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Another Tough overlapping problem

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Another Tough overlapping problem

by Mo2men » Sun Jun 18, 2017 12:28 pm
At Mnemosyne Middle School, there are 700 students: all the students are boys or girls in the 4th or 5th grade. There are 320 students in the 4th grade, and there are 210 girls in the 5th grade. Fifty percent of the 5th graders and 40% of the 4th graders take Mandarin Chinese. Ninety 5th grade boys do not take Mandarin Chinese. The number of 4th grade girls taking Mandarin Chinese is less than half of the number of 5th grade girls taking Mandarin Chinese. Which of the following could be the number of 4th grade boys in Mandarin Chinese?

(A) 10

(B) 40

(C) 70

(D) 100

(E) 130

Source: Magoosh
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by DavidG@VeritasPrep » Sun Jun 18, 2017 3:41 pm
Mo2men wrote:At Mnemosyne Middle School, there are 700 students: all the students are boys or girls in the 4th or 5th grade. There are 320 students in the 4th grade, and there are 210 girls in the 5th grade. Fifty percent of the 5th graders and 40% of the 4th graders take Mandarin Chinese. Ninety 5th grade boys do not take Mandarin Chinese. The number of 4th grade girls taking Mandarin Chinese is less than half of the number of 5th grade girls taking Mandarin Chinese. Which of the following could be the number of 4th grade boys in Mandarin Chinese?

(A) 10

(B) 40

(C) 70

(D) 100

(E) 130

Source: Magoosh
Were' given: There are 320 kids in the 4th grade, 40% of whom take Mandarin, so there are .40 * 320 = 128 kids taking Mandarin in 4th grade.

There are 700-320 = 380 kids in the 5th grade. We're told there are are 210 girls, so there are 170 boys. 50% of the class takes Mandarin, so that's 190 kids. We're told that 90 boys don't take Mandarin, so the other 80 boys do take Mandarin, leaving 190 - 80 = 110 girls who take Mandarin.

If the number of girls who take Mandarin in 4th grade is less than half the number who take Mandarin in the fifth grade, then there are fewer than 55 girls taking Mandarin in the 4th grade. (110/2 = 55.) So the number of girls taking Mandarin is between 0 and 54. If 0 girls take Mandarin, all 128 of the kids taking Mandarin in 4th grade would be boys. If 54 girls take Mandarin, there'd be 128 - 54 = 74 boys taking Mandarin. So the number of boys taking Mandarin is between 74 and 128. Only D first the bill.
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by Matt@VeritasPrep » Thu Jun 22, 2017 9:36 pm
A (relatively) quick approach would be to backsolve. We know that at most 128 fourth graders are taking Chinese, so the answer must be < 128.

From there, 10 and 128 seem the most likely answers (min and max of the remaining options). I'd try one, see if it fits the parameters, and if it doesn't, try the other. That gets you there relatively quickly with a pretty good shot of hitting the right choice.
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