Decimals

Study Sheet

Decimals

Everything you need to master decimal place value, operations, and word problems

Decimal Place Value; Reading & Writing Decimals

Concept
Places to the Right of the Point

A decimal point separates the whole-number part from the fractional part. Each place to the right of the point is worth ten times less than the place before it:

3⏟ones . 4⏟tenths 5⏟hundredths 6⏟thousandths\underbrace{3}_{\text{ones}}\,.\,\underbrace{4}_{\text{tenths}}\ \underbrace{5}_{\text{hundredths}}\ \underbrace{6}_{\text{thousandths}}

So 3.456=3+410+5100+610003.456 = 3 + \dfrac{4}{10} + \dfrac{5}{100} + \dfrac{6}{1000}. The place names go tenths, hundredths, thousandths, ten-thousandths, …\dots --- notice every fractional place name ends in “-ths.”

Example
Reading a decimal out loud

Read 52.0752.07 aloud. Read the whole part, say “and” for the point, read the fractional digits as a whole number, then name the last place.

  • Whole part: fifty-two.
  • The digits after the point are 0707, and the last digit (77) sits in the hundredths place.
  • So 52.07=52.07 = “fifty-two and seven hundredths.”
Example
Writing words as a decimal

Write “nine and thirteen thousandths” as a decimal. The word “and” marks the point, and “thousandths” means the last digit must land in the third place.

nine→9thirteen thousandths→0.013(fill to 3 places: 013)⇒ 9.013\begin{aligned} \text{nine} &\rightarrow 9 \\ \text{thirteen thousandths} &\rightarrow 0.013 \quad(\text{fill to 3 places: } 013)\\ \Rightarrow\ &9.013 \end{aligned}
Tip

The word “and” is reserved for the decimal point. So 305305 is “three hundred five,” but 3.053.05 is “three and five hundredths.” Never sprinkle extra “ands” into a whole number.

Comparing & Ordering Decimals

Concept
Line Up the Points, Compare Left to Right

To compare two decimals, stack them so the decimal points line up and add trailing zeros so both have the same number of places. Then compare digit by digit from the left --- the first place where they differ decides which is larger.

Example
Which is greater, 0.60.6 or 0.580.58?

Give them the same number of decimal places by writing 0.6=0.600.6 = 0.60.

0.60vs0.58tenths: 6>5\begin{aligned} 0.60 \quad&\text{vs}\quad 0.58 \\ \text{tenths: } 6 &> 5 \end{aligned}

The tenths digit already decides it: 0.60>0.580.60 > 0.58, so 0.6>0.580.6 > 0.58. A longer decimal is not automatically bigger!

Example
Ordering from least to greatest

Order   0.3, 0.29, 0.305, 0.31\;0.3,\ 0.29,\ 0.305,\ 0.31. Write each with three places:

0.300,0.290,0.305,0.3100.300,\quad 0.290,\quad 0.305,\quad 0.310

Now compare as if they were the whole numbers 300,290,305,310300, 290, 305, 310. Least to greatest:

0.29<0.3<0.305<0.31.0.29 < 0.3 < 0.305 < 0.31.
Tip

Adding zeros to the end of a decimal never changes its value: 0.6=0.60=0.6000.6 = 0.60 = 0.600. This trick lets you compare, add, and subtract decimals of different lengths safely.

Rounding Decimals

Concept
Look at the Next Digit

To round to a given place, find that place and look at the single digit just to its right:

  • If that digit is 55 or more, round up (add one to the rounding place).
  • If it is 44 or less, round down (leave the rounding place alone).

Then drop every digit after the rounding place.

Example
Round 7.3487.348 to the nearest hundredth

The hundredths digit is 44. The digit just to its right is 88.

7.348‾⇒8≥5, round up⇒ 7.35\begin{aligned} 7.34\underline{8} \quad&\Rightarrow\quad 8 \ge 5,\ \text{round up}\\ &\Rightarrow\ 7.35 \end{aligned}

Round 7.3487.348 to the nearest tenth instead: the tenths digit is 33, the next digit is 44, so it stays: 7.37.3.

Tip

Rounding up a 99 carries, just like in addition: 2.982.98 rounded to the nearest tenth is 3.03.0, because the 99 tenths becomes 1010 tenths. Keep the trailing zero to show the place you rounded to.

Adding & Subtracting Decimals

Concept
Line Up the Decimal Points

Write the numbers in a column so the decimal points sit directly above one another (this automatically lines up tenths with tenths, hundredths with hundredths). Fill empty places with zeros, then add or subtract exactly as with whole numbers. Bring the decimal point straight down into the answer.

