1. State the problem: There are 70 students in total. - 44 brought a packed lunch. - 13 visited the gift shop. - 21 did neither (no packed lunch, no gift shop). 2. Define variables for the four groups: Let: - $A$ = number who brought a packed lunch. - $B$ = number who visited the gift shop. - $x$ = number who brought a packed lunch and visited the gift shop. - $y$ = number who brought a packed lunch and did not visit the gift shop. - $z$ = number who did not bring a packed lunch but visited the gift shop. - $w$ = number who neither brought a packed lunch nor visited the gift shop. 3. From the problem: - $A = 44$ - $B = 13$ - $w = 21$ - Total $= 70$ 4. Use the total to relate groups: $$x + y + z + w = 70$$ 5. Use the number who brought packed lunch: $$x + y = 44$$ 6. Use the number who visited the gift shop: $$x + z = 13$$ 7. Use the number who did neither: $$w = 21$$ 8. Substitute $w=21$ into the total: $$x + y + z + 21 = 70 \implies x + y + z = 49$$ 9. From step 5, $y = 44 - x$. 10. From step 6, $z = 13 - x$. 11. Substitute $y$ and $z$ into $x + y + z = 49$: $$x + (44 - x) + (13 - x) = 49$$ $$44 + 13 - x = 49$$ $$57 - x = 49$$ $$x = 57 - 49 = 8$$ 12. Now find $y$: $$y = 44 - x = 44 - 8 = 36$$ 13. Find $z$: $$z = 13 - x = 13 - 8 = 5$$ 14. Summary of the frequency tree: - Brought packed lunch and visited gift shop: $x = 8$ - Brought packed lunch and did not visit gift shop: $y = 36$ - Did not bring packed lunch but visited gift shop: $z = 5$ - Neither brought packed lunch nor visited gift shop: $w = 21$ 15. Final answer: **Number of students who brought a packed lunch and did not visit the gift shop is $36$.**