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Place value, rounding and comparing
Every math question on the ASVAB is built from numbers, and every number is built from digits in places. Learn what each place is worth, how to round without slipping, and how to tell which of two decimals is bigger. Then the answer choices stop looking alike.
A digit is worth what its place says
There are only ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. We write every number with them. What a digit is worth depends on its place, the position where it sits in the number.
Here is one number with every place from millions down to thousandths. The decimal point splits it in two: on the left, whole things; on the right, parts of one.
| Place | Digit | Value |
|---|---|---|
| Millions | 6 | 6,000,000 |
| Hundred thousands | 3 | 300,000 |
| Ten thousands | 0 | 0 |
| Thousands | 5 | 5,000 |
| Hundreds | 1 | 100 |
| Tens | 2 | 20 |
| Ones | 8 | 8 |
| Tenths | 9 | 0.9 |
| Hundredths | 4 | 0.04 |
| Thousandths | 7 | 0.007 |
Each place is worth ten times the place on its right
Move one place to the left and the value is multiplied by 10. Move one place to the right and it is divided by 10. The value of a digit is the digit times its place. On a pay stub that says $2,437.58, the 2 is two thousand dollars, the 5 is five dimes (5 tenths of a dollar) and the 8 is eight pennies (8 hundredths of a dollar).
How to solve them
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1
Name every place, from millions to thousandths
Left of the decimal point, the places grow: ones, tens, hundreds, thousands, and so on. Commas split the whole part into groups of three digits, counted from the decimal point: the ones group, the thousands group and the millions group.
Right of the decimal point, the places shrink. Their names end in -ths, which means “parts of one”. A tenth is one of 10 equal parts, a hundredth is one of 100 and a thousandth is one of 1,000.
0.1One tenth: one of 10 equal parts.
a dime is 0.1 of a dollar0.01One hundredth: one of 100 equal parts.
a penny is 0.01 of a dollar0.001One thousandth: one of 1,000 equal parts.
gas at $3.879 shows thousandthsNext step: Words and digits- 48,206.73
- 40,000 + 8,206.73
- 40,000 + 8,000 + 206.73
- 40,000 + 8,000 + 200 + 6.73
- 40,000 + 8,000 + 200 + 6 + 0.73
- 40,000 + 8,000 + 200 + 6 + 0.7 + 0.03
Expanded form: each digit times its place, one piece at a time. The 0 in the tens place adds nothing, but it holds the 2 in the hundreds place. -
2
Numbers in words, and words back to digits
To read a whole number, read each group of three as a small number and then say the group’s name: “million”, “thousand”. The last group has no name.
The word “and” means the decimal point. Read the part after the point as a whole number, then say the name of its last place: 7.025 is “seven and twenty-five thousandths”, because the 5 is in the thousandths place.
Going from words to digits, every place must be filled. If a place has nothing in it, write a 0 there. That zero is called a placeholder.
Next step: RoundingDigits to words3,040,215
three million, forty thousand, two hundred fifteen
Decimal to words7.025
seven and twenty-five thousandths
Words to digitstwelve thousand, six
12,006: the hundreds and tens places get zeros
- 9,000,000
- 9,000,000 + 50,000
- 9,000,000 + 50,000 + 300
- 9,050,300
“Nine million, fifty thousand, three hundred”: write each group with all its zeros, then put them together. -
3
Rounding: look at one digit, just to the right
To round is to replace a number with a nearby number that is easier to use. “About $50” is easier than “$47”. The question always tells you the place to round to: the nearest ten, hundred, thousand.
1. Find the digit in that place. 2. Look at the one digit just to its right. 3. If that digit is 5 or more, add 1 to your digit. If it is 4 or less, leave your digit alone. 4. Every place to the right becomes 0.
Only that one digit matters. The digits after it do not change the answer.
Next step: Rounding decimalsExample47 to the nearest ten
- The tens digit is 4. The digit to its right is 7.
- 7 is 5 or more, so the 4 goes up to 5, and the ones place becomes 0.
- 47 is between 40 and 50, and closer to 50: 50.
