The Department of Foreign Languages of a liberal arts college conducted a survey of its recent graduates to determine the foreign language courses they had taken while undergraduates at the college. Of the 530 graduates
205 had at least one year of Spanish.
174 had at least one year of French.
144 had at least one year of German.
40 had at least one year of Spanish and French.
28 had at least one year of Spanish and German.
26 had at least one year of French and German.
8 had at least one year of all three languages.
(a) How many of the graduates had at least 1 yr of at least one of the three languages?
graduates
(b) How many of the graduates had at least 1 yr of exactly one of the three languages?
graduates
(c) How many of the graduates had less than 1 yr of any of the three languages?
graduates
(a) Well, it seems like we have a multilingual graduating class. To calculate the number of graduates who had at least 1 year of at least one of the three languages, we need to add up the number of graduates who had at least 1 year of Spanish, French, and German, and then subtract the overlaps. Let's do the math!
205 graduates had at least 1 year of Spanish.
174 graduates had at least 1 year of French.
144 graduates had at least 1 year of German.
Now, let's subtract the overlaps:
40 graduates had at least 1 year of both Spanish and French.
28 graduates had at least 1 year of both Spanish and German.
26 graduates had at least 1 year of both French and German.
8 graduates had at least 1 year of all three languages.
To calculate the number of graduates who had at least 1 year of at least one language, we can add up the individuals who studied Spanish, French, or German and then subtract the overlaps.
205 + 174 + 144 - (40 + 28 + 26 - 8) = ?
I'll let you do the math on this one! Don't worry, I'm not clowning around when it comes to math.
(b) To determine the number of graduates who had at least 1 year of exactly one of the three languages, we need to subtract the number of graduates who had multiple language studies from the total. In this case, we need to subtract the overlaps from the total number of graduates.
205 + 174 + 144 - 2(40 + 28 + 26) + 8 = ?
Remember to put on your math hat and calculate it!
(c) To find the number of graduates who had less than 1 year of any of the three languages, we subtract the number of graduates who had at least 1 year of any language from the total.
530 - (205 + 174 + 144 - 2(40 + 28 + 26) + 8) = ?
Put on your math clown nose and get that answer!
To solve this problem, we can use the principle of inclusion-exclusion. We'll break down the information given step-by-step to find the answers to each question:
(a) How many of the graduates had at least 1 yr of at least one of the three languages?
To find the number of graduates who had at least one year of at least one of the three languages, we need to add up the number of graduates who took each language individually, then subtract the number of graduates who took combinations of two languages, and finally add back the number of graduates who took all three languages.
Number of graduates who took at least one year of Spanish = 205
Number of graduates who took at least one year of French = 174
Number of graduates who took at least one year of German = 144
By adding these numbers, we get:
205 + 174 + 144 = 523
Now we need to subtract the number of graduates who took combinations of two languages:
Number of graduates who took at least one year of Spanish and French = 40
Number of graduates who took at least one year of Spanish and German = 28
Number of graduates who took at least one year of French and German = 26
By subtracting these numbers, we get:
523 - (40 + 28 + 26) = 429
Finally, we need to add back the number of graduates who took all three languages:
Number of graduates who took at least one year of all three languages = 8
By adding this number, we get the final answer:
429 + 8 = 437
Therefore, 437 of the graduates had at least one year of at least one of the three languages.
(b) How many of the graduates had at least 1 yr of exactly one of the three languages?
To find the number of graduates who had at least one year of exactly one of the three languages, we need to subtract the number of graduates who took combinations of two languages and those who took all three languages from the total number of graduates who took at least one year of any language.
From part (a), we found that 437 graduates had at least one year of at least one of the three languages.
By subtracting the number of graduates who took combinations of two languages and those who took all three languages, we get:
437 - (40 + 28 + 26 + 8) = 335
Therefore, 335 of the graduates had at least one year of exactly one of the three languages.
(c) How many of the graduates had less than 1 yr of any of the three languages?
To find the number of graduates who had less than one year of any of the three languages, we need to subtract the number of graduates who had at least one year of at least one of the three languages from the total number of graduates.
Total number of graduates = 530
From part (a), we found that 437 graduates had at least one year of at least one of the three languages.
By subtracting this number from the total, we get:
530 - 437 = 93
Therefore, 93 of the graduates had less than one year of any of the three languages.
To solve this problem, we can use a method called the Principle of Inclusion-Exclusion. Let's break down the given information and use it to find the answers to each question.
(a) How many of the graduates had at least 1 year of at least one of the three languages?
To find this answer, we need to add up the number of graduates who took at least one year of each language: Spanish, French, and German.
Let's start by finding the total number of graduates who took at least one year of Spanish, French, or German:
205 (Spanish) + 174 (French) + 144 (German)
However, we have counted some students multiple times because they took more than one of the languages. We need to subtract those overlaps:
- 40 (Spanish and French)
- 28 (Spanish and German)
- 26 (French and German)
Now, we have subtracted the overlaps twice, so we need to add back the triple overlap:
+ 8 (Spanish, French, and German)
Adding these up, we have:
205 + 174 + 144 - 40 - 28 - 26 + 8 = 437
Therefore, 437 graduates had at least one year of at least one of the three languages.
(b) How many of the graduates had at least 1 year of exactly one of the three languages?
To find this answer, we need to add up the number of graduates who took exactly one language and subtract the triple overlap because they are counted as taking all three languages.
Let's find the total number of graduates who took exactly one language:
(205 - 40 - 28) + (174 - 40 - 26) + (144 - 28 - 26)
Adding these up, we have:
137 + 108 + 90 = 335
Therefore, 335 graduates had at least one year of exactly one of the three languages.
(c) How many of the graduates had less than 1 year of any of the three languages?
To find this answer, we need to subtract the number of graduates who took at least one year of at least one of the three languages from the total number of graduates.
Total number of graduates = 530
Number of graduates who had at least one year of at least one of the three languages = 437 (from part a)
Number of graduates who had less than 1 year of any of the three languages = Total number of graduates - Number of graduates who had at least one year of at least one of the three languages.
Therefore, 530 - 437 = 93 graduates had less than 1 year of any of the three languages.