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Singapore MOE · Primary 6 · Mathematics

Ratio

Revision notes · 4 parts · 2026-07-16
1

Introduction to Ratios and Simplification

Ratios help us compare quantities. In this topic, you will learn how to read ratio notations, understand what they mean, and express them in their simplest form. We will also look at how to handle quantities with different units before forming a ratio.

❓ Before you read — have a guess: What is the main purpose of a ratio?

To compare two or more quantities.

⚡ Key points — the 5 things to remember
  • A ratio compares two or three whole number quantities.
  • The order of terms in a ratio is important.
  • Quantities must be in the same units before forming a ratio.
  • To simplify a ratio, divide all its terms by their Greatest Common Factor (GCF).
  • A ratio is in its simplest form when its terms have no common factor other than 1.

What you must be able to do

  • Interpret the notation a:b and a:b:c as a comparison of two or three whole number quantities.
  • Express the ratio of two or three given quantities in its simplest form.

What is a Ratio?

A ratio is a way to compare two or three quantities. A quantity is an amount or number of something. Each number in a ratio is called a term.

  • Ratios compare whole numbers only.
  • The order of the terms in a ratio matters.
  • A ratio can compare two quantities (a:b) or three quantities (a:b:c).
Ratio of Boys to Girls Boys 1 unit 1 unit Girls 1 unit 1 unit 1 unit Total students (5 units)
This bar model shows 2 units for boys and 3 units for girls, representing a ratio of 2:3.

When you compare parts of a whole, it's called a part:part ratio. For example, if there are 2 boys and 3 girls, the ratio of boys to girls is 2:3. If you compare a part to the total, it's a part:whole ratio. For example, the ratio of boys to total students is 2:5.

💡 Exam Tip Always read the question carefully to know which quantity comes first in the ratio. For example, 'ratio of apples to oranges' means apples:oranges.
📍 Singapore Focus Imagine a collection of items, like fruits. If you have 3 red apples and 5 green apples, the ratio of red apples to green apples is 3:5.

Simplifying Ratios

To express a ratio in its simplest form means to divide all its terms by their Greatest Common Factor (GCF) until the terms have no common factor other than 1.

  • You must divide all terms in the ratio by the same common factor.
  • The simplest form is like a fraction in its lowest terms.
  • Find the GCF of all terms to simplify in one step, or divide by common factors repeatedly.
Worked example

Express the ratio 12:18 in its simplest form.

  1. Find the common factors of 12 and 18.
    Why this step? We need to find numbers that can divide both 12 and 18 evenly to simplify the ratio.
  2. Identify the Greatest Common Factor (GCF).
    Why this step? Using the GCF allows us to simplify the ratio in the fewest steps and ensures it is in its simplest form.
  3. Divide both terms of the ratio by the GCF.
    Why this step? Dividing both terms by the same number keeps the comparison (ratio) the same, just in a smaller, easier-to-understand form.
2:3
✏️ Now you try

Express the ratio 20:30:40 in its simplest form.

  1. Find the Greatest Common Factor (GCF) of 20, 30, and 40. The common factors of 20, 30, and 40 are 1, 2, 5, 10. The GCF is 10.
  2. Divide all terms of the ratio by the GCF. 20 ÷ 10 : 30 ÷ 10 : 40 ÷ 10 = 2:3:4
Reveal the answer 2:3:4
Simplifying Ratio 12:18 to 2:3 Quantity A 6 units 6 units 12 Quantity B 6 units 6 units 6 units 18
The ratio 12:18 can be seen as 2 groups of 6 units to 3 groups of 6 units. Dividing by the common factor 6 simplifies it to 2:3.
⚠️ Common Mistake Simplifying a ratio incorrectly by dividing only one term instead of all terms by a common factor. — Correct: You must divide ALL terms in the ratio by the same common factor to keep the comparison accurate.
💡 Exam Tip Always check if your simplified ratio can be simplified further. If the terms still have a common factor (other than 1), you have not reached the simplest form.
💡 Exam Tip For ratios with three terms (a:b:c), ensure you find a common factor for all three terms.
🧠 Memory Aid GCF: Get Common Factors!

Ratios with Different Units

Before you can form or simplify a ratio, all the quantities must be expressed in the same units.

  • Convert all quantities to the same unit first (e.g., all to cm, or all to g).
  • It is usually easier to convert to the smaller unit (e.g., metres to centimetres, kilograms to grams).
  • Once units are the same, you can form the ratio and then simplify it.
Worked example

Find the ratio of 2 m to 50 cm in its simplest form.

  1. Convert 2 m to centimetres.
    Why this step? We must have both quantities in the same unit (centimetres) to make a fair comparison.
  2. Form the ratio using the same units.
    Why this step? Once units are the same, we can write the ratio directly.
  3. Simplify the ratio.
    Why this step? Simplifying ensures the ratio is in its most basic and easy-to-understand form.
4:1
✏️ Now you try

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

  1. Convert 1 kg to grams. 1 kg = 1000 g (Since 1 kg = 1000 g)
  2. Form the ratio using the same units. 1000 g : 250 g = 1000:250
  3. Simplify the ratio 1000:250. Divide both terms by their GCF, which is 250. 1000 ÷ 250 : 250 ÷ 250 = 4:1
Reveal the answer 4:1
⚠️ Common Mistake Failing to ensure that quantities are expressed in the same units before forming a ratio. — Correct: Always convert all quantities to the same unit first. A ratio compares 'like with like'.
💡 Exam Tip When you see different units in a ratio problem (e.g., metres and centimetres, kilograms and grams, hours and minutes), make unit conversion your very first step.
💡 Exam Tip Write down the conversion (e.g., 1 m = 100 cm) to avoid mistakes.
📍 Singapore Focus Imagine comparing the length of a desk (1.2 m) to the length of a pencil (15 cm). You must convert both to centimetres before finding their ratio.

