Understanding Ratios and Proportions

Ratios

What are Ratios?

A ratio is a way of comparing two or more quantities of the same kind. It shows the relative sizes of the quantities.

Ratios can be written in several ways:

  • Using a colon, e.g., 2:3 (read as "2 to 3").
  • Using the word "to", e.g., 2 to 3.
  • As a fraction, e.g., 2/3 (though this often implies a part-to-whole relationship, ratios can also be part-to-part).

For example, if a recipe calls for 2 cups of flour to 1 cup of sugar, the ratio of flour to sugar is 2:1.

Important Points:

  • The order of the numbers in a ratio is important. 2:3 is different from 3:2.
  • Ratios should usually be simplified to their lowest terms, just like fractions.
  • When comparing quantities in a ratio, they must be in the same units.

Working with Ratios: Examples

Visualizing Ratios with Boxes

Imagine you have a collection of square boxes:

4 Red Boxes:
2 Green Boxes:
6 Blue Boxes:

Question: What is the ratio of the Green, Red, Blue boxes in its simplest form?

  1. Write down the number of boxes for each color in the specified order (Green:Red:Blue):
    • Green boxes: 2
    • Red boxes: 4
    • Blue boxes: 6
    • Initial ratio: 2:4:6
  2. Find the highest common factor (HCF) of all the numbers in the ratio (2, 4, and 6).
    • Factors of 2 are: 1, 2.
    • Factors of 4 are: 1, 2, 4.
    • Factors of 6 are: 1, 2, 3, 6.
    • The Highest Common Factor (HCF) is 2.
  3. Divide each part of the ratio by the HCF (2):
    • Green: 2 ÷ 2 = 1
    • Red: 4 ÷ 2 = 2
    • Blue: 6 ÷ 2 = 3

Answer: The ratio of Green:Red:Blue boxes in its simplest form is 1:2:3.

Simplifying Ratios

Problem: Simplify the ratio 12:18.

  1. Find the highest common factor (HCF) of all the numbers in the ratio. To find the HCF of 12 and 18:
    • Factors of 12 are: 1, 2, 3, 4, 6, 12.
    • Factors of 18 are: 1, 2, 3, 6, 9, 18.
    • The common factors are 1, 2, 3, and 6. The Highest Common Factor is 6.
  2. Divide each part of the ratio by the HCF (6):
    • 12 ÷ 6 = 2
    • 18 ÷ 6 = 3

Answer: The simplified ratio is 2:3.

Sharing an Amount in a Given Ratio

Problem: Share £40 in the ratio 3:5.

  1. Add the parts of the ratio to find the total number of parts: 3 + 5 = 8 parts.
  2. Divide the total amount by the total number of parts to find the value of one part: £40 ÷ 8 = £5 per part.
  3. Multiply the value of one part by each number in the ratio:
    • First share: 3 parts × £5/part = £15.
    • Second share: 5 parts × £5/part = £25.
  4. Check: £15 + £25 = £40.

Answer: The amount is shared as £15 and £25.

Ratios with Different Units

Problem: Express the ratio of 50cm to 2 metres in its simplest form.

  1. Convert the quantities to the same unit. It's often easier to convert to the smaller unit.
    • 1 metre = 100 cm.
    • So, 2 metres = 2 × 100 = 200 cm.
  2. Write the ratio with the same units: 50cm : 200cm.
  3. Now simplify the ratio 50:200. The units can be dropped once they are the same.
    • HCF of 50 and 200 is 50. (Factors of 50: 1,2,5,10,25,50. Factors of 200: 1,2,4,5,8,10,20,25,40,50,100,200)
    • 50 ÷ 50 = 1
    • 200 ÷ 50 = 4

Answer: The ratio is 1:4.

Scaling Ratios / Finding Unknown Quantities

Problem 1: A recipe for 4 people uses 200g of flour and 100g of sugar. The ratio of flour to sugar is 200:100, which simplifies to 2:1. How much flour and sugar are needed for 6 people?

  1. Find the scaling factor: The recipe is for 4 people, and we need it for 6 people. Scaling factor = New quantity / Original quantity = 6 / 4 = 1.5.
  2. Multiply each ingredient quantity by the scaling factor:
    • Flour needed: 200g × 1.5 = 300g.
    • Sugar needed: 100g × 1.5 = 150g.
  3. The new ratio of flour to sugar for 6 people is 300:150, which still simplifies to 2:1.

Answer: 300g of flour and 150g of sugar are needed for 6 people.

Problem 2 (Cake Recipe): A cake recipe for 4 people requires the following ingredients:

  • Flour: 300g
  • Butter: 200g
  • Eggs: 2
How much of these ingredients are needed if you want to bake a cake for 7 people?

