Chapter 6: Rate and Ratio - Your Guide to Comparing the World!

Hey everyone! Welcome to the exciting world of Rate and Ratio. This might sound like a complicated topic, but don't worry! It's all about something we do every single day: comparing things.

Whether you're following a recipe, figuring out the best deal at a shop, or reading a map, you're using rates and ratios. By the end of these notes, you'll understand how to compare quantities like a pro and see how useful this part of Maths is in real life. Let's get started!


Part 1: Understanding Ratio

What is a Ratio?

A Ratio is a way to compare two or more quantities that have the same units. Think of it like a recipe for orange juice that tells you to mix 1 part concentrate with 3 parts water. You are comparing the amount of concentrate to the amount of water.

We can write this ratio in two ways:
1. Using a colon symbol --> 1 : 3
2. As a fraction --> $$ \frac{1}{3} $$
Both are read as "one to three".

Important: The order in a ratio matters! A ratio of 1 : 3 (concentrate to water) is very different from a ratio of 3 : 1 (which would make very strong juice!).

Simplifying Ratios

Just like fractions, ratios should always be written in their simplest form. To do this, we find the biggest number that can divide into all parts of the ratio and then divide by it.

Example: A class has 18 boys and 12 girls. Find the ratio of boys to girls.

Step 1: Write the initial ratio.
The ratio of boys to girls is 18 : 12.

Step 2: Find the highest common factor (HCF).
What is the biggest number that divides both 18 and 12? It's 6.
(If you can't find the biggest number, just keep dividing by smaller common numbers like 2 or 3 until you can't go any further!)

Step 3: Divide both parts of the ratio by the HCF.
$$ 18 \div 6 = 3 $$ $$ 12 \div 6 = 2 $$
Step 4: Write the simplified ratio.
The simplified ratio is 3 : 2. This means for every 3 boys, there are 2 girls.

Solving Problems with Ratios

Sometimes you'll be given a ratio and asked to find a missing amount or share something out.

Example: The ratio of red sweets to blue sweets in a bag is 4 : 5. If there are 20 blue sweets, how many red sweets are there?

Method 1: Using equivalent ratios
1. Write the ratios as fractions: $$ \frac{red}{blue} = \frac{4}{5} $$ 2. We know there are 20 blue sweets. So, $$ \frac{red}{20} = \frac{4}{5} $$ 3. To get from 5 to 20, you multiply by 4. So, we must do the same to the top number! 4. $$ 4 \times 4 = 16 $$ 5. There are 16 red sweets.

Quick Review: Ratios

What it is: Comparing quantities with the same units.
How to write it: `a : b` or as a fraction.
Key rule: Always simplify it!
Remember: Order is very important!


Part 2: Understanding Rate

What is a Rate?

A Rate is also a comparison, but this time it's between two quantities with different units. Because the units are different, we MUST write them down.

Think about these real-life examples:
- Your speed in a car: kilometres per hour (km/h)
- The price of bananas: dollars per kilogram ($/kg)
- Your heart beat: beats per minute

The key word is often "per", which means "for each". So 80 km/h means you travel 80 kilometres for each hour.

Memory Aid: Rate vs Ratio

Ratio has quantities of the same type. Both aRe alike.
Rate has quantities of diffeRent types.

Calculating with Rates

Calculating a rate is usually a simple division problem.

Example: A tap fills a 50-litre bucket in 5 minutes. What is the flow rate of the water?

Step 1: Identify the two different quantities.
We have litres (L) and minutes (min).

Step 2: Divide the first quantity by the second quantity.
$$ Rate = \frac{50 \text{ L}}{5 \text{ min}} $$
Step 3: Calculate the answer and include the units.
$$ Rate = 10 \text{ L/min} $$ The flow rate is 10 litres per minute.

Key Takeaway: Rate

A rate compares two things with different units (like distance and time). The units are essential and must be written as part of the answer, often using a slash `/` (e.g., m/s).


Part 3: Understanding Proportion

What is a Proportion?

A Proportion is a statement that two ratios or two rates are equal. It's the key to solving many real-world problems!

For example, if one apple costs $2, then two apples cost $4.
The rate of dollars to apples is the same: $$ \frac{$2}{1 \text{ apple}} = \frac{$4}{2 \text{ apples}} $$ This is a proportion! It shows an equal relationship.

Direct and Inverse Proportion

There are two main types of proportional relationships you need to know. Don't worry, the idea is quite simple!

1. Direct Proportion

This is when two quantities increase or decrease together at the same rate.
- The more hours you work, the more money you earn.
- The less paint you buy, the smaller the area you can cover.

Example: If 3 notebooks cost $12, how much will 7 notebooks cost?

Step 1: Find the cost of one item (the unit rate).
Cost of 1 notebook = $$ \frac{$12}{3} = $4 $$

Step 2: Multiply the unit rate by the new number of items.
Cost of 7 notebooks = $$ 7 \times $4 = $28 $$ So, 7 notebooks will cost $28.

2. Inverse Proportion

This is when one quantity goes up, the other goes down by the same rate.
- The faster you drive, the less time it takes to get there.
- The more people painting a wall, the less time it will take.

Example: If it takes 4 workers 6 hours to build a fence, how long will it take 8 workers?

Step 1: Find the total "work" required.
This is a bit different! Think about the total number of "worker-hours" needed.
Total work = 4 workers × 6 hours = 24 worker-hours.

Step 2: Divide the total work by the new number of workers.
Time for 8 workers = $$ \frac{24 \text{ worker-hours}}{8 \text{ workers}} = 3 \text{ hours} $$ It will take 8 workers 3 hours. (It makes sense - more workers means less time!)

Key Takeaway: Proportions

Direct Proportion: Both quantities move in the same direction (both up or both down).
Inverse Proportion: The quantities move in opposite directions (one up, one down).


Part 4: Real-World Application - Map Scales

Have you ever used a map? The scale on a map is a perfect example of a ratio!

A Scale on a plan or map compares the distance on the drawing to the actual distance in real life.

A scale might be written as 1 : 100 000.
This means 1 cm on the map represents 100 000 cm in the real world.

Example: On a map with a scale of 1 : 50 000, the distance between two towns is 4 cm. What is the actual distance in kilometres?

Step 1: Set up the proportion.
The ratio is map : real life.
So, 1 : 50 000.

Step 2: Find the actual distance in the same units (cm).
Actual distance = $$ 4 \text{ cm} \times 50 \text{ 000} = 200 \text{ 000 cm} $$
Step 3: Convert the units to what the question asks for (km).
Remember: 100 cm = 1 m, and 1000 m = 1 km.
First, convert cm to m: $$ 200 \text{ 000} \div 100 = 2000 \text{ m} $$ Next, convert m to km: $$ 2000 \div 1000 = 2 \text{ km} $$ The actual distance is 2 km.

Watch Out! A Common Mistake

Forgetting to convert units is the most common mistake with scale problems! Always check if you need to change from cm to m or km at the end.

Did you know? Model cars, architectural plans, and even the toys you play with are built using scales to make sure everything is in perfect proportion to the real thing!


You've made it! Rate, Ratio, and Proportion are powerful tools for understanding the world around you. Keep practising, and you'll find them everywhere you look. You've got this!