Switching Between Fractions and Decimals

The $7$ sits in the tenths place, so $0.7$ is seven tenths, which is $\boxed{{\displaystyle 7\mathrm{/}10}}$. Every terminating decimal works this way. The last filled place names the denominator, and the digits after the point read straight across as the numerator.

Two digits past the point means hundredths, so $0.45=\frac{45}{100}$. Since $45=3\times 3\times 5$ and $100=2\times 2\times 5\times 5$, the only shared prime is a single $5$, so $\gcd (45,100)=5$. $$\frac{45}{100}=\frac{45÷5}{100÷5}=\frac{9}{20}$$ $\boxed{{\displaystyle 9\mathrm{/}20}}$ The top $9=3\times 3$ and the bottom $20=2\times 2\times 5$ share no prime, which is how you know it is all the way reduced.

Read all the digits as one number, $25$. One digit sits past the point, so the denominator is $10^1=10$, giving $2.5=\frac{25}{10}$. Both share a factor of $5$, so $$\frac{25}{10}=\frac{25÷5}{10÷5}=\frac{5}{2}\mathrm{.}$$ Since $5>2$ this is already improper, which is what was asked. $\boxed{{\displaystyle 5\mathrm{/}2}}$ The whole part out front changes nothing. Only the count of digits after the point sets the power of ten underneath.

Convert the size first. The last digit of $1.72$ is in the hundredths place, so $1.72=\frac{172}{100}$. Since $\gcd (172,100)=4$, $$\frac{172}{100}=\frac{172÷4}{100÷4}=\frac{43}{25}\mathrm{.}$$ Put the minus sign back and you have $\boxed{{\displaystyle -43\mathrm{/}25}}$. It is fully reduced because $43$ is prime. The sign takes no part in the reducing, so set it aside and reattach it at the end.

A denominator of $100$ means hundredths, so the numerator fills two places to the right of the point. The $7$ drops into the hundredths place and the $4$ sits in the tenths place. Since $47<100$ there is no whole part, so a $0$ holds the ones place. That gives $\frac{47}{100}=\boxed{{\displaystyle 0.47}}$. This is the same place-value reading you used going the other way, just run backward. Two zeros in the denominator, two places after the point.

Since $5\times 2=10$, multiply top and bottom by $2$. $$\frac{3}{5}=\frac{3\times 2}{5\times 2}=\frac{6}{10}$$ Six tenths is exactly what the first place after the point holds, so the answer is $\boxed{{\displaystyle 0.6}}$. Multiplying top and bottom by the same number is multiplying by $\frac{2}{2}=1$, so the value never moves. Only the name changes.

$20$ is bigger than $10$, so aim for $100$. Since $100÷20=5$ with nothing left over, multiply top and bottom by $5$. $$\frac{11}{20}=\frac{11\times 5}{20\times 5}=\frac{55}{100}$$ That is $55$ hundredths, or $\boxed{{\displaystyle 0.55}}$. It should land just past $0.5$, since $\frac{10}{20}=0.5$ and you have one twentieth more than that.

$8=2^3$, so the bottom carries three $2$s and no $5$s. A power of ten pairs its $2$s and $5$s evenly, and $1,000=2^3\times 5^3$, so the denominator is short exactly three $5$s. Multiply top and bottom by $5^3=125$. $$\frac{7}{8}=\frac{7\times 125}{8\times 125}=\frac{875}{1000}$$ A denominator of $1,000$ means three decimal places, so $\boxed{{\displaystyle 0.875}}$. You only ever have to top up whichever of $2$ or $5$ the denominator is missing.

Factor the denominators. $16=2^4$, $40=2^3\times 5$, $25=5^2$, and $12=2^2\times 3$. A power of ten is $2^n\times 5^n$, so the first three can be scaled up to one, but the $3$ inside $12$ cannot. So ${\displaystyle \frac{5}{12}}$ is the fraction that never terminates, and its denominator is $\boxed{{\displaystyle 12}}$. Multiplying by more $2$s and $5$s only adds factors. It can never remove a $3$ that is already sitting on the bottom.

The $75$ sits in the hundredths place, so $5.75=\frac{575}{100}$. Both share a factor of $25$, and that reduces to $\frac{23}{4}$. Since $4\times 2=8$, multiply top and bottom by $2$. $$\frac{23}{4}=\frac{23\times 2}{4\times 2}=\frac{46}{8}$$ There are $\boxed{{\displaystyle 46}}$ eighths in $5.75$. Asking how many of a unit fit inside a number is just renaming the number with that unit on the bottom and reading the top.

$2.5=\frac{25}{10}$, which reduces to $\frac{5}{2}$. Flipping gives the reciprocal $\frac{2}{5}$. Multiply top and bottom by $2$ to reach tenths, $\frac{2}{5}=\frac{4}{10}$, and four tenths is $\boxed{{\displaystyle 0.4}}$. Flipping is easy on a fraction and awkward on a decimal, so switch names first. Check it with $2.5\times 0.4=1$.

$0.84$ means eighty-four hundredths, so $0.84=\frac{84}{100}$. Their $\gcd$ is $4$, so $$\frac{84}{100}=\frac{84÷4}{100÷4}=\frac{21}{25}\mathrm{.}$$ The division was already in simplest form, so the divisor had to be $25$, $50$, $75$, and so on. Only $25$ is under $30$, which puts $\boxed{{\displaystyle 21}}$ on top. The screen could show a clean ending decimal because $25=5\times 5$ is built from nothing but $5$s.

