Fractions and the correct order of operations
🎯 In this topic you will
- Carry out calculations involving fractions and mixed numbers using the correct order of operations
- Estimate the answers to calculations
You already know how to carry out a variety of calculations involving fractions and mixed numbers. In this section you will develop these skills, making sure you use the correct order of operations.
❓ EXERCISES
1. Copy and complete these calculations.
🔎 Reasoning Tip
Make sure you write your answer in its simplest form.
a. $5\dfrac{2}{3} + \Big(\dfrac{3}{5} - \dfrac{1}{2}\Big)$
👀 Show answer
Brackets: $\dfrac{3}{5} - \dfrac{1}{2} = \dfrac{6}{10} - \dfrac{5}{10} = \dfrac{1}{10}$
Addition: $5\dfrac{2}{3} + \dfrac{1}{10} = \dfrac{17}{3} + \dfrac{1}{10} = \dfrac{170}{30} + \dfrac{3}{30} = \dfrac{173}{30} = 5\dfrac{23}{30}$
b. $10 - \dfrac{5}{6} \times \dfrac{7}{10}$
👀 Show answer
Multiplication: $\dfrac{5}{6} \times \dfrac{7}{10} = \dfrac{35}{60} = \dfrac{7}{12}$
Rewrite 10: $10 = 9\dfrac{12}{12}$
Subtraction: $9\dfrac{12}{12} - \dfrac{7}{12} = 9\dfrac{5}{12}$
c. $5 \div \dfrac{3}{4} + \Big(\dfrac{2}{3}\Big)^2$
👀 Show answer
Brackets: $\Big(\dfrac{2}{3}\Big)^2 = \dfrac{2}{3} \times \dfrac{2}{3} = \dfrac{4}{9}$
Division: $5 \div \dfrac{3}{4} = 5 \times \dfrac{4}{3} = \dfrac{20}{3}$
Addition: $\dfrac{20}{3} + \dfrac{4}{9} = \dfrac{60}{9} + \dfrac{4}{9} = \dfrac{64}{9} = 7\dfrac{1}{9}$
2. Work out these calculations. Write each answer as a mixed number in its simplest form.
Show all the steps in your working.
a. $2\dfrac{1}{8} + \dfrac{1}{4} \times \dfrac{3}{4}$
👀 Show answer
$2\dfrac{1}{8}=\dfrac{17}{8}$
$\dfrac{1}{4}\times\dfrac{3}{4}=\dfrac{3}{16}$
$\dfrac{17}{8}+\dfrac{3}{16}=\dfrac{34}{16}+\dfrac{3}{16}=\dfrac{37}{16}=2\dfrac{5}{16}$
b. $\dfrac{9}{10}\times\dfrac{1}{2}+2\dfrac{4}{5}$
👀 Show answer
$\dfrac{9}{10}\times\dfrac{1}{2}=\dfrac{9}{20}$
$2\dfrac{4}{5}=\dfrac{14}{5}$
$\dfrac{9}{20}+\dfrac{14}{5}=\dfrac{9}{20}+\dfrac{56}{20}=\dfrac{65}{20}=\dfrac{13}{4}=3\dfrac{1}{4}$
c. $4\dfrac{1}{3}-\Big(5\dfrac{1}{2}-3\dfrac{1}{6}\Big)$
👀 Show answer
$5\dfrac{1}{2}=\dfrac{11}{2}, \quad 3\dfrac{1}{6}=\dfrac{19}{6}$
$\dfrac{11}{2}-\dfrac{19}{6}=\dfrac{33}{6}-\dfrac{19}{6}=\dfrac{14}{6}=\dfrac{7}{3}$
$4\dfrac{1}{3}=\dfrac{13}{3}$
$\dfrac{13}{3}-\dfrac{7}{3}=\dfrac{6}{3}=2$
d. $\dfrac{2}{3}\div\dfrac{4}{9}+2\dfrac{1}{4}$
👀 Show answer
$\dfrac{2}{3}\div\dfrac{4}{9}=\dfrac{2}{3}\times\dfrac{9}{4}=\dfrac{18}{12}=\dfrac{3}{2}=1\dfrac{1}{2}$
$2\dfrac{1}{4}=\dfrac{9}{4}$
$1\dfrac{1}{2}+\dfrac{9}{4}=\dfrac{3}{2}+\dfrac{9}{4}=\dfrac{6}{4}+\dfrac{9}{4}=\dfrac{15}{4}=3\dfrac{3}{4}$
🧠 Think like a Mathematician
Task: Explore how to estimate and calculate with fractions, then evaluate the accuracy of your estimation method.
