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Hint: We have to find the number of fruits in each box so that all the apples and oranges can be totally packed. Therefore, the minimum number of the required boxes should be a common factor of 96 and 112. However, as the minimum number of boxes required is to be found, we have to maximize the number of fruits in each box and therefore, we have to take the HCF of 96 and 112 as the number of fruits in each box. Thereafter we can count the number of boxes required for apples and oranges by dividing the total number of apples and oranges by the number of fruits in each box respectively.

Complete step-by-step answer:

As the boxes should contain the same variety of fruits, and all the fruits should be packed, the number of fruits in one box should be such that a multiple of that number should be equal to the total number of apples and another multiple of that should be equal to the total number of apples. Therefore, the number of fruits in a box should be a factor of 96 and 112.

Also, as we have to find the minimum number of boxes, the number of fruits in each box should be maximum. Thus, the number of fruits in each box should be equal to the highest common factor (H.C.F.) of 96 and 112.

We find that the prime factorization of 96 can be written as $96=2\times 2\times 2\times 2\times 2\times 3$

The prime factorization of 112 can be written as $112=2\times 2\times 2\times 2\times 7$

Thus the HCF of 112 and 96 is $2\times 2\times 2\times 2=16$.

Thus, the number of fruits in each box should be 16.

Then, the number of boxes required for the apples is $\dfrac{96}{16}=6$

And the number of boxes required for the oranges is $\dfrac{112}{16}=7$

Thus the minimum number of total boxes required to fill the fruits should be $6+7=13$

Note: We should be careful to take the HCF of 96 and 112 and not LCM as the fruits are being kept in the boxes. Therefore, the number of fruits in each box should be less than the total number of fruits. Also, we note that as we have taken the number of fruits in each box as the HCF of the total number of apples and oranges, the number of boxes required for apples and oranges came out as an integer.

Complete step-by-step answer:

As the boxes should contain the same variety of fruits, and all the fruits should be packed, the number of fruits in one box should be such that a multiple of that number should be equal to the total number of apples and another multiple of that should be equal to the total number of apples. Therefore, the number of fruits in a box should be a factor of 96 and 112.

Also, as we have to find the minimum number of boxes, the number of fruits in each box should be maximum. Thus, the number of fruits in each box should be equal to the highest common factor (H.C.F.) of 96 and 112.

We find that the prime factorization of 96 can be written as $96=2\times 2\times 2\times 2\times 2\times 3$

The prime factorization of 112 can be written as $112=2\times 2\times 2\times 2\times 7$

Thus the HCF of 112 and 96 is $2\times 2\times 2\times 2=16$.

Thus, the number of fruits in each box should be 16.

Then, the number of boxes required for the apples is $\dfrac{96}{16}=6$

And the number of boxes required for the oranges is $\dfrac{112}{16}=7$

Thus the minimum number of total boxes required to fill the fruits should be $6+7=13$

Note: We should be careful to take the HCF of 96 and 112 and not LCM as the fruits are being kept in the boxes. Therefore, the number of fruits in each box should be less than the total number of fruits. Also, we note that as we have taken the number of fruits in each box as the HCF of the total number of apples and oranges, the number of boxes required for apples and oranges came out as an integer.

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