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Lesson 128: Solving Problems Using Ratios, Rates and Proportions
In this lesson we'll learn more about using ratios and proportions to solve problems.
Before starting this lesson, make sure that you review Lesson 88 in which we introduce the concept of ratios and proportions.
Recall that a ratio is like a special type of fraction. It is a comparison of two numbers. Here is a typical ratio problem: In a jar, the ratio of red marbles to blue marbles is 5:3. There are 112 marbles in the jar. How many marbles of each color are there? What we know is that for ever 5 red marbles, there will be 3 blue marbles. If there are 10 red marbles, there will be 6 blue ones, and so on.
To solve this problem, we set up an equation like this: 5x + 3x = 112. We don't know what x is, but this equation is set up so that whatever value we choose for x, the red to blue ratio will still be 5:3. Once we find the value of x, we'll know how many marbles there are of each color. Combining we get 8x = 112, and dividing by 8 we get x = 14. We're not done yet. The 5x represents the number of blue marbles. We have to multiply 5 times 14 to get 70. There are 70 blue marbles. We can multiply 3 times 14 to see that there are 42 red marbles. As a check, we can add 70 + 42 to get 112 marbles.
Here is another typical problem. The angle measurements of a triangle are in the ratio 3:4:5. How many degrees are in the largest angle? We proceed the same as we did above, except this time, we have three values in ratio. We can set up our equation with 3x + 4x + 5x. No matter what x is, we are maintaining our ratio. What do we set the expression equal to? Recall that the sum of the angles in any triangle is 180 degrees. Therefore we'll write 3x + 4x + 5x = 180. Combining we get 12x = 180, and dividing we get x = 15.
We're not done. We need to determine the values of 3x, 4x, and 5x, to get the angle measurements. Substituting, we get 45, 60, and 75 degrees. As a check, we can see that these angles sum to 180 degrees. We're still not done. We have to answer exactly what was asked. The problem asked for the measure of the largest angle. The answer is 75 degrees. If you're careless and give all three angles in your answer, you will likely be marked wrong.
Recall that a proportion is an equation of two ratios. In Lesson 88, we saw that to solve a proportion problem that has a variable, just cross multiply and set the products equal to each other. Then use algebra to solve. For example: 5/7 = x/21. Cross multiply numerator of the first fraction times denominator of the second fraction to get 21 times 5 = 105. Cross multiply the other way to get 7x. Set them equal: 7x = 105. Divide to get x = 15. As always, check your answer.
Some proportion problems are more complicated, but just follow the same procedure. For example: (x - 2) / -2 = (x - 12) / 3. To solve this, just cross multiply like we did above. You'll need to use the distributive property to distribute the constants over the binomials. Be very careful with your signs, and with your rules of multiplying signed numbers. Try this one on your own, and contact me if you get stuck.
Recall that similar figures are in proportion. Remember that similar means "same shape, different size." You can use this information to solve proportion problems involving diagrams of similar shapes. The sides that are corresponding (i.e., in the same relative position in the shape) will be in proportion. In problems like this, just set up a proportion like we've worked with above, and cross multiply to find the unknown values.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.