  1. Determine the scaling factor: We are scaling from a recipe for 4 people to a recipe for 7 people.
    Scaling factor = (New number of people) / (Original number of people)
    Scaling factor = 7 / 4 = 1.75.
  2. Multiply each original ingredient quantity by the scaling factor (1.75):
    • Flour for 7 people: 300g × 1.75 = 525g.
    • Butter for 7 people: 200g × 1.75 = 350g.
    • Eggs for 7 people: 2 eggs × 1.75 = 3.5 eggs. (You might use 3 large eggs or 4 small eggs and adjust slightly, or use 3 eggs and a bit of extra liquid if the recipe is flexible).

Answer: For 7 people, you would need 525g of Flour, 350g of Butter, and 3.5 Eggs.

Proportions

Understanding Proportions

Proportion describes how two or more quantities are related to each other. If a ratio states that A:B is the same as C:D, then these quantities are in proportion. In simpler terms, it's about scaling: if you change one quantity, how does the other quantity change in relation?

Direct Proportion

Two quantities are in **direct proportion** if they increase or decrease at the same rate. This means that as one quantity increases, the other quantity increases by the same factor, and similarly for a decrease.

Think of it like this: if you buy more of an item, the total cost increases proportionally. If you buy twice as many items, it costs twice as much.

Example: Direct Proportion

Problem: If 3 apples cost £1.50, how much will 7 apples cost?

  1. Find the cost per apple (the unit rate):
    £1.50 ÷ 3 apples = £0.50 per apple.
  2. Multiply the cost per apple by the new number of apples:
    £0.50 x 7 apples = £3.50.

Answer: 7 apples will cost £3.50.

Key idea:

If A is directly proportional to B, we can write this as A $\propto$ B, or A = kB for some constant k. In the example, Cost = 0.50 x Number of apples.

Inverse Proportion

Two quantities are in **inverse proportion** (or inversely proportional) if an increase in one quantity causes a proportional decrease in the other, and vice versa. Their product remains constant.

Think of it like this: if more workers are assigned to a task, the time it takes to complete the task decreases. If you double the number of workers, the time taken is halved.

Example: Inverse Proportion

Problem: If 4 builders can build a wall in 6 days, how long would it take 8 builders to build the same wall (assuming they work at the same rate)?

  1. Calculate the total "builder-days" required for the job:
    4 builders x 6 days = 24 builder-days. (This is the constant product, representing the total work).
  2. Divide the total "builder-days" by the new number of builders to find the time:
    24 builder-days ÷ 8 builders = 3 days.

Answer: It would take 8 builders 3 days to build the wall.

Key idea:

If A is inversely proportional to B, we can write this as A $\propto$ 1/B, or A = k/B (which means AB = k) for some constant k. In the example, Builders x Days = 24.

Test Your Ratio Skills!

1. Simplify the ratio 24:36.

Answer: 2:3

Explanation: HCF of 24 and 36 is 12. (24÷12=2), (36÷12=3).

2. Share £70 in the ratio 2:5.

Answer: £20 and £50

Explanation: Total parts = 2+5=7. Value of one part = £70÷7 = £10. Shares: (2×£10=£20), (5×£10=£50).

3. Write the ratio of 250g to 1kg in its simplest form.

Answer: 1:4

Explanation: 1kg = 1000g. Ratio is 250g:1000g. Simplify 250:1000 (HCF is 250) to 1:4.

4. Direct Proportion: If 5 pens cost £3.50, how much will 8 pens cost?

Answer: £5.60

Workings:

  1. Find the cost per pen: £3.50 ÷ 5 pens = £0.70 per pen.
  2. Multiply the cost per pen by the new number of pens: £0.70 x 8 pens = £5.60.

5. Inverse Proportion: If 3 painters can paint a house in 12 days, how long would it take 6 painters to paint the same house?

Answer: 6 days

Workings:

  1. Calculate total "painter-days" for the job: 3 painters x 12 days = 36 painter-days.
  2. Divide total "painter-days" by the new number of painters: 36 painter-days ÷ 6 painters = 6 days.

6. Inverse Proportion: A car travels at 60 mph and takes 4 hours to reach its destination. How long would it take if it travels at 80 mph?

Answer: 3 hours

Workings:

  1. Calculate the total distance (constant): 60 mph x 4 hours = 240 miles.
  2. Divide the total distance by the new speed: 240 miles ÷ 80 mph = 3 hours.

Exam-Style Ratio Problems

1. A map has a scale of 1:50000. The distance between two towns on the map is 6cm. What is the actual distance between the two towns in kilometres?

