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Fractions, decimals and percents

Pythagoras comparing an orange slice, a few coins and a half-full glass at a sunny café table

One half, 0.5 and 50% are the same amount with three different names. A question can use all three names at once, so learn to switch between them in seconds, learn the eight benchmarks by heart, and you will always pick the form that makes the math short.

15 min read · 15 practice questions · 12 flashcards · Free

Three names for the same amount

40 of these 100 squares are filled: 40%, 0.4 and 2/5 are three names for this one amount.

A fraction like 2/5 splits a whole into equal parts. The bottom number, the denominator, says how many equal parts there are. The top number, the numerator, says how many of them you have. A decimal like 0.4 uses place value: the first place after the point is tenths, the second is hundredths, the third is thousandths. A percent like 40% means 40 out of 100.

Fraction2/5
Decimal0.4
Percent40%

A real example: the fuel gauge on a truck shows the tank 2/5 full. The driver writes 0.4 in the logbook, and a phone app says 40%. It is the same fuel. Only the name changes.

Key idea

Switch to the form that makes the work easy

Decimals are best for comparing. Fractions are best for taking a part of a number: 50% of 18 is 1/2 of 18, which is 9. Percents are how questions talk. You will move between the three all the time, so the moves have to be automatic.

Next: Learn

How to solve them

  1. 1

    Percent and decimal: move the point two places

    Percent means “per hundred”, so a percent is a number divided by 100. Dividing by 100 moves the decimal point two places left. To go back, multiply by 100: the point moves two places right.

    Percent → decimaldivide by 100point 2 places left
    Decimal → percentmultiply by 100point 2 places right

    If there are not enough digits, fill the empty places with zeros. A whole number has its point at the end, even when you don’t see it: 7 is 7.0.

    1. 7%
    2. 7 ÷ 100
    3. 0.07
    Two places left. 7 has only one digit, so a zero fills the empty place: 0.07, not 0.7.
    1. 0.4
    2. 0.4 × 100
    3. 40%
    Two places right. 0.4 has one digit after the point, so a zero fills the second place: 40%.
    More than 100%

    125% as a decimal

    1.25: more than one whole

    Part of a percent

    12.5% as a decimal

    0.125

    Above 1

    1.6 as a percent

    160%

    Next step: Fraction → decimal
  2. 2

    Fraction to decimal: the fraction bar means divide

    A fraction is a division waiting to happen. The top number is the number you divide; the bottom number is the number you divide by: 3/8 means 3 ÷ 8, never 8 ÷ 3. Give the top number a decimal point and some zeros, and divide the way you always do.

    1. 3/8
    2. 3 ÷ 8
    3. 3.000 ÷ 8
    4. 0.375
    8 goes into 30 three times (24), with 6 left. Into 60 seven times (56), with 4 left. Into 40 five times, with nothing left. Then 0.375 is 37.5%.

    There is often a faster way. If the bottom number fits evenly into 10, 100 or 1,000, multiply the top and the bottom by the same number to get there. The value does not change, because you multiply by a form of 1.

    Example

    7/20 as a decimal and a percent

    1. 20 × 5 = 100, so multiply the top and the bottom by 5.
    2. 7/20 = 35/100.
    3. 35 hundredths is 0.35, and 35 out of 100 is 35%.

    Answer 0.35 = 35%

    Shortcut

    Check the size first

    When the top number is smaller than the bottom, the fraction is less than 1 whole. Its decimal starts with 0., and its percent is under 100%. An answer that breaks this rule is wrong before you calculate.

    Next step: Back to a fraction
  3. 3

    Decimal or percent to fraction: say it, write it, simplify

    Read the decimal out loud with its place value, and it tells you the fraction. 0.45 is “45 hundredths”, so it is 45/100. A percent is already “out of 100”: 64% is 64/100.

    The last place tells you the bottom number
    DecimalRead it asFraction
    0.77 tenths7/10
    0.4545 hundredths45/100
    0.125125 thousandths125/1000

    Then simplify: divide the top and the bottom by the same number, again and again, until only 1 divides both. A fraction like that is in lowest terms, the simplest way to write it.

