Decimals Percentages and Fractions for the ESAT

Definitions and Principles

Fractions, decimals, and percentages are all mathematical tools used to describe a proportion of a whole, a number, or a group. Understanding the relationship between them is fundamental for numerical reasoning.

A terminating decimal is a decimal that contains a finite number of digits.

A recurring decimal contains digits or patterns of digits that repeat infinitely. To identify which type a fraction will produce, simplify the fraction to its lowest terms and examine the prime factors of the denominator. If the only prime factors are $2$ and/or $5$, the decimal will be terminating. If there are any other prime factors, the decimal will be recurring.

Converting Between Mixed Numbers and Improper Fractions

To convert a mixed number to an improper fraction, you must transform the integer part into a fraction with the same denominator as the fractional part, then sum them.

Example: Write $4\frac{5}{12}$ as an improper fraction.

$4\frac{5}{12} = 4 + \frac{5}{12} = \frac{4 \times 12}{12} + \frac{5}{12} = \frac{48}{12} + \frac{5}{12} = \frac{53}{12}$

To convert an improper fraction into a mixed number, divide the numerator by the denominator. The quotient is the whole number, and the remainder is placed over the original denominator.

Example: Write $\frac{17}{5}$ as a mixed number.

Divide the numerator by the denominator: $17 \div 5 = 3$ remainder $2$.

This gives the mixed number $3\frac{2}{5}$.

Cancelling and Equivalent Fractions

To cancel a fraction to its lowest terms, divide both the numerator and the denominator by their highest common factor (HCF). Alternatively, use prime factorisation.

Example: Cancel $\frac{630}{1638}$ to its lowest terms.

Method 1: Prime Factorisation

$\frac{630}{1638} = \frac{2 \times 3 \times 3 \times 5 \times 7}{2 \times 3 \times 3 \times 7 \times 13} = \frac{5}{13}$

Method 2: Repeated Division

You can also cancel by dividing by smaller common factors such as $2$ or $3$ until no further simplification is possible.

Equivalent fractions are found by multiplying or dividing the numerator and denominator by the same non-zero value. This allows you to find unknown values within equivalent ratios.

Example: Find the value of $x$ where $\frac{105}{168} = \frac{x}{120}$

You can determine the scale factor required to convert $168$ to $120$:

This gives $x = 75$. Alternatively, use cross-multiplication: $105 \times 120 = 168x$, which leads to $x = \frac{105 \times 120}{168} = 75$.

Converting Fractions to Decimals

There are two main methods to convert a fraction into a decimal: using powers of $10$ or using division.

1. Denominator as a power of 10: If possible, use equivalent fractions to make the denominator $10, 100, 1000,$ etc.

Example: Convert $\frac{7}{20}$ to a decimal.

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

1. Short Division: Divide the numerator by the denominator.

Example: Convert $\frac{5}{8}$ to a decimal.

Using short division: $8$ into $5.000$ gives $0.625$.

Converting Between Decimals and Percentages

To convert a decimal to a percentage, multiply by $100$ and add the percentage symbol. To convert a percentage to a decimal, divide by $100$.

Example: Convert $35\frac{1}{5}\%$ to a decimal.

$35\frac{1}{5}\% = 35.2\% = \frac{35.2}{100} = 0.352$

Example: Convert $5.5$ to a percentage.

$5.5 \times 100 = 550\%$

Converting Terminating Decimals to Fractions

Identify the place value of the final digit to determine the denominator. Use the digits as the numerator and then simplify.

Example: Convert $0.725$ to a fraction in its lowest terms.

The digit $5$ is in the thousandths column, so the denominator is $1000$.

$\frac{725}{1000} = \frac{145}{200} = \frac{29}{40}$

Converting Fractions to Recurring Decimals

Fractions with denominators like $9, 99,$ or $999$ are easily converted. For others, use division.

Example: Convert $\frac{3}{11}$ to a recurring decimal.

$\frac{3}{11} = \frac{27}{99} = 0.272727... = 0.\dot{2}\dot{7}$

Example: Convert $\frac{47}{90}$ to a recurring decimal.

Divide numerator and denominator by $10$: $\frac{4.7}{9}$.

Using division: $9$ into $4.700...$ gives $0.5222... = 0.5\dot{2}$. Note that after the first $2$, the remainder is always $2$, leading to an infinite string of $2$s.

Converting Recurring Decimals to Fractions

Use an algebraic method to eliminate the recurring part.

Example: Convert $0.\dot{4}\dot{2}$ to a fraction in its lowest terms.

Let $d = 0.\dot{4}\dot{2}$. Since two digits recur, multiply by $10^2$ (which is $100$):

$100d = 42.\dot{4}\dot{2}$

$d = 0.\dot{4}\dot{2}$

Subtracting the two equations: $99d = 42$.

$d = \frac{42}{99} = \frac{14}{33}$

Example: Convert $0.13\dot{2}$ to a fraction in its lowest terms.

Let $d = 0.13\dot{2}$. Multiply by $10$ to shift the decimal:

$10d = 1.3\dot{2}$

Subtract $d$ from $10d$: $10d - d = 1.3222... - 0.1322...$

$9d = 1.19$

$d = \frac{1.19}{9} = \frac{119}{900}$

Ordering Fractions, Decimals, and Percentages

To order different forms, convert them all into a single consistent form, usually decimals.

Example: Write in ascending order: $\frac{4}{11}, 37\%, \frac{7}{20}, 0.3\dot{6}$

$\frac{4}{11} = \frac{36}{99} = 0.\dot{3}\dot{6} = 0.3636...$

$37\% = 0.370$

$\frac{7}{20} = \frac{35}{100} = 0.350$

$0.3\dot{6} = 0.3666...$

Comparing these values: $0.350 < 0.3636... < 0.3666... < 0.370$.

Thus, the ascending order is: $\frac{7}{20}, \frac{4}{11}, 0.3\dot{6}, 37\%$.