Answer 50
- 3,962
- 39 hundreds + 62
- 40 hundreds
- 4,000
3,962 to the nearest hundred. The tens digit, 6, is 5 or more, so one more hundred: 39 hundreds become 40 hundreds. When a 9 goes up, it carries, just like counting. -
4
Rounding decimals and money
Decimals round with the same rule. The only difference: after the decimal point you do not write zeros on the end, you just stop. 4.351 to the nearest tenth is 4.4, not 4.400.
With money, the nearest dollar is the nearest whole number and the nearest cent is the nearest hundredth, because a cent is one hundredth of a dollar.
Next step: ComparingNearest whole number18.62
The tenths digit is 6, so 18 goes up: 19
Nearest tenth4.351
The hundredths digit is 5, so 3 tenths go up: 4.4
Nearest cent$3.879 a gallon
The thousandths digit is 9: $3.88
- 7.096
- 709 hundredths + 6 thousandths
- 710 hundredths
- 7.10
$7.096 to the nearest cent. The 6 thousandths push 709 hundredths up to 710. With money, keep the 0: $7.10. -
5
Comparing two numbers: the first place that differs decides
is greater than>9 > 4is less than<4 < 9is equal to=0.6 = 0.60The open side of > or < always faces the bigger number.
Whole numbers: the one with more digits is bigger. If they have the same number of digits, compare from the left. The first place where the digits differ decides.
Decimals: line up the decimal points. Add zeros on the end so both have the same number of places; a zero on the end does not change the value. Then compare from the left, the same way.
Next step: Ordering- 0.9 ? 0.85
- 0.90 ? 0.85
- 90 hundredths ? 85 hundredths
- 0.9 > 0.85
0.85 has more digits, but 0.9 is bigger: 9 tenths is more than 8 tenths. Example89,995 or 90,013?
- Both have five digits, so compare from the left.
- Ten thousands place: 8 and 9. They differ, so this place decides.
- 9 is more than 8: 90,013 > 89,995. The rest of the digits do not matter.
Answer 90,013 is greater
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6
Ordering a list: line up the decimal points
To put several decimals in order, write them in a column with the decimal points lined up. Fill in zeros so they all end in the same place. Now read them as whole numbers of thousandths and order those.
ShortcutLonger is not bigger
0.275 has more digits than 0.4, but 0.4 is the same as 0.400, and 400 thousandths is more than 275 thousandths. Count by place, never by length.
Next step: Three examples2.3, 2.035, 2.31 and 2.4, lined upNumber With zeros In thousandths 2.3 2.300 2,300 2.035 2.035 2,035 2.31 2.310 2,310 2.4 2.400 2,400 From least to greatest: 2,035, then 2,300, then 2,310, then 2,400 thousandths. So the order is 2.035, 2.3, 2.31, 2.4.
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7
Three worked examples, from easy to test level
Each one starts with the first try many people make, crossed out, and then the way that works. Read the wrong try too: it is the kind of answer the test puts in the choices.
The test-level one asks two things in a row: first which number is the biggest and which is the smallest, then how to round each one. Do one step at a time and write each result down.
Example · EasyRound 6,749 to the nearest hundred.
First try
6,749 → 6,750 → 6,800- Rounding in steps, starting from the end, lets the 9 push everything up. That is the mistake: only one digit counts.
- The hundreds digit is 7. The one digit to its right is 4.
- 4 is less than 5, so the 7 stays.
- The tens and ones places become 0: 6,700.
Answer 6,700
Example · MediumPut 0.406, 0.46, 0.4 and 0.064 in order from least to greatest.
First try
0.4, 0.46, 0.064, 0.406- The first try reads them as 4, 46, 64 and 406, so the shortest comes first. Length is not size.
- Fill in zeros so all of them have three places: 0.406, 0.460, 0.400, 0.064.
- Read them as thousandths: 406, 460, 400 and 64.
- Order those: 64, 400, 406, 460. So the order is 0.064, 0.4, 0.406, 0.46.
Answer 0.064, 0.4, 0.406, 0.46
Example · Test levelFour checkpoints on a road march are 4.25, 4.5, 4.205 and 4.052 miles from the start. The squad leader reports the farthest one to the nearest whole mile and the closest one to the nearest tenth of a mile. What two numbers does he report?