    1. 0.45
    2. 45/100
    3. (45 ÷ 5)/(100 ÷ 5)
    4. 9/20
    5 divides both 45 and 100. After that, 9 and 20 share no factor but 1, so you stop.
    Example

    64% as a fraction

    1. 64% means 64 out of 100: 64/100.
    2. Both numbers are even, so halve them: 32/50. Both are still even, so halve again: 16/25.
    3. 16 and 25 share no factor but 1. Stop: 16/25.

    Answer 16/25

    Next step: Benchmarks
  4. 4

    The benchmark table: learn these by heart

    A few fractions come up again and again in math problems. A benchmark is one you know so well that you never work it out. Learn this table and many questions take seconds.

    Benchmarks
    FractionDecimalPercent
    1/20.550%
    1/30.333…33 1/3%
    2/30.666…66 2/3%
    1/40.2525%
    3/40.7575%
    1/50.220%
    1/80.12512.5%
    1/100.110%

    Build the others from the table. Multiply the top number by the benchmark of its family:

    Fifths

    4/5

    4 × 0.2 = 0.8 = 80%

    Eighths

    3/8

    3 × 0.125 = 0.375 = 37.5%

    Tenths

    9/10

    9 × 0.1 = 0.9 = 90%

    Shortcut

    Halves of halves

    1/2 is 50%. Half of that is 1/4, which is 25%. Half again is 1/8, which is 12.5%. Each step halves the fraction and the percent together.

    Next step: Repeating decimals
  5. 5

    Repeating decimals: when the division never ends

    Some divisions never come out even. In 2 ÷ 3, the same remainder comes back every time, so the same digit comes back too. The decimal is 0.666…, and the three dots mean the 6s never stop. This is a repeating decimal. Some books draw a bar over the digit that repeats instead of the dots.

    1. 2/3
    2. 2 ÷ 3
    3. 2.000 ÷ 3
    4. 0.666…
    3 goes into 20 six times (18), with 2 left. You bring down a zero, get 20 again, and it never ends.

    You can tell before you divide. Write the fraction in lowest terms and look at the bottom number. If it is made only of 2s and 5s multiplied together (2, 4, 5, 8, 10, 16, 20, 25…), the decimal ends. If it has any other factor, like 3, 7 or 11, the decimal repeats.

    Ends

    7/40

    40 = 2 × 2 × 2 × 5, only 2s and 5s: 0.175

    Repeats

    1/6

    6 = 2 × 3, and the 3 makes it repeat: 0.1666…

    Repeats

    5/9

    Ninths repeat the top number: 0.555…

    Example

    5/6 as a decimal and a percent

    1. Split it into benchmarks: 5/6 is 3/6 + 2/6, which is 1/2 + 1/3.
    2. Add the decimals: 0.5 + 0.333… is 0.833…
    3. As a percent, that is 83 1/3%, or about 83%.

    Answer 0.833…, about 83%

    Shortcut

    Round at the end

    If you must round, do it last. 2/3 of 900 is exactly 600. If you round 2/3 to 0.67 first, 0.67 × 900 = 603: close, but the test may list both numbers.

    Next step: Easiest form
  6. 6

    Choose the easiest form for the job

    You never have to stay in the form the question uses. Switch to the one that makes the work short.

    To compareuse decimalssame number of places
    To take a partuse a benchmark fraction75% of 44 → 3/4 of 44
    Any other percentuse a decimal18% → 0.18

    To compare decimals, give them all the same number of places by adding zeros at the end. Zeros at the end do not change a decimal: 0.4 is 0.400. Now 0.400 is more than 0.375, because 400 thousandths are more than 375 thousandths.

    Example

    Order from least to greatest: 0.32, 3/8, 7/25, 30%

    1. Make them all decimals with three places.
    2. 0.32 = 0.320. 3/8 = 0.375 (a benchmark family). 7/25 = 28/100 = 0.280. 30% = 0.300.
    3. Now compare 280, 300, 320 and 375 thousandths.