Questions:
👀 show answer
- a) Round the fractions: $6\dfrac{4}{5}\approx 7$, $3\dfrac{1}{4}\approx 3$, $5\dfrac{2}{3}\approx 6$, $2\dfrac{7}{10}\approx 3$. Then: $7+3-(6-3)=7+3-3=7$. Estimate ≈ 7.
- b) Exact calculation: $6\dfrac{4}{5} = \dfrac{34}{5}$, $3\dfrac{1}{4} = \dfrac{13}{4}$, $5\dfrac{2}{3} = \dfrac{17}{3}$, $2\dfrac{7}{10} = \dfrac{27}{10}$. Inside brackets: $\dfrac{17}{3} - \dfrac{27}{10} = \dfrac{170-81}{30} = \dfrac{89}{30}$. Whole expression: $\dfrac{34}{5} + \dfrac{13}{4} - \dfrac{89}{30}$. Common denominator 60: $\dfrac{408}{60} + \dfrac{195}{60} - \dfrac{178}{60} = \dfrac{425}{60}$. Simplify: $\dfrac{425}{60} = \dfrac{85}{12} = 7\dfrac{1}{12}$.
- c) The estimate (7) is very close to the exact answer (7 1/12). The method of rounding fractions to whole numbers worked well. To improve, keep one fraction closer to its actual value rather than rounding everything.
- d) Best method: round fractions sensibly, especially those close to a whole number, while keeping others accurate. This balances speed with reliability.
❓ EXERCISES
4. Work out i an estimate ii the accurate answer to these calculations.
Show all the steps in your working.
a. $8\dfrac{9}{10} - \Big(2\dfrac{1}{5} + 3\dfrac{5}{8}\Big)$
👀 Show answer
Estimate: $9 - (2 + 4) = 3$
Accurate:
$8\dfrac{9}{10}=\dfrac{89}{10}, \quad 2\dfrac{1}{5}=\dfrac{11}{5}, \quad 3\dfrac{5}{8}=\dfrac{29}{8}$
$\dfrac{11}{5}+\dfrac{29}{8}=\dfrac{88}{40}+\dfrac{145}{40}=\dfrac{233}{40}$
$\dfrac{89}{10}-\dfrac{233}{40}=\dfrac{356}{40}-\dfrac{233}{40}=\dfrac{123}{40}=3\dfrac{3}{40}$
b. $7\dfrac{2}{3} + \Big(2\dfrac{5}{12}\times\dfrac{7}{8}\Big)$
👀 Show answer
Estimate: $8 + (2\times1) = 10$
Accurate:
$7\dfrac{2}{3}=\dfrac{23}{3}, \quad 2\dfrac{5}{12}=\dfrac{29}{12}$
$\dfrac{29}{12}\times\dfrac{7}{8}=\dfrac{203}{96}$
Common denominator with $\dfrac{23}{3}$: $\dfrac{23}{3}=\dfrac{736}{96}$
Total $=\dfrac{736}{96}+\dfrac{203}{96}=\dfrac{939}{96}=9\dfrac{75}{96}=9\dfrac{25}{32}$
c. $5\dfrac{1}{9}+2\dfrac{1}{3}\times 16$
👀 Show answer
Estimate: $5 + (2\times16)=37$
Accurate:
$5\dfrac{1}{9}=\dfrac{46}{9}, \quad 2\dfrac{1}{3}=\dfrac{7}{3}$
$\dfrac{7}{3}\times 16=\dfrac{112}{3}$
Common denominator: $\dfrac{46}{9}+\dfrac{112}{3}=\dfrac{46}{9}+\dfrac{336}{9}=\dfrac{382}{9}=42\dfrac{4}{9}$
d. $15\dfrac{3}{4}-\dfrac{7}{12}\times\dfrac{1}{2}$
👀 Show answer
Estimate: $16 - (1\times0.5)=15.5$
Accurate:
$15\dfrac{3}{4}=\dfrac{63}{4}$
$\dfrac{7}{12}\times\dfrac{1}{2}=\dfrac{7}{24}$
Common denominator: $\dfrac{63}{4}=\dfrac{378}{24}$
So $\dfrac{378}{24}-\dfrac{7}{24}=\dfrac{371}{24}=15\dfrac{11}{24}$
5. This is part of Fiona’s homework. Her method is to change all the mixed numbers into improper fractions, then work through the solution and change back to a mixed number at the very end.
