Mathematics (Syllabus D) 4024/12 — May/June 2022

Walkthrough

When adding fractions, the most critical rule is that you can only directly add the top numbers (numerators) if the bottom numbers (denominators) are identical. This is because the denominator tells us what size the pieces are being divided into. Here, we have thirds ($\frac{2}{3}$) and sixths ($\frac{1}{6}$). A sixth is a smaller piece than a third. To make them comparable, we need to express both in terms of sixths.

We know that $3 \times 2 = 6$, so we multiply the fraction $\frac{2}{3}$ by $\frac{2}{2}$ (which equals $1$ and does not change its value):

Now our problem is simply:

Since the denominators match, we just add the tops: $4 + 1 = 5$. The result is $\frac{5}{6}$.

Key Takeaways

• Always find a Lowest Common Multiple (LCM) for the denominators before adding or subtracting fractions.
• Multiply both the numerator and the denominator by the same number to create an equivalent fraction.
• Check your final answer to see if it can be simplified (reduced) by dividing out any common factors.

Common Mistakes

• Adding the numerators and denominators directly: $\frac{2+1}{3+6} = \frac{3}{9}$. This is incorrect because the 'sizes' of the pieces are different.
• Failing to simplify the final answer. While $\frac{5}{6}$ is already simplest form, a mistake like getting $\frac{6}{8}$ would require simplifying to $\frac{3}{4}$.

Things to Be Careful About

• Ensure you perform the multiplication on BOTH the top and bottom of the fraction when converting. For example, changing $\frac{2}{3}$ to $\frac{4}{6}$ requires multiplying the $2$ by $2$ AND the $3$ by $2$.

Walkthrough

Multiplication of decimals follows the same rules as whole numbers, but we must carefully track the decimal point. Think of $0.4$ as "four tenths" and $0.2$ as "two tenths".

When you multiply four tenths by two tenths, you are calculating:

Eight hundredths is written as $0.08$.

A quick way to do this mentally without fractions is:

1. Multiply $4 \times 2$ to get $8$.
2. Count the digits behind the decimal in the question: there is one in $0.4$ and one in $0.2$, making two total.
3. Start at the right of the number $8$ and move the decimal point two steps to the left: $.08$, which becomes $0.08$.

Key Takeaways

• The number of decimal places in the product is always the SUM of the decimal places in the factors.
• If the result of the integer multiplication has fewer digits than the required decimal places, pad with zeros on the left (e.g., $0.5 \times 0.02 \rightarrow 5 \times 2 = 10 \rightarrow$ need 3 decimal places $\rightarrow 0.010$).

Common Mistakes

• Writing the answer as $0.8$: forgetting to account for the second decimal place.
• Writing the answer as $8$: treating the decimals as whole numbers and ignoring the decimal point entirely.
• Placing the decimal point incorrectly, such as $0.008$ (adding too many zeros).

Things to Be Careful About

• Pay close attention to the exact number of decimal places in each factor. In this case, both $0.4$ and $0.2$ have exactly one decimal place.