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Decimals
Prices, pay, gas, miles: most numbers in real life have a decimal point. You need only four rules to handle them by hand. Learn where the point goes in each operation and the answer choices stop being a guessing game.
A decimal is a fraction of 10, 100 or 1,000
The decimal point separates whole things from parts of a thing. Left of the point are whole numbers: ones, tens, hundreds. Right of the point are parts: tenths (out of 10), hundredths (out of 100) and thousandths (out of 1,000). Each place is 10 times smaller than the one to its left.
| Digit | Place | Value |
|---|---|---|
| 4 | tens | 40 |
| 6 | ones | 6 |
| 2 | tenths | 0.2 |
| 5 | hundredths | 0.05 |
| 8 | thousandths | 0.008 |
Money is the decimal you already know
A dollar is 100 cents, so cents are hundredths. $3.75 is 3 whole dollars and 75 hundredths of a dollar. When a decimal feels strange, read it as money: 0.5 is half a dollar, 50 cents. Every rule in this lesson is something a cashier does every day.
How to solve them
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1
Compare: give both numbers the same number of places
To see which decimal is bigger, do not count digits. Add zeros at the end until both numbers have the same number of places after the point. Then compare them like whole numbers.
Zeros at the end change nothing: 0.5, 0.50 and 0.500 are the same amount, like 50 cents and half a dollar. Zeros right after the point do change the number: 0.05 is five cents, not fifty.
Next step: Add & subtractExampleWhich is bigger: 0.7 or 0.65?
- 0.65 has two places and 0.7 has one. Add a zero to 0.7: it becomes 0.70.
- Now compare 70 hundredths with 65 hundredths. 70 is more.
- So 0.7 is bigger, even though 0.65 has more digits.
Answer 0.7
- 0.7 ? 0.65
- 0.70 ? 0.65
- 0.70 > 0.65
Same number of places first, then compare. -
2
Add and subtract: line up the points
point under point, then add as usual
Write the numbers in a column with the decimal points straight under each other. Fill the empty places with zeros so every row is the same length. Add or subtract from the right, carrying and borrowing as you do with whole numbers, and drop the point straight down into the answer.
A whole number has a point too: it is hiding at the end. 20 is 20.00.
Next step: Multiply- 12.5 + 3.75
- 12.50 + 3.75
- 12.50 + 3.75 = 16.25
Fill the empty hundredths place with a zero. - 20 − 6.35
- 20.00 − 6.35
- 20.00 − 6.35 = 13.65
Show the hidden point and its zeros, then borrow. -
3
Multiply: ignore the points, then count the places
You do not line up the points to multiply. Multiply the numbers as if there were no points at all. Then count how many digits are after the point in both numbers together. The answer gets that many places, counted from the right.
- 2.4 × 0.3
- 24 × 3 = 72 (points ignored)
- 1 place + 1 place: two places
- 2.4 × 0.3 = 0.72
Multiply the digits, then put back the places. Next step: DivideExample4.9 × 2.1, with an estimate first
- Estimate: 4.9 is close to 5 and 2.1 is close to 2, so the answer is close to 10.
- Ignore the points: 49 × 21 is 49 × 20 plus 49, which is 980 + 49 = 1,029.
- One place in 4.9 and one in 2.1: two places. 1,029 becomes 10.29.
- It is close to 10, so the point is in the right place.
Answer 10.29
ShortcutTimes 10, 100 or 1,000
Move the point one place right for each zero. 3.75 × 10 is 37.5; 3.75 × 100 is 375.
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4
Divide by a decimal: move both points
Nobody divides by 0.15 in their head. So change the problem: move the point in the number you divide by until it is a whole number. Then move the point in the other number the same number of places. The answer does not change.
- 4.5 ÷ 0.15
- 4.50 ÷ 0.15 (same places)
- 450 ÷ 15 (both points moved two places)
- 450 ÷ 15 = 30
Both numbers moved together, so the answer stays the same. Why it works: $4.50 split into 15-cent pieces is the same as 450 cents split into 15-cent pieces. You only changed the unit.
Next step: RoundExampleDividing by a whole number: 8.4 ÷ 4
- The number you divide by, 4, is already whole. Nothing to move.
- Divide as usual and put the point straight up: 8 ÷ 4 is 2, then 4 tenths ÷ 4 is 1 tenth.
