A ratio is one thing or value compared with or related to another thing or value; it is just a statement or an expression, and can only perhaps be simplified or reduced.
On the other hand, a proportion is two ratios which have been set equal to each other; a proportion is an equation that can be solved.
When I say that a proportion is two ratios that are equal to each other, I mean this in the sense of two fractions being equal to each other. For instance, equals .
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Solving a proportion means that we have been given an equation containing two fractions which have been set equal to each other, and we are missing one part of one of the fractions; we then need to solve for that one missing value. For instance, suppose we are given the following equation:
We already know, by just looking at this equation and comparing the two fractions, that x must be equal to 5, but let's suppose for the moment that we hadn't noticed this. We can solve the given equation by multiplying through on both sides by 10 (or, if one prefers, ) to clear the denominators:
Verifying what we already knew, we get a solution of x = 5.
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Often times, students are asked to solve proportions before they've learned how to solve rational equations, which can be a bit of a problem. If one hasn't yet learned about rational expressions (that is, polynomial fractions), then it will be necessary to "get by" with "cross-multiplication".
To cross-multiply, we start with an equation in which two fractions are set equal to each other. Then we take each denominator and move it aCROSS the "equals" sign and then MULTIPLY it against the other fraction's numerator. The cross-multiplication solution of the above exercise looks like this:
Then we would solve the resulting linear equation by dividing through by 2 to again arrive at x = 5.
Note the process in the above. We multiplied the left-hand side's denominator by the right-hand side's numerator, and then divided by the right-hand side's denominator. You may see this process explicitly applied for the solving of proportions. The method of solution would then by to cross-multiply the numbers (that is, in the direction that does not involve the variable), and then divide by the remaining number. In very informal notation, the process looks like this:
The green arrow pointing northeast (that is, from bottom left to upper right) indicates the multiplication step; the purple looping arrow that ends up pointing at the variable indicates the division step.
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The variable in this proportion is in the denominator of the right-hand side's fraction, but that's okay. I can still cross-multiply and solve.
The solution of the proportion is the value of the variable, so my answer is:
y = 39
If I'd done the shorthand method (shown with the green and purple arrows above), the computations would have been:
(13 × 18) ÷ 6 = 39 = y
It's harder to "show your work" using the shorthand method, but the shorthand method is easier to plug into your calculator. Use whatever method works well for you.
I'll cross-multiply, and then divide:
Hm... Can proportions have fractional solutions? Yes, definitely; they can! I mustn't let the only-whole-number exercises and examples mislead me into thinking that proportions must always have whole-number answers. They don't. My fractional answer is perfectly fine.
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Proportions wouldn't be of much use if you only used them for reducing fractions. A more typical use would be something like the following:
They've asked me to solve for an unknown value, so I'll need an equation with a variable. They've given me a ratio, so my equation will be a proportion.
I'll let "G" stand for the unknown number of geese. I'll clearly label the orientation of my ratios (so I don't confuse which number stands for what), and then I'll set up my proportional equation:
I'll cross-multiply to solve for the value of G:
16G = 1728
G = 108
I labelled things clearly at the beginning, so I know that "G" stands for "the number of geese in the park". So my answer is:
108 geese
"Cross-multiplying" is standard classroom language, in that it is very commonly used by students and instructors, but it is not technically a mathematical term. You might not see "cross-multiplication" mentioned in your textbook, but you will almost certainly hear it in your class or study group.
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Notice how, when I was setting up my equation at the beginning of my solution above, I prefaced my proportion by writing out my ratio in words; namely:
This is not standard notation (in the sense of your textbook being likely to use it), but it can be very helpful for setting up proportions. By clearly labelling which values are represented by the numerators and denominators, respectively, you will help yourself keep track of what each number stands for; you won't mix up which number or unit goes where.
In other words, using this method will help you set up your proportions correctly. If you do not set up the ratios consistently (for instance, if, in the above example, I'd mixed up where the values for the "ducks" and the "geese" were supposed to go in the various fractions), I'd have gotten an incorrect answer. Clarity in your set-up is crucially important when working with proportions. We will return to this subject later.
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