1. **Problem Statement:** James has 4 cards with different numbers. Given: - The mean of the first two numbers is 12. - The mean of the next three numbers is 18. - The mean of all four numbers is 17. - One of the numbers is 18. Find the numbers on each card. 2. **Define variables:** Let the four numbers be $a$, $b$, $c$, and $d$. 3. **Write equations from the means:** - Mean of first two numbers: $\frac{a+b}{2} = 12 \implies a+b=24$ - Mean of next three numbers: $\frac{b+c+d}{3} = 18 \implies b+c+d=54$ - Mean of all four numbers: $\frac{a+b+c+d}{4} = 17 \implies a+b+c+d=68$ 4. **Use the equations to find values:** - From $a+b=24$ and $a+b+c+d=68$, subtracting gives $c+d=68-24=44$ - From $b+c+d=54$ and $c+d=44$, subtracting gives $b=54-44=10$ 5. **Known number:** One number is 18. Since $b=10$, $b$ is not 18. - Check if $a=18$: then $a=18$, $b=10$, so $a+b=28$ which contradicts $a+b=24$. - Check if $c=18$: then $c=18$, so $c+d=44 \implies d=26$. - Check if $d=18$: then $d=18$, so $c=44-18=26$. 6. **Verify which fits:** - Using $c=18$, $d=26$, $b=10$, and $a+b=24 \implies a=14$. - Check all means: - First two: $\frac{14+10}{2}=12$ correct. - Next three: $\frac{10+18+26}{3} = \frac{54}{3} = 18$ correct. - All four: $\frac{14+10+18+26}{4} = \frac{68}{4} = 17$ correct. 7. **Final answer:** The numbers on the cards are $14$, $10$, $18$, and $26$. **Summary:** - $a=14$ - $b=10$ - $c=18$ - $d=26$