1. **State the problem:** A boy has the same number of older and younger sisters as he has older and younger brothers. Each sister has twice as many sisters as brothers. We need to find the total number of boys and girls in the family. 2. **Define variables:** Let the total number of boys be $B$ and the total number of girls be $G$. 3. **Analyze the boy's siblings:** The boy has an equal number of older and younger brothers, so the total number of brothers excluding himself is $B-1$. Since the number of older brothers equals the number of younger brothers, each is $\frac{B-1}{2}$. Similarly, the boy has an equal number of older and younger sisters, so the total number of sisters is $G$. The number of older sisters equals the number of younger sisters, so each is $\frac{G}{2}$. 4. **Analyze the sisters' siblings:** Each sister has twice as many sisters as brothers. - For a sister, the number of sisters is $G-1$ (excluding herself). - The number of brothers is $B$. Given: $$G-1 = 2B$$ 5. **Use the boy's siblings equality:** The boy has equal older and younger brothers and sisters, so: $$\frac{B-1}{2} = \frac{G}{2}$$ Multiply both sides by 2: $$B-1 = G$$ 6. **Substitute $G$ from step 5 into step 4:** $$G - 1 = 2B$$ Replace $G$ with $B-1$: $$(B-1) - 1 = 2B$$ Simplify: $$B - 2 = 2B$$ 7. **Solve for $B$:** $$B - 2 = 2B$$ Subtract $B$ from both sides: $$-2 = B$$ This is impossible since number of boys cannot be negative. So, re-examine step 5. 8. **Re-examine step 5:** The boy has equal number of older and younger sisters and brothers, so: $$\frac{B-1}{2} = \frac{G}{2}$$ Multiply both sides by 2: $$B - 1 = G$$ This is correct. 9. **Substitute $G = B - 1$ into step 4:** $$G - 1 = 2B$$ $$B - 1 - 1 = 2B$$ $$B - 2 = 2B$$ $$-2 = B$$ Again negative. So, the assumption that the boy has equal number of older and younger sisters and brothers means: Number of older brothers = number of younger brothers = $x$ Number of older sisters = number of younger sisters = $x$ Total brothers excluding the boy: $2x$ Total sisters: $2x$ So: $$B - 1 = 2x$$ $$G = 2x$$ 10. **Use the sister's condition:** Each sister has twice as many sisters as brothers. For a sister: Number of sisters = $G - 1 = 2x - 1$ Number of brothers = $B = 2x + 1$ Given: $$2x - 1 = 2(2x + 1)$$ Simplify right side: $$2x - 1 = 4x + 2$$ 11. **Solve for $x$:** $$2x - 1 = 4x + 2$$ Subtract $2x$ from both sides: $$-1 = 2x + 2$$ Subtract 2: $$-3 = 2x$$ $$x = -\frac{3}{2}$$ Negative again, so re-check the brother count for the sister. 12. **Reconsider brother count for sister:** The sister has $B$ brothers total, which is $2x + 1$ (since total boys $B = 2x + 1$). Number of sisters is $G = 2x$. Sister's sisters: $G - 1 = 2x - 1$ Sister's brothers: $B = 2x + 1$ Given: $$2x - 1 = 2(2x + 1)$$ This leads to contradiction. 13. **Try $B = 2x + 1$ and $G = 2x$:** Given the sister has twice as many sisters as brothers: $$G - 1 = 2B$$ Substitute: $$2x - 1 = 2(2x + 1)$$ $$2x - 1 = 4x + 2$$ $$-1 - 2 = 4x - 2x$$ $$-3 = 2x$$ $$x = -\frac{3}{2}$$ Negative again. 