First try
Farthest: 4.205, which rounds to 4. Closest: 4.5- The first try thinks the longest number is the biggest one. Line them up instead: 4.250, 4.500, 4.205, 4.052.
- In thousandths: 4,250, 4,500, 4,205 and 4,052. The farthest is 4.5 and the closest is 4.052.
- Farthest, to the nearest whole mile: the tenths digit is 5, which is 5 or more, so 4 goes up to 5.
- Closest, to the nearest tenth: 4.052 has 0 tenths and the digit to the right is 5, so 0 tenths go up to 1 tenth: 4.1.
- He reports 5 miles and 4.1 miles.
Answer 5 miles and 4.1 miles
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Longer looks bigger
Which is greater, 0.8 or 0.75?
Wrong0.75, because 75 is more than 8
Right0.80 > 0.75: 80 hundredths beats 75
Rounding in steps
Round 2,451 to the nearest thousand:
Wrong2,451 → 2,500 → 3,000
Right2,000: the hundreds digit is 4
Forgetting the placeholder zero
Write “five thousand, thirty” in digits:
Wrong530
Right5,030
Tens or tenths
Round 64.38 to the nearest ten:
Wrong64.4 (that is the nearest tenth)
Right60
15 questions in three levels
Warm-up first, then core, then test level. Pick an answer, then read why.
Place value is a tool you use inside other problems: a word problem may ask you to round an answer to the nearest cent, or to pick the largest of several decimals. Answer choices that differ only in where the decimal point is are easy to mix up.
There is no calculator, and on the computer test you cannot go back to a question.
A practical tip: when a question asks you to round, find the place, put your finger on the one digit to its right, and decide from that digit only. When choices look alike, line up their decimal points on your scratch paper before you pick.
What is the value of the digit 8 in 3,815,407?
Show the answer
D. Name the places from the right: 7 ones, 0 tens, 4 hundreds, 5 thousands, 1 ten thousand, 8 hundred thousands. So the 8 is worth 8 × 100,000 = 800,000. 8 is only the digit, not its value. 8,000 reads the 8 as thousands because it sits next to a comma, and 80,000 is one place off. The value is 800,000.
Which digit is in the hundredths place of 92.638?
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A. After the decimal point the places are tenths, hundredths, thousandths. In 92.638 that is 6 tenths, 3 hundredths and 8 thousandths. 6 is the tenths digit, 8 is the last digit (thousandths), and 9 is in the tens place, on the other side of the point. The hundredths digit is 3.
Round 5,276 to the nearest hundred.
Show the answer
D. The hundreds digit is 2. The digit just to its right is 7, which is 5 or more, so the 2 goes up to 3 and the tens and ones become 0. 5,000 rounds to the nearest thousand, 5,280 rounds to the nearest ten, and 5,200 just chops off the end without looking at the 7. The answer is 5,300.
Write “forty thousand, six hundred nine” in digits.
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B. Forty thousand is 40,000. Six hundred nine is 609: 6 hundreds, 0 tens, 9 ones. Together: 40,609. 4,609 writes forty thousand as four thousand. 40,690 puts the 9 in the tens place, which is “six hundred ninety”. 400,609 adds an extra zero in the gap and makes four hundred thousand. The answer is 40,609.
How many hundredths are in 0.4?
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C. 0.4 is 4 tenths. Each tenth is 10 hundredths, so 4 tenths is 4 × 10 = 40 hundredths. Write it as 0.40 and you can see the 40. 4 counts tenths, not hundredths, 400 counts thousandths, and 0.04 is the number “4 hundredths”, not how many there are. There are 40.
Round $14.386 to the nearest cent.
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C. A cent is a hundredth, so look at the hundredths digit, 8, and the digit to its right, 6. 6 is 5 or more, so 8 hundredths go up to 9: $14.39. $14.38 chops off the 6 without rounding, $14.40 rounds to the nearest dime (tenth), and $14.00 rounds to the nearest dollar. The answer is $14.39.
Round 7.254 to the nearest tenth.
Show the answer
D. The tenths digit is 2. The digit just to its right is 5, which is 5 or more, so 2 tenths go up to 3 tenths: 7.3. 7.2 looks at the last digit, 4, instead of the one next to the tenths. 7.25 rounds to the nearest hundredth, and 7 rounds to the nearest whole number. The answer is 7.3.
