Decimal Place Value: Practice

The leftmost $5$ is in the tenths place (value $\tfrac{5}{10}$) and the rightmost $5$ is in the thousandths place (value $\tfrac{5}{1000}$). Divide the two values.

$\frac{5/10}{5/1000} = \frac{1000}{10} = 100$

So the leftmost $5$ is worth $100$ times the rightmost $5$.

Pad to three places and compare from the left. The tenths digit is $4$ for all but $0.390$, which has $3$. Among the $4$-tenths group, compare hundredths: $0.400$ and $0.401$ have $0$, while $0.410$ has $1$.

$0.41 > 0.401 > 0.4 > 0.39$

So $0.41$ is greatest; more digits does not mean larger.

Three digits after the point means the denominator is $1000$. The greatest common factor of $875$ and $1000$ is $125$, so divide the top and bottom by it.

$0.875 = \frac{875}{1000} = \frac{875 \div 125}{1000 \div 125} = \frac{7}{8}$

Rewrite $0.4$ with the same denominator as one hundredth. Since $\tfrac{4}{10} = \tfrac{40}{100}$, it is forty hundredths.

$0.4 = \frac{40}{100}$

So $0.4$ contains $40$ hundredths.

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

$0.\underbrace{0}_{\text{tenths}}\underbrace{4}_{\text{hundredths}}\underbrace{7}_{\text{thousandths}} = 0.047$

A $0$ in the tenths place makes the value less than any choice with $4$ or $7$ there.

Pad to three places: $0.060$, $0.600$, $0.606$, $0.660$. Compare the tenths first ($0$ for $0.06$, $6$ for the rest), then settle the $6$-tenths group on the hundredths and thousandths.

$0.06 < 0.6 < 0.606 < 0.66$

A value of $\tfrac{6}{100}$ puts the $6$ in the hundredths place, the second place after the point.

$6 \times \frac{1}{100} = \frac{6}{100}$

Only $0.36$ has the $6$ as its second digit after the point. In $0.6$ and $0.63$ the $6$ is in the tenths place (worth $\tfrac{6}{10}$), and in $0.006$ it is in the thousandths place (worth $\tfrac{6}{1000}$).

As tenths, $0.5 = \tfrac{5}{10}$, so it is five tenths. Rewriting over a hundred, $\tfrac{5}{10} = \tfrac{50}{100}$, so it is fifty hundredths.

$0.5 = \frac{5}{10} = \frac{50}{100}$

So $0.5$ is $5$ tenths and $50$ hundredths.

The last digit, $8$, sits in the thousandths place, and the digits "408" read as a whole number named by that last place.

$0.408 = \frac{408}{1000}$

So it reads "four hundred eight thousandths." The $4$ is worth $\tfrac{4}{10}$, the $8$ is thousandths, and $0.408 \ne 0.48$, so the other statements are false.

Write both as fractions: $0.7 = \tfrac{7}{10}$ and $0.07 = \tfrac{7}{100}$. Divide one by the other.

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

So $0.7$ is ten times $0.07$, just as each place is ten times the place to its right.

To make a decimal large, put the largest digit in the highest place after the point (the tenths), then continue in decreasing order.

$0.\underbrace{9}_{\text{tenths}}\underbrace{2}_{\text{hundredths}}\underbrace{0}_{\text{thousandths}} = 0.920$

Putting $9$ in the tenths place outweighs any arrangement with a smaller tenths digit.

Pad to three places and compare each candidate with $0.620$ and $0.630$.

$0.620 < 0.625 < 0.630$

The others fall outside: $0.600$ and $0.619$ are below $0.62$, and $0.631$ is above $0.63$.