Example
Adding 12.5+3.4712.5 + 3.47

Line up the points and write 12.512.5 as 12.5012.50:

12.50+ 3.4715.97\begin{aligned} \begin{array}{r} 12.50 \\ +\ 3.47 \\ \hline 15.97 \end{array} \end{aligned}

The point in the answer sits right below the points above it: 12.5+3.47=15.9712.5 + 3.47 = 15.97.

Example
Subtracting 8−2.658 - 2.65

Write the whole number 88 as 8.008.00 so both have two decimal places, then subtract with borrowing:

8.00− 2.655.35\begin{aligned} \begin{array}{r} 8.00 \\ -\ 2.65 \\ \hline 5.35 \end{array} \end{aligned}

So 8−2.65=5.358 - 2.65 = 5.35.

Tip

A whole number hides a decimal point at its right end: 8=8.0=8.008 = 8.0 = 8.00. Writing it in lets you line up the points and borrow correctly.

Multiplying Decimals

Concept
Multiply, Then Count the Places

Ignore the decimal points at first and multiply the numbers as whole numbers. Then count the total number of decimal places in both factors together, and place the decimal point in the answer so it has that many places.

Example
Multiplying 1.2×0.41.2 \times 0.4

Multiply as whole numbers: 12×4=4812 \times 4 = 48. Now count decimal places: 1.21.2 has 11 place and 0.40.4 has 11 place, for a total of 22.

1.2×0.4 ⇒ 12×4=482 decimal places ⇒ 0.48\begin{aligned} 1.2 \times 0.4 \ &\Rightarrow\ 12 \times 4 = 48 \\ 2\ \text{decimal places} \ &\Rightarrow\ 0.48 \end{aligned}

So 1.2×0.4=0.481.2 \times 0.4 = 0.48.

Example
When you must add a leading zero

Compute 0.3×0.020.3 \times 0.02. Multiply: 3×2=63 \times 2 = 6. Total decimal places: 1+2=31 + 2 = 3. The answer needs 33 places, so pad with zeros: 0.0060.006.

0.3×0.02=0.006.0.3 \times 0.02 = 0.006.
Tip

When multiplying, you do not line up the decimal points. Just count total decimal places in the factors and give the product that many. Unlike addition, the answer can have more places than either factor.

Dividing Decimals

Concept
Dividing by a Whole Number

When the divisor is a whole number, place the decimal point in the quotient directly above the point in the dividend, then divide normally. Add zeros to the dividend if you need to keep going.

9.6÷3=3.2,7÷4=1.75 (using 7.00).9.6 \div 3 = 3.2, \qquad 7 \div 4 = 1.75 \ (\text{using } 7.00).
Concept
Dividing by a Decimal --- Move the Point

When the divisor is a decimal, slide its point to the right until it becomes a whole number, and slide the dividend's point the same number of places (add zeros if needed). The answer is unchanged because you multiplied both by the same power of ten.

2.4÷0.6=24÷6=4.2.4 \div 0.6 = 24 \div 6 = 4.
Example
Dividing by a decimal, step by step

Compute 1.44÷0.121.44 \div 0.12. The divisor 0.120.12 has two decimal places, so move both points two places right:

1.44÷0.12 ⇒ 144÷12=12.\begin{aligned} 1.44 \div 0.12 \ &\Rightarrow\ 144 \div 12 \\ &= 12. \end{aligned}

So 1.44÷0.12=121.44 \div 0.12 = 12.

Tip

Whatever you do to the divisor's point, do the same to the dividend's point. Move both the same number of places --- never just one of them.

Converting Between Fractions & Decimals

Concept
Fraction →\to Decimal: Divide

To turn a fraction into a decimal, divide the numerator by the denominator. To turn a “nice” fraction into a decimal quickly, rewrite it with a denominator of 1010, 100100, or 10001000.

34=3÷4=0.75,710=0.7,920=45100=0.45.\frac{3}{4} = 3 \div 4 = 0.75, \qquad \frac{7}{10} = 0.7, \qquad \frac{9}{20} = \frac{45}{100} = 0.45.
Concept
Decimal →\to Fraction: Use the Place Value

Write the decimal's digits over the place value of the last digit, then simplify.

0.6=610=35,0.125=1251000=18.0.6 = \frac{6}{10} = \frac{3}{5}, \qquad 0.125 = \frac{125}{1000} = \frac{1}{8}.
Concept
Terminating vs. Repeating

When you divide, one of two things happens:

  • Terminating: the division ends with a remainder of 00. Example: 38=0.375\tfrac{3}{8} = 0.375.
  • Repeating: a block of digits repeats forever. Mark it with a bar. Example: 13=0.333⋯=0.3‾\tfrac{1}{3} = 0.333\dots = 0.\overline{3} and 211=0.1818⋯=0.18‾\tfrac{2}{11} = 0.1818\dots = 0.\overline{18}.

