Symbol Sentence B928C7

1. **State the problem:** We need to fill in the blanks in the expression $ (4 \ __ \ 3) \ __ \ (16 \ __ \ 8) \ __ \ 7 = 9 $ using each of the symbols $+, -, \times, \div$ exactly once to make the equation true. 2. **Understand the symbols and order:** The symbols are addition ($+$), subtraction ($-$), multiplication ($\times$), and division ($\div$). Each must be used once. 3. **Try possible placements:** We test combinations respecting the order of operations (multiplication and division before addition and subtraction). 4. **Check one valid solution:** - Let the first blank be $\times$: $$ (4 \times 3) = 12 $$ - Let the second blank be $-$: $$ 12 - (16 \div 8) \ __ \ 7 = 9 $$ - Calculate $16 \div 8$: $$ 16 \div 8 = 2 $$ - Now the expression is: $$ 12 - 2 \ __ \ 7 = 9 $$ - The remaining symbols are $+$ and $\div$, but only one blank remains, so the third blank must be $+$ or $\div$. - Try $\div$: $$ 12 - 2 \div 7 = 9 $$ Calculate $2 \div 7 = \frac{2}{7} \approx 0.2857$: $$ 12 - 0.2857 = 11.7143 \neq 9 $$ - Try $+$: $$ 12 - 2 + 7 = 17 \neq 9 $$ - So try switching the second blank to $+$ and the third to $-$: $$ 12 + 2 - 7 = 7 \neq 9 $$ - Try second blank $\div$ and third blank $-$: $$ 12 \div 2 - 7 = 6 - 7 = -1 \neq 9 $$ - Try first blank $+$: $$ (4 + 3) = 7 $$ - Second blank $\times$: $$ 7 \times (16 \div 8) \ __ \ 7 = 9 $$ Calculate $16 \div 8 = 2$: $$ 7 \times 2 \ __ \ 7 = 9 $$ - Third blank $-$: $$ 14 - 7 = 7 \neq 9 $$ - Third blank $+$: $$ 14 + 7 = 21 \neq 9 $$ - Third blank $\div$: $$ 14 \div 7 = 2 \neq 9 $$ - Try first blank $-$: $$ (4 - 3) = 1 $$ - Second blank $+$: $$ 1 + (16 \div 8) \ __ \ 7 = 9 $$ Calculate $16 \div 8 = 2$: $$ 1 + 2 \ __ \ 7 = 9 $$ - Third blank $\times$: $$ 3 \times 7 = 21 \neq 9 $$ - Third blank $-$: $$ 3 - 7 = -4 \neq 9 $$ - Third blank $+$: $$ 3 + 7 = 10 \neq 9 $$ - Third blank $\div$: $$ 3 \div 7 = 0.4286 \neq 9 $$ - Try first blank $\div$: $$ (4 \div 3) \ __ \ (16 \times 8) \ __ \ 7 = 9 $$ Calculate $4 \div 3 = \frac{4}{3}$ and $16 \times 8 = 128$: $$ \frac{4}{3} \ __ \ 128 \ __ \ 7 = 9 $$ - Second blank $-$ and third blank $\div$: $$ \frac{4}{3} - 128 \div 7 = 9 $$ Calculate $128 \div 7 \approx 18.2857$: $$ \frac{4}{3} - 18.2857 = -16.9524 \neq 9 $$ - Second blank $+$ and third blank $-$: $$ \frac{4}{3} + 128 - 7 = 9 $$ Calculate: $$ \frac{4}{3} + 121 = 9 $$ No. - Second blank $\times$ and third blank $-$: $$ \frac{4}{3} \times 128 - 7 = 9 $$ Calculate: $$ \frac{4}{3} \times 128 = \frac{512}{3} \approx 170.67 $$ $$ 170.67 - 7 = 163.67 \neq 9 $$ - Second blank $-$ and third blank $+$: $$ \frac{4}{3} - 128 + 7 = 9 $$ No. 5. **Try the solution:** $$ (4 \times 3) - (16 \div 8) + 7 = 9 $$ Calculate stepwise: $$ 4 \times 3 = 12 $$ $$ 16 \div 8 = 2 $$ $$ 12 - 2 + 7 = 10 + 7 = 17 \neq 9 $$ No. 6. **Try:** $$ (4 \times 3) \div (16 \div 8) - 7 = 9 $$ Calculate: $$ 4 \times 3 = 12 $$ $$ 16 \div 8 = 2 $$ $$ 12 \div 2 - 7 = 6 - 7 = -1 \neq 9 $$ 7. **Try:** $$ (4 + 3) \times (16 \div 8) - 7 = 9 $$ Calculate: $$ 4 + 3 = 7 $$ $$ 16 \div 8 = 2 $$ $$ 7 \times 2 - 7 = 14 - 7 = 7 \neq 9 $$ 8. **Try:** $$ (4 + 3) - (16 \div 8) \times 7 = 9 $$ Calculate: $$ 4 + 3 = 7 $$ $$ 16 \div 8 = 2 $$ $$ 2 \times 7 = 14 $$ $$ 7 - 14 = -7 \neq 9 $$ 9. **Try:** $$ (4 - 3) + (16 \div 8) \times 7 = 9 $$ Calculate: $$ 4 - 3 = 1 $$ $$ 16 \div 8 = 2 $$ $$ 2 \times 7 = 14 $$ $$ 1 + 14 = 15 \neq 9 $$ 10. **Try:** $$ (4 - 3) \times (16 \div 8) + 7 = 9 $$ Calculate: $$ 4 - 3 = 1 $$ $$ 16 \div 8 = 2 $$ $$ 1 \times 2 + 7 = 2 + 7 = 9 $$ This works! **Final answer:** $$ (4 - 3) \times (16 \div 8) + 7 = 9 $$ Symbols used once: $-$, $\times$, $\div$, $+$.