Converting Decimals Percentages and Fractions

Definitions and Overview

Fractions, decimals, and percentages are different ways to describe a proportion of a group, number, or whole. Understanding their relationships is essential for solving problems in the TMUA. There are two main types of decimals that we must be able to manipulate:

1. A terminating decimal is a decimal which has a finite number of digits.
2. A recurring decimal is a decimal which has repeating digits or a repeating pattern of digits that continue infinitely.

Standard Conversion Methods

To move between these different forms, follow these standard procedures:

• Mixed Number to Improper Fraction: Convert the integer part into a fraction with the same denominator as the fractional part, then add it to the fractional part.
• Improper Fraction to Mixed Number: Divide the numerator by the denominator. The quotient is the whole number part and the remainder is written over the original denominator.
• Cancelling a Fraction: Divide both the numerator and the denominator by their highest common factor (HCF) to reach the lowest terms.
• Equivalent Fractions: These are found by multiplying or dividing both the numerator and the denominator by the same non-zero value.
• Fraction to Decimal: Divide the numerator by the denominator. Alternatively, use equivalent fractions to write the fraction over a denominator that is a power of 10, then use place value.
• Decimal to Percentage: Multiply the decimal by 100.
• Percentage to Decimal: Divide the percentage by 100.
• Terminating Decimal to Fraction: Use the smallest place value to identify the denominator and write the digits of the decimal as the numerator. For example, $0.0307 = \frac{307}{10000}$ because the 7 is in the ten thousandths place.

The Prime Factor Rule for Decimals

A simplified fraction will result in a terminating decimal if and only if the only prime factors in the denominator are 2, 5, or both. If any other prime factors (such as 3, 7, or 11) exist in the simplified denominator, the decimal will be recurring.

Worked Examples: Basic Conversions

Converting Mixed Numbers and Improper Fractions

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

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

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

Divide the numerator by the denominator: $17 \div 5 = 3$ remainder 2. This gives $3\frac{2}{5}$.

Cancelling to Lowest Terms

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

We can use two methods to simplify this. Method 1 involves 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 involves repeated cancelling:

Finding Equivalent Fractions

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

We can use cross multiplication: $105 \times 120 = 168x$. Rearranging gives $x = \frac{105 \times 120}{168} = 75$.

Worked Examples: Decimals and Percentages

Fraction to Decimal

Convert $\frac{7}{20}$ and $\frac{5}{8}$ to decimals.

For $\frac{7}{20}$, we use equivalent fractions: $\frac{7}{20} = \frac{35}{100} = 0.35$.

For $\frac{5}{8}$, use short division: $5.000 \div 8 = 0.625$.

Decimals and Percentages

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

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

Convert 5.5 to a percentage.

$5.5 \times 100\% = 550\%$.

Terminating Decimal to Fraction

Convert 0.725 to a fraction in its lowest terms.

The 5 is in the thousandths column. $\frac{725}{1000} = \frac{145}{200} = \frac{29}{40}$.

Recurring Decimals

Fraction to Recurring Decimal

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

$\frac{3}{11} = \frac{27}{99} = 0.\dot{2}\dot{7}$. The dots show that the digits 2 and 7 repeat indefinitely: $0.272727...$

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

Divide 4.7 by 9. The division results in $0.5222...$, which is $0.5\dot{2}$. Only the 2 is recurring because each step in the division results in a remainder of 2.

Recurring Decimal to Fraction

To convert a recurring decimal to a fraction, we define the decimal as $d$ and multiply by a power of 10 to shift the decimal point past the recurring digits, then subtract to eliminate them.

Convert $0.\dot{4}\dot{2}$ to a fraction.

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

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

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

Subtracting gives $99d = 42$, so $d = \frac{42}{99} = \frac{14}{33}$.

Convert $0.13\dot{2}$ to a fraction.

Let $d = 0.13222...$. Multiply by 10 to get $10d = 1.3222...$. Subtracting $d$ from $10d$ gives $9d = 1.19$. Thus $d = \frac{1.19}{9} = \frac{119}{900}$.

Ordering Mixed Types

To order numbers in different formats, convert them all to a single form, usually decimals.

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

$\frac{4}{11} = 0.3636...$

$37\% = 0.370$

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

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

Ascending order: $\frac{7}{20}, \frac{4}{11}, 0.3\dot{6}, 37\%$.