Everyday Math Demystified, 2nd edition
Stan Gibilisco
Explanations for Quiz Answers in Chapter 9
1. Truncation involves simply "chopping off" all the digits to the right of a certain place. In this case, we want to remove just enough right-hand digits to leave us with four significant figures (that is, four digits in the whole expression, ignoring the position of the decimal point). Choice C gives us the correct expression. Choices A and B won't work, because they go to four decimal places but six significant figures. Choice D is wrong because it shows us the result of rounding; this question asks us to truncate. Again, the correct choice is C.
2. Rounding involves finding the value that gives us the best possible approximation, up to a certain number of significant figures or decimal places. This question asks us to round the number 13.578793 to four decimal places (not significant figures). When we do that, we get 13.5788. The answer is B. Choice A gives us the result that we would derive from truncation, but again, this question specifically asks us to round the value, not truncate it. Choices C and D give us results to four significant figures, but only two decimal places. Again, the correct choice is B.
3. Multiplication always precedes addition. In the expression 3 x 4 + 5 x 6, we should find the products 3 x 4 and 5 x 6 first. When we carry off that task, the expression "boils down" to 12 + 30, which equals 42. The correct choice is C.
4. In this situation, we should take the product 4 x 5 first. When we rewrite the expression with 20 in place of 4 x 5, we get 3 + 20 + 6. These numbers add from left to right to yield 29. The answer is A.
5. To begin, let's multiply the coefficients together. A calculator gives us the product 2.567 x 1.7 = 4.3639. We can "legally" claim only two significant figures here, so we must round this value to 4.4. Next, we add the powers of 10, giving us 10(5+5) = 1010. We can now assemble the final product by combining the new coefficient and the new power of 10 to get 4.4 x 1010. That's choice A.
6. First, we find the quotient of the coefficients, getting 2.567 / 1.7 = 1.51. We can claim only two significant figures, so we must round it to 1.5. Then, because we're finding a quotient, we should subtract the powers of 10. That operation yields 10(5-5) = 100 = 1. The final answer is therefore 1.5 x 1, or simply 1.5. The correct choice is C.
7. The first step in solving this problem involves finding the cube (third power) of the coefficient. When we do that, we get 2.567 x 2.567 x 2.567 = 16.915218263. We're entitled to four significant figures in this case, so we should round this value to 16.92. Next, we must multiply the power of 10 by the "external exponent" of 3, getting 10(5x3) = 1015 as the power of 10. We can now put the coefficient together with the power of 10 to get 16.92 x 1015, but that's not in proper scientific notation! We must move the coefficient's decimal point one place to the left (in effect dividing it by 10). Then, in order to cancel out that effect, we must multiply 1015 by 10, getting 1016. The final result, written in proper scientific notation, is therefore 1.692 x 1016. That's choice D.
8. Without a doubt, this problem is the "messiest" one in our quiz. Before we start, we had better make sure that we know what the -3 power means. When we take any number to a negative integer power, it's the equivalent of taking the reciprocal of that number to the positive power. We start by finding (2.567)-3. That's the same thing as 1 divided by (2.567)3. In the solution to the previous problem, we found that (2.567)3 = 16.915218263. When we use a calculator to find the reciprocal of 16.915218263, we get 0.0591183622 (and some more digits that we needn't worry about because we'll round off to only four significant figures to conclude the process). To find the power of 10 for our "preliminary answer," we multiply 5 x (-3) to get -15. We can now say that the value we seek equals approximately 0.0591183622 x 10-15. However, that's not in good scientific notation! We must move the decimal place two spots to the right in the coefficient, so that the coefficient becomes 5.91183622. Then, because we went to the right in the coefficient, we must reduce the power of 10 by 2, from 10-15 to 10-17. When we round the coefficient down to four significant figures, we get 5.912 x 10-17 as our final answer. The correct choice is B.
9. In this situation, the first quantity, 3.7887x 1078, vastly exceeds all of the other three quantities in the sum. We can use that first value, precisely as it stands (to all five significant figures), as our answer. The correct choice is A.
10. Here, the first two quantities are vastly smaller than the last two. The last two numbers in the sum both go to three significant figures, and they add neatly to give us 366 + 102 = 468. We can therefore take 468 directly as our answer. The correct choice is C.