The $3$ sits one spot right of the point, in the tenths place, so $0.3=\frac{3}{10}$, which is $\boxed{{\displaystyle 3\mathrm{/}10}}$. It is already in lowest terms, since $3$ is prime and $10=2\times 5$ shares nothing with it.

The $6$ is in the tenths place, so $0.6$ means six tenths, $\frac{6}{10}$. Both are even and $\gcd (6,10)=2$, so $$\frac{6}{10}=\frac{6÷2}{10÷2}=\frac{3}{5}\mathrm{.}$$ The bottle is $\boxed{{\displaystyle 3\mathrm{/}5}}$ full. Simplest form is the point where the top and bottom share no factor bigger than $1$, and $3$ and $5$ do not.

Two digits past the point means hundredths, so $0.85=\frac{85}{100}$. Since $85=5\times 17$ and $100=5\times 20$, the greatest common divisor is $5$. $$\frac{85}{100}=\frac{85÷5}{100÷5}=\frac{17}{20}$$ The poll result is $\boxed{{\displaystyle 17\mathrm{/}20}}$. $17$ is prime and does not divide $20$, so nothing more comes out.

The last digit lands in the thousandths place, so $0.375=\frac{375}{1,000}$. Since $375=125\times 3$ and $1,000=125\times 8$, the gcd is $125$, and dividing both by it gives $\boxed{{\displaystyle 3\mathrm{/}8}}$. Check it by going back the other way, $3÷8=0.375$.

Convert the size first. The $6$ is in the tenths place, so $2.6=\frac{26}{10}$. Since $\gcd (26,10)=2$, that reduces to $\frac{26÷2}{10÷2}=\frac{13}{5}$. Put the sign back for $\boxed{{\displaystyle -13\mathrm{/}5}}$. It is improper and in lowest terms already, since $13$ is prime and bigger than $5$.

A denominator of $100$ counts hundredths, so the numerator fills the first two places after the point. Drop $73$ into the tenths and hundredths spots with a $0$ in front, and the gauge reads $\boxed{{\displaystyle 0.73}}$ of a liter. When the denominator is already a power of ten there is nothing to divide. You just read the numerator into its places.

Since $5\times 2=10$, multiply top and bottom by the same $2$, so $\frac{4}{5}=\frac{8}{10}$. Eight tenths sits in the first place after the point, giving $\boxed{{\displaystyle 0.8}}$. The decimal stops after a single digit because $5$ divides evenly into $10$.

Since $20\times 5=100$, multiply top and bottom by $5$. $$\frac{13}{20}=\frac{13\times 5}{20\times 5}=\frac{65}{100}$$ Sixty-five hundredths is $0.65$, so the jar is $\boxed{{\displaystyle 0.65}}$ full. It terminates because $20=2^2\times 5$ carries only $2$s and $5$s.

$8=2^3$, so the bottom holds three $2$s and no $5$s. The smallest power of ten using three $2$s is $10^3=2^3\times 5^3=1,000$, so supply $5^3=125$. $$\frac{5}{8}=\frac{5\times 125}{8\times 125}=\frac{625}{1,000}$$ Six hundred twenty-five thousandths is $\boxed{{\displaystyle 0.625}}$. Check it another way. $\frac{4}{8}=0.5$, and one more eighth is $0.125$, so $0.5+0.125=0.625$.

Factor each denominator. $8=2^3$ and $40=2^3\times 5$ use only $2$s and $5$s, so both terminate. But $15=3\times 5$ carries a $3$, and the fraction is already in lowest terms, so nothing cancels it. The answer is $\boxed{{\displaystyle 15}}$. A power of ten is $2^n\times 5^n$, so a denominator holding any other prime can never be scaled into one.

The last digit of $1.25$ is in the hundredths place, so $1.25=\frac{125}{100}$, which reduces to $\frac{5}{4}$. Flipping gives $\frac{4}{5}$, and multiplying top and bottom by $2$ gives $\frac{8}{10}$. The low-power factor is $\boxed{{\displaystyle 0.8}}$. Check it with $1.25\times 0.8=1$, which is what a reciprocal pair has to do.

Since $50\times 2=100$, multiply top and bottom by $2$. $$\frac{17}{50}=\frac{17\times 2}{50\times 2}=\frac{34}{100}$$ A denominator of $100$ means two decimal places, so the glaze is $\boxed{{\displaystyle 0.34}}$ cobalt. $50=2\times 5^2$ holds only $2$s and $5$s, so a clean decimal was guaranteed.

Two digits sit after the point, so $3.25=\frac{325}{100}$. Both share a factor of $25$, which reduces it to $\frac{13}{4}$. Since $4\times 2=8$, multiply top and bottom by $2$. $$\frac{13}{4}=\frac{13\times 2}{4\times 2}=\frac{26}{8}$$ So the shelf holds $\boxed{{\displaystyle 26}}$ eighths of a foot. Renaming does not change the length, only the size of the piece you count with. Check it with $26÷8=3.25$.

$0.65$ means sixty-five hundredths, so start from $\frac{65}{100}$. Since $65=5\times 13$ and $100=5\times 20$, that reduces to $\frac{13}{20}$. The divisor is the bottom number, and $20$ is indeed under $25$, so the number divided was $\boxed{{\displaystyle 13}}$. $13$ and $20$ share no factor bigger than $1$, which confirms the lowest-terms clue.