Converting Between Fractions and Decimals: Core practice

Sixteen copies make eleven, so the number is eleven sixteenths, $\frac{11}{16}$, which means eleven divided by sixteen.

Sixteen fits six times into $110$ tenths, leaving $14$ tenths.

Those become $140$ hundredths, giving eight with $12$ hundredths left.

Those become $120$ thousandths, giving seven with $8$ thousandths left.

Those become $80$ in the next place, which gives five, with no remainder, in the fourth decimal place.

Thus

The multiplication check gives

Multiply the denominator by $2$ to reach $100$.

Multiplying the numerator by $2$ preserves the fraction’s value.

The new fraction names twenty-six hundredths, which reads directly as the decimal $0.26$.

So the form records $0.26$.

Divide the total amount by the number of jars.

Eight hundredths converts and reduces as follows.

Dividing numerator and denominator by $4$ gives

Nine shares of $0.08$ liters total $0.72$ liters, which checks the division.

Two fifths is twenty-four sixtieths and a twelfth is five sixtieths, so the total length is

This fraction is already in lowest terms, because $29$ is prime and does not divide $60$.

Dividing $29$ by $60$, sixty fits four times into $290$ tenths, leaving $50$.

Sixty fits eight times into $500$ hundredths, leaving $20$.

Sixty fits three times into $200$ thousandths, leaving $20$ again.

The remainder $20$ has returned, and the same remainder forces the same next digit forever, so the digit $3$ repeats after $0.48$.

The total length is exactly $0.48\overline{3}$ meters; stopping at any finite string of threes would not give the exact length.

The lit share is $52$ out of $320$.

Their greatest common factor is $4$.

The denominator $80=2\times2\times2\times2\times5$ contains no prime other than $2$ and $5$, so the decimal ends.

Multiplying numerator and denominator by $125$ gives

This is $0.1625$.

The fraction is in lowest terms because $13$ and $80$ have no common factor greater than one.

Reducing first kept the scaling short, since $320$ itself divides no power of ten smaller than a million.

The fraction with denominator $125=5\times5\times5$ ends.

Multiplying top and bottom by $8$ gives

It is exactly $0.136$ g.

The fraction $\frac{14}{45}$ is already reduced, and $45=3\times3\times5$ contains a prime other than $2$ or $5$.

Its decimal must repeat.

Dividing $140$ by $45$ gives the first digit $3$ and remainder $5$.

Each following step divides $50$ by $45$, giving digit $1$ and remainder $5$ again.

The scale cannot show the second weight exactly, because its decimal never ends.

As written, $150=2\times3\times5\times5$ and $175=5\times5\times7$ contain the primes $3$ and $7$, so each fraction must be reduced before the prime-factor test applies.

Divide the first numerator and denominator by $6$.

Divide the second numerator and denominator by $7$.

Both reduce to the same fraction.

Its denominator contains only the prime $5$, and multiplying top and bottom by $4$ gives

Thus both are exactly $0.12$, and Lena is correct.

Dividing $3$ by $11$ produces $2$ with remainder $8$, then $7$ with remainder $3$.

The starting remainder returns, so

The terminating decimal $0.2727$ begins with the given digits but has zeros in every later place.

The fraction has a $2$ in the fifth decimal place.

Therefore the beginning alone does not establish equality; a continuing pattern needs justification, such as a returning remainder.

Redo the division and record each remainder.

Thirteen fits into $50$ three times, leaving $11$, and into $110$ eight times, leaving $6$.

It fits into $60$ four times, leaving $8$, and into $80$ six times, leaving $2$.

It fits into $20$ once, leaving $7$, and into $70$ five times, leaving $5$.

The remainder after the sixth digit is $5$, the amount the division started with.

Each digit is decided entirely by the remainder before it, so the next six steps repeat the first six, and so on forever.

Since $5$ and $13$ share no factor larger than $1$, $\frac{5}{13}$ is in lowest terms, so the prime $13$ in its denominator, which is neither $2$ nor $5$, already shows that the decimal repeats; the returning remainder $5$ shows which block, $384615$, and no further dividing is needed.