Both terminating and repeating decimals are rational numbers.

Example
A repeating-decimal conversion

Convert 56\tfrac{5}{6} to a decimal. Divide 5÷65 \div 6:

5.000÷6=0.8333…=0.83‾.\begin{aligned} 5.000 \div 6 &= 0.8333\dots \\ &= 0.8\overline{3}. \end{aligned}

The 88 appears once; only the 33 repeats, so the bar goes over just the 33.

Tip

A fraction in lowest terms gives a terminating decimal exactly when its denominator's only prime factors are 22 and 55 (the factors of 1010). So 78\tfrac{7}{8} terminates (8=238 = 2^3) but 16\tfrac{1}{6} repeats (6=2×36 = 2 \times 3).

Word Problems with Decimals (Money & Measurement)

Concept
Translate, Then Compute

Decide which operation the situation calls for, line up units, and keep money to two decimal places (cents). Common signals:

  • “total / altogether / sum” →\rightarrow add.
  • “how much more / change / difference” →\rightarrow subtract.
  • “each / per / times as much” →\rightarrow multiply.
  • “split evenly / per item / how many fit” →\rightarrow divide.
Example
Making change (money)

A book costs $12.75\$12.75 and you pay with a $20\$20 bill. How much change do you get? “Change” means subtract; write $20\$20 as $20.00\$20.00:

20.00− 12.757.25\begin{aligned} \begin{array}{r} 20.00 \\ -\ 12.75 \\ \hline 7.25 \end{array} \end{aligned}

Your change is $7.25\$7.25.

Example
Total cost then per-person (multi-step)

Four friends each buy a smoothie for $3.50\$3.50. What is the total, and if they split it evenly among 22 cars, how much does each car pay?

Total:4×$3.50=$14.00Per car:$14.00÷2=$7.00\begin{aligned} \text{Total:} \quad 4 \times \$3.50 &= \$14.00 \\ \text{Per car:} \quad \$14.00 \div 2 &= \$7.00 \end{aligned}

The total is $14.00\$14.00 and each car pays $7.00\$7.00.

Tip

Always attach the units to your answer and check that it makes sense. Money answers should show two decimal places ($7.00\$7.00, not $7\$7), and a length in meters should not suddenly turn into centimeters without converting.

Going Deeper: Advanced Decimal Ideas

Concept
Why Only 22 and 55 Make a Decimal Terminate

A decimal terminates exactly when it can be written with a denominator that is a power of ten. Every power of ten factors as 10n=2n×5n10^n = 2^n \times 5^n, so its only prime factors are 22 and 55. If a fraction in lowest terms has a denominator built only from 22s and 55s, you can multiply top and bottom by the “missing” factor to reach a power of ten:

740=723×5=7×5223×53=1751000=0.175.\frac{7}{40} = \frac{7}{2^3 \times 5} = \frac{7 \times 5^2}{2^3 \times 5^3} = \frac{175}{1000} = 0.175.

Any other prime (like 33 or 77) in the reduced denominator can never be cancelled into a power of ten, so the decimal must repeat instead.

Example
Predicting terminate vs. repeat by factoring

Without dividing, decide whether each fraction terminates. Factor each reduced denominator and check for primes other than 22 and 55.

980:80=24×5 ⇒ terminates512:12=22×3 ⇒ repeats (the 3)13200:200=23×52 ⇒ terminates\begin{aligned} \tfrac{9}{80}:\quad 80 &= 2^4 \times 5 \ \Rightarrow\ \textbf{terminates} \\ \tfrac{5}{12}:\quad 12 &= 2^2 \times 3 \ \Rightarrow\ \textbf{repeats}\ (\text{the } 3) \\ \tfrac{13}{200}:\quad 200 &= 2^3 \times 5^2 \ \Rightarrow\ \textbf{terminates} \end{aligned}

The number of decimal places when it terminates equals the larger of the two exponents: 980\tfrac{9}{80} needs 44 places (0.11250.1125), since 242^4 beats 515^1.

Concept
The 10x−x10x - x Trick for Repeating Decimals

A repeating decimal is an infinite sum, but a little algebra collapses it to a fraction. Set xx equal to the decimal, then multiply by the power of ten that shifts one full repeating block past the point. Subtracting the original cancels the infinite tail, leaving an ordinary equation to solve. If the repeating block has kk digits, multiply by 10k10^k.

