Ratio (Junior Cert Mathematics): Revision Notes

Example 1: Sharing Chocolates Imagine you have 5 chocolates, and you want to share them between two people in a ratio of 3:2. This means:

• The first person gets 3 parts of the chocolates.
• The second person gets 2 parts of the chocolates. The key idea in ratios is that the numbers represent parts of a whole. So, in a 3:2 ratio, the first person gets more because their part of the ratio is larger.

Example 2: Calculating Total Parts Let's continue with the 3:2 ratio:

1. Add the numbers in the ratio: $3 + 2 = 5$ This tells you that the quantity is divided into 5 parts.

2. Each part represents a fraction of the total amount:

• The first person's share is 3 out of 5 parts, which can be written as the fraction $\frac{3}{5} .$
• The second person's share is 2 out of 5 parts, written as $\frac{2}{5} .$

Example: Dividing Money Let's work through a detailed example to understand how to use ratios to divide a quantity.

Question: Divide €200 between Alan and Brian in the ratio 3:2.

Step 1: Understand the Ratio

The ratio given is 3:2.

This means for every 3 parts Alan gets, Brian gets 2 parts.

Step 2: Calculate the Total Number of Parts

Add the numbers in the ratio: $3 + 2 = 5$

So, there are 5 parts in total.

Step 3: Determine the Value of One Part

To find out how much one part is worth, divide the total amount of money €200 by the total number of parts 5: $\frac{200 \, \text{euros}}{5} = 40 \, \text{euros per part}$

Step 4: Calculate Each Person's Share

• Alan's Share: Multiply the number of parts Alan gets by the value of one part: $3 \times 40 \, \text{euros} = 120 \, \text{euros}$
• Brian's Share: Multiply the number of parts Brian gets by the value of one part: $2 \times 40 \, \text{euros} = 80 \, \text{euros}$

Explanation:

• We divided the total amount €200 into 5 parts because the sum of the ratio is 5.
• Alan receives 3 parts out of the 5, so he gets €120.
• Brian receives 2 parts out of the 5, so he gets €80. So, Alan gets €120, and Brian gets €80.

Example: Sharing Sweets Among Friends Question: A pack of 60 sweets is to be shared between three friends, Sarah, John, and Emily, in the ratio 4:3:3. How many sweets does each person get?

Step 1: Understand the Ratio

The ratio given is 4:3:3.

This means for every 4 parts Sarah gets, John and Emily each get 3 parts.

Step 2: Calculate the Total Number of Parts

Add the numbers in the ratio: $4 + 3 + 3 = 10$

So, there are 10 parts in total.

Step 3: Determine the Value of One Part

To find out how much one part is worth, divide the total number of sweets 60 by the total number of parts 10: $\frac{60 \, \text{sweets}}{10} = 6 \, \text{sweets per part}$

Step 4: Calculate Each Person's Share

• Sarah's Share: Multiply the number of parts Sarah gets by the value of one part: $4 \times 6 \, \text{sweets} = 24 \, \text{sweets}$
• John's Share: Multiply the number of parts John gets by the value of one part: $3 \times 6 \, \text{sweets} = 18 \, \text{sweets}$
• Emily's Share: Multiply the number of parts Emily gets by the value of one part: $3 \times 6 \, \text{sweets} = 18 \, \text{sweets}$

Explanation:

• We divided the total number of sweets 60 into 10 parts because the sum of the ratio is 10.
• Sarah receives 4 parts out of the 10, so she gets 24 sweets.
• John receives 3 parts out of the 10, so he gets 18 sweets.
• Emily receives 3 parts out of the 10, so she gets 18 sweets. So, Sarah gets 24 sweets, John gets 18 sweets, and Emily gets 18 sweets.

Try it out! Question 1: A painter needs to mix red and blue paint to create purple paint. The ratio of red to blue paint must be 7:5. If the painter has 84 litres of red paint, how much blue paint does he need to mix to maintain the ratio?

Question 2: Three friends, Alice, Bob, and Charlie, start a business together and agree to share the profit in the ratio 5:3:2. After a successful year, the business makes a profit of €25,000. How much does each friend get?

Solutions:

Question 1: A painter needs to mix red and blue paint to create purple paint. The ratio of red to blue paint must be 7:5. If the painter has 84 litres of red paint, how much blue paint does he need to mix to maintain the ratio?

Step 1: Understand the Ratio

The ratio given is 7:5. This means for every 7 parts of red paint, there are 5 parts of blue paint.

Step 2: Set Up the Proportion

• Let $x$ be the amount of blue paint needed.

• The ratio of red to blue is set up as: $\frac{7}{5} = \frac{84}{x}$

• Here, 7 parts of red correspond to 5 parts of blue, and 84 litres of red correspond to $x$ litres of blue. Step 3: Solve the Proportion

• To solve for $x ,$ cross multiply: $7x = 5 \times 84$

• Multiply: $7x = 420$

• Divide by 7 to solve for $x :$ $x = \frac{420}{7} = 60 \, \text{litres}$

Explanation:

The painter needs to mix 60 litres of blue paint with 84 litres of red paint to maintain the 7:5 ratio.

Answer: The painter needs 60 litres of blue paint.

Question 2: Three friends, Alice, Bob, and Charlie, start a business together and agree to share the profit in the ratio 5:3:2. After a successful year, the business makes a profit of €25,000. How much does each friend get?

Step 1: Understand the Ratio

The ratio given is 5:3:2. This means Alice will get 5 parts, Bob will get 3 parts, and Charlie will get 2 parts of the total profit.

Step 2: Calculate the Total Number of Parts

• Add the numbers in the ratio to find the total number of parts: $5 + 3 + 2 = 10 \, \text{parts}$
• So, the total profit is divided into 10 parts. Step 3: Determine the Value of One Part

To find out how much one part is worth, divide the total profit €25,000 by the total number of parts 10: $\frac{25,000 \, \text{euros}}{10} = 2,500 \, \text{euros per part}$

Step 4: Calculate Each Person's Share

• Alice's Share: Multiply the number of parts Alice gets by the value of one part: $5 \times 2,500 \, \text{euros} = 12,500 \, \text{euros}$
• Bob's Share: Multiply the number of parts Bob gets by the value of one part: $3 \times 2,500 \, \text{euros} = 7,500 \, \text{euros}$
• Charlie's Share: Multiply the number of parts Charlie gets by the value of one part: $2 \times 2,500 \, \text{euros} = 5,000 \, \text{euros}$

Explanation:

• We divided the total profit €25,000 into 10 parts because the sum of the ratio is 10.
• Alice receives 5 parts out of the 10, so she gets €12,500.
• Bob receives 3 parts out of the 10, so he gets €7,500.
• Charlie receives 2 parts out of the 10, so he gets €5,000. Answer: Alice gets €12,500, Bob gets €7,500, and Charlie gets €5,000.