3) In MAPEH competition, 90 medals are awarded in different categories: medals in sports, medals
in arts, medals in music and medals in music, art and sports.
33 got medals in arts
12 got medals in music and arts
45 got medals in sports
18 got medals in sports and music
55 got medals in music
17 got medals in arts and sports
4 got medals in art, music and sports
How many got medals in music only? How many got medals in art only? How many got medals in sports only?
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Answer:
a.) 29 got medals in music only
b.) 8 got medals in arts only
c.) 14 got medals in sports only
Step-by-step explanation:
3-CIRCLE VENN DIAGRAM ILLUSTRATION:
Given:
Total number in each circle:
The total number of medals in the overlapping areas shared by two circles:
Total number in the overlapping areas shared by three circles:
Each set is represented by a circle:
- One circle for arts
- One circle for sports
- One circle for music
To illustrate, draw three overlapping circles to explain how the elements of three sets are related.
POPULATING THE AREAS:
1.) The easiest way to start populating the areas is to write the total number of medals shared by three circles, which is 4. (See attached image #1)
2.) Next, we populate the area shared by two circles:
Formula: (arts and music) – (arts, music, and sports) = (arts and music ONLY)
= 12 – 4 = 8 (arts and music only)
Formula: (sports and music) – (arts, music, and sports) = (sports and music ONLY)
= 18 – 4 = 14 (sports and music only)
Formula: (arts and sports) – (arts, music, and sports) = (arts and sports ONLY)
= 17 – 4 = 13 (arts and sports only)
3.) Lastly, we populate the arts only, sports only, and music only areas.
Formula: (Total number of medals in music) – [(arts, music, and sports) + (arts and music only) + (sports and music only)]
= 55 – (4 + 8 + 14)
= 55 – 26
= 29
Therefore, 29 got medals in music only
Formula: (Total number of medals in arts) – [(arts, music, and sports) + (arts and music only) + (arts and sports only)]
= 33 – (4 + 8 + 13)
= 33 – 25
= 8
Therefore, 8 got medals in arts only
Formula: (Total number of medals in sports) – [(arts, music, and sports) + (sports and music only) + (arts and sports only)
= 45 – (4 + 14 + 13)
= 45 – 31
= 14
Therefore, 14 got medals in sports only.
4.) Adding all the numbers in all the three circles, we’ll get:
14 + 14 + 4 + 13 + 8 + 8 + 29 = 90 medals
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