    Answer 7/25, 30%, 0.32, 3/8

    Example

    75% of 44

    1. 75% is the benchmark 3/4.
    2. A quarter of 44 is 11, so three quarters is 3 × 11 = 33.

    Answer 33

    Next step: Three examples
  7. 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 mixes all three forms in one problem. Put everything in the same form first, then do the math one step at a time.

    Example · Easy

    Write 9/25 as a percent.

    First try25 ÷ 9 ≈ 2.78, so 278%

    1. The fraction bar means top ÷ bottom: 9 ÷ 25, not 25 ÷ 9. The top is smaller than the bottom, so the answer must be under 100%.
    2. 25 × 4 = 100, so multiply the top and the bottom by 4: 9/25 = 36/100.
    3. 36 out of 100 is 36%.

    Answer 36%

    Example · Medium

    Order from least to greatest: 0.7, 2/3, 68%, 11/16

    First tryBy the size of the digits: 0.7, 2/3, 11/16, 68%

    1. Make every number a decimal with the same number of places.
    2. 0.7 = 0.7000 and 68% = 0.6800.
    3. 2/3 is a benchmark: 0.666…, about 0.6667.
    4. 1/16 is half of 1/8: half of 0.125 is 0.0625. 10/16 = 5/8 = 0.625 (eighths family), so 11/16 = 0.625 + 0.0625 = 0.6875.
    5. Use four places: 0.6667, 0.6800, 0.6875, 0.7000. The order is 2/3, 68%, 11/16, 0.7.

    Answer 2/3, 68%, 11/16, 0.7

    Example · Test level

    A supply sergeant has 240 meals to hand out. 3/8 of them go to first platoon, 0.25 of them go to second platoon and 30% of them go to third platoon. How many meals are left?

    First try240 − 90 − 60 = 90; 30% of 90 = 27; 90 − 27 = 63

    1. Every share is a share of all 240 meals, not of what is left. Put all three in the same form: percents.
    2. 3/8 = 37.5% (three times 12.5%). 0.25 = 25%. 30% stays 30%.
    3. Together: 37.5% + 25% + 30% = 92.5%. What is left: 100% − 92.5% = 7.5%.
    4. 7.5% of 240: 10% is 24, so 5% is 12 and 2.5% is 6. Then 7.5% is 12 + 6 = 18.
    5. Check with meals: 90 + 60 + 72 = 222, and 240 − 222 = 18.

    Answer 18 meals

Next: Watch out

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.

Trap

Moving the point only one place

Writing 4% as a decimal:

Wrong4% = 0.4

Right4% = 0.04

Trap

Putting a point between the numbers

Writing 4/5 as a decimal:

Wrong4/5 = 4.5

Right4/5 = 4 ÷ 5 = 0.8

Trap

Treating 1/3 as 33%

Finding 1/3 of 600 people:

Wrong33% of 600 = 198

Right1/3 of 600 = 600 ÷ 3 = 200

Trap

Comparing by how long the number looks

Which is bigger, 0.125 or 0.5?

Wrong0.125, because 125 is more than 5

Right0.5 = 0.500, and 500 thousandths are more than 125

Next: Practice

15 questions in three levels

Warm-up first, then core, then test level. Pick an answer, then read why.

On test day

Conversions often come inside word problems: a share of a budget written as a fraction, a discount written as a percent, a measure written as a decimal. On the computer test, Arithmetic Reasoning has 15 questions in 55 minutes and Mathematics Knowledge has 15 in 31 minutes. On paper, Arithmetic Reasoning is 30 questions in 36 minutes and Mathematics Knowledge is 25 in 24 minutes.

No calculator is allowed. The benchmark table is your calculator: 12.5% is 1/8, 33 1/3% is 1/3, 0.2 is 1/5.

