1.2.1: Preparation N.2

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Below is a table that describes some large numbers in words, standard decimal notation, and scientific notation. Notice how scientific notation keeps track of the number of decimal places to the right of the first digit but before the decimal point. Keeping track of the number of decimal places in quantities can be useful for some estimation problems. Recall that if a number does not have a decimal point displayed, then the decimal point is assumed to be at the right end of the number.

Quantity in Words	Decimal Notation	Scientific Notation	Number of places the decimal point was moved
One hundred twenty-two and one tenth	122.1	1.221 × 10²	2
One thousand forty-five	1,045	1.045 × 10³	3
Three million, four hundred seventy-five thousand, three hundred twenty and seventy-one hundredths	3,475,320.71	3.47532071 × 10⁶	6
One billion, four hundred seventy-two million, three hundred forty-one thousand, nine hundred eighty-two and three tenths	1,472,341,982.3	1.4723419823 × 10⁹	9
One Trillion	1,000,000,000,000	1 × 10¹²	12

How to Round Numbers

To estimate with large numbers, we often round to a convenient place and then use that value in subsequent calculations. Rounding to a certain place is done in two steps:

First, we need to identify the place value to round;
Then, If the number to the immediate right of the place value you wish to round to is 5 or higher, you increase the digit you are rounding to by one and change all of the numbers to the right of it to zeroes. If the number to the immediate right of the place value you wish to round to is 4 or lower, you simply change to zeroes all of the numbers to the right of the place value to which you are rounding.

For example,

12,650 rounded to the nearest thousand is 13,000
12,499 rounded to the nearest thousand is 12,000
65,452,120 rounded to the nearest million is 65,000,000
12,811,120 rounded to the nearest million is 13,000,000

Tens and Hundreds of Thousands/Millions/Billions/Trillions

When writing numbers using words it is often helpful to look at numbers as products of powers of 10 listed above: thousands, millions, billions, trillions.

For example,

65,000 can be written as 65 * 1,000 or simply 65 thousand
95,000 can be written as 95 * 1,000 or simply 95 thousand
12,000 can be written as 12 * 1,000 or simply 12 thousand

The three numbers listed above are in tens of thousands.

Similarly,

850,000,000 can be written as 850 * 1,000,000 or simply 850 million
122,000,000 can be written as 122 * 1,000,000 or simply 122 million
650,000,000 can be written as 650 * 1,000,000 or simply 650 million

All three numbers listed above are in hundreds of millions.

For example, suppose we wanted a quick estimate of the yearly total expenses for a project in China that is expected to cost approximately $10 for every person in the country per day. By using estimates, we may be able to perform the calculations by hand reasonably quickly, as follows, to obtain a rough answer. We could estimate the population of China as 1.5 billion, use the ten dollars per person, and estimate the number of days in a year as 350. The product of these three numbers is:

350 × 10 × 1.5 × 10⁹

If we leave off the 10⁹ we get:

= 350 × 15

which is 10 times 350 plus another half of that: 3500 + 1750 = 5250, so now we have:

= 5250 × 10⁹

which is 5,250 billion or 5.25 trillion. So the cost would be about $5.25 trillion.

Recall that, when we say something like “5250 × 10⁹ is 5,250 billion” we are using these facts:

Number	Power of Ten	Words
1,000	10³	Thousand
1,000,000	10⁶	Million
1,000,000,000	10⁹	Billion
1,000,000,000,000	10¹²	Trillion

Now, use the information above to match the given numbers to tens or hundreds of thousands/millions/billions/trillions.

15,000,000
250,000,000,000
76,000
879,000,000
876,000,000,000,000
58,000,000,000,000

tens of thousands
tens of trillions
hundreds of millions
hundreds of trillions
hundreds of billions
tens of millions

(1) Use the above estimation strategy, together with any others you are familiar with, to decide if the following quantities are in the millions, billions, trillions, or some other size, such as tens of millions. Note: For some of these questions you could find the exact values, but for others, such as populations, any number would be an estimate.

(a) The number of seconds you have been alive if you are 23 years old.

(b) The amount of money needed to send all adults in the United States to a public college for four years. Assume there are about 258 million adults in the United States and that the cost of tuition and fees to a public college is about $25,000.

(2) Refine your above estimates by using a calculator and obtaining any missing information you need.

(a) The number of seconds you have been alive.
For example, for a 10-year-old: 60 * 60 * 24 * 365 * 10 = 315,360,000 or about 315 million seconds.

(c) The number of words in the novel War and Peace. Using an internet search, we found that sample pages of an e-book version of War and Peace have about 300 words on each page. This e-book version is 1307 pages long.

Preparations are also available online. The online versions of Preparations contain one or more Learning sections and a Questions section. The Learning sections in online Preparations match what is in this workbook and no score is given for these sections. The Questions section in online Preparations contains similar questions, but the numbers and figures vary from your workbook, so they are good for extra practice. Completing the Questions section will generate a score.

Tip: You can use the print version or the Learning sections online for practice. Then you can do the Questions section for additional practice and a score.

After Preparation N.2 (survey)

You should be able to do the following things for the next collaboration. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident).

Before beginning Collaboration N.2, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can …	Rating from 1 to 5
convert between millions, billions, and trillions.
compute basic percentages and ratios.
express one number as a percent of another.

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