Decimals: Chapter Review

Vocabulary and notation

Decimal point

The mark separating the whole part from the fractional part. It is not itself a place: ones sits immediately left of it, tenths immediately right.

Expanded form

A decimal written as the sum of each digit times its place value: $6.209 = 6 + \tfrac{2}{10} + \tfrac{9}{1000}$.

Terminating decimal

A decimal whose digits stop, which is what happens when long division reaches a remainder of $0$: $\tfrac{7}{16} = 0.4375$.

Repeating decimal, $0.\overline{3}$

A decimal whose digits never stop but cycle through a fixed block. The overline marks the block, so $0.1\overline{6}$ is $0.1666\ldots$.

Dividend and divisor

In $a \div b$, the dividend is $a$ and the divisor is $b$.

Rounding place and deciding digit

Rounding to a place replaces a number with the nearest multiple of that place's value. The rounding place is the one named; the deciding digit is the single digit immediately to its right.

Estimation

Rounding the numbers of a calculation before computing, for a fast approximate answer and a size check on an exact one.

Front-end estimation

Keeping only each number's leading digit and zeroing the rest. For a sum it runs low, since chopping only discards value: $381 + 264 \approx 500$, against $645$.

Formulas and theorems

• Place values to the right of the point

  $$1, \quad \frac{1}{10}, \quad \frac{1}{100}, \quad \frac{1}{1000}, \quad \ldots$$

  Text description

  A place value chart for 6.209 naming ones, tenths, hundredths and thousandths, with each place worth one tenth of the place to its left.

  Use when On both sides of the point, every place is one tenth of the place to its left. First after the point is tenths, then hundredths, then thousandths.

  e.g. In $0.404$ the left $4$ is $\tfrac{4}{10}$ and the right $4$ is $\tfrac{4}{1000}$, so the left is $100$ times the right.

• Decimal to fraction by place

  $$\begin{gathered} 0.7 = \frac{7}{10} \\ 0.49 = \frac{49}{100} \\ 0.013 = \frac{13}{1000} \end{gathered}$$

  Use when Terminating decimals only. Digits after the point equal zeros in the denominator, and those digits are the numerator; then reduce by the GCF. A whole part stays put, so $2.6 = 2\tfrac{3}{5}$.

• Reading and writing a decimal in words

  Read the whole part, say "and" at the point, read the digits after it as one whole number, then name the place of the LAST digit.

  Use when That last place names the denominator for the whole string, so $0.41$ is "forty-one hundredths". From words, the named place fixes how many digits follow the point, sometimes needing a placeholder zero: "two and seven hundredths" is $2.07$, not $2.7$.

• Trailing zeros versus placeholder zeros

  $$0.3 = 0.30 = 0.300$$

  Use when Zeros PAST THE POINT and right of the last nonzero digit only: a trailing zero on a whole number is a place, so $3 \neq 30$. A zero between the point and a nonzero digit holds that digit out one place, so removing it multiplies the value by ten: $0.05$ against $0.5$.

• Adding and subtracting decimals

  $$\begin{array}{r} 9.40 \\ -\ 2.65 \\ \hline 6.75 \end{array}$$

  Use when Addition and subtraction only. Line up the points, pad with trailing zeros so no column is empty, work right to left, and drop the point straight down.

• Multiplying decimals: the places add

  $$\begin{gathered} \text{places in the product} \\ = \text{places in one factor} \\ + \text{places in the other} \end{gathered}$$

  Use when The points are NOT lined up: multiply as whole numbers, then count. Prepend zeros when the product is too short to hold the point; a trailing zero the count creates may be dropped, so $0.06 \times 0.5 = 0.030 = 0.03$.

  e.g. $1.2 \times 0.35$: $12 \times 35 = 420$, and $1 + 2 = 3$ places give $0.42$.

• Multiplying or dividing by $10$, $100$, $1000$

  $$\begin{gathered} 0.048 \times 100 = 4.8 \\ 52.7 \div 1000 = 0.0527 \end{gathered}$$

  Use when Count the zeros: that is how many places the point moves, right to multiply and left to divide. Append zeros wherever the digits run out.

• Dividing by a decimal: shift both

  $$a \div b = (a \times 10) \div (b \times 10)$$

  Use when $b \neq 0$. Shift the divisor's point right until it is whole and the dividend's the same number of places, leaving the quotient unchanged. An already-whole divisor needs no shift. Append zeros to the dividend when its digits run out.

  e.g. $6.5 \div 0.13 = 650 \div 13 = 50$, while $3.5 \div 4 = 0.875$ shifts nothing.