Question: Work out $5 \dfrac{2}{3} - \left(\dfrac{3}{5} + 2 \dfrac{5}{6}\right)$
Answer:
1) Change to improper fractions:
$ \dfrac{17}{3} - \left(\dfrac{8}{5} + \dfrac{17}{6}\right) $
2) Work out brackets:
$ \dfrac{8}{5} + \dfrac{17}{6} = \dfrac{48}{30} + \dfrac{85}{30} = \dfrac{133}{30} $
3) Work out subtraction:
$ \dfrac{17}{3} - \dfrac{133}{30} = \dfrac{170}{30} - \dfrac{133}{30} = \dfrac{37}{30} $
4) Simplify:
$ \dfrac{37}{30} = 1 \dfrac{7}{30} $
a. Critique Fiona’s method.
b. Can you think of a better/easier method to use to answer this type of question?
👀 Show answer
a. Fiona’s method is correct but lengthy. Changing all numbers to improper fractions at the start often makes calculations harder to follow and involves larger numbers.
b. A simpler approach is to simplify inside the brackets first (add $1\dfrac{3}{5}+2\dfrac{5}{6}$ directly as mixed numbers), then subtract from $5\dfrac{2}{3}$. Working with mixed numbers step by step reduces the size of the fractions and is easier to check for mistakes.
6. The diagram shows the lengths of two sides of a triangle. The triangle has a perimeter of $25$ m.
a. Write the calculation you must do to work out the length of the third side of the triangle.
b. Zara estimates the third side to be about $15\dfrac{1}{2}$ m. What do you think of Zara’s estimate? Explain.
c. Work out the length of the third side. Was your answer to part b correct?
👀 Show answers
a. $25-\!\Big(5\dfrac{1}{9}+8\dfrac{7}{15}\Big)$
b. Too big. $5\dfrac{1}{9}\approx5.11$ and $8\dfrac{7}{15}\approx8.47$; their sum is about $13.58$, so the third side should be about $25-13.58\approx11.4$ m, not $\sim15.5$ m.
c. Exact length: $\;5\dfrac{1}{9}=\dfrac{46}{9},\;8\dfrac{7}{15}=\dfrac{127}{15}.$ Sum $=\dfrac{230}{45}+\dfrac{381}{45}=\dfrac{611}{45}.$ Third side $=25-\dfrac{611}{45}=\dfrac{1125-611}{45}=\dfrac{514}{45}=11\dfrac{19}{45}\text{ m}.$
7. Holly has three bags of apples. First bag $=2\dfrac{4}{5}$ kg. Second bag is twice the first. Total of three bags $=11\dfrac{13}{20}$ kg. Work out the mass of the third bag.
👀 Show answer
First $=\dfrac{14}{5}$, second $=2\cdot\dfrac{14}{5}=\dfrac{28}{5}.$ Total $=\dfrac{233}{20}.$ Third $=\dfrac{233}{20}-\dfrac{14}{5}-\dfrac{28}{5}=\dfrac{233-56-112}{20}=\dfrac{65}{20}=\dfrac{13}{4}=3\dfrac{1}{4}\text{ kg}.$
8. Copy and complete the workings to calculate $6\div\dfrac{4}{5}+3\dfrac{1}{4}\times5$.
👀 Show answer
Division: $6\div\dfrac{4}{5}=6\times\dfrac{5}{4}=\dfrac{30}{4}=\dfrac{15}{2}=7\dfrac{1}{2}$
Multiplication: $3\dfrac{1}{4}\times5=\dfrac{13}{4}\times5=\dfrac{65}{4}=16\dfrac{1}{4}$
Addition: $\dfrac{15}{2}+\dfrac{65}{4}=\dfrac{30}{4}+\dfrac{65}{4}=\dfrac{95}{4}=23\dfrac{3}{4}$
9. Work out the area of each shape. Show all your working.
a. Parallelogram with base $1\dfrac{2}{3}+ \dfrac{5}{6}$ m and height $2\dfrac{3}{4}$ m.
b. Triangle with base $4\dfrac{7}{8}$ cm and height $\dfrac{10}{3}$ cm.
c. Circle with radius $\dfrac{7}{11}$ m. Use $\pi=\dfrac{22}{7}$.
