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Proportions - Problem 2
MA, Stanford University
Teaching in the San Francisco Bay Area
Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts
A proportion is two equal fractions, or two equal ratios. Since the fractions are equal to each other, the variable must have a value that would make the two fractions equivalent. If one part of the fraction has a variable expression, use cross-multiplication to solve this problem -- multiply the numerator of the first fraction by the denominator of the second and the denominator of the first by the numerator of the second fraction. Remember to distribute properly. Solve for the variable. Check your answer by substituting in the value for the variable into the original proportion.
I know this problem is a proportion because it's two equal fractions. But it's a little bit harder than what I've seen before in proportions before because now I have not just plain old m up there, I have m plus 3 in the numerator. It's like a whole expression. So in order to solve this problem, I'm going to use cross multiplying and the reason I picked cross multiplying is because it's not obvious to me what I'm multiplying 5 by to get 4, like it's not clear to me how I'm getting from one fraction to the other. And I could also use solving techniques where I multiply both sides by 5, that would work also. I want to show you guys how cross multiplying works though because my guess is that it might be your favorite method, it is for most students.
Cross multiplying is where the products of the diagonals are equal to each other. So the product of 4 and m plus 3 is going to be equal to the product of 5 and 12. Just be super careful that you remember to distribute. So often I see students write 4m plus 3 right here. They forget to distribute. That's incorrect, 4 gets multiplied by the m and also by the 3 so make sure you guys are using the distributive property correctly. Once I have this set up let's simplify the left hand side, 4m plus 12, that was distributing, equals 60, subtract 12 from both sides, so now I have 4m equals 48.
The last step is to divide both sides by 4, m is going to be 48 divided by 4 which is 12. This is the cross multiplying method. If you guys wanted to, you might have used solving or even maybe multiplying by whatever number you multiply by to get from 5 to 4 which is 4/5.
I also want to show you before we move on how you could check your work in this situation because you did a lot of Math and you might have made a little mistake. The way you would check your work is by taking this m value that you got for 12, substitute it back in there and make sure you have two equal fractions. Is it true that 12 pus 3, let's see that's 15. Is it true that 15 over 5 is equal 12/4? Yeah because those are both equal to 3 if I reduce them.
By the way, before you started this problem you might have already seen that 12/4 reduces to 3 so you might not have solved this problem using proportion techniques at all, you might have just simplified that to 3, multiplied both sides by 5 and gone about your business like that. But if you use this method you still get the right answer. You can check it using substitution and I think you guys are going to get some awesome proportion problems that you can do.
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