Ratio, Proportions And Rates
Ratio, Proportions And Rates
Proportions are gotten from ratios. A "ratio" is merely a comparison between two distinct things. For example, someone can look at a group of people, count noses, and compute the "ratio of men to women" in the group. Assuming there are thirty-five people, fifteen of whom are men. Then the ratio of men to women is 15 to 20.
Observe that, in the expression "the ratio of men to women", "men" came first. This order is extremely significant, and ought to be respected: whichever word came first, its number ought to appear first. If the expression had been "the ratio of women to men", then the numbers ought to have been "20 to 15".
Expressing the ratio of men to women as "15 to 20" is expressing the ratio in words. There are two other notations for this "15 to 20" ratio:
Odds notation: 15 : 20
Fractional notation: 15/20
You ought to be able to recognize all three notations; you will probably be required to know them for your test.
There is a few terminology connected to proportions that you may require to know. In the proportion:
a⁄b=c⁄d
...the values in the "b" and "c" positions are known as the "means" of the proportion, while the values in the "a" and "d" positions are known as the "extremes" of the proportion. A basic defining property of a proportion is that the product of the means is equal to the product of the extremes. In other words, provided with the proportional statement:
a⁄b=c⁄d
...you can make conclusions that ad = bc. This is, in effect, the cross-multiplication shown earlier. This relationship is occasionally turned into a homework question like:
is 24⁄140 proportional to 30⁄176
For these ratios to be proportional which means for them to be a true proportion when they are set equal to each other, I have to be able to illustrate that the product of the means is equal to the product of the extremes. In other words, they are requiring you to find the product of 140 and 30 and the product of 24 and 176, and then see if these products are equal. Therefore, you’ll check:
140 × 30 = 4200
24 × 176 = 4224
Although these values are close, they are not equal, therefore, I know the original fractions cannot be proportional to each other. Therefore, the answer is that they are not proportional.
The other technical exercise that is based on terminology is the finding of the "mean proportional" between two numbers. Mean proportionals are a special class of proportions, where the means of the proportion are equal to each other. An example ought to be:
.1⁄2 = 2⁄4 Copyright © 2001-2011 All Rights Reserved
Due to the fact that the means are both "2" while the extremes are 1 and 4. This tells you that 2 is the "mean proportional" between 1 and 4. You may be given two values and be asked to obtain the mean proportional between them.
Find the mean proportional of 3 and 12.
I'll let "x" be the number that we are asked to find. Since x will as well be both of the means. We will set up our proportion with 3 and 12 as the extremes, and x as both means:
3⁄x = x⁄12
Currently we'll solve for x:
3⁄x = x⁄12
3 × 12 = x2
36 = x2
± 6 = x
Since we are looking for the mean proportional of 3 and 12, it is easy to decipher that we will require to take the positive answer, in order that the mean proportional would be merely the 6. Nevertheless, taking into consideration the fractions, either value would work:
3⁄-6 = -6⁄12
3⁄6 = 6⁄12
Therefore, actually, there are two mean proportionals: –6 and 6
Your book (or instructor) may wish to just consider the positive mean proportional, since the positive value is between 3 and 12.
Find the mean proportional of –3 and –12.
I'll establish this the same way as before, and solve:
-3⁄x = x⁄-12
(–3)(–12) = x2
36 = x2
± 6 = x
Therefore, there are again two mean proportionals: –6 and 6
Your book or teacher may just "–6" as an answer.
Find the mean proportional of –3 and 12.
Observe that the difference is signs; this problem is different from the ones that preceded it. But I can establish the proportion in the same precise way:
-3⁄x = x⁄12
Now we'll solve for x:
-3⁄x = x⁄12
(–3)(12) = x2
–36 = x2
Due to the fact that we can't take the square root of a negative number, then there is no solution for the mean proportion of the two given values.
Find the mean proportional of 3/2 and 3/8.
At first glance, you may think that it is not possible, but it is. You ought to establish the proportion with the use of fractions within fractions, and move ahead as you would normally:
Therefore, the two mean proportionals are –3⁄4 and 3⁄4.
Your text or teacher may only be asking for " 3/4 ".
Given a pair of numbers, you ought to be able obtain the ratios. For instance:
There are 16 ducks and 9 geese in a certain park. Express the ratio of ducks to geese in all three formats.
16:9,16⁄9, 16 to 9
Consider the above park. Express the ratio of geese to ducks in all three formats.
16:9,16⁄9, 16 to 9
The numbers were the same in every one of the exercises above, but the order in which they were listed varied according in relation to the order in which the elements of the ratio were expressed. In ratios, order is highly significant.
Let's go back to the 15 men and 20 women in our original group. We had expressed the ratio as a fraction, 15⁄20. This fraction can be reduced to 3⁄4. This means that you can as well express the ratio of men to women as 3/4, 3 : 4, or "3 to 4".
