Decimals (OCR GCSE Maths): Revision Notes

• Examples: $0.5, 0.125, 0.75$

• Examples: $0.\overline{3}$ (which means $0.3333…$), $0.1\overline{6}$(which means $0.16666…$)

• Examples: $\pi = 3.14159\ldots, \ \sqrt{2} = 1.41421\ldots$

Example 1: Convert $0.\overline{3}$ into a fraction.

1. Let $x=0.\overline{3}$, so $x=0.3333$…
2. Multiply both sides by $10: 10x =3.3333…$
3. Subtract the original equation from this: $10x−x = 3.3333…−0.3333…$, which simplifies to $9x=3$.
4. Solve for $x: x = \frac{3}{9} = \frac{1}{3}.$

So, $0.\overline{3} = \frac{1}{3}.$

Example 2: Convert $0.1\overline{6}$ into a fraction. 5. Let $x=0.1\overline{6}$ 6. Multiply by $10$ to shift the decimal: $10x=1.6666…$. 7. Multiply by $10$ again to cover the repeating part: $100x=16.6666….$ 8. Subtract the first multiplied equation from the second: $100x−10x=16.6666…−1.6666…,$ giving $90x=15$.

9. Solve for $x: x = \frac{15}{90} = \frac{1}{6}.$ So, $0.1\overline{6} = \frac{1}{6}.$

Example 1: Adding Decimals

Example 2: Subtracting Decimals

• Problem: Subtract $0.62−0.0159$
• Steps:

1. Align the decimals: Write the numbers in a column, ensuring that the decimal points are aligned.
2. Perform the subtraction:

3. Result: The difference is $0.6041.$

Example 1: Simple Multiplication

Example 2: Multiplying Larger Decimals

• Problem: Multiply $0.32×0.528$
• Steps:

4. Multiply ignoring the decimals:

5. Count total decimal places: There are $5$ decimal places ($2$ from $0.32$ and $3$ from $0.528$).
6. Place the decimal:

7. Result: The product is $0.16896.$

Example:

• Problem: Divide $75.92÷1.3$
• Steps:

8. Multiply both numbers by $10$ to eliminate the decimal in the divisor:

9. Divide as whole numbers:

10. Result: The quotient is $58.4$.

• Example: Convert $0.364$ to a percentage.
• Steps:

1. Multiply the decimal by $100$:

2. Result: $0.364$ as a percentage is $36.4$%

• Example: Convert $8.3$% to a decimal.
• Steps:

3. Divide the percentage by $100$:

4. Result: $8.3$% as a decimal is $0.083$.

• Example: Convert $0.16$ to a fraction.
• Steps:

5. Write the decimal as a fraction with the denominator as $100$ (since there are two decimal places):

6. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is $4$ in this case:

7. Result: $0.16$ as a fraction is $\frac{4}{25}.$

• Example: Convert $\frac{13}{20}$ to a decimal.
• Steps:

8. Adjust the fraction so that the denominator is $100$ (multiply both the numerator and the denominator by $5$):

9. Convert the fraction to a decimal by dividing the numerator by the denominator:

10. Result: $\frac{13}{20}$ as a decimal is $0.65$.

• Example: Convert $\frac{5}{8}$ to a percentage.
• Steps:

11. First, convert the fraction to a decimal by dividing the numerator by the denominator:

12. Then, convert the decimal to a percentage by multiplying by $100$:

13. Result: $\frac{5}{8}$ as a percentage is $62.5$.

• Example: Convert $12.5$% to a fraction.
• Steps:

14. Start by writing the percentage as a fraction over $100$:

15. To remove the decimal, multiply both the numerator and the denominator by $10$:

16. Simplify the fraction by dividing both the numerator and the denominator by their GCD, which is $25$:

17. Result: $12.5$% as a fraction is $\frac{1}{8}$.

Example: Convert $0.\overline{615}$ (which means $0.165165165…$) into a fraction.

Step 1: Set up the Equation

• Let $x=0.165165165…$

Step 2: Eliminate the Recurrence

• To remove the repeating decimal part, multiply $x$ by a power of $10$ that matches the length of the repeating block. In this case, the block "$165$" has three digits, so multiply by $1000$:

• Now, you have two equations:

Step 3: Subtract the Equations

• Subtract equation (i) from equation (ii) to eliminate the recurring part:

• Simplifying the subtraction:

Step 4: Solve for $x$

• Divide both sides by $999$ to isolate $x$:

• Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is $3$:

Step 5: Verify the Result

• You can check your result by converting $\frac{55}{333}$ back into a decimal using long division or a calculator. You should get $0.\overline{165}$ confirming that the conversion is correct.

Example: $\frac{1}{8}$ $=0.125$ (since $8=2³$).

Example: $\frac{2}{3}$ = $0.\overline{6}$ (since $3$ is a prime factor that is neither $2$ nor $5$).