Interchanging Fractions Decimals and Percentages

Using Fractions Decimals and Percentages Interchangeably

Many mathematical problems involve numerical values provided in various formats. To solve these efficiently for the TMUA, you must select the most suitable form: fractions, decimals, or percentages. Choosing the correct representation often depends on the operations required. For instance, when multiplying a decimal by a fraction, it is generally simpler to convert both values into fractions or both into decimals before proceeding.

Equivalent Fractions

Equivalent fractions represent the same value using different numerators and denominators. To find a fraction equivalent to a given fraction, you must either multiply both the numerator and the denominator by the same non zero number, or divide both by the same non zero number. This is expressed algebraically as:

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

Worked Example: Sale Price Calculations

In this example, we see how combining different numerical types can lead to a percentage result. Consider the following problem:

The sale price of a chair is $\frac{3}{4}$ of the price of the chair 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 price before the sale is the final day price?

Step 1: Define the original price. Let the original price be $x$.

Step 2: Calculate the first sale price. The sale price is $\frac{3}{4}$ of the original, so it is written as $\frac{3x}{4}$.

Step 3: Convert the final day reduction to a fraction. The final reduction is given as a decimal, $0.6$. To work with the fraction $\frac{3x}{4}$, it is easier to change $0.6$ into the fraction $\frac{3}{5}$.

Step 4: Calculate the final price. Multiply the two fractions: $\frac{3}{5} \times \frac{3x}{4} = \frac{9x}{20}$.

Step 5: Convert the result to a percentage. The final price is $\frac{9}{20}$ of the original price. To find the percentage: $\frac{9}{20} \times 100 = \frac{900}{20} = 45\%$.

Worked Example: Comparing Proportions in Context

When data is presented in a mix of percentages, fractions, and decimals, you can use three different methods to find a remaining part.

Problem: The pupils in school year 8 are asked how they travel to school. $52\%$ use the bus, $\frac{1}{5}$ walk, and $0.1$ cycle. The rest come by car. What fraction of Year 8 come by car?

Method 1: Working in Fractions

1. Convert $52\%$ to a simplified fraction: $52\% = \frac{52}{100} = \frac{13}{25}$.
2. Convert $0.1$ to a fraction: $0.1 = \frac{1}{10}$.
3. Find a common denominator ($50$) and add: $\frac{26}{50} + \frac{10}{50} + \frac{5}{50} = \frac{41}{50}$.
4. Subtract from the whole: $1 - \frac{41}{50} = \frac{9}{50}$.

Method 2: Working in Percentages

1. Convert $\frac{1}{5}$ to a percentage: $\frac{1}{5} = 20\%$.
2. Convert $0.1$ to a percentage: $0.1 = 10\%$.
3. Add the percentages: $52\% + 10\% + 20\% = 82\%$.
4. Find the remaining percentage: $100\% - 82\% = 18\%$.
5. Convert $18\%$ back to a fraction: $\frac{18}{100} = \frac{9}{50}$.

Method 3: Working in Decimals

1. Convert $52\%$ to a decimal: $0.52$.
2. Convert $\frac{1}{5}$ to a decimal: $0.2$.
3. Add the decimals: $0.52 + 0.2 + 0.1 = 0.82$.
4. Find the remaining decimal: $1 - 0.82 = 0.18$.
5. Convert $0.18$ back to a fraction: $\frac{18}{100} = \frac{9}{50}$.