The Four Operations and Formal Written Methods

• Step 1: Find the LCM of the denominators 3 and 5. The LCM is 15.

• Step 2: Create equivalent fractions by multiplying the numerator and denominator by the same value.

  • $\frac{2 \times 5}{3 \times 5} = \frac{10}{15}$
  • $\frac{1 \times 3}{5 \times 3} = \frac{3}{15}$

• Step 3: Add the numerators and keep the denominator exactly the same. $\frac{10}{15} + \frac{3}{15} = \frac{13}{15}$.

• Step 4: Check for simplification. 13 is a prime number, so $\frac{13}{15}$ is the simplest form.

Worked Example: Subtraction of Mixed Numbers

Calculate $5\frac{1}{10} - 2\frac{7}{15}$

• Step 1: Convert both mixed numbers to improper fractions.

  • $5\frac{1}{10} = \frac{(5 \times 10) + 1}{10} = \frac{51}{10}$

Fractions and Mixed Numbers: Multiplication and Division

Unlike addition, multiplication and division do not require a common denominator. However, there is a strict universal rule: any mixed number must be converted into an improper fraction before you begin.

For division, you must use the Reciprocal method. A reciprocal is the result of "flipping" a fraction so the numerator and denominator switch places.

Worked Example: Multiplication of Mixed Numbers

Calculate $4\frac{2}{3} \times 2\frac{1}{5}$

• Step 1: Convert both values to improper fractions: $4\frac{2}{3} = \frac{14}{3}$ and .

Worked Example: Division of Mixed Numbers

Calculate $3\frac{1}{4} \div \frac{3}{8}$

• Step 1: Convert the mixed number to an improper fraction: $3\frac{1}{4} = \frac{13}{4}$.

• Apply the reciprocal method (Keep-Change-Flip). Keep , change division to multiplication, and flip the second fraction: .

Formal Arithmetic Methods: Decimals

When operating on decimals, you must use a Formal Written Method, which provides a structured layout for examiners to follow. The Column Method involves arranging numbers vertically based on their Place value.

A critical step in decimal arithmetic is keeping decimal points perfectly aligned. You must also use a Placeholder zero to fill empty place value positions, ensuring numbers have the same number of decimal places before calculating.

Worked Example: Addition of Decimals

Calculate $14.7 + 5.82$

• Step 1: Align the decimal points vertically. Add a placeholder zero to 14.7 so it becomes 14.70.

• Step 2: Add the hundredths column: $0 + 2 = 2$.

• Step 3: Add the tenths column: $7 + 8 = 15$. Write 5 and carry 1 to the units column.

• Step 4: Add the units column: $4 + 5 + 1 \text{ (carried)} = 10$. Write 0 and carry 1 to the tens.

Worked Example: Subtraction of Decimals

Calculate $12.4 - 3.56$

• Step 1: Align the decimal points and add a placeholder zero to make it $12.40 - 03.56$.

• Step 2: Subtract the hundredths. You cannot do $0 - 6$, so you must use Exchanging. Take 1 from the tenths column (4 becomes 3) to make 10 hundredths. $10 - 6 = 4$.

• Step 3: Subtract the tenths. You cannot do , so exchange from the units. .

Formal Arithmetic Methods: Long Multiplication

When multiplying larger numbers, long multiplication breaks the process into manageable stages, generating a Partial Product for each digit of the multiplier. When you multiply by the tens digit, you must insert a placeholder zero on the second row of working to shift the place value correctly.

Worked Example: Integer Long Multiplication

Calculate $426 \times 37$

• Step 1: Multiply by the units digit (7). $426 \times 7 = 2982$. This is the first partial product.

• Step 2: Multiply by the tens digit (3). First, add a placeholder 0. Then calculate $426 \times 3 = 1278$. Write this as $12780$.

• Step 3: Add the partial products together: .

Worked Example: Decimal Long Multiplication

Calculate $3.64 \times 2.5$

• Step 1: Ignore the decimals temporarily and calculate $364 \times 25$.

• Step 2: Row 1 calculation: $364 \times 5 = 1820$.

• Step 3: Row 2 calculation: Add a placeholder 0, then $364 \times 2 = 728$. Write as .

Formal Arithmetic Methods: Division

The Bus Stop Method (short division) is used to calculate how many times a Divisor fits into a Dividend, producing a Quotient (the answer) and sometimes a Remainder.

Worked Example: Short Division with a Fractional Remainder

Calculate $150 \div 8$

• Step 1: Set up the bus stop. How many 8s go into 1? 0, with a remainder of 1.

• Step 2: Carry the 1 to the next digit to make 15. How many 8s go into 15? 1, with a remainder of 7.

• Step 3: Carry the 7 to make 70. How many 8s in 70? 8 ($8 \times 8 = 64$), with a remainder of 6.

• Step 4: Express the remainder as a fraction. The remainder is the numerator and the divisor is the denominator: $18\frac{6}{8}$.

Worked Example: Long Division with a Decimal Remainder

Calculate $435 \div 12$

• Step 1: Set up division. How many 12s in 43? 3 ($3 \times 12 = 36$). The remainder is $43 - 36 = 7$.

• Step 2: Bring down the 5 to make 75. How many 12s in 75? 6 ($6 \times 12 = 72$). The remainder is .