- 8.4 ÷ 4 = 2.1
Answer 2.1
ShortcutCheck by multiplying back
If 4.5 ÷ 0.15 is 30, then 30 × 0.15 must give 4.5 back. It does: 30 × 0.15 = 4.5.
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5
Round: look at one digit to the right
Find the place you are rounding to. Look at the one digit just to its right. If it is 5 or more, the digit you are rounding goes up by 1. If it is 4 or less, it stays the same. Every digit after it is dropped.
Nearest whole numberRound 3.846
The digit after the ones is 8: 4
Nearest tenthRound 3.846
The digit after the tenths is 4: 3.8
Nearest hundredthRound 3.846
The digit after the hundredths is 6: 3.85
Next step: MoneyExampleRound $2.996 to the nearest cent
- Cents are hundredths. The hundredths digit is 9.
- The digit to its right is 6, so the 9 goes up. 9 + 1 is 10: write 0 and carry 1 to the tenths.
- The tenths 9 also becomes 10: write 0 and carry 1 to the dollars. 2 becomes 3.
Answer $3.00
ShortcutNever round twice
Round 2.449 to the nearest tenth. Look only at the digit after the tenths, a 4: the answer is 2.4. Rounding first to 2.45 and then to 2.5 is a mistake.
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6
Money problems: estimate, then think in cents
Money always has two places, so money problems are decimal problems. Two habits make them safe. First, estimate with whole dollars so you know about where the answer is. Second, when the cents get messy, think in cents: $4.65 is 465 cents, a whole number.
To make change, count up from the price to the bill, the way a cashier does. It avoids borrowing.
Next step: Three examplesExampleA $8.75 meal and a $1.90 drink, paid with $20
- Estimate: about $9 + $2 = $11, so the change is about $9.
- The total: 8.75 + 1.90 = 10.65.
- Count up to 20: 10.65 + 0.35 = 11, and 11 + 9 = 20. The change is 9 + 0.35.
Answer $9.35
ShortcutRound up, then take back
3 items at $1.99: think 3 × $2 = $6, then take back the 3 cents you added. The total is $5.97.
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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 needs three moves in a row: multiply, divide, then round to the cent. Write each result down before you start the next one.
Example · EasyWhat is 5.4 + 0.86?
First try
Lined up on the right: 54 + 86 → 1.40- The last digits are in different places: the 4 is tenths and the 6 is hundredths. Line up the points instead.
- Write 5.4 as 5.40 so both numbers have two places.
- 5.40 + 0.86 = 6.26.
- Check: about 5 + 1 = 6. The crossed-out 1.40 is less than 5.4 alone, so it cannot be right.
Answer 6.26
Example · MediumWhat is 0.35 × 0.6?
First try
35 × 6 = 210, so 2.10- 35 × 6 = 210 is right. The mistake is where the point went.
- Count the places: two in 0.35 and one in 0.6, three in all.
- Three places from the right of 210: 0.210, which is 0.21.
- Check: multiplying by 0.6 makes 0.35 smaller, and 2.10 is bigger than both numbers.
Answer 0.21
Example · Test levelA squad orders 6 pizzas at $11.75 each and splits the bill equally among 8 soldiers. To the nearest cent, how much does each soldier pay?
First try
6 × 11.75 → 66.75, so 66.75 ÷ 8 → about $8.34- The first try multiplied only the dollars: 6 × 11 = 66, and then it just stuck the 75 cents on.
- Multiply both parts: 6 × 11 = 66, and 6 × 75 cents = 450 cents = 4.50. The bill is 66 + 4.50 = 70.50.
- Split it: 70.50 ÷ 8. 8 goes into 70 eight times (64), 6 left. Now you reach the decimal point in 70.50, so put a point in the answer right after the 8. Bring down the 5: 65 ÷ 8 is 8 (64), 1 left. Bring down the 0: 10 ÷ 8 is 1, 2 left. Write one more 0: 20 ÷ 8 is 2. So far 8.812…, and that is enough to round to the cent.
- To the nearest cent: the digit after the hundredths is 2, so the 1 stays. Each soldier pays $8.81.
Answer $8.81
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.