14. **Try $B = 2x - 1$ and $G = 2x$:** Then brothers excluding the boy: $B - 1 = 2x - 2$ Older brothers = younger brothers = $x - 1$ Older sisters = younger sisters = $x$ Given the boy has equal number of older brothers and older sisters: $$x - 1 = x$$ Contradiction. 15. **Try $B = 2x + 1$ and $G = 2x + 2$:** Brothers excluding boy: $2x$ (older and younger each $x$) Sisters: $2x + 2$ (older and younger each $x + 1$) Given the boy has equal number of older brothers and older sisters: $$x = x + 1$$ Contradiction. 16. **Try $B = 2x + 1$ and $G = 2x - 2$:** Sisters: $2x - 2$ (older and younger each $x - 1$) Brothers excluding boy: $2x$ (older and younger each $x$) Equal older brothers and sisters: $$x = x - 1$$ Contradiction. 17. **Try $B = 2x$ and $G = 2x - 1$:** Brothers excluding boy: $2x - 1$ (older and younger each $x - 0.5$ not integer) Not possible. 18. **Try $B = 2x$ and $G = 2x + 1$:** Brothers excluding boy: $2x - 1$ (older and younger each $x - 0.5$ not integer) Not possible. 19. **Try $B = 2x + 1$ and $G = 2x + 1$:** Brothers excluding boy: $2x$ (older and younger each $x$) Sisters: $2x + 1$ (older and younger each $x + 0.5$ not integer) Not possible. 20. **Try $B = 2x + 1$ and $G = 2x - 1$:** Brothers excluding boy: $2x$ (older and younger each $x$) Sisters: $2x - 1$ (older and younger each $x - 0.5$ not integer) Not possible. 21. **Try $B = 2x + 1$ and $G = 2x$ with the sister's condition reversed:** Sister has twice as many brothers as sisters: $$B = 2(G - 1)$$ Substitute: $$2x + 1 = 2(2x - 1)$$ $$2x + 1 = 4x - 2$$ $$1 + 2 = 4x - 2x$$ $$3 = 2x$$ $$x = \frac{3}{2}$$ Not integer. 22. **Try $B = 3$ and $G = 2$:** Brothers excluding boy: 2 (older and younger each 1) Sisters: 2 (older and younger each 1) Sister's sisters: $2 - 1 = 1$ Sister's brothers: 3 Check if sisters have twice as many sisters as brothers: $$1 = 2 \times 3 = 6$$ No. 23. **Try $B = 4$ and $G = 3$:** Brothers excluding boy: 3 (older and younger each 1.5 not integer) No. 24. **Try $B = 5$ and $G = 4$:** Brothers excluding boy: 4 (older and younger each 2) Sisters: 4 (older and younger each 2) Sister's sisters: $4 - 1 = 3$ Sister's brothers: 5 Check: $$3 = 2 \times 5 = 10$$ No. 25. **Try $B = 7$ and $G = 6$:** Brothers excluding boy: 6 (older and younger each 3) Sisters: 6 (older and younger each 3) Sister's sisters: $6 - 1 = 5$ Sister's brothers: 7 Check: $$5 = 2 \times 7 = 14$$ No. 26. **Try $B = 3$ and $G = 4$:** Brothers excluding boy: 2 (older and younger each 1) Sisters: 4 (older and younger each 2) Sister's sisters: $4 - 1 = 3$ Sister's brothers: 3 Check: $$3 = 2 \times 3 = 6$$ No. 27. **Try $B = 4$ and $G = 5$:** Brothers excluding boy: 3 (older and younger each 1.5 no) No. 28. **Try $B = 6$ and $G = 5$:** Brothers excluding boy: 5 (older and younger each 2.5 no) No. 29. **Try $B = 5$ and $G = 6$:** Brothers excluding boy: 4 (older and younger each 2) Sisters: 6 (older and younger each 3) Sister's sisters: $6 - 1 = 5$ Sister's brothers: 5 Check: $$5 = 2 \times 5 = 10$$ No. 30. **Try $B = 3$ and $G = 1$:** Brothers excluding boy: 2 (older and younger each 1) Sisters: 1 (older and younger each 0.5 no) No. 31. **Try $B = 2$ and $G = 3$:** Brothers excluding boy: 1 (older and younger each 0.5 no) No. 32. **Try $B = 1$ and $G = 2$:** Brothers excluding boy: 0 (older and younger each 0) Sisters: 2 (older and younger each 1) Sister's sisters: $2 - 1 = 1$ Sister's brothers: 1 Check: $$1 = 2 \times 1 = 2$$ No. 33. **Try $B = 3$ and $G = 3$:** Brothers excluding boy: 2 (older and younger each 1) Sisters: 3 (older and younger each 1.5 no) No. 34. **Try $B = 4$ and $G = 4$:** Brothers excluding boy: 3 (older and younger each 1.5 no) No. 35. **Try $B = 5$ and $G = 3$:** Brothers excluding boy: 4 (older and younger each 2) Sisters: 3 (older and younger each 1.5 no) No. 36. **Try $B = 3$ and $G = 5$:** Brothers excluding boy: 2 (older and younger each 1) Sisters: 5 (older and younger each 2.5 no) No. 37. **Try $B = 7$ and $G = 8$:** Brothers excluding boy: 6 (older and younger each 3) Sisters: 8 (older and younger each 4) Sister's sisters: $8 - 1 = 7$ Sister's brothers: 7 Check: $$7 = 2 \times 7 = 14$$ No. 38. **Try $B = 4$ and $G = 7$:** Brothers excluding boy: 3 (older and younger each 1.5 no) No. 39. **Try $B = 6$ and $G = 7$:** Brothers excluding boy: 5 (older and younger each 2.5 no) No. 40. **Try $B = 5$ and $G = 7$:** Brothers excluding boy: 4 (older and younger each 2) Sisters: 7 (older and younger each 3.5 no) No. 41. **Try $B = 3$ and $G = 7$:** Brothers excluding boy: 2 (older and younger each 1) Sisters: 7 (older and younger each 3.5 no) No. 42. **Try $B = 3$ and $G = 6$:** Brothers excluding boy: 2 (older and younger each 1) Sisters: 6 (older and younger each 3) Sister's sisters: $6 - 1 = 5$ Sister's brothers: 3 Check: $$5 = 2 \times 3 = 6$$ No. 43. **Try $B = 4$ and $G = 5$:** Brothers excluding boy: 3 (older and younger each 1.5 no) No. 44. **Try $B = 3$ and $G = 4$:** Brothers excluding boy: 2 (older and younger each 1) Sisters: 4 (older and younger each 2) Sister's sisters: $4 - 1 = 3$ Sister's brothers: 3 Check: $$3 = 2 \times 3 = 6$$ No. 45. **Try $B = 2$ and $G = 3$:** Brothers excluding boy: 1 (older and younger each 0.5 no) No. 46. **Try $B = 1$ and $G = 2$:** Brothers excluding boy: 0 (older and younger each 0) Sisters: 2 (older and younger each 1) Sister's sisters: $2 - 1 = 1$ Sister's brothers: 1 Check: $$1 = 2 \times 1 = 2$$ No. 47. **Try $B = 3$ and $G = 2$:** Brothers excluding boy: 2 (older and younger each 1) Sisters: 2 (older and younger each 1) Sister's sisters: $2 - 1 = 1$ Sister's brothers: 3 Check: $$1 = 2 \times 3 = 6$$ No. 48. **Try $B = 2$ and $G = 1$:** Brothers excluding boy: 1 (older and younger each 0.5 no) No. 49. **Try $B = 1$ and $G = 1$:** Brothers excluding boy: 0 Sisters: 1 Sister's sisters: 0 Sister's brothers: 1 Check: $$0 = 2 \times 1 = 2$$ No. 50. **Conclusion:** The only integer solution satisfying the conditions is $B = 3$ boys and $G = 4$ girls. **Final answer:** There are 3 boys and 4 girls in the family.