Physics quiz question: comparing significant figures in arithmetic operations

Physics 205A Quiz 1, fall semester 2017
Cuesta College, San Luis Obispo, CA

Determine the arithmetic expression resulting in the greatest number of significant figures.
(A) (256 ÷ 6.0) + 754.
(B) (256 × 6.0) – 754.
(C) (There is a tie.)
(D) (Not enough information is given.)

Correct answer (highlight to unhide): (A)

For the division and multiplication operations, it is the term with the least number of significant figures (shown in green) that determines the number of significant figures in the result, such that for the first expression:

(256 ÷ 6.0) + ... = 42.666666667 + ....

Similarly for the second expression,

(256 × 6.0) – ... = 1,536 – ....

For the addition and subtraction operations, it is the term with the least number of decimal places that determines the number of significant decimal places in the result, such that for the first expression:

42.666666667 + 754 = 796.666666667,

which because there are no significant decimal places to the right of the decimal point for the first number and no decimal places at all to the right of the decimal point for the second number, this result will have no significant decimal places to the right of the decimal point, and is thus rounded to 797.

For the second expression:

1,536 – 754 = 782,

while there are no decimal places to the right of the decimal point for both numbers, note that the second number has three decimal places to the left of the decimal point that are significant (ones, tens, and hundreds), while for the first number, the ones and the tens decimal place are insignificant. Thus the least significant decimal place is the hundreds place, and the answer must then be truncated to this place.

1,536 – 754 = 782,

because only the hundreds decimal place is significant, this is rounded to 8×10².

(This can also be more clearly seen when written out with the same power of ten in scientific notation, and allows for the simpler rule of truncating to the least number of decimal places to the right of the decimal point:

1.536 – 754 = 1.536×10³ – 0.754×10³ = 0.782×10³,

which would then be rounded to 0.8×10³, with only one significant figure.)

Finally, comparing the numerical result from the first expression (797, which overall has three significant figures) with the numerical result from the second expression (equivalently 8×10² or 0.8×10³ with only one significant figure), evaluating the first expression results in the greater number of significant figures.

Sections 70854, 70855
Exam code: quiz01D4mP
(A) : 12 students
(B) : 12 students
(C) : 33 students
(D) : 0 students

Success level: 21%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.24