Everyday Math Demystified, 2nd edition
Stan Gibilisco
Explanations for Quiz Answers in Chapter 1
1. If you arrange a lot of objects into a square array measuring a hundred wide by a hundred deep by one tall, then you'll have 100 x 100, or 10,000 objects. The correct choice is D. This fact holds true in the decimal numeration system (base ten). Here's a bit of "food for thought": Does this same notion work in other bases? For example, is it true that 100 x 100 = 10,000 in the octal system, or in the hexadecimal system, or in the binary system? Try these "extra credit" exercises and see for yourself! (Here's a hint: Convert all values to base-ten numerals, then multiply those, and then convert them back to expressions in the original base.)
2. If you arrange a lot of objects into a cubical array measuring a hundred wide by a hundred deep by a hundred tall, then you'll end up with a total of 100 x 100 x 100, or 1,000,000 objects. The correct choice is B. For "extra credit," you can figure out whether or not this same principle works in the binary, octal, or hexadecimal systems.
3. The Roman numeral M represents the decimal 1000, the Roman C represents the decimal 100, the Roman X represents the decimal 10, and the Roman I represents the decimal 1. When we write these four Roman symbols one after the other as MCXI, we get the decimal quantity 1000 + 100 + 10 + 1, or 1111. The correct choice is A.
4. In an octal numeral, the digit farthest to the right represents a multiple of the decimal quantity 1. The next digit to the left represents a multiple of the decimal quantity 8. To the left of that, we have the multiple of the decimal quantity 8 x 8, or 64. To the left of that, we have the multiple of the decimal quantity 8 x 8 x 8, or 512. In the octal numeral 5103, the digit 5 occupies the place for the multiple of the decimal quantity 512, so it represents 5 x 512, or 2560, in decimal terms. The correct choice is A.
5. We represent the decimal quantity 50 as the Roman numeral L, and we represent the decimal quantity 1 as the Roman numeral I. In order to denote the decimal numeral 52 in Roman form, we write LII, which stands for 50 + 1 + 1. The correct choice is C.
6. In a decimal numeral, the digit farthest to the right indicates a multiple of 1, and the next digit to the left tells us a multiple of 10. To the left of that, we find a digit that gives us a multiple of 100. In the decimal numeral 335,427, the digit 4 lies in the spot that indicates a multiple of 100, so it represents 4 x 100, or 400. The answer is B.
7. In the octal system, we have the digits 0, 1, 2, 3, 4, 5, 6, and 7 only. After 7, we encounter the octal quantity 10. When we see the digit 7 at the extreme right-hand end of an octal numeral and we want to increase the value by 1, we must replace the 7 with a digit 0, and then increase the digit immediately to its left by 1. According to this rule, the octal 70 follows the octal 67. Arithmetically, 67 + 1 = 70 in octal terms. The correct choice is A.

8. Let's convert the Roman numeral to a decimal numeral, and then convert the decimal numeral to a binary numeral. The Roman symbol V represents the decimal quantity 5, and the Roman symbol I represents the decimal quantity 1. When we see the Roman numeral VIII, we can add the decimal values to get 5 + 1 + 1 + 1 = 8. In the binary system, the rightmost digit tells us a multiple of the decimal quantity 1; the next digit to the left gives us a multiple of the decimal quantity 2; going one more place to the left, we encounter a multiple of the decimal quantity 4; after that we get a multiple of the decimal quantity 8. When we write the binary numeral 1000, we get a decimal value of 8 + 0 + 0 + 0, which equals 8. The answer is B.

9. The binary numeral 101010101 has nine digits. Let's track the decimal-equivalent place values from right to left:

  • The rightmost digit is a multiple of decimal 1
  • The second-rightmost digit is a multiple of the decimal 2
  • The third-rightmost digit is a multiple of the decimal 4
  • The fourth-rightmost digit is a multiple of the decimal 8
  • The fifth-rightmost digit is a multiple of the decimal 16
  • The sixth-rightmost digit is a multiple of the decimal 32
  • The seventh-rightmost digit is a multiple of the decimal 64
  • The eighth-rightmost digit is a multiple of the decimal 128
  • The ninth-rightmost digit (or the leftmost one; that's the one we want) is a multiple of the decimal 256

We can see that the leftmost digit is 1, so we know that it represents the decimal quantity 1 x 256, or 256. The answer is C.
10. The Roman numeration system has no symbol for zero. The correct choice is D.