tomrocksmaths

Oxbridge Professor takes a 159 year old Maths Exam

Dr Tom takes the New York State Arithmetic Exam from the year 1866

7 comments

  1. 3) In multiplying by more than one figure, where is the first figure in each partial product written, and why is it so written?

    example: 21 x 12 using long multiplication

    21

    x12

    _______

    4 2 First partial product, 2 x 21. The first figure (2) of the first partial product is written in the one’s place (because 2 x 1 = 2)

    2 1— Second partial product 1 x 21 (x 10) = 210. The first figure (1) of the second partial product is written in the ten’s place (because 10 x 1 = 10)

    ___________

    2 6 2 Giving a final product .

    I think the question is focused on the shift to the left by one ten’s place of each successive partial product.

    Bryan

    Austin (Texas, USA) Independent School District

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  2. (3) Where does the first number in each partial product go and why?

    /

    I believe the answer they are looking for is:

    The first digit of the first partial product in long multiplication goes in the one’s place.

    The first digit of the second partial product in long multiplication goes in the ten’s place.

    The first digit of the nth partial product in long multiplication goes in the 10^(n-1) place.

    Example. 12 x 21.

    1 2

    x 2 1

    ____________

    1 2 The first digit (2) of the first partial product goes in the one’s place,

    because 1 x 12 is 12 with the 2 in the one’s place.

    2 4 — The first digit (4) of the second partial product goes in the ten’s place,

    because 20 times 12 is 24_0 with the 4 in the ten’s place.

    ______________

    2 5 2 The total product.

    In general:

    The 1st digit of the nth partial product goes in the 10^{n-1}’s place for a partial product because each partial product is the product of the multiplicand and the nth digit in the multiplier working from right to left from the least significant digit in the multiplier in the long multiplication (boat/ship) algorithm, that is, if we always multiply (from the right) taking successive digits of the multiplier, the digit is in the 1’s, 10’s, 100’s, etc. place so the partial product (working from right to left) starts in the corresponding place.

    Bryan Link Hunt-Murrieta,

    Austin (Texas, USA) Independent School District

    of interest, I was in the last class to compete in the state slide-rule competition and the first class to compete in the calculator competition of the university interscholastic league competition for primary and secondary students.

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  5. (7) cotton does not refer to the fabric, but the weight of the picked and cleaned cotton itself, typically sold in bales of 480-500 lb depending on moisture content.

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  6. (12) rule for multiplication of decimals. Multiply the numerical portions and then shift the decimal to the left by the number of the sum of the digits after the decimal point in each of the two multiplicands.

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