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Edexcel GCSE Maths (Higher) Study Notes

3.1.1 Understanding Ratios

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.

Ratio

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: 2010:3010:4010=2:3:4\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: 4515:6015:9015=3:4:6\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=92 + 3 + 4 = 9

2. Value per Part: £1209=£13.33\frac{£120}{9} = £13.33

3. Distribution:

  • Person 1: 2×£13.33=£26.672 \times £13.33 = £26.67
  • Person 2: 3×£13.33=£40.003 \times £13.33 = £40.00
  • Person 3: 4×£13.33=£53.334 \times £13.33 = £53.33

Divide £180 in the ratio 3:2:5

  1. Total Parts: 3+2+5=103 + 2 + 5 = 10
  2. Value per Part: £18010=£18.00\frac{£180}{10} = £18.00
  3. Distribution:
    • Part 1: 3×£18.00=£54.003 \times £18.00 = £54.00
    • Part 2: 2×£18.00=£36.002 \times £18.00 = £36.00
    • Part 3: 5×£18.00=£90.005 \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 g900 \ g / 2 = 450 \ g per part.
  3. Required quantities:
    • Second ingredient: 3×450 g=1350 g3 \times 450 \ g = 1350 \ g.
    • Third ingredient: 4×450 g=1800 g4 \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=53 + 2 = 5.
  2. Total Paint: 500 ml
  3. Value per Part: 500ml5=100ml\frac{500ml}{5} = 100ml
  4. Quantities:
    • Yellow paint: 3×100ml=300ml3 \times 100ml = 300ml
    • Blue paint: 2×100ml=200ml2 \times 100ml = 200ml

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