Decimal Expansions

Decimal Expansions exercise

• What is the place value of the digit $4$ in each of the numbers?

  Hints

  Place value charts are helpful when deciding the value of a digit.

  For example, $162.57$ has $1$ hundred, $6$ tens, $2$ ones, $5$ tenths and $7$ hundredths.

  There are three numbers in each group.

  Solution

  $4$ ones

  • $4.8$
  • $4.93$
  • $24.7$

  $4$ tenths

  • $0.421$
  • $5.47$
  • $8.4$

  $4$ hundredths

  • $0.84$
  • $8.04$
  • $22.142$

• Round each number to the given accuracy.

  Hints

  Rounding to the nearest tenth means the number is written up to the tenths place value.

  We have to decide whether to round up or down each time. If the digit in the hundredths place is greater than $5$ we round up. If the digit is less than $5$ it stays the same.

  For example, what is $3.86$ to the nearest tenth.

  We know that it is between $3.8$ and $3.9$, which is it nearest to?

  First, look at the digit in the tenths place, this is an $8$.

  Next, look at the digit in the hundredths place, this is $6$.

  As $6$ is greater than $5$ so we round up to get $3.9$.

  Solution

  $2.43$ to the nearest tenth is $2.4$.

  $0.238$ to the nearest hundredth is $0.24$

  $2.43$ to the nearest whole number is $2$

  $24.281$ to the nearest tenth is $24.3$

  $0.241$ to the nearest hundredth is $0.24$

• Convert each fraction to its decimal expansion.

  Hints

  To convert a fraction to a decimal we divide the numerator by the denominator.

  For example, $\frac{2}{5} = 2 \div 5 = 0.4$

  For numbers greater than $1$, work out the fraction first.

  For example, for $3 \frac{1}{8}$.

  $\frac{1}{8} = 0.125$ so $3 \frac{1}{8} = 3.125$.

  Solution

  $\frac{1}{4} = 0.25$ because $1 \div 4 = 0.25$.

  $1\frac{1}{5} =1.2$ because $1 \div 5 = 0.2$.

  $\frac{3}{4} = 0.75$ because $3 \div 4 = 0.75$.

  $2\frac{1}{2} = 2.5$ because $1 \div 2 = 0.5$.

• Convert $\frac{2}{3}$ to a decimal.

  Hints

  To convert a fraction to a decimal, divide the numerator by the denominator.

  Terminating decimals have an end point: $0.5, 3.47, 6.7032$ etc.

  Recurring decimals continue forever: $0.222..., 8.4545...$ etc.

  Recurring decimals are written with a dot over the repeating part: $0.\dot 2$, $8.\dot 4 \dot 5$

  Our calculators will usually display a recurring decimal with the repeating digits repeated, for example, $\frac {1}{9} = 0.11111111....$.

  When a question asks for the full calculator display this is how we write it. The three dots at the end indicate that it continues forever.

  Solution

  Divide $\bf{2}$ by $\bf{3}$.

  The result is $\bf{0.66666.....}$, a decimal that goes on forever.

  We represent this with a repeating symbol over the $\bf{6}$.

  Written as a recurring decimal $\frac{2}{3} =$ $\bf{0.\dot 6}$

• Determine the smallest number.

  Hints

  Write the numbers in a place value chart to help you.

  The digits in a place value chart increase in value as we move to the left and decrease as we move to the right.

  For example, comparing $5$ and $0.5$ in a place value chart, we can see that $0.5$ is less than $5$.

  Solution

  $0.006$ is the smallest number. It has a value of $6$ thousandths.

• Order the decimals from smallest to largest.

  Hints

  To start, convert the fractions to decimals.

  Be careful with recurring decimals, remember the digits are repeated. So $0.\dot4$ can be written as $0.444444...$, this helps place it in the right position.

  It is helpful to write all the numbers to the same number of decimal places. Use zero as placeholders to do this, so $0.2$ can be written as $0.20$.

  Write all the numbers in a place value chart to help you compare them.

  Solution

  The correct order is,

  • $0.13$
  • $0.31$
  • $\frac{1}{3}$
  • $1.03$
  • $1\frac{3}{5}$

  The image shows the conversion of the fractions to decimals. Then if we compare all the values to two decimal places we will have:

  • $0.13$
  • $0.31$
  • $0.33$
  • $1.03$
  • $1.60$