Example
Converting 0.27‾0.\overline{27} to a fraction

The block “2727” has k=2k = 2 digits, so multiply by 102=10010^2 = 100. Let x=0.27‾x = 0.\overline{27}:

100x=27.272727…−x=27.0.272727…99x=27x=2799=311.\begin{aligned} 100x &= 27.272727\dots \\ -\quad x &= \phantom{27.}0.272727\dots \\ \hline 99x &= 27 \\ x &= \frac{27}{99} = \frac{3}{11}. \end{aligned}

So 0.27‾=3110.\overline{27} = \tfrac{3}{11}. Check: 3÷11=0.2727…3 \div 11 = 0.2727\dots Good. The denominator 9999 is always 9⋯9⏟k\underbrace{9\cdots9}_{k} for a block of length kk.

Example
A repeating decimal with a delay: 0.166‾0.16\overline{6}

Here one digit (11) sits before the repeating part, so shift twice. Let x=0.16666…x = 0.16666\dots; multiply by 1010 and by 100100 so the two versions line up on the repeating 66s:

100x=16.6666…−10x=11.6666…90x=15x=1590=16.\begin{aligned} 100x &= 16.6666\dots \\ -\quad 10x &= \phantom{1}1.6666\dots \\ \hline 90x &= 15 \\ x &= \frac{15}{90} = \frac{1}{6}. \end{aligned}

So 0.16‾=160.1\overline{6} = \tfrac{1}{6}, matching the division 1÷6=0.1666…1 \div 6 = 0.1666\dots

Tip

The famous identity 0.9‾=10.\overline{9} = 1 falls straight out of the trick: if x=0.9‾x = 0.\overline{9} then 10x=9.9‾10x = 9.\overline{9}, so 10x−x=910x - x = 9, giving 9x=99x = 9 and x=1x = 1. A repeating 99 is just another name for the whole number above it.

Concept
Scientific Notation: A Bridge for Tiny and Huge Decimals

Scientific notation writes a number as a single nonzero digit, a decimal part, and a power of ten: a×10na \times 10^n with 1≤a<101 \le a < 10. A positive exponent shifts the point right (big numbers); a negative exponent shifts it left (small decimals).

53,000=5.3×104,0.00047=4.7×10−4.53{,}000 = 5.3 \times 10^{4}, \qquad 0.00047 = 4.7 \times 10^{-4}.

The exponent is just a bookkeeping count of how many places the decimal point moved --- the same “move the point” idea from decimal multiplication and division.

Concept
Rounding Error and Bounds

When you round, the true value lives in a band around your rounded answer. Rounding to a place means the exact value is within half of that place. If xx rounds to 4.74.7 at the tenths place, then

4.65≤x<4.75,4.65 \le x < 4.75,

so the rounding error is at most 0.050.05. This is why measurements are reported with a place that reflects their precision: writing 4.704.70 instead of 4.74.7 claims a tighter band, 4.695≤x<4.7054.695 \le x < 4.705.

Concept
Decimals Are Dense: Always Room in Between

Between any two different decimals there is always another decimal --- in fact infinitely many. To find one, just add a place and pick a digit strictly between them, or average the two:

between 0.4 and 0.5:0.4+0.52=0.45,then 0.41, 0.42, …\text{between } 0.4 \text{ and } 0.5:\quad \frac{0.4 + 0.5}{2} = 0.45, \quad \text{then } 0.41,\ 0.42,\ \dots

Unlike whole numbers (nothing sits between 44 and 55), the decimals have no “next” number. This density is a key difference between counting numbers and the numbers on the full number line.

Tip

Terminating and repeating decimals are exactly the rational numbers (ratios of whole numbers). A decimal that goes on forever without ever repeating --- like π=3.14159…\pi = 3.14159\dots or 2=1.41421…\sqrt{2} = 1.41421\dots --- is irrational and can never be written as a fraction. Every such number still has a home on the number line.

Formulas, Proofs & Tips

Tip
Decimals, fractions and place value
0.d1d2…dn⏟n places=d1d2…dn10n0.\underbrace{d_1d_2\ldots d_n}_{n\ \text{places}} = \frac{d_1d_2\ldots d_n}{10^{n}}

What it means. Each place is a power of ten, so any terminating decimal is a fraction over a power of 1010.

Example. 0.24=24100=6250.24=\tfrac{24}{100}=\tfrac{6}{25}.

Why it works. The digits after the point count tenths, hundredths, thousandths — that is d110+d2100+⋯\tfrac{d_1}{10}+\tfrac{d_2}{100}+\cdots, which over the common denominator 10n10^n is the whole digit string over 10n10^n.

Tip. Multiplying decimals: multiply as whole numbers, then give the answer as many decimal places as the two factors had together.