Fractions, Decimals, and Significant Figures Revision Notes for Leaving Cert Mathematics

Fractions, Decimals, and Significant Figures

Understanding fractions, decimals, and significant figures forms the foundation of arithmetic calculations. These concepts are essential for accurate mathematical work and appear frequently in Leaving Cert examinations.

Fractions

Equivalent fractions are fractions that represent the same value despite having different numerators and denominators. For example, $\frac{12}{20}$ and $\frac{3}{5}$ are equivalent fractions because they both represent the same portion of a whole.

The simplest form of a fraction occurs when the numerator and denominator share no common factors other than 1. To find the simplest form, divide both the numerator and denominator by their greatest common factor. In the example above, $\frac{3}{5}$ represents the simplest form of $\frac{12}{20}$ .

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Finding equivalent fractions is like scaling a recipe - you multiply or divide both ingredients (numerator and denominator) by the same amount to maintain the same proportional relationship.

Ordering fractions

When comparing fractions, rewrite them as equivalent fractions with a common denominator. This allows you to compare the numerators directly to determine which fraction is larger or smaller.

Converting fractions to decimals

Any fraction can be converted to a decimal by dividing the numerator by the denominator. A calculator proves very useful for this process.

For example: $\frac{3}{8} = 3 \div 8 = 0.375$

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Worked Example: Practical Fraction Problems

Problem 1: Express 125 metres as a fraction of 1 km.

Solution: There are 1000 metres in 1 km. Therefore: $125$ m as a fraction of 1 km = $\frac{125}{1000} = \frac{1}{8}$

Problem 2: $\frac{3}{7}$ of a sum of money is €360. Find the sum of money.

Solution:

$\frac{3}{7} = €360$
$\frac{1}{7} = €120$ (divide by 3)
$\frac{7}{7} = €120 \times 7 = €840$

Therefore, the sum of money is €840.

Decimals

The decimal system uses place value to represent numbers. Each position to the right of the decimal point represents a fraction with a denominator that is a power of 10.

For example: $2.347 = 2 + \frac{3}{10} + \frac{4}{100} + \frac{7}{1000}$

Converting decimals to fractions

Any decimal can be converted to a fraction by using the place value of the final digit.

Example: $0.35 = \frac{35}{100} = \frac{7}{20}$

The values 0.35 and $\frac{7}{20}$ are equivalent.

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When converting decimals to fractions, count the number of decimal places to determine the denominator. One decimal place means tenths, two decimal places means hundredths, and so on.

Recurring decimals

Not all fractions have an exact decimal equivalent. When a fraction produces a decimal where one or more digits repeat indefinitely, this creates a recurring decimal.

For example: $\frac{2}{3} = 0.66666...$

This is written as $0.\overline{6}$ where the bar indicates the recurring digit.

Similarly, $\frac{2}{11} = 0.181818... = 0.\overline{18}$

Rounding decimals

Decimals can be rounded to a specified number of decimal places for practical use:

6.4837 = 6.484 (correct to 3 decimal places)
6.4837 = 6.48 (correct to 2 decimal places)
6.4837 = 6.5 (correct to 1 decimal place)

Significant figures

Significant figures indicate the precision of a measurement or calculation. They represent the meaningful digits in a number.

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Understanding significant figures is crucial for scientific calculations and measurements. They tell you how precise your answer should be and prevent false precision in your final results.

Rules for whole numbers

When expressing whole numbers to a given number of significant figures, zeros at the end are not counted but must be included in the final result. All other zeros are significant.

Example with 52,764:

52,760 (correct to 4 significant figures)
52,800 (correct to 3 significant figures)
53,000 (correct to 2 significant figures)
50,000 (correct to 1 significant figure)

For the number 70,425 = 70,400 (correct to 3 significant figures). Notice that the zero between 7 and 4 is significant, but the two final zeros are not.

Rules for decimal numbers

If a number is less than 1, zeros immediately after the decimal point are not significant. They are placeholders only.

Examples:

0.07406 = 0.0741 (correct to 3 significant figures)
0.00892 = 0.0089 (correct to 2 significant figures)

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Common Mistake to Avoid: Leading zeros (zeros at the beginning of decimal numbers) never count as significant figures. For example, in 0.0045, only the digits 4 and 5 are significant.

Exam tips for significant figures

Always count from the first non-zero digit
Trailing zeros in whole numbers don't count unless specified
Leading zeros in decimals never count
Check your final answer has the correct number of digits

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When in doubt about significant figures, remember this simple rule: if you're measuring with a ruler marked in millimetres, you can't claim precision to the nearest micrometre!

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Key Points to Remember:

Equivalent fractions represent the same value - find them by multiplying or dividing numerator and denominator by the same number
Convert fractions to decimals by dividing numerator by denominator using a calculator
Recurring decimals occur when division doesn't produce an exact answer - use bar notation to show repeating digits
Significant figures show precision - count meaningful digits only, ignoring leading zeros in decimals and trailing zeros in whole numbers
Always simplify fractions to their lowest terms by finding the greatest common factor

Fractions, Decimals, and Significant Figures (Leaving Cert Mathematics): Revision Notes