Unit 3 Notes: Rates, Ratios, and Proportions
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| The Golden Ratio - A "Divine" Proportion |
Rates and Unit Rates
- Ratios are a comparison of two values. These can be written in three ways:
- a:b
- a to b
- a/b
- Rates are a special kind of ratio. They are a comparison of two different types of values. These are more common than you think. In fact, you see rates posted almost everywhere you go. Miles per Gallon, Miles per Hour, Price per Pound, Price per Gallon, etc.
- Unit Rates are a special kind of rate that creates a rate, and then simplifies that rate to be a comparison to 1. For example:
If it costs $6.00 for 4 candy bars, we would write the original rate as $6.00/4 candy bars. We would then simplify by dividing both the numerator and denominator by 4, because we want to find the price per 1 item. 6/4 equals 1.5, which means each item costs $1.50. This is a Unit Rate, AND a Unit Cost, or the Cost of 1 item.
Links:
- Multiple Videos and Lessons Related to Rates, Ratios, and Proportions
- Quick Ratio Introduction Notes
- More Videos and Lessons by the Khan Academy
- Unit Rate Simple Notes
- Rates and Ratios Notes
- Unit Rate Notes
Setting Up Proportions
- Proportions are a set of two equivalent fractions. Proportional relationships show the same relationships between the two numerators and denominators, and the same relationship between each numerator and its corresponding denominator. The relationships look like this.
- What you should be able to see is that each fractions parts are connected to each other simply by being part of the same ratio. This makes a lot more sense when we relate it to real world situations, but we should always be able to see that the relationships on each fraction have to be the same, and the relationships between both numerators and denominators have to be the same.
- The relationship between the 3 and the 4 on the fraction on the left is the same as the relationship between the 12 and 16 on the right. Likewise, the relationship between the numerators is the same as the relationship between the denominators.
- If you take each part of the starting fraction and multiply it by 4 you can see it create the new equivalent fraction. This number you multiply by is called the Scale Factor. The most important thing to understand about Scale Factors is that we always have to Multiply! Even if the number gets smaller, we are still looking for a Multiplicative Scale Factor. It is the number that helps us create other parts of this proportion. Scale Factors can go left to right AND up and down. If we look at this proportion, it is hard to see the relationship between each numerator and its matching denominator. We can easily see the Scale Factor of 4, but the Scale Factor of the numerator to it's denominator will take an extra step to find.
- Think about what we've done already, but think about it backwards. What can we do to get from 12 to 3? Of course we can divide by 4. That tells us that we can try to same thing to the 4 to see what we multiplied the 3 by. What that really means is if we want to find the Scale Factor of something, we can Divide the New Value by the Original Value. Since I am changing the 3 into the 4, we should divide 4 by the 3. This ends up equaling 1 1/3. That is our scale factor. If we tried the same thing with 16 and 12 we would get the same answer.
- So now we can see that each numerator is related with its denominator, as well as the matching numerator on the other fraction. Look now at the diagonal lines. Are they related to each other somehow? Can we find a matching scale factor between them? I can see that 4 x 3 is 12, but does that work the same way with 3 and 16? Well, we can find out by dividing 16 by 3. 3 times 5 is 15, so we know it's close to that, but 16 divided by 3 is already not going to be the same. It actually would equal 5 and 1/3.
- If our diagonals are not related, but the rest of the relationships are there, this is a correctly set up proportion. Remember, however, that there are many other ways to use these 4 numbers to correctly set up a proportion. The main goal is to have everything related in those straight lines boxed in with our rectangles, and the diagonals NOT RELATED. This will let us solve many proportion problems soon!
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