Solving a Ratio with and without a DiagramDate: 05/12/2003 at 12:56:28 From: Winona Subject: Ratio In an auditorium, the ratio of the number of girls to the number of boys was 5:9. When 203 girls entered the auditorium, the new ratio of the number of girls to the number of boys became 4:3. How many pupils were in the auditorium at first? How can I solve this without using a diagram? Date: 05/12/2003 at 18:30:13 From: Doctor Ian Subject: Re: Ratio Hi Winona, I'd probably use a diagram, but if you don't want to, you might approach it this way. Suppose the number of boys is B, and the number of girls is G. Then we know that G 5 - = - B 9 After 203 girls enter the auditorium, we have G+203 4 ----- = - B 3 How does this help? Well, from the orignal situation, we know that 5 G = - * B 9 Do you see why? So we can substitute for G in the modified situation: (5/9)B + 203 4 ------------ = - B 3 Can you take it from here? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 05/13/2003 at 09:58:51 From: Winona Subject: Thank you (Ratio) Thanks Dr. Ian, My mom wants to know which way is easier, using a diagram or the way that you showed me? Bye, Winona Date: 05/13/2003 at 10:50:33 From: Doctor Ian Subject: Re: Thank you (Ratio) Hi Winona, It's sort of like asking: Which is easier, tennis or golf? It depends on who you are, and what you find easy. It also depends on what kind of diagram you draw. I didn't draw one, because you told me not to use one; and I didn't see the one that you drew. I like diagrams because they help me keep my information straight, and a good one can help me solve a problem without having to set up and solve equations. But every problem is different, and there's no point in making a diagram unless it's going to be helpful. Let's look at the problem again: In an auditorium, the ratio of the number of girls to the number of boys was 5:9. When 203 girls entered the auditorium, the new ratio of the number of girls to the number of boys became 4:3. How many pupils were in the auditorium at first? The first thing I notice here is that I'd like both the ratios to be 'something:9', since that makes them easier to compare. So I'd change the second ratio from 4:3 to 12:9, which has the same meaning: In an auditorium, the ratio of the number of girls to the number of boys was 5:9. When 203 girls entered the auditorium, the new ratio of the number of girls to the number of boys became 12:9. How many pupils were in the auditorium at first? Now, without doing anything more, I can see that I must have added 7 girls for each group of 9 boys. This means that 203 must be a multiple of 7, and in fact, 203 = 7 * 29. So I must have had 29 groups of 9 boys to begin with, and 29 groups of 5 girls. Would a diagram help? Originally, I can divide the students into groups, where each group has 5 girls and 9 boys: ggggg bbbbbbbbb ggggg bbbbbbbbb . . ggggg bbbbbbbbb Afterward, each group has 12 girls; that is, there are 7 extra girls in each group: gggggGGGGGGG bbbbbbbbb gggggGGGGGGG bbbbbbbbb . . gggggGGGGGGG bbbbbbbbb Now, there are 203 new girls, which is 29 groups of 7 girls: 7 ------- * GGGGGGG | GGGGGGG | . | 29 . | GGGGGGG | So I must have started with 29 of these: ggggg bbbbbbbbb In this case, the diagram didn't add anything. It just took extra time to draw. In other cases, a diagram can make the solution very easy to see without doing any calculations: Changing the Concentration of a Solution http://mathforum.org/library/drmath/view/60742.html But even when diagrams would he useful, some people have a hard time making them, and prefer to use equations whenever possible. There are lots of ways to solve any problem, and which one is 'best' depends on who you are. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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