Sharing ratios

Sharing in a ratio is a key mathematical concept used to divide a quantity into parts that maintain a specified relationship. It is often used in practical contexts, such as dividing money or splitting resources. Understanding how to share in a ratio ensures fair and proportional distribution.

1. Steps for Sharing in a Ratio:

To share a quantity in a given ratio, follow these steps:

1. Add the parts of the ratio: Find the total number of parts by adding the terms of the ratio.
2. Divide the total amount: Divide the quantity by the total number of parts to find the value of one part.
3. Allocate the parts: Multiply the value of one part by each term in the ratio to find the share for each part.

2. Example 1: Sharing Money

Divide £60 in the ratio \(2 : 3\):

• Step 1: Add the parts of the ratio: \(2 + 3 = 5\).
• Step 2: Divide £60 by 5 to find the value of one part: \(\frac{60}{5} = 12\).
• Step 3: Allocate the parts:
  • First part: \(2 \times 12 = 24\)
  • Second part: \(3 \times 12 = 36\)

The £60 is shared as £24 and £36.

3. Example 2: Sharing Items

Share 48 sweets in the ratio \(3 : 5\):

• Step 1: Add the parts of the ratio: \(3 + 5 = 8\).
• Step 2: Divide 48 by 8 to find the value of one part: \(\frac{48}{8} = 6\).
• Step 3: Allocate the parts:
  • First part: \(3 \times 6 = 18\)
  • Second part: \(5 \times 6 = 30\)

The 48 sweets are shared as 18 and 30.

4. Using Ratios to Solve Problems:

Ratios can also be used to solve reverse problems, where you are given one share and asked to find the total or other shares. For example:

Example 3: If £40 represents the larger share in a \(2 : 3\) split, what is the total amount?

• Step 1: Recognise that the larger share corresponds to \(3\) parts.
• Step 2: Find the value of one part: \(\frac{40}{3} \approx 13.33\).
• Step 3: Find the total by adding all parts: \(5 \times 13.33 \approx 66.67\).

The total amount is approximately £66.67.

5. Summary:

• To share a quantity in a ratio, add the parts of the ratio, divide the total by the number of parts, and allocate accordingly.
• Ratios can be used to solve forward and reverse problems involving division and proportional relationships.