Fractions, mixed numbers, decimals and percentages — 0580
Convert equivalent representations without changing the quantity.
Core & Extended · Syllabus C1.4, E1.4
Notes and key methods
A mixed number \(\displaystyle a+ \frac{b}{c}\) equals \(\displaystyle \frac{ac+ b}{c}\) when \(\displaystyle 0\le b<c\). Simplify a fraction by dividing numerator and denominator by their HCF. To convert a fraction to a decimal, divide; to express it as a percentage, multiply its value by 100 and attach %.
A percentage such as 125% represents 1.25, not 125. Keep track of whether an answer is a number or a percentage. Terminating decimals can be checked using a denominator of 10, 100, 1000 and so on.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Fractions, mixed numbers, decimals and percentages
Let \(\displaystyle M=4\,\frac{2}{5}\) be a mixed number, and let \(\displaystyle F=\)\(\displaystyle \frac{2}{5}\).
Convert M to one fraction in lowest terms.
Give F as a decimal and as a percentage.
Express F as a percentage. Enter the numerical percentage.
Find the numerical value of F as an exact decimal.
Show the reasoning
(a)
The unsimplified numerator is \(4(5)+2=22\). Divide numerator and denominator by their HCF to obtain \(\displaystyle \frac{22}{5}\).
Answer: \(\displaystyle \frac{22}{5}\)
(b)
Dividing 2 by 5 gives 0.4. Multiplying that value by 100 gives the numerical percentage \(40\).
Answer: 0.4; \(40\)%.
(c)
Convert the mixed number with ac+b, simplify using the HCF, and use the correct decimal/percentage scale.
\(40\)
Answer: \(40\).
(d)
Convert the mixed number with ac+b, simplify using the HCF, and use the correct decimal/percentage scale.
\(\displaystyle \frac{2}{5}\)
Answer: \(\displaystyle \frac{2}{5}\).
Example 2 · Fractions, mixed numbers, decimals and percentages
Let \(\displaystyle M=5\,\frac{7}{8}\) be a mixed number, and let \(\displaystyle F=\)\(\displaystyle \frac{7}{8}\).
Convert M to one fraction in lowest terms.
Give F as a decimal and as a percentage.
Express F as a percentage. Enter the numerical percentage.
Find the numerical value of F as an exact decimal.
Show the reasoning
(a)
The unsimplified numerator is \(5(8)+7=47\). Divide numerator and denominator by their HCF to obtain \(\displaystyle \frac{47}{8}\).
Answer: \(\displaystyle \frac{47}{8}\)
(b)
Dividing 7 by 8 gives 0.875. Multiplying that value by 100 gives the numerical percentage \(\displaystyle \frac{175}{2}\).
Answer: 0.875; \(\displaystyle \frac{175}{2}\)%.
(c)
Convert the mixed number with ac+b, simplify using the HCF, and use the correct decimal/percentage scale.
\(\displaystyle \frac{175}{2}\)
Answer: \(\displaystyle \frac{175}{2}\).
(d)
Convert the mixed number with ac+b, simplify using the HCF, and use the correct decimal/percentage scale.
\(\displaystyle \frac{7}{8}\)
Answer: \(\displaystyle \frac{7}{8}\).
Example 3 · Fractions, mixed numbers, decimals and percentages
Let \(\displaystyle M=1\,\frac{9}{10}\) be a mixed number, and let \(\displaystyle F=\)\(\displaystyle \frac{9}{10}\).
Convert M to one fraction in lowest terms.
Give F as a decimal and as a percentage.
Express F as a percentage. Enter the numerical percentage.
Find the numerical value of F as an exact decimal.
Show the reasoning
(a)
The unsimplified numerator is \(1(10)+9=19\). Divide numerator and denominator by their HCF to obtain \(\displaystyle \frac{19}{10}\).
Answer: \(\displaystyle \frac{19}{10}\)
(b)
Dividing 9 by 10 gives 0.9. Multiplying that value by 100 gives the numerical percentage \(90\).
Answer: 0.9; \(90\)%.
(c)
Convert the mixed number with ac+b, simplify using the HCF, and use the correct decimal/percentage scale.
\(90\)
Answer: \(90\).
(d)
Convert the mixed number with ac+b, simplify using the HCF, and use the correct decimal/percentage scale.
\(\displaystyle \frac{9}{10}\)
Answer: \(\displaystyle \frac{9}{10}\).
Common mistakes
Keep intermediate values exact until the final step.