Lining up the last digits
Adding 4.5 and 1.25 with both numbers pushed to the right:
Wrong4.5 + 1.25 → 1.70
Right4.50 + 1.25 = 5.75
Too few decimal places
Multiplying 0.4 by 0.2:
Wrong0.4 × 0.2 → 0.8
Right4 × 2 is 8, two places: 0.08
Moving only one point
Dividing 6 by 0.3:
Wrong6 ÷ 3 → 2
Right60 ÷ 3 = 20
More digits is not bigger
Comparing 0.5 and 0.125:
Wrong0.125 is bigger: it has more digits
Right0.500 is more than 0.125, so 0.5 is bigger
15 questions in three levels
Warm-up first, then core, then test level. Pick an answer, then read why.
You will use decimals in Arithmetic Reasoning word problems about prices, pay, gas and distances. On the computer test that section has 15 questions in 55 minutes; on paper, 30 questions in 36 minutes. The same skills carry into Mathematics Knowledge (15 questions in 31 minutes on the computer, 25 in 24 minutes on paper).
No calculator. Estimate first with whole numbers: a point in the wrong place makes an answer 10 times too big or too small, and the estimate catches it before you pick a choice.
On the computer test you cannot go back to a question. In money problems, keep every cent while you work and round to the nearest cent only at the end.
What is 3.6 + 2.75?
Show the answer
D. Line up the points and write 3.6 as 3.60. Add from the right: 0 and 5 make 5 hundredths; 6 and 7 make 13 tenths, so write 3 and carry 1; then 3 and 2 and the 1 you carried make 6. 3.11 lines up the last digits instead of the points, 5.35 forgets the carry, and 5.81 reads 3.6 as 3.06. The sum is 6.35.
What is 9 − 4.28?
Show the answer
A. Write 9 as 9.00 so both numbers have two places. You cannot take 8 hundredths from 0, so borrow: 9.00 is the same as 8 ones, 9 tenths and 10 hundredths. Now subtract column by column: 10 take away 8 is 2, 9 take away 2 is 7, 8 take away 4 is 4. 5.72 forgets that the 9 became 8, 5.28 takes away only the 4 and keeps the 28 hundredths, and 13.28 adds. The difference is 4.72.
What is 0.6 × 0.4?
Show the answer
B. Ignore the points: 6 × 4 is 24. Count the places: one in 0.6 and one in 0.4, two in all. So 24 becomes 0.24. 2.4 counts only one place, 0.024 counts three, and 1.0 adds instead of multiplying. The product is 0.24.
Round 7.468 to the nearest tenth.
Show the answer
C. The tenths digit is 4. Look at the one digit to its right, the 6. It is 5 or more, so the 4 goes up to 5 and every digit after it is dropped. 7.4 just cuts off the digits, 7.47 rounds to the nearest hundredth, and 8 rounds to the nearest whole number. The answer is 7.5.
Which number is the largest?
Show the answer
D. Give them all three places: 0.073, 0.300, 0.307 and 0.370. Now compare them like whole numbers of thousandths: 73, 300, 307 and 370. The largest is 370 thousandths. 0.307 looks bigger because it has more digits, but more digits do not make a number bigger. The largest is 0.37.
What is 2.5 × 0.4?
Show the answer
B. Ignore the points: 25 × 4 is 100. There is one place in 2.5 and one in 0.4, so the answer needs two: 1.00, which is 1. 10 counts only one place, 0.1 counts three, and 100 forgets the points. Check: half of 2.5 is 1.25, and 0.4 is a little less than half, so the answer is a little less than 1.25. The product is 1.
What is 3.6 ÷ 0.04?
Show the answer
D. Make the number you divide by whole: 0.04 becomes 4 when its point moves two places right. Move the point in 3.6 two places right too: 3.6 becomes 360. Now 360 ÷ 4 is 90. 9 moves the point in 3.6 only one place, 0.9 divides by 4 without moving the point in 3.6, and 0.144 multiplies instead of dividing. The answer is 90.
You buy $31.74 of groceries and pay with a $50 bill. How much change do you get?
Show the answer
A. Count up from the price, the way a cashier does: 31.74 + 0.26 = 32, and 32 + 18 = 50. Together you counted up 18 dollars and 26 cents. $19.26 forgets a borrow, $21.74 takes the smaller digit from the bigger one in every column, and $81.74 adds. The change is $18.26.
