Decimal Place Value: Additional Practice (Set 2)

Place each digit in its named slot after the point: tenths $5$, hundredths $0$, thousandths $9$. The placeholder zero in the hundredths place keeps the $9$ in the thousandths place.

$0.\underbrace{5}_{\text{tenths}}\underbrace{0}_{\text{hundredths}}\underbrace{9}_{\text{thousandths}} = 0.509$

In $0.25$ the $5$ is in the hundredths place, worth $\tfrac{5}{100}$. Ten times that is $\tfrac{5}{10}$, the tenths place.

$10 \times \frac{5}{100} = \frac{5}{10}$

Only $0.5$ has its $5$ in the tenths place. In $0.15$ the $5$ is still hundredths, and in the others it is thousandths.

In $0.43$ the $3$ is in the hundredths place, worth $\tfrac{3}{100}$. Ten times that is $\tfrac{3}{10}$, which is the tenths place.

$10 \times \frac{3}{100} = \frac{3}{10}$

Only $0.3$ has its $3$ in the tenths place. In $0.13$ the $3$ is still hundredths, and in the others it is thousandths.

Rewrite $0.7$ over a hundred. Since $\tfrac{7}{10} = \tfrac{70}{100}$, it is seventy hundredths.

$0.7 = \frac{70}{100}$

So $0.7$ contains $70$ hundredths.

Both decimals share $3$ in the tenths place, so the hundredths place decides it: the blank must be less than $6$.

$\square < 6 \;\Rightarrow\; \square = 5 \text{ at most}$

With $5$, $0.35 < 0.36$. Any larger digit makes it at least $0.36$.

Pad to three places: $0.080$, $0.800$, $0.808$, $0.880$. Compare the tenths first ($0$ for $0.08$, $8$ for the rest), then settle the $8$-tenths group on the hundredths and thousandths.

$0.08 < 0.8 < 0.808 < 0.88$

The first $2$ is in the tenths place (value $\tfrac{2}{10}$) and the second $2$ is in the thousandths place (value $\tfrac{2}{1000}$). Divide the values.

$\frac{2/10}{2/1000} = \frac{1000}{10} = 100$

So the first $2$ is worth $100$ times the second.

The last digit, $5$, sits in the thousandths place, and the digits "205" read as a whole number named by that last place.

$0.205 = \frac{205}{1000}$

So it reads "two hundred five thousandths." The $2$ is worth $\tfrac{2}{10}$, the $5$ is thousandths, and $0.205 \ne 0.25$, so the other statements are false.

Pad to three places and compare each candidate with the two bounds $0.300$ and $0.400$.

$0.300 < 0.351 < 0.400$

The others fall outside: $0.29$ is below $0.3$, and $0.41$ and $0.43$ are above $0.4$ in the tenths place.

Write both as fractions: $0.4 = \tfrac{4}{10}$ and $0.04 = \tfrac{4}{100}$. Divide one by the other.

$\frac{4/10}{4/100} = \frac{100}{10} = 10$

So $0.4$ is ten times $0.04$, just as each place is ten times the place to its right.

A $0$ in the tenths place forces the form $0.0\square$, and the digits must sum to $7$, so the hundredths digit is $7 - 0 = 7$.

$0 + 7 = 7 \;\Rightarrow\; 0.07$

Only $0.07$ has a $0$ in the tenths place with digits summing to $7$.

The leftmost $3$ is in the tenths place (value $\tfrac{3}{10} = \tfrac{300}{1000}$) and the rightmost $3$ is in the thousandths place (value $\tfrac{3}{1000}$). Subtract over a common denominator.

$\frac{300}{1000} - \frac{3}{1000} = \frac{297}{1000}$