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

1. Total Parts: $3 + 2 + 5 = 10$
2. Value per Part: $\frac{£180}{10} = £18.00$
3. 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:

1. Given Ratio: 2:3:4.
2. Total parts of the given ingredient: $900 \ g / 2 = 450 \ g$ per part.
3. 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:

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