $144
b) 5 feet to make 4 mobiles. To make 18 mobiles she will need 5 x 18/4 = 22.5 feet
c). 5 mags cost $14.25. 8 mags should cost $14.25 x 8/5 = $22.80
d) 4 caps cost $25. You can therefore buy 4 x 43.75 / 25 caps = 7 caps
e) no answer - The fact that two helicopters were used in three hours does not mean that 2 x 12 /3 =8 helicopters are used in 12 hours. The same two helicopters may be used for the entire 12 hour period. There may only be 2 helicopters.



1. Doesn't seem fair 
because he told the truth 

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