Fractions Decimals and Percentages for the ESAT

Use fractions, decimals and percentages interchangeably in calculations

In many mathematical problems, values are provided in a variety of forms. To solve these efficiently, you must choose the most appropriate representation for your calculations. Whether you use fractions, decimals, or percentages often depends on which form makes the arithmetic simpler or avoids recurring decimals.

When you are required to multiply a decimal by a fraction, there are two primary strategies. You can either convert both numbers into fractions, which is usually the easier option, or convert both into decimals. If a decimal does not terminate, such as $0.333...$, converting to a fraction is almost always necessary to maintain precision.

Understanding Equivalent Fractions

To find fractions that are equivalent to a given fraction, you must either multiply or divide both the numerator and the denominator by the same non-zero number. This maintains the ratio between the two parts of the fraction. This principle is expressed as:

$\frac{x}{y} = \frac{nx}{ny} = \frac{\left(\frac{x}{n}\right)}{\left(\frac{y}{n}\right)}$

Simplifying a fraction to its lowest terms involves dividing both parts by their highest common factor until no further common factors exist.

Fractions, decimals and percentages in calculations

Consider a problem where different formats are combined. Suppose the sale price of a chair is $\frac{3}{4}$ of its price before the sale. On the final day of the sale, the price is reduced to $0.6$ of the sale price. What percentage of the original price is the final day price?

Let the original price be $x$.

The sale price is $\frac{3}{4}$ of this, represented as $\frac{3x}{4}$.

The next piece of information is given as a decimal, $0.6$. To calculate $0.6$ of the sale price, it is easier to convert $0.6$ into the fraction $\frac{6}{10}$, which simplifies to $\frac{3}{5}$.

Now, calculate the final price by multiplying the two fractions:

$\frac{3}{5} \times \frac{3x}{4} = \frac{9x}{20}$

To find what percentage this represents, multiply the resulting fraction by $100$:

$\frac{9}{20} \times 100 = \frac{900}{20} = 45\%$

Comparing methods for multi-format problems

When a problem contains percentages, fractions, and decimals, you can choose the method that feels most natural. Consider a school year group where pupils name their main way of travelling to school: $52\%$ use the bus, $\frac{1}{5}$ walk, and $0.1$ cycle. The remaining pupils come by car. What fraction of the year group come by car?

Method 1: Working in fractions

First, convert all values to fractions with a common denominator:

$52\% = \frac{52}{100} = \frac{13}{25}$

$0.1 = \frac{1}{10}$

The walking fraction is $\frac{1}{5}$. To add these, find a common denominator of $50$:

$\frac{13}{25} + \frac{1}{5} + \frac{1}{10} = \frac{26}{50} + \frac{10}{50} + \frac{5}{50} = \frac{41}{50}$

The fraction who come by car is the remainder from the whole ($1$):

$1 - \frac{41}{50} = \frac{9}{50}$

Method 2: Working in percentages

Convert all values to percentages and sum them:

$\frac{1}{5} = 20\%$

$0.1 = 10\%$

$52\% + 20\% + 10\% = 82\%$

The percentage coming by car is $100\% - 82\% = 18\%$.

To provide the answer as a fraction, convert $18\%$ back:

$18\% = \frac{18}{100} = \frac{9}{50}$

Method 3: Working in decimals

Convert all values to decimals and sum them:

$52\% = 0.52$

$\frac{1}{5} = 0.2$

$0.52 + 0.2 + 0.1 = 0.82$

The decimal of the year group coming by car is $1 - 0.82 = 0.18$.

As a fraction, $0.18 = \frac{18}{100} = \frac{9}{50}$.