Tip: when the answer choices mix forms, change all of them to decimals with the same number of places before you compare. On the computer test you cannot go back, so check that your answer makes sense: a part of a whole is less than 1 and less than 100%.

Question 1 of 15 · Warm-up

Write 0.09 as a percent.

Show the answer

C. Decimal to percent: multiply by 100, which moves the point two places right. 0.09 becomes 9. 0.9% moves the point only one place, 90% reads 0.09 as if it were 0.9, and 0.009% moves the point the wrong way. 0.09 is 9%.

Question 2 of 15 · Warm-up

Write 3% as a decimal.

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B. Percent to decimal: divide by 100, so the point moves two places left. 3 is 3.0. One jump gives 0.3, and the second jump gives 0.03. 0.3 stops after one place, 0.003 goes three places, and 3 forgets to move the point at all. 3% is 0.03.

Question 3 of 15 · Warm-up

Write 1/8 as a decimal.

Show the answer

A. The fraction bar means divide: 1 ÷ 8. Write 1 as 1.000. 8 goes into 10 once, with 2 left. Into 20 twice (16), with 4 left. Into 40 five times. That gives 0.125, a benchmark worth knowing by heart. 0.18 and 1.8 just put a point between the digits, and 8 divides the wrong way, 8 ÷ 1. 1/8 is 0.125.

Question 4 of 15 · Warm-up

Write 0.9 as a fraction.

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C. Read it out loud: 0.9 is “9 tenths”, so it is 9/10. 9 and 10 share no factor but 1, so it is already in lowest terms. 9/100 reads it as hundredths, 1/9 flips the fraction, and 9 forgets the decimal point. 0.9 is 9/10.

Question 5 of 15 · Warm-up

Write 3/10 as a percent.

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C. 3/10 is 3 tenths, which is 0.3. To make a percent, move the point two places right: 0.3 becomes 30. Another way: multiply the top and the bottom by 10 to get 30/100. 0.3% forgets to multiply by 100, 3% moves the point only one place, and 300% multiplies the top by 100 and forgets to divide by 10. 3/10 is 30%.

Question 6 of 15 · Core

Write 7/8 as a percent.

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C. Use the benchmark: 1/8 is 12.5%, so 7/8 is 7 times 12.5%. 7 × 12 = 84 and 7 × 0.5 = 3.5, so the total is 84 + 3.5 = 87.5. Or think of 7/8 as 1 whole minus 1/8: 100% − 12.5% = 87.5%. 0.875% is the decimal with a % sign stuck on, 78% writes the digits, and 114.3% divides 8 by 7. 7/8 is 87.5%.

Question 7 of 15 · Core

Write 0.24 as a fraction in lowest terms.

Show the answer

B. 0.24 is “24 hundredths”: 24/100. Both numbers divide by 4: 24 ÷ 4 = 6 and 100 ÷ 4 = 25. 6 and 25 share no factor but 1, so stop. 3/125 reads it as thousandths (24/1000), 12/5 reads it as tenths (24/10), and 2/4 just uses the two digits. 0.24 is 6/25.

Question 8 of 15 · Core

Which of these is the greatest?

Show the answer

D. Turn all four into decimals with three places. 2/5 = 0.400. 0.43 = 0.430. Ninths repeat the top number, so 4/9 = 0.444… 45% = 0.450. The biggest is 0.450. 4/9 looks big because 9 is the largest digit you see, but 0.444… is just below 0.450. The greatest is 45%.

Question 9 of 15 · Core

What is 12.5% of 72?

Show the answer

C. 12.5% is the benchmark 1/8, so find 1/8 of 72: 72 ÷ 8 = 9. That is much easier than 0.125 × 72. 90 and 0.9 put the decimal point in the wrong place, and 6 treats 12.5% as 1/12 (72 ÷ 12 = 6). 12.5% of 72 is 9.

Question 10 of 15 · Core

Write 1/6 as a percent.

