This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Define the sets and the universal set. Let C represent students who take Cocacola. Let F represent students who take Fanta. Let S represent students who take Sprite. The universal set, U, represents the total number of students in the class, which is 50. Step 2: Fill in the known values for the regions of the Venn diagram. • Students who take all three drinks (C F S): 7 • Students who take none of the drinks: 6 Step 3: Calculate the values for the "only two" intersections. • Students who take Cocacola and Fanta (C F) = 16. Students who take Cocacola and Fanta only = (C F) - (C F S) = 16 - 7 = 9. • Students who take Cocacola and Sprite (C S) = 12. Students who take Cocacola and Sprite only = (C S) - (C F S) = 12 - 7 = 5. • Students who take Fanta and Sprite only (F S only) = 8. (This is given directly as "only") Step 4: Fill in the values for the "only one" regions using the given expressions. • Students who take Cocacola only (C only) = m+3. • Students who take Sprite only (S only) = m. Step 5: Calculate the value for the remaining "only one" region (Fanta only). The sum of all regions inside the circles plus those who take none must equal the total number of students. Total students = (C only) + (F only) + (S only) + (C F only) + (C S only) + (F S only) + (C F S) + (None) 50 = (m+3) + (F only) + m + 9 + 5 + 8 + 7 + 6 50 = 2m + (F only) + (3+9+5+8+7+6) 50 = 2m + (F only) + 38 (F only) = 50 - 38 - 2m (F only) = 12 - 2m Step 6: Present the information in a Venn diagram. Draw a rectangle for the universal set. Inside, draw three overlapping circles labeled C (Cocacola), F (Fanta), and S (Sprite). ` +-----------------------------------------------------------------+ | | | | | +-----------------------+ | | | C | | | | | | | | m+3 | | | | \ | | | | \ | | | | 9 | | | | / \ | | | | / \ | | | | / \ | | | | / 7 \ | | | | / \ | | | | / \ | | | | 5 8 | | | | / \ | | | | / \| | | | / \ | | | / \ | | | +-----------------------+---------------------+ | | | S | F | | | | | | | | | m | 12-2m | | | | | | | +-----------------------+---------------------+ | | | | | 6 (None) | | | +-----------------------------------------------------------------+ ` (Note: The diagram above is a textual representation. A proper Venn diagram would show three circles with the numbers in the correct overlapping regions.) Here is a description of the regions for a standard three-circle Venn diagram: • Center (C F S): 7 • C F only: 9 • C S only: 5 • F S only: 8 • C only: m+3 • F only: 12-2m • S only: m • Outside all circles (None): 6 That's 2 down. 3 left today — send the next one.