Rate and Ratio

Q: What is a ratio, and how is it written?

A ratio compares two or more quantities of the same units without writing the units. It can be written using a colon (e.g., $ 1 : 3 $) or as a fraction (e.g., $ \frac{1}{3} $).

Q: How do you simplify a ratio to its simplest form?

Divide all parts of the ratio by their Highest Common Factor (HCF) .

Q: What is a rate, and how is it different from a ratio?

A rate compares two quantities with different units , so its units must always be written down (e.g., km/h, \$/kg).

Q: What is direct proportion and what is its algebraic form?

Two quantities are in direct proportion if they increase or decrease together at a constant ratio, represented by $ y = kx $ or $ \frac{y}{x} = k $ (where $ k $ is a constant).

Q: What is inverse proportion and what is its algebraic form?

Two quantities are in inverse proportion if one increases while the other decreases such that their product is constant, represented by $ xy = k $ or $ y = \frac{k}{x} $.

Practice

Mathematics · Junior Secondary · 6 min read · Updated: Aug 17, 2026

Chapter 6: Rate, Ratio and Proportion - 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 divide all parts of the ratio by their Highest Common Factor (HCF).

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).
The HCF of 18 and 12 is 6.

Step 3: Divide both parts of the ratio by 6.
$ 18 \div 6 = 3 $
$ 12 \div 6 = 2 $

Step 4: Write the simplified ratio.
The simplified ratio is 3 : 2.

3-Term Ratios ($ a : b : c $)

Ratios can compare three or more quantities at once! For example, dividing \$180 among Alan, Betty, and Carl in the ratio 2 : 3 : 4.

Step 1: Find the total number of parts.
Total parts = $ 2 + 3 + 4 = 9 \text{ parts} $.

Step 2: Find the value of 1 part.
$ \text{Value of 1 part} = \frac{\$180}{9} = \$20 $.

Step 3: Multiply each share by the value of 1 part.
Alan gets: $ 2 \times \$20 = \$40 $
Betty gets: $ 3 \times \$20 = \$60 $
Carl gets: $ 4 \times \$20 = \$80 $

Solving Problems with Equivalent Ratios

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?

1. Write the ratios as fractions: $ \frac{\text{red}}{\text{blue}} = \frac{4}{5} $
2. Substitute the known value: $ \frac{\text{red}}{20} = \frac{4}{5} $
3. Multiply both sides by 20: $ \text{red} = \frac{4}{5} \times 20 = 16 $
4. There are 16 red sweets.

Quick Review: Ratios

What it is: Comparing quantities with the same units.
How to write it: $ a : b $ or $ a : b : c $.
Key rule: Ratios do not have units written on them. Always simplify!

Part 2: Understanding Rate

What is a Rate?

A Rate is a comparison between two quantities with different units. Because the units are different, we MUST write them down.

Think about these real-life examples:
- Speed: kilometres per hour (km/h)
- Unit price: dollars per kilogram (\$/kg)
- Heart rate: beats per minute

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

Calculating with Rates

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

$ \text{Rate} = \frac{50 \text{ L}}{5 \text{ min}} = 10 \text{ L/min} $
The flow rate is 10 litres per minute.

Part 3: Understanding Proportion

What is a Proportion?

A Proportion is an equation stating that two ratios or rates are equal.

1. Direct Proportion

Two quantities $ x $ and $ y $ are in direct proportion if they increase or decrease together at a constant ratio.
- Algebraic form: $ y = kx $ or $ \frac{y}{x} = k $ (where $ k $ is a non-zero constant).

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

Unit Rate Method:
Cost of 1 notebook = $ \frac{\$12}{3} = \$4 $
Cost of 7 notebooks = $ 7 \times \$4 = \$28 $.

2. Inverse Proportion

Two quantities $ x $ and $ y $ are in inverse proportion if one increases while the other decreases such that their product remains constant.
- Algebraic form: $ xy = k $ or $ y = \frac{k}{x} $ (where $ k $ is a non-zero constant).