Three friends split a $20 bill equally. To the nearest cent, how much is each share?
Show the answer
C. 3 goes into 20 six times with 2 left over, so each share is 6 dollars and a bit. Write 20 as 20.000 and keep dividing: the share is 6.666, with the 6 repeating forever. To the nearest cent, look at the thousandths digit: it is 6, so the hundredths go up from 6 to 7. $6.66 cuts off without rounding, $6.60 stops at the tenths, and $7.00 rounds to the dollar. Each share is $6.67.
Gas costs $3.40 a gallon. How much do 4.5 gallons cost?
Show the answer
D. Drop the end zero: $3.40 is 3.4. Ignore the points: 34 × 45. 34 × 40 is 1,360 and 34 × 5 is 170, so 34 × 45 is 1,530. One place in 3.4 and one in 4.5 make two: 15.30. Estimate: about 3 × 5 = 15, so it fits. $1.53 counts three places, $13.60 pays for only 4 gallons, and $7.90 adds. The cost is $15.30.
A soldier buys 3 phone chargers at $4.65 each and pays with a $20 bill. How much change does she get?
Show the answer
A. First the cost. Think in cents: 465 × 3. 400 × 3 is 1,200 and 65 × 3 is 195, so it is 1,395 cents, or $13.95. Then the change: count up, 13.95 + 0.05 = 14, and 14 + 6 = 20, so 6.05. $13.95 stops after the first step, $15.35 pays for only one charger, and $7.05 forgets a borrow. Her change is $6.05.
A 37.5-foot roll of paracord is cut into pieces 2.5 feet long. Each piece makes a bracelet that sells for $1.20 at a unit fundraiser. If every bracelet sells, how much money is raised?
Show the answer
B. First the number of pieces: 37.5 ÷ 2.5. Move both points one place: 375 ÷ 25. 25 goes into 300 twelve times, and 75 is three more 25s, so there are 15 pieces. Then the money: 15 × 1.20 is 15 × 1 plus 15 × 0.20, which is 15 + 3 = 18. $180.00 moves only one point (375 ÷ 2.5 gives 150 pieces), $45.00 multiplies the length by the price, and $1.80 puts the point in the wrong place. The fundraiser raises $18.00.
The base grocery store (the commissary) sells coffee in three bags: 2 pounds for $5.70, 1.5 pounds for $4.05, and 2.5 pounds for $7.00. What is the lowest price per pound?
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A. Divide each price by its pounds. 5.70 ÷ 2 is 2.85. For 4.05 ÷ 1.5, move both points one place: 40.5 ÷ 15. 15 goes into 40 twice (30), 10 left; 105 tenths ÷ 15 is 7 tenths (15 × 7 = 105), so 2.7. For 7.00 ÷ 2.5, move both points: 70 ÷ 25. 25 goes into 70 twice (50), 20 left; 200 tenths ÷ 25 is 8 tenths, so 2.8. The lowest is 2.70. $4.05 is the cheapest bag, not the cheapest pound. The lowest price per pound is $2.70.
Gas costs $3.459 per gallon. Ana buys exactly 12 gallons. What does she pay, rounded to the nearest cent?
Show the answer
C. Multiply first, in two easy parts: 3.459 × 12 = 3.459 × 10 + 3.459 × 2. Times 10 moves the point one place right: 34.59. Times 2 is 6.918. Add them: 34.59 + 6.918 = 41.508. Round only at the end: the digit after the cents is 8, so 41.50 goes up to 41.51. $41.52 rounds the price to $3.46 first (3.46 × 12 = 41.52), $41.40 cuts the price to $3.45 (3.45 × 12 = 41.40), and $3.46 is the price of one gallon. She pays $41.51.
Pvt. Lee has $40. He buys a shirt for $12.85 and two pairs of socks at $6.49 a pair. How much money does he have left?
Show the answer
A. First the total. A pair of socks is 1 cent less than $6.50, so two pairs are 2 cents less than $13.00: 12.98. Shirt and socks: 12.85 + 12.98 = 25.83. Then what is left, counting up to 40: 25.83 + 0.17 = 26, and 26 + 14 = 40, so 14.17. $25.83 stops after the total, $20.66 pays for only one pair of socks, and $15.17 forgets a borrow. What he has left: $14.17.
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