Mathematics (Syllabus D) 4024/12 — October/November 2020

Walkthrough

When subtracting fractions, the bottom numbers (denominators) must be the same so that we are counting pieces of the same size.

First, we look at the denominators 5 and 3. The smallest number both 5 and 3 divide into evenly is 15. This becomes our common denominator.

Next, we convert $\frac{4}{5}$. To turn the bottom 5 into 15, we multiply by 3. We must do the same to the top: $4 \times 3 = 12$. So $\frac{4}{5}$ becomes $\frac{12}{15}$.

Then, we convert $\frac{2}{3}$. To turn the bottom 3 into 15, we multiply by 5. We multiply the top by 5 as well: $2 \times 5 = 10$. So $\frac{2}{3}$ becomes $\frac{10}{15}$.

Finally, we subtract the top numbers (numerators): $12 - 10 = 2$. The bottom number stays 15. The answer is $\frac{2}{15}$.

Key Takeaways

• Always find a common denominator before adding or subtracting fractions.
• Whatever you multiply the denominator by, you must also multiply the numerator by to keep the fraction's value the same.

Common Mistakes

• Subtracting the numerators and denominators directly ($\frac{4-2}{5-3} = \frac{2}{2} = 1$), which is incorrect because the 'pieces' are different sizes.
• Forgetting to change one of the fractions when finding the common denominator.

Things to Be Careful About

• Ensure the final fraction is in its simplest form if possible. In this case, 2 and 15 share no common factors, so $\frac{2}{15}$ is the simplest form.

Walkthrough

To multiply decimals by hand, first treat them as whole numbers. Multiply $27$ by $2$ to get $54$.

Next, determine where the decimal point goes. Look at the original numbers: $2.7$ has one digit after the decimal point, and $0.2$ has one digit after the decimal point. Adding these together ($1 + 1$) tells us the answer must have 2 digits after the decimal point.

Starting from the right of the number $54$, move the decimal point two places to the left. Since there are only two digits, we add a zero placeholder: $0.54$.

Key Takeaways

• The number of decimal places in the answer is the sum of the decimal places in the numbers being multiplied.
• If there aren't enough digits to place the decimal point, add zeros to the front.

Common Mistakes

• Placing the decimal point incorrectly (e.g., writing $5.4$ or $54.0$).
• Miscounting the total decimal places required.

Things to Be Careful About

• Remember that trailing zeros in the factors (like in $0.20$) still count towards the decimal place total, though simplifying them first can help avoid errors.