Operations on Integers Decimals and Fractions

Place Value

Understanding place value is essential for performing arithmetic with integers and decimals. Every digit in a number has a value determined by its position relative to the decimal point. Each place value is ten times larger than the place to its right and one tenth the size of the place to its left.

The standard place values are:

1. Millions: $1,000,000$
2. Hundred Thousands: $100,000$
3. Ten Thousands: $10,000$
4. Thousands: $1,000$
5. Hundreds: $100$
6. Tens: $10$
7. Units: $1$
8. Decimal Point: $\cdot$
9. Tenths: $0.1$
10. Hundredths: $0.01$
11. Thousandths: $0.001$

For example, consider the digit $8$ in the following numbers:

• In $76,890$, the $8$ represents $8$ hundreds.
• In $23.986$, the $8$ represents $8$ hundredths.
• In $0.008$, the $8$ represents $8$ thousandths.

Addition and Subtraction of Integers and Decimals

To add or subtract, numbers must be aligned according to their place value. This is achieved by lining up the decimal points. Any empty positions should be filled with zeros, which is especially vital for subtraction.

Example: Addition

Calculate $23.69 + 9.043$.

Align the decimal points and fill the thousandths column with a zero:

$23.690 + 09.043 = 32.733$

Starting from the right, $0 + 3 = 3$. In the hundredths column, $9 + 4 = 13$: write $3$ and carry $1$ to the tenths. The final result is $32.733$.

Example: Subtraction

Calculate $63.79 - 9.036$.

Align the points and use zeros: $63.790 - 09.036$.

We cannot take $6$ from $0$, so we convert one hundredth into $10$ thousandths. Then $10 - 6 = 4$. Next, $8 - 3 = 5$ in the hundredths column. In the units, we cannot take $9$ from $3$, so we borrow from the tens: $13 - 9 = 4$. The final result is $54.754$.

Addition and Subtraction of Fractions

To add or subtract fractions, you must find a common denominator. For mixed numbers, you can either handle the integer and fractional parts separately or convert the entire number into an improper fraction first.

Example: Addition of Fractions

Find $\frac{2}{3} + \frac{3}{5}$.

The lowest common multiple (LCM) of $3$ and $5$ is $15$. Convert both fractions:

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

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

Example: Subtraction of Fractions

Find $1\frac{1}{4} - \frac{7}{8}$.

Convert $1\frac{1}{4}$ to the improper fraction $\frac{5}{4}$. Using a common denominator of $8$:

$\frac{10}{8} - \frac{7}{8} = \frac{3}{8}$

Multiplication of Integers and Decimals

Method 1: Formal Column Multiplication

Calculate $123 \times 46$.

1. Multiply $123$ by $6$ units to get $738$.
2. Multiply $123$ by $40$ (put a $0$ in the units column then multiply by $4$) to get $4920$.
3. Add $738 + 4920 = 5658$.

Method 2: Partitioning (Box Method)

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

• $100 \times 40 = 4000$
• $100 \times 6 = 600$
• $20 \times 40 = 800$
• $20 \times 6 = 120$
• $3 \times 40 = 120$
• $3 \times 6 = 18$

Summing $4000 + 600 + 800 + 120 + 120 + 18 = 5658$.

Method 3: Bones (Lattice Multiplication)

This method uses a grid where each box is split diagonally. Digits are multiplied and placed in the boxes, then diagonal sums are calculated.

Multiplying Decimals

Calculate $12.3 \times 0.46$.

First, multiply as integers: $123 \times 46 = 5658$. Count the total decimal places in the original numbers: $12.3$ has one and $0.46$ has two. The answer must have $1 + 2 = 3$ decimal places. Thus, $5658 \div 1000 = 5.658$.

Division of Integers and Decimals

The number being divided is the dividend, the number dividing is the divisor, and the result is the quotient.

Example: Integer Division

Calculate $23856 \div 6$.

Using the bus stop method:

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

The answer is $3976$.

Example: Decimal Division

Calculate $23.856 \div 0.06$.

Multiply both the dividend and divisor by a power of $10$ to make the divisor an integer. Multiplying both by $100$ gives $2385.6 \div 6$.

Performing the division gives $397.6$.

Multiplication and Division of Fractions

Multiplying Fractions

Convert mixed numbers to improper fractions. Multiply numerators together and denominators together. Simplify before or after multiplying.

Example: $1\frac{3}{5} \times 3\frac{3}{4} = \frac{8}{5} \times \frac{15}{4}$.

Simplifying: $\frac{8}{4} = 2$ and $\frac{15}{5} = 3$. The calculation becomes $2 \times 3 = 6$.

Dividing Fractions

Invert the divisor (the second fraction) and then multiply.

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