Answer: 3 km

Workings:

  1. The scale 1:50000 means 1cm on the map represents 50000cm in actual distance.
  2. Actual distance in cm: 6cm (map) × 50000 = 300000 cm.
  3. Convert cm to metres: There are 100cm in 1 metre.
    300000 cm ÷ 100 = 3000 metres.
  4. Convert metres to kilometres: There are 1000 metres in 1 kilometre.
    3000 metres ÷ 1000 = 3 kilometres.

2. To make a fruit punch, you need orange juice, pineapple juice, and lemonade in the ratio 3:2:5. If you want to make 5 litres of fruit punch in total, how many millilitres of pineapple juice do you need? (1 litre = 1000 ml)

Answer: 1000 ml (or 1 litre)

Workings:

  1. Total parts in the ratio: 3 + 2 + 5 = 10 parts.
  2. Total amount of punch to make: 5 litres = 5 × 1000 = 5000 ml.
  3. Value of one part: 5000 ml ÷ 10 parts = 500 ml per part.
  4. Pineapple juice corresponds to 2 parts in the ratio.
  5. Amount of pineapple juice needed: 2 parts × 500 ml/part = 1000 ml.

3. In a bag of sweets, the ratio of red sweets to green sweets is 4:7. If there are 28 red sweets, how many green sweets are there?

Answer: 49 green sweets

Workings:

  1. The ratio of red to green is 4:7. This means for every 4 red sweets, there are 7 green sweets.
  2. We know there are 28 red sweets. This corresponds to the '4' part of the ratio.
  3. Find the value of one 'part' by dividing the number of red sweets by its ratio number: 28 red sweets ÷ 4 = 7. So, one part represents 7 sweets.
  4. The green sweets correspond to the '7' part of the ratio.
  5. Calculate the number of green sweets: 7 parts × 7 sweets/part = 49 green sweets.

4. To make pink paint, a decorator mixes red paint and white paint in the ratio 2:5. If the decorator needs to make 17.5 litres of pink paint, how many litres of red paint and white paint are needed?

Red (2 parts)
White (5 parts)

Answer: 5 litres of Red Paint and 12.5 litres of White Paint

Workings:

  1. The ratio of red paint to white paint is 2:5.
  2. Add the parts of the ratio to find the total number of parts: 2 (red) + 5 (white) = 7 total parts.
  3. Divide the total amount of pink paint needed by the total number of parts to find the value of one part:
    17.5 litres ÷ 7 parts = 2.5 litres per part.
  4. Multiply the value of one part by each number in the ratio to find the amount for each color:
    • Red paint needed: 2 parts × 2.5 litres/part = 5 litres.
    • White paint needed: 5 parts × 2.5 litres/part = 12.5 litres.
  5. Check your answer: 5 litres (red) + 12.5 litres (white) = 17.5 litres (total pink paint).

5. A carpenter is making a wooden gate using the design below. The gate has a total width of 2m.

The gate will need:

  • 3 horizontal wooden battens of length 2m.
  • 2 tall vertical posts of length 1.5m.
  • 3 medium vertical slats.
  • 2 short vertical slats.

The heights of the tall, medium, and short vertical pieces are in the ratio 5:4:3.

How many metres of wood in total does the carpenter need to make the gate?

Answer: 14.4 metres

Workings:

  1. Find the value of one part of the ratio.
    The ratio for Tall:Medium:Short is 5:4:3. We know a tall post is 1.5m, which corresponds to the '5' part of the ratio.
    Value of one part = 1.5m ÷ 5 = 0.3m.
  2. Calculate the lengths of the medium and short slats.
    Medium slat length (4 parts) = 4 parts × 0.3m/part = 1.2m.
    Short slat length (3 parts) = 3 parts × 0.3m/part = 0.9m.
  3. Calculate the total length of all vertical pieces.
    2 Tall posts: 2 × 1.5m = 3.0m.
    3 Medium slats: 3 × 1.2m = 3.6m.
    2 Short slats: 2 × 0.9m = 1.8m.
    Total vertical length = 3.0m + 3.6m + 1.8m = 8.4m.
  4. Calculate the total length of all horizontal pieces.
    3 Horizontal battens: 3 × 2m = 6m.
  5. Calculate the total wood needed.
    Total length = Total vertical + Total horizontal
    Total = 8.4m + 6m = 14.4m.

Interactive Tool: Share Amount in Ratio

Enter a total amount, select a unit, and provide a ratio (e.g., 2:3 or 1:2:4) to see how the amount is shared.

Shared Amounts: ?

Interactive Tool: Ratio Simplifier

Enter a ratio with whole numbers (e.g., 12:18 or 10:20:30) to see its simplest form.

Simplified Ratio: ?

Key Ratio & Proportion Facts