Operations with Integers Decimals and Fractions

Place value

To perform arithmetic accurately, you must understand and use place value for both integers and decimals. Each place value is a factor of $10$ different from its immediate neighbour. For instance, hundreds multiplied by $10$ are thousands. The following table illustrates standard place values:

$1,000,000$	$100,000$	$10,000$	$1,000$	$100$	$10$	$1$	$\bullet$	$0.1$	$0.01$	$0.001$
millions	hundred thousands	ten thousands	thousands	hundreds	tens	units	decimal point	tenths	hundredths	thousandths

Example: Identifying place value

By placing numbers in the table, we can identify the value of specific digits:

1. The $8$ in $76,890$ represents $8$ hundreds.
2. The $8$ in $23.986$ represents $8$ hundredths.
3. The $8$ in $0.008$ represents $8$ thousandths.

$1,000,000$	$100,000$	$10,000$	$1,000$	$100$	$10$	$1$	$\bullet$	$0.1$	$0.01$	$0.001$
		$7$	$6$	$8$	$9$	$0$	$\bullet$
					$2$	$3$	$\bullet$	$9$	$8$	$6$
						$0$	$\bullet$	$0$	$0$	$8$

Addition and subtraction of integers and decimals

To add or subtract, numbers must be aligned according to their place value. Line up the decimal points vertically to ensure digits of the same value are in the same column.

Example: Addition of decimals

Calculate $23.69 + 9.043$

Fill any blanks with zeros. In addition, start from the rightmost column. For $9 + 4 = 13$, the $1$ is moved (carried) to the next column as $10$ hundredths, which equals $1$ tenth.

tens ($10$)	units ($1$)	$\bullet$	tenths ($0.1$)	hundredths ($0.01$)	thousandths ($0.001$)
$2$	$3$	$\bullet$	$6$	$9$	$0$
$0$	$9$	$\bullet$	$0$	$4$	$3$
$3$	$2$	$\bullet$	$7$	$3$	$3$

Example: Subtraction of decimals

Calculate $63.79 - 9.036$

Filling blanks with zeros is critical here. Start from the right. Since $6$ cannot be taken from $0$, convert one hundredth into $10$ thousandths. $10 - 6 = 4$. Then, subtract $3$ from $8$ and $0$ from $7$. For the units, since $9$ cannot be taken from $3$, convert one ten into $10$ units, giving $13 - 9 = 4$. Finally, $5 - 0 = 5$.

tens ($10$)	units ($1$)	$\bullet$	tenths ($0.1$)	hundredths ($0.01$)	thousandths ($0.001$)
$5$ (from $6$)	$10+3$	$\bullet$	$7$	$8$ (from $9$)	$10+0$
$0$	$9$	$\bullet$	$0$	$3$	$6$
$5$	$4$	$\bullet$	$7$	$5$	$4$

Addition and subtraction of fractions

To add or subtract fractions, they must have a common denominator. For mixed numbers, you can process the integer and fractional parts separately.

Example: Addition of fractions

Find $\frac{2}{3} + \frac{3}{5}$. The lowest common multiple (LCM) of $3$ and $5$ is $15$.

$\frac{2}{3} = \frac{10}{15}$ and $\frac{3}{5} = \frac{9}{15}$

Sum: $\frac{10}{15} + \frac{9}{15} = \frac{19}{15} = 1\frac{4}{15}$

Example: Subtraction of fractions

Find $1\frac{1}{4} - \frac{7}{8}$. Convert the mixed number: $1\frac{1}{4} = \frac{5}{4}$. Using the LCM of $4$ and $8$, we get $\frac{10}{8} - \frac{7}{8} = \frac{3}{8}$.

Multiplication of integers and decimals

Place value is key in multiplication. There are three primary methods.

Method 1: Formal column multiplication

To calculate $123 \times 46$, write the numbers in columns. First, multiply $123$ by $6$ units (carrying hundreds and tens). Then, multiply by $40$ by placing a $0$ in the units column and multiplying $123$ by $4$. Finally, add the results: $738 + 4920 = 5658$.

Method 2: Boxes (partitioning)

Split $123$ into $100 + 20 + 3$ and $46$ into $40 + 6$. Multiply every part and sum the results:

$\times$	$40$	$6$	Total
$100$	$4000$	$600$	$4600$
$20$	$800$	$120$	$920$
$3$	$120$	$18$	$138$
Total			$5658$

Method 3: Bones

This method uses diagonal lines to sum products of individual digits. Multiply the digits relating to each box, writing the answer as two digits (e.g., $4 \times 1 = 04$). Sum the diagonals starting from the bottom right.

Example: Multiplication of decimals

To calculate $12.3 \times 0.46$, first calculate $123 \times 46 = 5658$. Since $12.3 = 123 \div 10$ and $0.46 = 46 \div 100$, the final answer must be divided by $10 \times 100 = 1000$. Thus, $5658 \div 1000 = 5.658$. Always estimate to check: $12 \times 0.5 = 6$, confirming our answer is reasonable.

Division of integers and decimals

In a division, the dividend is the number being divided, the divisor is the number you divide by, and the quotient is the result. For example, in $360 \div 6 = 60$, $360$ is the dividend, $6$ is the divisor, and $60$ is the quotient.

Example: Division of integers (Bus stop method)

Calculate $23856 \div 6$. Place the dividend under the 'bus stop' and the divisor to the left.

1. $2 \div 6 = 0$ remainder $2$. Carry the $2$ to the next column ($23$).
2. $23 \div 6 = 3$ remainder $5$. Carry the $5$ ($58$).
3. $58 \div 6 = 9$ remainder $4$. Carry the $4$ ($45$).
4. $45 \div 6 = 7$ remainder $3$. Carry the $3$ ($36$).
5. $36 \div 6 = 6$.

The quotient is $3976$.

Example: Division of decimals

Calculate $23.856 \div 0.06$. To divide by a decimal, multiply both the dividend and divisor by the same power of $10$ until the divisor is an integer. $(23.856 \times 100) \div (0.06 \times 100) = 2385.6 \div 6$.

Performing the division gives $397.6$. Check with an approximation: $2400 \div 6 = 400$.

Multiplication of fractions

To multiply fractions, convert mixed numbers to improper fractions, then multiply numerators and denominators independently.

Example: $\frac{3}{4} \times \frac{2}{3}$. Multiplying directly gives $\frac{6}{12} = \frac{1}{2}$. Alternatively, simplify before multiplying by dividing the top $3$ and bottom $3$ by $3$, and the top $2$ and bottom $4$ by $2$, yielding $\frac{1}{2} \times \frac{1}{1} = \frac{1}{2}$.

Example: $1\frac{3}{5} \times 3\frac{3}{4} = \frac{8}{5} \times \frac{15}{4} = \frac{8 \times 15}{5 \times 4}$. Simplifying before multiplying ($8 \div 4 = 2$ and $15 \div 5 = 3$) results in $2 \times 3 = 6$.

Division of fractions

To divide fractions, invert the divisor (the second fraction) and multiply. Convert mixed numbers to improper fractions first.

Example 1: $15 \div \frac{3}{8} = \frac{15}{1} \times \frac{8}{3} = 5 \times 8 = 40$.

Example 2: $15 \div 1\frac{7}{8} = 15 \div \frac{15}{8} = \frac{15}{1} \times \frac{8}{15} = 8$.