Decimal Place Value: Additional Practice (Set 1)

The leftmost $7$ is in the tenths place (value $\tfrac{7}{10}$) and the middle $7$ is in the hundredths place (value $\tfrac{7}{100}$). Each place is ten times the place to its right.

$\frac{7/10}{7/100} = \frac{100}{10} = 10$

So the leftmost $7$ is worth $10$ times the middle $7$.

One digit after the point means the denominator is $10$. Then reduce by the greatest common factor of $6$ and $10$, which is $2$.

$0.6 = \frac{6}{10} = \frac{3}{5}$

Pad to three places and compare from the left. All have $7$ tenths except $0.690$, which has $6$. Among the $7$-tenths group, compare hundredths: $0.700$ and $0.701$ have $0$, while $0.710$ has $1$.

$0.71 > 0.701 > 0.7 > 0.69$

So $0.71$ is greatest; more digits does not mean larger.

Place each digit in its named slot after the point: tenths $8$, hundredths $0$, thousandths $3$. The placeholder zero in the hundredths place keeps the $3$ in the thousandths place.

$0.\underbrace{8}_{\text{tenths}}\underbrace{0}_{\text{hundredths}}\underbrace{3}_{\text{thousandths}} = 0.803$

Both decimals share $5$ in the tenths place, so the hundredths place decides it: the blank must be less than $4$.

$\square < 4 \;\Rightarrow\; \square = 3 \text{ at most}$

With $3$, $0.53 < 0.54$. Any larger digit makes it at least $0.54$.

To make a decimal small, put the smallest digit in the highest place after the point (the tenths), then continue in increasing order.

$0.\underbrace{0}_{\text{tenths}}\underbrace{3}_{\text{hundredths}}\underbrace{6}_{\text{thousandths}} = 0.036$

A $0$ in the tenths place makes the value less than any choice with $3$ or $6$ there.

The first $4$ is in the tenths place (value $\tfrac{4}{10}$) and the second $4$ is in the thousandths place (value $\tfrac{4}{1000}$). Divide the values.

$\frac{4/10}{4/1000} = \frac{1000}{10} = 100$

So the first $4$ is worth $100$ times the second.

As tenths, $0.8 = \tfrac{8}{10}$, so it is eight tenths. Rewriting over a hundred, $\tfrac{8}{10} = \tfrac{80}{100}$, so it is eighty hundredths.

$0.8 = \frac{8}{10} = \frac{80}{100}$

So $0.8$ is $8$ tenths and $80$ hundredths.

A $0$ in the tenths place forces the form $0.0\square$, and the digits must sum to $9$, so the hundredths digit is $9 - 0 = 9$.

$0 + 9 = 9 \;\Rightarrow\; 0.09$

Only $0.09$ has a $0$ in the tenths place with digits summing to $9$.

Three digits after the point means the denominator is $1000$. The greatest common factor of $375$ and $1000$ is $125$, so divide the top and bottom by it.

$0.375 = \frac{375}{1000} = \frac{375 \div 125}{1000 \div 125} = \frac{3}{8}$

Pad to three places and compare each candidate with $0.470$ and $0.480$.

$0.470 < 0.472 < 0.480$

The others fall outside: $0.400$ and $0.469$ are below $0.47$, and $0.481$ is above $0.48$.

The leftmost $6$ is in the tenths place (value $\tfrac{6}{10} = \tfrac{600}{1000}$) and the rightmost $6$ is in the thousandths place (value $\tfrac{6}{1000}$). Subtract over a common denominator.

$\frac{600}{1000} - \frac{6}{1000} = \frac{594}{1000}$