Which list is in order from least to greatest?
Show the answer
B. Fill in zeros so all four have three places: 0.800, 0.840, 0.084, 0.408. As thousandths they are 800, 840, 84 and 408. From least to greatest: 84, 408, 800, 840. The first list orders them by length, the third puts 0.8 first because it is the shortest, and the last one goes from greatest to least. The answer is 0.084, 0.408, 0.8, 0.84.
Which number is “six and fourteen thousandths”?
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B. “And” is the decimal point, so the whole part is 6. “Fourteen thousandths” must end in the thousandths place, the third place after the point: 0.014. Together: 6.014. 6.14 is “six and fourteen hundredths”, 6.0014 has one zero too many, and 6,014 ignores the “and” and makes six thousand fourteen. The answer is 6.014.
A soldier’s pay stub shows $2,418.75. To the nearest hundred dollars, about how much is that?
Show the answer
A. The hundreds digit is 4. The digit just to its right is the tens digit, 1, which is 4 or less, so the 4 stays: $2,400. $2,418 chops off the cents, $2,419 rounds to the nearest dollar, and $2,500 looks at the 8 in the ones place instead of the tens. The answer is $2,400.
A supply clerk weighs four packages: 2.35 lb, 2.305 lb, 2.4 lb and 2.53 lb. The lightest one goes on top. What does the lightest package weigh, rounded to the nearest tenth of a pound?
Show the answer
B. First find the lightest. Fill in zeros: 2.350, 2.305, 2.400, 2.530. As thousandths: 2,350, 2,305, 2,400 and 2,530, so the lightest is 2.305. Now round it to the nearest tenth: the tenths digit is 3 and the next digit is 0, so it stays 2.3. The choice 2.4 comes from picking 2.4 because it looks shortest, or from picking 2.35 and rounding it. 2.31 rounds to the nearest hundredth, and 2 rounds to the nearest whole pound. The answer is 2.3 lb.
A town has 73,846 people. A news report rounds that to the nearest thousand. How much greater is the rounded number than the real number?
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C. First round: the thousands digit is 3 and the next digit is 8, so it goes up to 74,000. Then count up from 73,846 to 74,000: plus 4 is 73,850, plus 50 is 73,900, plus 100 is 74,000. That is 4 + 50 + 100 = 154. 4 rounds to the nearest ten, 46 rounds to the nearest hundred (73,800) and measures the gap below, and 846 rounds down to 73,000. The answer is 154.
Three companies bid on a barracks repair: $48,950, $49,095 and $48,905. The lowest bid wins. What is the winning bid, rounded to the nearest hundred dollars?
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A. First find the lowest. $49,095 has 49 thousands, the others have 48, so it is out. Compare $48,950 and $48,905 from the left: the first place that differs is the tens, 5 and 0, so $48,905 is lower. Round it to the nearest hundred: the hundreds digit is 9 and the next digit is 0, so it stays: $48,900. $49,000 picks $48,950 by comparing the last digits, $49,100 rounds the highest bid, and $48,910 rounds to the nearest ten. The answer is $48,900.
A base’s yearly fuel budget is two million, sixty thousand, five hundred dollars. What is that rounded to the nearest hundred thousand dollars?
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C. First write it in digits: 2,000,000 + 60,000 + 500 = 2,060,500. The hundred thousands digit is 0 and the next digit, the ten thousands, is 6. 6 is 5 or more, so 0 goes up to 1: $2,100,000. $2,000,000 writes sixty thousand as 6,000. $2,600,000 writes it as 600,000. $2,060,000 rounds to the nearest ten thousand. The answer is $2,100,000.
A machined part passes inspection if its length, rounded to the nearest tenth of an inch, is 3.7 inches. Which length passes?
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C. Round each one to the nearest tenth by looking only at the hundredths digit. 3.65: 6 tenths, and the next digit is 5, so it goes up to 3.7. It passes. 3.64 stays 3.6. 3.648 also stays 3.6: rounding it in steps (3.648 → 3.65 → 3.7) is the trap. 3.751 starts with 3.7, but its next digit is 5, so it goes up to 3.8. The answer is 3.65 in.
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