So many percent more

Updated with an answer... see below

I'm continuing to catch up after vacation, and spotted a good discussion about problems with percent, at MathNotations. (via Let's Play Math blog).

Here's a problem to solve, first of all:

There are 20% more girls than boys in the senior class.
What percent of the seniors are girls?

The answer is NOT that 40% are boys and 60% are girls...

You see, let's say there were 40 boys and 60 girls, 100 students total. If there are 40 boys, then 20% more than that would be 40 + 4 + 4 = 48 girls and not 60!

Try solve it. I'll let you think a little before answering it myself. Don't just rush over to the Mathnotations blog either! Use your thinking caps! I've already given you a big hint!

Update:
You can easily solve this problem by taking any example number for the number of boys. Like I did above, if you have 40 boys, you'd need 48 girls and there'd be 88 students total. What percent of the seniors are girls then? It'd be 48/88 = 0.545454... ≈ 54.55%. And 45.45% are boys.

But the same works even if you choose that there'd be 10 boys, which then means that there are 12 girls, and then the percent of girls in the class is 12/22 * 100% = 54.55%.

NOTE that this problem includes two DIFFERENT "wholes". First of all, it says "There are 20% more girls than boys in the senior class." This is a comparison, and the total number of boys is the "one whole" or the "100%". The number of girls is 20% more, or 120% of the boys.

In algebra terms, if there are p boys, then there will be 1.2p girls.

Then the final question involves a totally different "one whole" or 100%: it asks how many percent of the seniors are girls.

So the group of seniors becomes the 100% or the "whole", and all percent calculations are based on that. Therefore one will then compare the number of girls to the total number of seniors.

In algebra terms, the final answer as a decimal is 1.2p/(p + 1.2p) = 1.2/2.2

See also Denise's post about searching for the 100% in percent problems. Anonymous said…
I asked my kids to try the problem... Our 18 yod, headed to LeTourneau University to major in engineering next week, solved it easily by USING YOUR HINT! She took the 40 boys and the 48 girls in your example and quickly figured out that the girls were 54% of 188. :) Anonymous said…
I tried to post how I solved the problem, but I had trouble with posting. I'm the same "anonymous" that wrote about our 18 yod.

It took me a few efforts at setting up equations-- trial and error is my style! Finally, I came up with x + 1.2x = 100, using x to represent the number of boys in a class of 100.

We are hoping to work on "application" or word problems more this year, as it is a weakness for us. Usually we avoid our weaknesses, but we're trying a different tactic now! Any similar problems you send out will be used in our house, with a 7th grader, a 9th grader doing Alg 1, and an 11th grader doing Alg 2/geometry. Anonymous said…
In Japanese mathematics textbooks, these problems are often represented by what they call segment diagrams. Basically, there are two segments, one for boys and one for girls. The one for the girls is 1/5 longer than the one for the boys since there are 20% (1/5) more girls than boys. Thus, the total relative amount is 11 while the relative amount for girls is 6. So, the percent of the girls in a class can be determined by 6 divided by 11. I think a major part of the reasons that Japanese students outperform US students is that they have such experiences BEFORE they study algebraic equations.

If anyone is interested in looking at a Japanese elementary textbook series, a translated series is available from Global Education Resources (www.globaledresources.com). Anonymous said…
Hi,

Here is how I approched it (using algebra)

boys = x
girls = 20% more than boys so
girls = (x = 20%x)

seniors = boys + girls

To get the percent
=>boys + girls = 100

x+(x+20%x) = 100
2x+(1/5)x = 100
10x + 1x = 500
11x = 500
x = 500/11 = 45.45455

so thats boys: 500/11

girls = 500/11 + (500/11 * 20%)
which comes out to be: 54.545

it was interesting. H M Dasti said…
Hi,

I tried your given percent problem below:

Here is how I approched it (using algebra)

boys = x
girls = 20% more than boys so
girls = (x = 20%x)

seniors = boys + girls

To get the percent
=>boys + girls = 100

x+(x+20%x) = 100
2x+(1/5)x = 100
10x + 1x = 500
11x = 500
x = 500/11 = 45.45455

so thats boys: 500/11

girls = 500/11 + (500/11 * 20%)
which comes out to be: 54.545

it was interesting.