• Fraction to decimal: the bar means divide

  $$\frac{a}{b} = a \div b$$

  Use when $b \neq 0$, and the numerator is the dividend: top divided by bottom, never bottom by top. Append zeros after the point in the dividend until the remainder clears or a block of digits recurs.

  e.g. $\tfrac{7}{16} = 7 \div 16 = 0.4375$.

• Scaling shortcut to a decimal

  $$\frac{a}{b} = \frac{a \times k}{b \times k}$$

  Use when Available only when a whole $k$ makes $b \times k$ equal $10$, $100$, or $1000$, that is when $b$ divides one of them. The new numerator is then read off as the decimal digits.

  e.g. $\tfrac{11}{25} = \tfrac{44}{100} = 0.44$.

• Terminating test

  $$\begin{gathered} \frac{a}{b} \text{ in lowest terms terminates} \\ \iff \text{every prime factor of } b \\ \text{is } 2 \text{ or } 5 \end{gathered}$$

  Use when Reduce first: the test reads the REDUCED denominator. Both directions hold there, so any surviving prime other than $2$ or $5$ forces a repeat, and a denominator of $1$ has no prime factors at all and terminates.

  e.g. $40 = 2 \times 2 \times 2 \times 5$, so $\tfrac{9}{40} = 0.225$; $12 = 2 \times 2 \times 3$, so $\tfrac{7}{12} = 0.58\overline{3}$.

• The rounding rule

  $$\begin{gathered} \text{deciding digit} \ge 5 \;\Rightarrow\; \text{up} \\ \text{deciding digit} < 5 \;\Rightarrow\; \text{down} \end{gathered}$$

  Use when Exactly one digit is consulted; everything beyond it is ignored. The answer keeps the rounding place, trailing zero included. An exact tie is settled by agreement, and round half up sends it up.

  e.g. $7.85$ to the nearest tenth is $7.9$, and $19.98$ to the nearest tenth is $20.0$.

• Which numbers round to a given value

  $$L - \frac{u}{2} \;\le\; x \;<\; L + \frac{u}{2}$$

  Use when $L$ is the rounded result and $u$ is one unit of the rounding place. Under round half up the lower midpoint is included and the upper one is not.

Problem types, step by step

Exam traps

• Trap Lining the points up to multiply, so $0.3 \times 0.2$ comes out as $0.6$.

  Fix Alignment belongs to addition and subtraction. Multiply as whole numbers and add the places: $3 \times 2 = 6$ with $1 + 1 = 2$ places, giving $0.06$.

• Trap Ranking decimals by how many digits they show, so $0.45$ looks bigger than $0.5$.

  Fix Pad to equal length and read from the left: $0.45$ against $0.50$ is settled at the tenths, so $0.45 < 0.5$.

• Trap Dropping the leading zero a product needs, writing $0.25 \times 0.3 = 0.75$.

  Fix Three places were counted, so the point sits three digits in from the right of $75$: it is $0.075$, ten times smaller.

• Trap Sliding the point in the divisor only, turning $9.6 \div 0.04$ into $9.6 \div 4 = 2.4$.

  Fix Both points move the same number of places, so it is $960 \div 4 = 240$. Moving one alone rescales the quotient, here by $100$.

• Trap Assuming multiplication always enlarges and division always shrinks.

  Fix A positive factor below $1$ shrinks ($0.4 \times 0.4 = 0.16$) and dividing a positive by one below $1$ enlarges ($9.6 \div 0.04 = 240$). A size check must allow both.

• Trap Dividing bottom by top, computing $\tfrac{3}{4}$ as $4 \div 3$.

  Fix The bar is top divided by bottom, so $3 \div 4 = 0.75$. The reversed order gives $1.33\ldots$, above $1$ where the fraction is below it.

• Trap Applying the $2$s-and-$5$s test before reducing, calling $\tfrac{3}{6}$ a repeater on the strength of its $3$.

  Fix Reduce first: $\tfrac{3}{6} = \tfrac{1}{2}$, whose denominator is $2$, so it terminates at $0.5$.

• Trap Rounding digit by digit from the right, so $6.249$ to the nearest tenth becomes $6.3$.

  Fix Only the hundredths digit is consulted, and $4 < 5$, so the answer is $6.2$. The $9$ never gets a vote.

• Trap Writing $\tfrac{1}{3} = 0.33$.

  Fix That is a rounded value, not an equality. The exact decimal is $0.\overline{3}$, whose $3$s never stop.

• Trap Rounding a $9$ in place, turning $2.96$ to the nearest tenth into $2.10$.

  Fix Ten tenths is one whole, so the carry lands in the ones: $3.0$, keeping the trailing zero to show the tenths was the target.

• Trap Rounding every input first when the question asks for an exact answer.

  Fix Compute exactly and round the final result once. Rounding first is for estimates, where the lost accuracy buys speed.