This says something very crucial about ratios: the numbers used in the ratio might not be the absolute measured values. The ratio "15 to 20" refers to the absolute numbers of men and women, in that order, in the group of thirty-five people. The simplified or reduced ratio "3 to 4" specifies to you just that, for every three men, there are four women.
The simplified ratio as well tells you that, in any representative set of seven people (3 + 4 = 7) from this group, three will be men. In other words, the men is made up of e 3/7 of the people in the group. These relationships and reasoning are what you make use of to solve a lot of word problems:
In a particular class, the ratio of passing grades to failing grades is 7 to 5. How many of the 36 students failed the course?
The ratio, "7 to 5" (or 7 : 5 or 7/5), shows that out of every 7 + 5 = 12 students, five failed. That is, 5/12 of the class flunked. Then ( 5/12 )(36) = 15 students failed.
In the park specified above, the ratio of ducks to geese is 16 to 9. How many of the 300 birds are geese?
The ratio shows that, of every 16 + 9 = 25 birds, 9 are geese. That is, 9/25 of the birds are geese. Then there are ( 9/25 )(300) = 108 geese.
In general, ratio problems will merely be a matter of stating ratios or simplifying them. For example:
Express the ratio in simplest form: N10 to N45
This exercise asks that we write the ratio as a reduced fraction:
.10⁄45 = 2⁄9.
This reduced fraction is the ratio's expression in simplest fractional form. Observe that the units (the "dollar" signs) "canceled" on the fraction, since the units, "N", were equivalent on both values. When both values in a ratio have the same unit, there ought to in general be no unit on the reduced form.
Express the ratio in simplest form: 240 miles to 8 gallons
When I simplify, I get (240 miles) / (8 gallons) = (30 miles) / (1 gallon), or, in more common language, 30 miles per gallon.
In contrast to the answer to the earlier exercise, this exercise's answer did require to have units on it, due to the fact that the units on the two parts of the ratio, the "miles" and the "gallons", do not "cancel" with each other.
Conversion factors are simplified ratios, therefore, they may be covered around the same time that you're being taught ratios and proportions. For example, assuming you are asked how many feet long a Nigerian football field is. You are aware that its length is 1000 yards. You would then make use of the relationship of 3 feet to 1 yard, and multiply by 3 to obtain 300 feet.
Ratios and proportions are related mathematical concepts. Ratios compare similar things, such as distances or time. Proportions are two equivalent ratios, like traveling 80 miles in 90 minutes is proportional to traveling 160 miles in three hours. You calculate ratio and proportion by division.
Ratios are the comparison of one thing to another (miles to gallons, feet to yards, ducks to geese, and so on. But their true importance is in the establishment and computation of proportions.
A ratio is one thing compared to or related to another thing; it is merely a statement or an expression. A proportion is two ratios that have been set equal to each other; a proportion is an equation that can be calculated. By saying that a proportion is two ratios that are equal to each other, it is meant in the sense of two fractions that are equal to each other. For example, 5/10 equals 1/2. Computing a proportion means that you are missing one part of one of the fractions, and you are required to solve for that missing value. For example, assuming you were given the following equation:
x⁄10 = 1⁄2
You already know, by merely looking at this equation and comparing the two fractions, that x ought to be 5, but suppose you hadn't observed this. You can solve the equation by multiplying through on either sides by 10 to clear the denominators:
x⁄10 = 1⁄2
10(x⁄10) = 10(1⁄2)
x = 5
Checking what we already knew, we obtain x = 5.
Frequently, students are asked to solve proportions before they've been thought to solve rational equations, which can be a bit of a problem. If you haven't yet learned about rational expressions (that is, polynomial fractions), then you will be required to go over this through the use of "cross-multiplication".
To cross-multiply, you take each denominator ACROSS the "equals to" sign and MULTIPLY it on the other fraction's numerator. The cross-multiplication solution of the above question will look as shown below:
x⁄10 = 1⁄2
2(x)=10(1)
Then you would solve the resulting linear equation by dividing through by 2.
Proportions wouldn't be of much use if you only make use of them for reducing fractions. A more typical use ought to be something like the following:
Consider those ducks and geese we counted back at the park. Their ratio was 16 ducks to 9 geese. Suppose that there are 192 ducks. How many geese are there?
Let "G" stand for the unknown number of geese. Then clearly label the orientation of the ratios, and establish your proportional equation:
ducks⁄geese:16⁄9 = 192⁄G
Multiply through on both sides by the G to get it up to the left-hand side, out of the denominator, and then solve for the value of G:
16⁄9 = 192⁄G
G(sup>16⁄9) = G(192⁄G)
16G⁄9 = 192
9(16G⁄9) = 9(192)
16G=1728
16G⁄16=1728⁄16
G=108
Thus, there are 108 geese.
To solve the proportion above with cross-multiplication, you would do the following:
16⁄9 = 192⁄G
16G=(192)(9)
"Cross-multiplying" is standard language, in that it is frequently used, but it is not technically a mathematical term. You may not see it in your text book, but you will almost certainly hear it in your class or study group.
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