Ratios | Mathematics for GCSE/IGCSE - Year 11 PDF Download

What is a ratio?

• A ratio is a method of comparing one part of a whole to another
• A ratio can be represented as a fraction (of the whole)
• Ratios are commonly used when illustrating how things are divided or in situations involving scale factors
• For instance, if a pizza is divided into 8 slices and shared in the ratio 6:2, it implies that person A gets 6 slices, and person B gets 2 slices

How do I simplify a ratio or find an equivalent ratio?

• When we talk about ratios, we are essentially comparing two quantities. For instance, if we have a pizza cut into 8 slices and they are shared in the ratio 6:2, we are comparing the amounts received by two individuals.
• It's important to understand that ratios can be simplified. For example, the ratio 6:2 can also be expressed as 3:1 by dividing both sides of the ratio by 2. This simplification does not change the relative amounts each person receives.
• Simplifying ratios is akin to simplifying fractions. Just like 1/2 is equivalent to 2/4, the ratio 6:2 is equivalent to 3:1. The key is to ensure that the ratio's proportions remain the same.
• When determining an equivalent ratio, it involves either multiplying or dividing both sides of the ratio by the same number.
• For example, consider a giant pizza cut into 800 slices. Person A might get 600 slices, while Person B would receive 200 slices. In this scenario, both sides of the ratio have been scaled by a factor of 100.
• Scaling ratios is a flexible process. It allows us to maintain proportionality, such as ensuring that Person A consistently receives three times the amount that Person B receives.
• Scaling ratios is akin to finding equivalent fractions or simplifying them. However, it's crucial to note that a ratio like 1:4 is not the same as 1/4.

How do I use a ratio to find a fraction?

• Ratios can be used to calculate a fraction of a total amount. 
• For instance, if 8 pizza slices are shared in the ratio 6:2, Person A would receive 6 out of 8 slices, which is 6/8 or 3/4 of the pizza.
• Similarly, Person B would get 2 out of 8 slices, which is 2/8 or 1/4 of the pizza. 
• These fractions can be converted to percentages if necessary.

Working with Ratios

How do I share an amount into a ratio?

• If we have $200 to divide between individuals A and B in the ratio 5:3
• This ratio translates to a total of 8 parts, with A receiving 5 parts and B receiving 3 parts
• To split $200 into 8 parts, each part is valued at $25 ($200 ÷ 8 = $25)
• Illustrating this division with a simple diagram can aid understanding:
  
• Person A gets 5 parts, each valued at $25, totaling $125 for person A (5 × 25 = $125)
• Person B gets 3 parts, each valued at $25, totaling $75 for person B (3 × 25 = $75)
• Ensure that the sum of the amounts for each person equals the total amount: $125 + $75 = $200

What do I do when given the difference in a ratio problem?

• When faced with a scenario involving ratios, rather than being provided with the total quantity, you might be given the discrepancy between two separate shares.
• For instance, consider a situation in a parking lot where the ratio of blue cars to silver cars is 3:5, with the added information that there are 12 more silver cars than blue cars.
• Some students find it helpful to show this in a simple diagram
  
• The difference in the number of parts of the ratio is 2 (5 – 3 = 2)
  
• The difference in the number of cars is 12
  
• This means that 2 parts = 12 cars
• We can simplify this to 1 part = 6 cars (by dividing both sides by 2)
  
• Now that we know how much 1 part is worth, we can find how many cars of each colour there are, and the total number of cars
  
  • 3 parts are blue
    3 × 6 = 18 blue cars
  • 5 parts are silver
    5 × 6 = 30 silver cars

  • 8 parts in total
    8 × 6 = 48 cars in total

Given one part of a ratio, how can I find the other part?

• Instead of being provided with the total quantity to be divided, you might be given the value of one side of the ratio.
• For instance, consider a fruit drink prepared by blending concentrate with water in the ratio 2:3. We aim to determine the volume of water required to be added to 5 litres of concentrate.
• Some students find it helpful to show this in a simple diagram
  
• We are told that there are 5 litres of concentrate, and it must be mixed in the ratio 2:3
• This means that the two parts on the left, are equivalent to 5 litres
  
• This means that 1 part must be equal to 2.5 litres (5 ÷ 2 = 2.5)
  
• Now that we know how much 1 part is worth, we can find how many litres of water are required, and the total amount of fruit drink produced
  
  3 parts are water3 × 2.5 = 7.5 litres of water
  5 parts in total
  5 × 2.5 = 12.5 litres of fruit drink produced in total

How do I combine two ratios to make a 3-part ratio?

• Consider a scenario where you are presented with two distinct ratios that interconnect, forming a 3-part ratio.
• Imagine a farm housing a total of 85 animals.
  • Ratio of cows to sheep: 2:3
  • Ratio of sheep to pigs: 6:7
  • Objective: Determine the quantity of each animal on the farm.
  • Directly distributing 85 animals based on the ratios 2:3 or 6:7 isn't feasible as these ratios do not encompass all the animals independently.
    • It's necessary to establish a consolidated, 3-part ratio illustrating the proportional shares of all animals collectively.
    • Key Insight: Leveraging the common entity, sheep, in both ratios as a connecting link.
      • Given C:S = 2:3 and S:P = 6:7, scale the C:S ratio by 2 to standardize both ratios concerning 6 sheep.
      • Resulting in C:S = 4:6 and S:P = 6:7, which can be amalgamated into a comprehensive ratio, C:S:P = 4:6:7.
    • Allocation of Animals
      • Utilize the unified ratio 4:6:7 to apportion the 85 animals accordingly.
      • With a total of 17 parts (4 + 6 + 7 = 17) in the ratio, each part corresponds to 5 animals (85 ÷ 17 = 5).
      • Consequently, there are 20 cows (4 x 5), 30 sheep (6 x 5), and 35 pigs (7 x 5) on the farm.