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

Constant Product Method:
$ k = 4 \text{ workers} \times 6 \text{ hours} = 24 \text{ worker-hours} $
$ \text{Time for 8 workers} = \frac{24 \text{ worker-hours}}{8 \text{ workers}} = 3 \text{ hours} $.

Part 4: Real-World Application - Map Scales and Area Scales

Linear Scale (Length Ratio)

A Scale compares distance on a map to actual distance in real life.
Scale = $ \text{Map Length} : \text{Actual Length} = 1 : n $.

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?

$ \text{Actual distance} = 4 \text{ cm} \times 50\,000 = 200\,000 \text{ cm} $
Convert to metres: $ 200\,000 \div 100 = 2000 \text{ m} $
Convert to kilometres: $ 2000 \div 1000 = 2 \text{ km} $.

Area Scale

When dealing with areas on a map, remember that area scales with the square of the linear scale factor:
If the linear scale is $ 1 : n $, then:
$ \text{Area Scale} = 1 : n^2 $
$ \frac{\text{Map Area}}{\text{Actual Area}} = \left(\frac{\text{Map Length}}{\text{Actual Length}}\right)^2 $

Example: On a map with scale 1 : 10 000, a park has an area of 3 cm$^2$. What is the actual area in m$^2$?

Since $ 1 \text{ cm} \text{ on map} = 10\,000 \text{ cm} = 100 \text{ m} \text{ in reality} $:
$ 1 \text{ cm}^2 \text{ on map} = (100 \text{ m})^2 = 10\,000 \text{ m}^2 \text{ in reality} $.
$ \text{Actual Area} = 3 \times 10\,000 \text{ m}^2 = 30\,000 \text{ m}^2 $.

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!

References

Mathematics Education - Curriculum Documents — Education Bureau (EDB), Hong Kong
Comparison of the Content of Revised Mathematics Curriculum and Current Mathematics Curriculum — Education Bureau (EDB), Hong Kong
Junior Secondary Mathematics Curriculum Overview & Scheme of Work — Ning Po No. 2 College / EDB Local Implementation

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is a ratio, and how is it written?

A ratio compares two or more quantities of the same units without writing the units. It can be written using a colon (e.g., $ 1 : 3 $) or as a fraction (e.g., $ \frac{1}{3} $).

How do you simplify a ratio to its simplest form?

Divide all parts of the ratio by their Highest Common Factor (HCF).

How do you divide a given total amount into a multi-term ratio such as $ a : b : c $?

Find the total number of parts ($ a + b + c $), divide the total amount by that sum to find the value of 1 part, and multiply each share by the value of 1 part.

How can equivalent ratios be used to find an unknown quantity?

Set the ratio as a fraction equation with the unknown value and solve by multiplying across to find the unknown.

What is a rate, and how is it different from a ratio?

A rate compares two quantities with different units, so its units must always be written down (e.g., km/h, \$/kg).

What is direct proportion and what is its algebraic form?

Two quantities are in direct proportion if they increase or decrease together at a constant ratio, represented by $ y = kx $ or $ \frac{y}{x} = k $ (where $ k $ is a constant).

What is inverse proportion and what is its algebraic form?

Two quantities are in inverse proportion if one increases while the other decreases such that their product is constant, represented by $ xy = k $ or $ y = \frac{k}{x} $.

How do you convert between a map distance and actual distance using a linear scale of $ 1 : n $?

Multiply the map distance by $ n $ to find the actual distance, then convert the result to appropriate units (e.g., metres or kilometres).

What is the relationship between a map's linear scale and its area scale?

If the linear scale is $ 1 : n $, then the area scale is $ 1 : n^2 $, or $ \frac{\text{Map Area}}{\text{Actual Area}} = \left(\frac{\text{Map Length}}{\text{Actual Length}}\right)^2 $.

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