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Using four different digits, form the subtraction 5A - 4B to make the answer as small as possible and as large as possible.

Mathematics

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Step-by-step solution

4 steps

Here are the solutions for both parts of the problem.

  1. Determine the digits for the smallest answer.

    To make the subtraction (50+A)−(40+B)=10+A−B(50+A) - (40+B) = 10 + A - B as small as possible, we need to choose the smallest possible digit for AA and the largest possible digit for BB. The digits 5 and 4 are already used, so we cannot use them for AA or BB. The available digits from 1 to 9 (excluding 4 and 5) are {1,2,3,6,7,8,9}\{1, 2, 3, 6, 7, 8, 9\}.

    • Smallest AA: Choose A=1A=1.
    • Largest BB: Choose B=9B=9. The four different digits used are 5,1,4,95, 1, 4, 9.
  2. Calculate the smallest answer.

    Substitute A=1A=1 and B=9B=9 into the subtraction:

    51−49\begin{array}{c} 5 \quad 1 \\ - 4 \quad 9 \\ \hline \end{array} 51−49=251 - 49 = 2

    The completed subtraction for the smallest answer is:

    51−4902\boxed{ \begin{array}{c} 5 \quad 1 \\ - 4 \quad 9 \\ \hline 0 \quad 2 \end{array} }
  3. Determine the digits for the largest answer.

    To make the subtraction (50+A)−(40+B)=10+A−B(50+A) - (40+B) = 10 + A - B as large as possible, we need to choose the largest possible digit for AA and the smallest possible digit for BB. Again, the digits 5 and 4 are already used. The available digits from 1 to 9 (excluding 4 and 5) are {1,2,3,6,7,8,9}\{1, 2, 3, 6, 7, 8, 9\}.

    • Largest AA: Choose A=9A=9.
    • Smallest BB: Choose B=1B=1. The four different digits used are 5,9,4,15, 9, 4, 1.
  4. Calculate the largest answer.

    Substitute A=9A=9 and B=1B=1 into the subtraction:

    59−41\begin{array}{c} 5 \quad 9 \\ - 4 \quad 1 \\ \hline \end{array} 59−41=1859 - 41 = 18

    The completed subtraction for the largest answer is:

    59−4118\boxed{ \begin{array}{c} 5 \quad 9 \\ - 4 \quad 1 \\ \hline 1 \quad 8 \end{array} }
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