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 550 graduates
210 had at least one year of Spanish.
179 had at least one year of French.
140 had at least one year of German.
41 had at least one year of Spanish and French.
31 had at least one year of Spanish and German.
28 had at least one year of French and German.
9 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?
(b) How many of the graduates had at least 1 yr of exactly one of the three languages?
(c) How many of the graduates had less than 1 yr of any of the three languages?
Draw your Venn diagram and place your numbers. That is always a great start : )
To find the answers to the questions, we can use a method called Venn diagrams. Venn diagrams are graphical representations that can help organize and calculate information about sets and their relationships.
First, let's draw a Venn diagram with three intersecting circles to represent the three languages: Spanish, French, and German.
(a) To find the number of graduates who had at least 1 year of at least one of the three languages, we need to find the total number of graduates in the three circles combined.
Starting with the individual circles, we have:
- The number of graduates who had at least 1 year of Spanish (210)
- The number of graduates who had at least 1 year of French (179)
- The number of graduates who had at least 1 year of German (140)
Next, we need to consider the overlapping regions:
- The number of graduates who had at least 1 year of Spanish and French (41)
- The number of graduates who had at least 1 year of Spanish and German (31)
- The number of graduates who had at least 1 year of French and German (28)
Finally, we have the center region where all three languages overlap. This represents the number of graduates who had at least 1 year of all three languages (9).
To calculate the total number of graduates with at least 1 year of at least one of the three languages, we add up the numbers inside the circles and subtract the repetitions in the overlapping regions:
210 + 179 + 140 - 41 - 31 - 28 + 9 = 438 graduates.
Therefore, (a) 438 graduates had at least 1 year of at least one of the three languages.
(b) To find the number of graduates who had at least 1 year of exactly one of the three languages, we need to sum the numbers in the individual circles and subtract the repetitions in the overlapping regions, including the center region:
(210 - 41 - 31 + 9) + (179 - 41 - 28 + 9) + (140 - 31 - 28 + 9) = 127 + 119 + 90 = 336 graduates.
Therefore, (b) 336 graduates had at least 1 year of exactly one of the three languages.
(c) To find the number of graduates who had less than 1 year of any of the three languages, we need to subtract the total number of graduates with at least 1 year of at least one of the three languages from the total number of graduates (550):
550 - 438 = 112 graduates.
Therefore, (c) 112 graduates had less than 1 year of any of the three languages.