👀 Show answers
a. Base $=1\dfrac{2}{3}+\dfrac{5}{6}=\dfrac{5}{3}+\dfrac{5}{6}=\dfrac{15}{6}=\dfrac{5}{2}.$ Height $=2\dfrac{3}{4}=\dfrac{11}{4}.$ Area $=\text{base}\times\text{height}=\dfrac{5}{2}\cdot\dfrac{11}{4}=\dfrac{55}{8}=6\dfrac{7}{8}\;\text{m}^2$
b. $A=\dfrac{1}{2}\,bh=\dfrac{1}{2}\cdot\dfrac{39}{8}\cdot\dfrac{10}{3}=\dfrac{195}{24}=\dfrac{65}{8}=8\dfrac{1}{8}\;\text{cm}^2$
c. $A=\pi r^2=\dfrac{22}{7}\cdot\Big(\dfrac{7}{11}\Big)^2=\dfrac{22}{7}\cdot\dfrac{49}{121}=\dfrac{154}{121}=\dfrac{14}{11}=1\dfrac{3}{11}\;\text{m}^2$
🧠 Think like a Mathematician
Task: Explore two different methods for squaring a mixed number. Compare the results and identify the correct approach.
Discussion Statements:
Questions:
👀 show answer
- a) - Marcus’s method: Convert $1\dfrac{1}{2}$ to $\dfrac{3}{2}$. Then $(\dfrac{3}{2})^2 = \dfrac{9}{4} = 2\dfrac{1}{4}$. - Arun’s method: Square 1 to get 1, and square $\dfrac{1}{2}$ to get $\dfrac{1}{4}$. Add them: $1\dfrac{1}{4}$. This is incorrect because it leaves out the cross term from expansion. ✔ Marcus is correct.
- b) General rule: Always convert a mixed number into an improper fraction before squaring. This ensures the calculation includes all terms correctly.
❓ EXERCISES
11. Work out the answers to these calculations.
a. $\Big(2\dfrac{1}{2}\Big)^2-2\dfrac{1}{2}$
👀 Show answer
$\Big(\dfrac{5}{2}\Big)^2-\dfrac{5}{2}=\dfrac{25}{4}-\dfrac{10}{4}=\dfrac{15}{4}=3\dfrac{3}{4}$
b. $9\times 3\dfrac{1}{3}-\Big(\dfrac{2}{3}\Big)^2$
👀 Show answer
$9\times\dfrac{10}{3}-\dfrac{4}{9}=30-\dfrac{4}{9}=\dfrac{270-4}{9}=\dfrac{266}{9}=29\dfrac{5}{9}$
c. $4\dfrac{1}{5}+10\times\Big(1\dfrac{1}{5}\Big)^2$
👀 Show answer
$\dfrac{21}{5}+10\Big(\dfrac{6}{5}\Big)^2=\dfrac{21}{5}+10\cdot\dfrac{36}{25}=\dfrac{105}{25}+\dfrac{360}{25}=\dfrac{465}{25}=18\dfrac{3}{5}$
12. The diagram shows a compound shape made of a square joined to a rectangle.
a. Write the calculation you must do to work out the total area of the shape.
👀 Show answer
Let the common height be $2\dfrac{1}{3}$ m. Area $=\big(2\dfrac{1}{3}\big)^2+\big(2\dfrac{1}{3}\big)\times\big(5\dfrac{1}{2}\big)$.
b. Work out the area of the shape.
👀 Show answer
Square side $=\dfrac{7}{3}$, so $A_{\text{sq}}=\Big(\dfrac{7}{3}\Big)^2=\dfrac{49}{9}$.
Rectangle: $h=\dfrac{7}{3},\ w= \dfrac{11}{2}$, so $A_{\text{rect}}=\dfrac{7}{3}\cdot\dfrac{11}{2}=\dfrac{77}{6}$.
Total $=\dfrac{49}{9}+\dfrac{77}{6}=\dfrac{98}{18}+\dfrac{231}{18}=\dfrac{329}{18}=18\dfrac{5}{18}\ \text{m}^2$.
⚠️ Be careful! Fractions and Order of Operations
- Follow BODMAS/BIDMAS strictly: Do brackets first, then multiplication/division, then addition/subtraction. Do not just work left to right.
- Convert mixed numbers carefully. Always change them into improper fractions before multiplying or dividing.
- Use common denominators only when adding or subtracting fractions — not when multiplying or dividing.
- Don’t combine steps too soon. Work out bracketed parts separately to avoid mistakes (e.g. $\tfrac{3}{4}+\tfrac{4}{5}$ must be done before subtracting from $3\tfrac{1}{2}$).
- Simplify at the end, writing answers as mixed numbers in simplest form.
- Estimate first. Roughly round fractions to check your final answer makes sense (e.g. if you add a small fraction, your total should only increase slightly).