Mathematics (Syllabus D) 4024/12 — May/June 2019

Walkthrough

When we encounter a division problem involving two fractions, such as $\frac{a}{b} \div \frac{c}{d}$, the standard method is to keep the first fraction unchanged, change the division sign to multiplication, and flip the second fraction (find its reciprocal).

Here, we have $\frac{4}{7} \div \frac{5}{8}$.

Step 1: Change $\div$ to $\times$ and flip $\frac{5}{8}$ to become $\frac{8}{5}$.
Now we have $\frac{4}{7} \times \frac{8}{5}$.

Step 2: Multiply straight across. Top numbers (numerators) go together, bottom numbers (denominators) go together.
$4 \times 8 = 32$
$7 \times 5 = 35$

Step 3: Combine them to get $\frac{32}{35}$.

Step 4: Check if it can be simplified. We look for a number that divides both $32$ and $35$ exactly. Since $32 = 2^5$ and $35 = 5 \times 7$, they have no common factors. Thus, $\frac{32}{35}$ is the final answer.

Key Takeaways

• Dividing by a fraction is equivalent to multiplying by its reciprocal.
• To multiply fractions, multiply the numerators together and the denominators together.
• Always check if the final fraction can be simplified by cancelling common factors.

Common Mistakes

• Cross-multiplying: Students sometimes multiply diagonally ($4 \times 5$ and $7 \times 8$) instead of top-by-top and bottom-by-bottom. This is incorrect for multiplication.
• Adding denominators: A common error is to add the denominators ($7+5=12$) instead of multiplying them when performing multiplication.
• Forgetting to flip: Simply multiplying $\frac{4}{7} \times \frac{5}{8}$ without inverting the second fraction yields the wrong answer.

Things to Be Careful About

• Ensure you simplify the final answer if possible. In this case, $\frac{32}{35}$ cannot be simplified further.
• On the non-calculator component, showing the inversion step clearly helps secure method marks even if the final arithmetic is wrong.

Walkthrough

This question tests your knowledge of square and cube roots.

1. Square Root: $\sqrt{64}$ asks "what number squared gives 64?" You should know your square tables up to at least $12^2$. Since $8^2 = 64$, $\sqrt{64} = 8$.

2. Cube Root: $\sqrt[3]{125}$ asks "what number cubed gives 125?" This means the number multiplied by itself twice more ($x \cdot x \cdot x$). Checking small integers: $1^3=1$, $2^3=8$, $3^3=27$, $4^3=64$, $5^3=125$. So, $\sqrt[3]{125} = 5$.

3. Subtraction: Finally, subtract the cube root result from the square root result: $8 - 5 = 3$.

Key Takeaways

• Memorize perfect squares ($1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...$) and perfect cubes ($1, 8, 27, 64, 125, 216...$) for quick recall.
• The symbol $\sqrt{}$ denotes the principal (positive) square root.
• The symbol $\sqrt[3]{}$ denotes the cube root.

Common Mistakes

• Confusing roots: Thinking $\sqrt{64}$ might be $4$ (since $4+4+4+4+4+4+4+4=32$, no, maybe confusing with $4^2=16$?) or thinking $\sqrt[3]{125}$ is related to $25$ (since $5^2=25$).
• Arithmetic errors: Subtracting incorrectly, e.g., $5-8=-3$ or $8+5=13$.
• Wrong root type: Calculating $\sqrt{125}$ (which is irrational) instead of the cube root because the index '3' was missed.

Things to Be Careful About

• Ensure you identify the correct root index. $\sqrt{x}$ is square root, $\sqrt[3]{x}$ is cube root.
• Verify your roots: Does $8 \times 8 = 64$? Yes. Does $5 \times 5 \times 5 = 125$? Yes.