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.

  • $5\frac{1}{10} = \frac{(5 \times 10) + 1}{10} = \frac{51}{10}$
  • $2\frac{7}{15} = \frac{(2 \times 15) + 7}{15} = \frac{37}{15}$

• Step 2: Find the LCM of 10 and 15, which is 30.

• Step 3: Create equivalent fractions with a denominator of 30.

  • $\frac{51 \times 3}{10 \times 3} = \frac{153}{30}$
  • $\frac{37 \times 2}{15 \times 2} = \frac{74}{30}$

• Step 4: Subtract the numerators: $\frac{153 - 74}{30} = \frac{79}{30}$.

• Step 5: Convert the result back to a mixed number. $79 \div 30 = 2$ with a remainder of $19$, giving a final answer of $2\frac{19}{30}$.

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 $2\frac{1}{5} = \frac{11}{5}$.

• Step 2: Multiply the numerators together, and the denominators together: $\frac{14 \times 11}{3 \times 5} = \frac{154}{15}$.

• Step 3: Convert the improper fraction back into a mixed number. $154 \div 15 = 10$ remainder $4$, giving a final answer of $10\frac{4}{15}$.

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: $3\frac{1}{4} = \frac{13}{4}$.

• Step 2: Apply the reciprocal method (Keep-Change-Flip). Keep $\frac{13}{4}$, change division to multiplication, and flip the second fraction: $\frac{13}{4} \times \frac{8}{3}$.

• Step 3: Cross-simplify before multiplying. The 4 and 8 share a common factor of 4. The calculation becomes $\frac{13}{1} \times \frac{2}{3}$.

• Step 4: Multiply the numerators and denominators: $\frac{13 \times 2}{1 \times 3} = \frac{26}{3}$.

• Step 5: Convert to a mixed number. $26 \div 3 = 8$ remainder $2$, giving a final answer of $8\frac{2}{3}$.

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.

• Step 5: Add the tens column: $1 + 0 + 1 \text{ (carried)} = 2$. Place the decimal point in the answer directly below the others to get $20.52$.

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 $3 - 5$, so exchange from the units. $13 - 5 = 8$.

• Step 4: Subtract the units. Exchange from the tens. $11 - 3 = 8$.

• Step 5: The final answer is $8.84$.

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: $2982 + 12780 = 15762$.

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 $7280$.

• Step 4: Add the rows together: $1820 + 7280 = 9100$.

• Step 5: Count the total number of decimal places in the original question (two in 3.64, one in 2.5, making 3 in total). Place the decimal point 3 spaces in from the right: $9.100$, which simplifies to $9.1$.

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: $18\frac{6}{8}$.

• Step 5: Simplify the fraction to get a final answer of $18\frac{3}{4}$.

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 $75 - 72 = 3$.

• Step 3: To express as a decimal, add a decimal point and a zero to the dividend ($435.0$). Bring down the 0 to make 30. Place a decimal point in the quotient.

• Step 4: How many 12s in 30? 2 ($2 \times 12 = 24$). The remainder is $30 - 24 = 6$.

• Step 5: Add another zero ($435.00$) and bring it down to make 60. How many 12s in 60? Exactly 5.

• Step 6: The final answer is $36.25$.