Show the answer

D. 1/6 is half of 1/3, and 1/3 is the benchmark 33 1/3%. Half of 33 1/3: half of 32 is 16, and half of 1 1/3 is 2/3, so you get 16 2/3%. Check by dividing: 100 ÷ 6 is 16, because 6 × 16 = 96, with 4 left over, and 4/6 = 2/3. 16% drops the 2/3, the same mistake as calling 1/3 33%. 6% uses the bottom number as the percent. 0.16% cuts the decimal 0.1666… short and sticks a % sign on it without moving the point. 1/6 is 16 2/3%.

Question 11 of 15 · Test level

A truck’s fuel tank holds 50 gallons. The driver’s log says the tank is 0.45 full. How many gallons does it take to fill the tank?

Show the answer

D. First, how much fuel is in the tank: 0.45 of 50. 0.45 is 45%. 10% of 50 is 5, so 40% is 20, and 5% is 2.5: 20 + 2.5 = 22.5 gallons. The empty part: 50 − 22.5 = 27.5. Faster: the empty part is 1 − 0.45 = 0.55 of the tank, and 0.55 of 50 is half of 55. 22.5 is the fuel already in the tank, 5 takes 45 gallons away from 50, and 0.55 is the empty part of the tank, not gallons. Gallons to fill the tank: 27.5.

Question 12 of 15 · Test level

A $45 jacket is on sale at three stores. Store A takes 1/3 off, Store B takes 30% off, and Store C takes $14 off. What is the lowest sale price?

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B. Use the easiest form for each store. Store A: 1/3 of 45 is 15, and 45 − 15 = 30. Store B: you pay 70%. 10% of 45 is 4.50, so 70% is 7 × 4.50: 7 × 4 = 28 and 7 × 0.50 = 3.50, so 28 + 3.50 = 31.50. Store C: 45 − 14 = 31. Treating 1/3 as 30% makes A and B look the same, but 1/3 is 33 1/3%, a bigger discount. $15.00 is the discount at Store A, not a price, and $31.50 and $31.00 are the other two stores. The lowest price is $30.00.

Question 13 of 15 · Test level

One water tank is 8/25 full. An identical tank is 3/8 full. How much more does the second tank hold, as a percent of a full tank?

Show the answer

B. Put both in percents. 25 × 4 = 100, so 8/25 = 32/100 = 32%. 3/8 is three times the benchmark 12.5%: 37.5%. The difference: 37.5% − 32% = 5.5%. 6% rounds 37.5% up to 38% too early, 55% is ten times too big (as a decimal the difference is 0.055, and moving the point two places right gives 5.5%, not 55%), and 0.055% is the decimal with a % sign stuck on. The second tank holds 5.5%.

Question 14 of 15 · Test level

A unit has a $3,200 budget. It spends 1/5 on fuel and 12.5% on parts. How much of the budget is left?

Show the answer

A. Both shares are taken from the whole $3,200. Fuel: 1/5 of 3,200 is 3,200 ÷ 5 = 640. Parts: 12.5% is the benchmark 1/8, and 3,200 ÷ 8 = 400. Left: 3,200 − 640 − 400 = 2,160. $2,240 takes the 12.5% from what is left after fuel (2,560 ÷ 8 = 320), $2,560 forgets the parts, and $2,800 forgets the fuel. Left: $2,160.

Question 15 of 15 · Test level

A trail is 12 miles long. A hiker covers 3/8 of the trail before lunch and 0.4 of the trail after lunch. How many miles are left?

Show the answer

A. Both parts are parts of the whole 12 miles. Before lunch: 1/8 of 12 is 1.5, so 3/8 is 3 × 1.5 = 4.5 miles. After lunch: 0.4 is 4 tenths, and a tenth of 12 is 1.2, so 4 × 1.2 = 4.8 miles. Hiked: 4.5 + 4.8 = 9.3. Left: 12 − 9.3 = 2.7. 9.3 is what was hiked, 7.5 forgets the miles after lunch, and 4.8 is only the miles after lunch. Miles left: 2.7.

Next: Flashcards

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