Everyday Math Demystified, 2nd edition Stan Gibilisco Explanations for Quiz Answers in Chapter 3 1. Each of the quantities given in choices A, B, and C differs from 2 by an order of magnitude (power of 10). The quantity 20 equals 10 times 2, the quantity 2/10 equals 1/10 times 2, and the quantity 0.2 is the same as 2/10, which equals 1/10 times 2. The correct choice is D, "All of the above." 2. If we move the decimal point two places to the right in the numeral 23.456, we get 2345.6, which exceeds the original quantity by a factor of 102, or 100. (That's two orders of magnitude, by the way.) The answer is C. 3. If we move the decimal point four places to the left in the numeral 78,003.22, we get 7.800322, which is smaller than the original quantity by a factor of 10,000 or 104. That's four orders of magnitude smaller. The correct choice is B. 4. The quantity 3456 / 10 is one order of magnitude (a factor of 101) smaller than 3456. We should therefore move the decimal point one place to the left. The numeral 3456 has an "implied decimal point" after the digit 6, so when we move it one place to the left, we get 345.6. The correct choice is A. 5. The quantity 3456 / 1000 is three orders of magnitude (a factor of 103) smaller than 3456. We should therefore move the decimal point three places to the left. When we do that, we get 3.456. The answer is C. 6. When we see a fraction whose denominator comprises a finite string of 9s and the numerator is less than the denominator, we know that the fraction converts to an endless repeating decimal expression with 0 to the left of the point, and a repeating sequence of digits to the right of the point. We must make sure that the repeating sequence to the right of the point contains the same number of individual digits as the denominator has. According to that rule, we have 2371 / 9999 = 0.237123712371... The sequence 2371 repeats over and over again, endlessly, to the right of (after) the decimal point. The digit to the left of (before) the decimal point is a cipher. The correct choice is B. 7. The fraction 23 / 9999 has only two digits in the numerator, but has four digits in the denominator. We should change the numerator to 0023 before we make any attempt to convert this fraction to a repeating decimal. Once we've done that, we can use the conversion rule for fractions whose denominators comprise entirely 9s. We have 23 / 9999 = 0023 / 9999 = 0.002300230023... The correct choice is C. 8. This problem illustrates an interesting phenomenon. When we look at the endless repeating decimal expression 0.888..., we can think of it as an endless sequence of 8s, or an endless sequence of 88s, or an endless sequence of 888s. Therefore, the expression can equal any of the three fractions listed here, and infinitely many more! The answer is D, "All of the above." If we think of 0.888... as an endless sequence of single 8s, then 0.888... = 8 / 9 If we think of it as an endless sequence of 88s, then 0.888... = 88 / 99 If we think of it as an endless sequence of 888s, then 0.888... = 888 / 999 We can go on with this extrapolation indefinitely. We might, for example, say that 0.888... = 888,888,888,888 / 999,999,999,999 You can divide all of these fractions out with your extended-digit calculator (such as the one on a personal computer, set to work in the scientific mode), and you'll see that they all produce a decimal expression with 0 to the left of the point and an endless string of 8s to the right. 9. The decimal expression shown here is endless, but it's not strictly repeating. We can identify a pattern of sorts, but no specific sequence of digits that repeats in an identical fashion every time. If we convert each of the three choices A, B, and C given here according to the rules we've learned, we get 1 / 99 = 0.010101... 1 / 9999 = 0.000100010001... 1 / 99,999,999 = 0.000000010000000100000001... None of these equal the quantity under scrutity, which is 0.0100100001000000001... so the correct choice is D, "None of the above." 10. A pure mathematician would consider 24.56 and 24.560 as precisely the same quantity in all respects, so the correct choice is D. A physicist or engineer would see the extra cipher in the second expression as an additional significant figure, indicating greater precision. (If this question had asked for the opinion of a physicist or engineer, both A and C would have been correct. In no event, however, would B hold true!)