Question bank — Converting - fractions ↔ decimals ↔ percentages

$0.5=5\%$ ::: False. $0.5$ means five tenths $=\frac{50}{100}=50\%$. To reach percent you multiply by 100, not by 10 — so $0.5\times 100=50\%$. Half a thing is obviously half of 100, i.e. $50\%$. $\frac{1}{4}=1.4\%$ ::: False. You cannot glue the numerator and denominator into a decimal. $\frac{1}{4}$ is a division: $1÷4=0.25$, then $\times 100=25\%$. A percentage can never be more than $100\%$ ::: False. $100\%$ is one whole, but nothing stops you having more than one whole. $150\%=\frac{150}{100}=1.5$ — one and a half wholes. Prices rise "$200\%$", populations grow past $100\%$. $0.7$ and $0.70$ are the same number ::: True. Trailing zeros after the last non-zero decimal digit add nothing: $\frac{7}{10}=\frac{70}{100}$. Both equal $70\%$. (See Decimal place value.) $\frac{3}{4}=\frac{6}{8}$, so they convert to different percentages ::: False. Equivalent fractions are the same number, so they give the same percentage: both are $0.75=75\%$. Simplifying never changes the value (see Fractions - simplifying and equivalent fractions). To turn any decimal into a percentage you "move the point two places right" ::: True in mechanics, dangerous in memory. Moving the point two right is $\times 100$. But students forget the direction. Anchor to meaning: percent asks "how many hundredths?", so you always multiply by 100 going to percent. $\frac{1}{3}=33\%$ exactly ::: False. $\frac{1}{3}=0.\overline{3}$ (the bar means the $3$ repeats forever), so $\times 100=33.\overline{3}\%=33\frac{1}{3}\%$. The value $33\%$ is only a rounded approximation (see Recurring decimals to fractions). $0\%=0=\frac{0}{b}$ for any $b\ne 0$ ::: True. Zero out of a hundred is zero; zero parts of anything is zero. Every form of the number zero collapses to the single point $0$ on the number line.

"$0.05=50\%$ because I move the point two places." ::: Error: wrong direction / miscount. $0.05=\frac{5}{100}=5\%$. Moving the point two places right on $0.05$ gives $5$, not $50$; so it's $5\%$. "$0.7=\frac{7}{100}$ because percent means out of 100." ::: Error: place value ignored. One digit after the point = tenths, so $0.7=\frac{7}{10}$. The number of decimal places sets the number of zeros in the denominator, not the word "percent". "$\frac{5}{8}$: since $8÷5=1.6$, we get $160\%$." ::: Error: division flipped. $\frac{a}{b}=a÷b$, so it's $5÷8=0.625=62.5\%$. Dividing the wrong way inverts the fraction (see Long division). "$0.35=\frac{35}{100}=\frac{7}{20}$… actually let me leave it as $\frac{35}{100}$, same thing." ::: Error: not fully simplified. Both are the same value, but "convert" usually demands lowest terms: since $\gcd (35,100)=5$ (the biggest number dividing both), $\frac{35}{100}=\frac{35÷5}{100÷5}=\frac{7}{20}$. "$150\%$ can't be right — you can't have more than the whole, so cap it at $100\%$." ::: Error: false ceiling. $100\%$ is one whole, not a maximum. $150\%=1.5$ is perfectly valid. "$\frac{1}{3}\times 100=33.3\%$, so $\frac{1}{3}=33.3\%$ exactly." ::: Error: rounded a recurring decimal. $\frac{1}{3}=0.\overline{3}$ (the bar = repeats forever), so exactly it's $33.\overline{3}\%=33\frac{1}{3}\%$. Writing $33.3$ silently threw away the endless tail. "$1.2$ as a fraction is $\frac{12}{100}=\frac{3}{25}$." ::: Error: miscounted places. $1.2$ has one decimal place $\Rightarrow \frac{12}{10}=\frac{6}{5}$. It's bigger than one, so the fraction must exceed $1$ — $\frac{3}{25}$ is far too small.

Why does turning a fraction into a decimal mean dividing? ::: Because $\frac{a}{b}$ is defined as "$a$ shared into $b$ equal parts", which is exactly the division $a÷b$. It's the meaning, not a trick. Why do we multiply by 100 (and not 10 or 1000) to get a percentage? ::: Because "per-cent" literally means "per hundred" — the whole is chopped into exactly $100$ pieces. Multiplying by 100 counts how many hundredths the number is, which is precisely what percent asks. Why does replacing "$\%$" with "$÷100$" always work? ::: Because "$\%$" is pure shorthand for "$÷100$" — they mean the same operation. So $75\%=75÷100=0.75$ by definition, not by coincidence. Why does the denominator's number of zeros match the decimal places? ::: Each decimal place is one more factor of ten smaller: tenths, hundredths, thousandths. $n$ places means the last digit sits over $10^n$, which has $n$ zeros. Why is keeping $\frac{1}{3}$ as a fraction "more honest" than writing $0.33$? ::: Because $0.33$ terminates and $\frac{1}{3}=0.\overline{3}$ does not — they are genuinely different numbers. The fraction stores the exact value; the decimal only approximates it. Why can two different-looking fractions give the same percentage? ::: Equivalent fractions occupy the same point on the number line, so they represent one number wearing different costumes — and one number has one percentage.

What is $0$ as a fraction, decimal and percent? ::: $\frac{0}{b}=0=0.0=0\%$ for any $b\ne 0$. All three costumes collapse onto the single point $0$. What is $1$ (one whole) as a percentage? ::: $1\times 100=100\%$. One whole is exactly one hundred hundredths — that's what $100\%$ means. How do you write a percentage bigger than $100\%$, e.g. $250\%$, as a mixed number? ::: $250\%=\frac{250}{100}=\frac{5}{2}=2\frac{1}{2}$ — two and a half wholes. The rule $p\%=\frac{p}{100}$ works unchanged past 100. Can a percentage be a fraction of a percent, like $12.5\%$? ::: Yes. $12.5\%=\frac{12.5}{100}=\frac{125}{1000}=\frac{1}{8}$. Percentages don't have to be whole numbers. What about a negative value, e.g. a $-20\%$ change? ::: $-20\%=\frac{-20}{100}=-0.2$. The $÷100$ rule ignores sign; the minus just travels along (see Percentage increase and decrease). How do you convert a recurring decimal like $0.\overline{6}$ to a fraction, and why does the method work? ::: Place value fails on an endless tail, so use the multiply-and-subtract trick: let $x=0.\overline{6}$. Multiply by $10$ to shift one whole repeat: $10x=6.\overline{6}$. Subtracting, the identical endless tails cancel: $10x-x=6.\overline{6}-0.\overline{6}=6$, so $9x=6$ and $x=\frac{6}{9}=\frac{2}{3}$. The subtraction is the whole point — it deletes the infinite tail (see Recurring decimals to fractions). Is $\frac{a}{b}$ with $b=0$ ever a valid number to convert? ::: No. Division by zero is undefined, so $\frac{a}{0}$ names no point on the number line — there is nothing to convert. When does a fraction give a terminating decimal versus a recurring one? ::: It terminates exactly when the simplified denominator's only prime factors (the primes you multiply to build it) are $2$ and $5$ — the primes hiding inside ten. E.g. $\frac{1}{8}$ has $8=2\times 2\times 2$, all $2$s, so it terminates ($0.125$); $\frac{1}{3}$ has the prime $3$, so long division never hits remainder $0$ and it recurs.