Introduction
- Ratios compare the number of one thing to another. They tell us how much of one quantity is present with respect to another.
Simplification:
- (Common Factor : 3) If we multiply or divide the ratio’s numbers, the ratio still remains the same.
- From this simplification using a common factor, we can say that there are 5 Apples for each 4 Mangoes.
Ratios - GCSE Maths
Ratios Representation
- Ratios in Algebra are used to solve problems by finding unknown quantities and understanding variables.
- Ratios can be expressed as fractions, using colons (e.g., $5:4$), or in words (e.g., “5 to 4”).
Types of Ratios:
Part to Part: Compares two or more parts of a whole (e.g., apples to mangoes).
Part to Whole: Compares parts to the whole quantity (e.g., apples to all fruits).
Part to Whole: Compares parts to the whole quantity (e.g., apples to all fruits).
Divide in the Ratios
We can divide a quantity in some ratio by following these steps:
1
Add the numbers present in the ratio to find the total number of parts.
2
Divide the total quantity by the sum of the ratio parts. This gives the value of a single part (the Unit Ratio).
3
Multiply each ratio part by the value of one part to find the specific quantities.
- Key Point: When one of the parts in the ratio is $1$, it is called a Unit Ratio.
Need help with Ratios?
Our tutors explain it step by step, matched to your exam board.
Book a Consultation →Solved Examples
Solved Example
Divide $120$ in the ratio $2:3$
SOLUTION
1
Addition: Add the numbers present in the ratio.
$$2 + 3 = 5$$
$$2 + 3 = 5$$
2
Division: Divide the quantity by the sum to find the value of one part.
$$\frac{120}{5} = 24$$
$$\frac{120}{5} = 24$$
3
Multiplication: Multiply each ratio part by the value of one part.
$$2 \times 24 = 48$$ $$3 \times 24 = 72$$
$$2 \times 24 = 48$$ $$3 \times 24 = 72$$
Final Answer: The quantity is divided into two parts: $48, 72$
Solved Example
Divide $200$ in the ratio $4:6$
SOLUTION
1
Addition:
$$4 + 6 = 10$$
$$4 + 6 = 10$$
2
Division:
$$\frac{200}{10} = 20$$
$$\frac{200}{10} = 20$$
3
Multiplication:
$$4 \times 20 = 80$$ $$6 \times 20 = 120$$
$$4 \times 20 = 80$$ $$6 \times 20 = 120$$
Final Answer: The quantity is divided into two parts: $80, 120$
Solved Example
Divide $1080$ in the ratio $4:8$
SOLUTION
1
Addition:
$$4 + 8 = 12$$
$$4 + 8 = 12$$
2
Division:
$$\frac{1080}{12} = 90$$
$$\frac{1080}{12} = 90$$
3
Multiplication:
$$4 \times 90 = 360$$ $$8 \times 90 = 720$$
$$4 \times 90 = 360$$ $$8 \times 90 = 720$$
Final Answer: The quantity is divided into two parts: $360, 720$
Solved Example
Divide $900$ in the ratio $1:3:5$
SOLUTION
1
Addition:
$$1 + 3 + 5 = 9$$
$$1 + 3 + 5 = 9$$
2
Division:
$$\frac{900}{9} = 100$$
$$\frac{900}{9} = 100$$
3
Multiplication:
$$1 \times 100 = 100$$ $$3 \times 100 = 300$$ $$5 \times 100 = 500$$
$$1 \times 100 = 100$$ $$3 \times 100 = 300$$ $$5 \times 100 = 500$$
Final Answer: The quantity $900$ is divided into three parts: $100, 300, 500$
Reasoning Problem
Suppose you and your friend have $51$ toffees and you want to share it with him in a $1:2$ ratio. How will you divide the quantity?
SOLUTION
1
Addition: Add the parts.
$$1 + 2 = 3$$
$$1 + 2 = 3$$
2
Division: Find the value of one part.
$$\frac{51}{3} = 17$$
$$\frac{51}{3} = 17$$
3
Multiplication: Find the value of each part of the ratio.
$$1 \times 17 = 17$$ $$2 \times 17 = 34$$
$$1 \times 17 = 17$$ $$2 \times 17 = 34$$
Final Answer: The $51$ toffees are divided into: $17, 34$
Reasoning Problem
Suppose there are $540$ crayons to be divided into the ratio $4:5$ for distribution between two children. How will you divide the crayons so that the ratio is satisfied?
SOLUTION
1
Addition:
$$4 + 5 = 9$$
$$4 + 5 = 9$$
2
Division: Find the value of one part.
$$\frac{540}{9} = 60$$
$$\frac{540}{9} = 60$$
3
Multiplication: Find the value of each part.
$$4 \times 60 = 240$$ $$5 \times 60 = 300$$
$$4 \times 60 = 240$$ $$5 \times 60 = 300$$
Final Answer: The $540$ crayons are divided into: $240, 300$
Reasoning Problem
Suppose there are $12$ classes in a school. Each class contains $40$ students and the ratio of girls to boys is $5:7$. Find how many girls and boys there are in the school.
SOLUTION
The total quantity is $12 \times 40 = 480$. The ratio of girls to boys is $5:7$.
1
Addition:
$$5 + 7 = 12$$
$$5 + 7 = 12$$
2
Division: Find the value of one part.
$$\frac{480}{12} = 40$$
$$\frac{480}{12} = 40$$
3
Multiplication: Find the value of each part.
$$5 \times 40 = 200 \text{ (Girls)}$$ $$7 \times 40 = 280 \text{ (Boys)}$$
$$5 \times 40 = 200 \text{ (Girls)}$$ $$7 \times 40 = 280 \text{ (Boys)}$$
Final Answer: There are $200$ girls and $280$ boys.
Reasoning Problem
If James and Jason divided the money they got in the ratio $6:8$ and James got $£150$, then how much did Jason get? And what was the total amount?
SOLUTION
1
Find the value of one part:
James’s share represents $6$ parts of the ratio, and we know his share is $£150$.
$$\frac{150}{6} = 25$$
James’s share represents $6$ parts of the ratio, and we know his share is $£150$.
$$\frac{150}{6} = 25$$
2
Find the value of the second part:
Jason’s share represents $8$ parts. Now we multiply the value of one part by $8$.
$$8 \times 25 = 200$$
Jason’s share represents $8$ parts. Now we multiply the value of one part by $8$.
$$8 \times 25 = 200$$
3
Find the total amount:
Add James’s and Jason’s shares together.
$$150 + 200 = 350$$ (Verification: $6 + 8 = 14$ total parts. $350 \div 14 = 25$, which matches our single part value!)
Add James’s and Jason’s shares together.
$$150 + 200 = 350$$ (Verification: $6 + 8 = 14$ total parts. $350 \div 14 = 25$, which matches our single part value!)
Final Answer: Jason got $£200$, and the total amount was $£350$.
Ready to test your knowledge?
You've reviewed the notes. Now try exam-style questions on Ratios.
Start Practice Questions →