Ratios are crucial in mathematics for comparing quantities. Simplifying ratios and dividing quantities according to ratios are essential skills for a variety of real-world applications.

**Simplifying Ratios**

Reducing ratios to their simplest form involves finding the greatest common divisor (GCD) of the numbers and dividing them by it.

**Simplify the ratio 20:30:40**

**1. GCD of 20, 30, 40 is 10.**

**2. Simplified Ratio:** $\frac{20}{10} : \frac{30}{10} : \frac{40}{10} = 2 : 3 : 4$

**Simplify the ratio 45:60:90**

**1. GCD of 45, 60, 90 is 15.**

**2. Simplified Ratio:** $\frac{45}{15} : \frac{60}{15} : \frac{90}{15} = 3 : 4 : 6$

**Dividing Quantities in a Ratio**

This involves calculating the value of a single part in the ratio and then distributing the total quantity accordingly.

**Divide £120 in the ratio 2:3:4**

**1. Total Parts: **$2 + 3 + 4 = 9$

**2. Value per Part: **$\frac{£120}{9} = £13.33$

**3. Distribution:**

**Person 1:**$2 \times £13.33 = £26.67$**Person 2:**$3 \times £13.33 = £40.00$**Person 3:**$4 \times £13.33 = £53.33$

**Divide £180 in the ratio 3:2:5**

**Total Parts:**$3 + 2 + 5 = 10$**Value per Part:**$\frac{£180}{10} = £18.00$**Distribution:****Part 1:**$3 \times £18.00 = £54.00$**Part 2:**$2 \times £18.00 = £36.00$**Part 3:**$5 \times £18.00 = £90.00$

**Worked Problems**

**Problem 1: Ratio Application in Recipes**

Suppose a recipe for a cake requires ingredients in the ratio 2:3:4. If you have 900g of the first ingredient, how much of the other two ingredients do you need?

#### Solution:

**Given Ratio: 2:3:4.****Total parts of the given ingredient:**$900 \ g / 2 = 450 \ g$**per part.****Required quantities:****Second ingredient:**$3 \times 450 \ g = 1350 \ g$**.****Third ingredient:**$4 \times 450 \ g = 1800 \ g$**.**

**Problem 2: Mixing Paints**

To get a particular shade of green, a painter mixes yellow and blue paint in the ratio 3:2. If the painter needs 500ml of green paint, how much of each colour does he use?

#### Solution:

**Total Ratio Parts:**$3 + 2 = 5$**.****Total Paint: 500 ml****Value per Part:**$\frac{500ml}{5} = 100ml$**Quantities:****Yellow paint:**$3 \times 100ml = 300ml$**Blue paint:**$2 \times 100ml = 200ml$