Mathematics (Syllabus D) 4024/11 — October/November 2020

Find a Common Denominator: We look for a number that both $7$ and $3$ divide into evenly. Since $7$ and $3$ are prime numbers relative to each other, their lowest common multiple is simply their product: $7 \times 3 = 21$.

Equivalent Fractions: We rewrite each fraction so it has $21$ as the denominator. To keep the value of the fraction the same, whatever we do to the denominator, we must also do to the numerator.

• For $\frac{4}{7}$, multiply top and bottom by $3$: $\frac{4 \times 3}{7 \times 3} = \frac{12}{21}$.
• For $\frac{1}{3}$, multiply top and bottom by $7$: $\frac{1 \times 7}{3 \times 7} = \frac{7}{21}$.

Subtract: Now that the denominators match, we subtract the numerators: $12 - 7 = 5$. The denominator stays $21$. So, the result is $\frac{5}{21}$.

Simplify: Check if the numerator ($5$) and denominator ($21$) have any common factors. $5$ is prime and does not divide $21$, so the fraction is already simplified.

Order of Operations: The expression is $2 - 1.2 \times 0.3$. You might be tempted to subtract $1.2$ from $2$ first, but multiplication takes priority over subtraction. Therefore, we must calculate $1.2 \times 0.3$ first.

Decimal Multiplication: Multiply $1.2$ by $0.3$. Treat them as whole numbers: $12 \times 3 = 36$. Count the total decimal places in the factors: $1.2$ has one, and $0.3$ has one. Total = 2 decimal places. Place the decimal point in the result so there are two digits to the right: $0.36$.

Decimal Subtraction: Now calculate $2 - 0.36$. It is crucial to align the decimal points. Think of $2$ as $2.00$ so that you have hundredths to subtract from.

• $0 - 6$: Can't do it, borrow from the left. The tenths place is $0$, so borrow from the units place ($2$). The $2$ becomes $1$, the tenths become $10$, then lend $1$ to the hundredths, becoming $9$. The hundredths become $10$.
• $10 - 6 = 4$.
• $9 - 3 = 6$.
• $1 - 0 = 1$.
  The answer is $1.64$.