🪤 Watch for these traps

The wrong answers examiners plant to catch you — and how to dodge them.

Tempting answer2:50
Looks right because The numbers 2 and 50 are directly taken from the problem '2 m to 50 cm'.
But actually The quantities are in different units (metres and centimetres) and must be converted to the same unit before forming the ratio.
✓ How to avoid it Always check for different units in the question. Convert them to the same unit first (e.g., 2 m = 200 cm), then form the ratio (200:50).
Tempting answer6:18 (for 12:18)
Looks right because 12 was divided by 2 to get 6, which is a common way to simplify numbers.
But actually Only one term (12) was divided by the common factor (2). For a ratio to be simplified correctly, ALL terms must be divided by the same common factor.
✓ How to avoid it After dividing one term, remember to divide ALL other terms in the ratio by the exact same common factor. If you divide 12 by 2, you must also divide 18 by 2, resulting in 6:9, which then needs further simplification.
Tempting answer3:2 (if the question asks for 'boys to girls' and numbers are given as 2 boys, 3 girls)
Looks right because The numbers 2 and 3 are present in the problem, and students might list them in the order they appear in the sentence.
But actually The question specifically asks for the ratio of 'boys to girls'. The first quantity mentioned (boys) must correspond to the first term in the ratio, and the second quantity (girls) to the second term.
✓ How to avoid it Underline the order of quantities requested in the question (e.g., 'boys to girls') and ensure your ratio terms match that exact order.

🔍 Question Decoder

What exam phrasings are really asking for.

When the question says… It is testing… Your move
What is the ratio of A to B? Understanding ratio notation and order. Write the ratio as A:B, ensuring the quantities are in the same units.
Express the ratio in its simplest form. Ability to find common factors and simplify ratios. Divide all terms of the ratio by their Greatest Common Factor (GCF) until no more common factors exist (other than 1).
Find the ratio of X (in unit 1) to Y (in unit 2). Ability to convert units before forming a ratio. Convert both X and Y to the same unit (usually the smaller unit), then form the ratio and simplify.

🧭 Understand → Plan → Work → Check

How to answer: Ratio problems involving unit conversion and simplification.

  1. 1Understand: Read the problem carefully. What quantities are being compared? What units are they in? What is the required form of the answer? 1 mark
  2. 2Plan: Decide if unit conversion is needed. Identify the common unit. Plan to find the GCF for simplification.
  3. 3Work: Show all steps: unit conversion, forming the ratio, and simplifying it. Carry units through conversion steps. 1 mark (for conversion), 1 mark (for ratio), 1 mark (for simplification)
  4. 4Check: Is the ratio in the correct order? Is it in its simplest form? Are the units consistent before forming the ratio? Is the answer reasonable?
Example: A piece of string is 1.2 metres long. Another piece of string is 40 centimetres long. Find the ratio of the length of the first piece to the length of the second piece in its simplest form.
✅ Earns the mark

“Understand: - First piece: 1.2 metres - Second piece: 40 centimetres - Find ratio of first piece : second piece in simplest form. Plan: - Convert metres to centimetres. - Form the ratio. - Simplify the ratio by finding the GCF. Work: 1. Convert 1.2 m to cm: 1.2 m = 1.2 × 100 cm = 120 cm 2. Form the ratio of first piece to second piece: 120 cm : 40 cm = 120:40 3. Simplify the ratio 120:40: GCF of 120 and 40 is 40. 120 ÷ 40 : 40 ÷ 40 = 3:1 Check: - Units were converted correctly (m to cm). - Ratio is in the correct order (first to second). - Ratio is in simplest form (3 and 1 have no common factors other than 1). Answer: The ratio of the length of the first piece to the length of the second piece is 3:1.”

This answer clearly shows all steps, including unit conversion and simplification. The 'Understand' and 'Plan' sections demonstrate clear thinking, and the 'Check' ensures accuracy. Units are handled correctly throughout.

❌ Loses the mark

“1.2 : 40 = 12:400 = 3:100”

This answer incorrectly converts 1.2 m to 12 cm (instead of 120 cm) or assumes 1.2 is 12 units of something. It also attempts to simplify without clear steps. The initial ratio setup is wrong due to unit conversion error, leading to an incorrect final ratio.

🗝️ Words that earn marks

Simplest form The question asks for the ratio to be reduced to its lowest terms, where terms have no common factor other than 1.
Common factor You are looking for a number that can divide into all terms of a ratio evenly.
Greatest Common Factor (GCF) You want to simplify a ratio in one step to its simplest form.
Same units You are comparing quantities that are initially given in different units (e.g., metres and centimetres) and need to convert them before forming a ratio.
🎯

Test Yourself

Generate practice questions on Introduction to Ratios and Simplification

2

Equivalent Ratios and Proportional Reasoning

This section teaches you how to work with ratios that represent the same comparison, even if the numbers look different. You will learn to find missing numbers in these ratios using simple scaling methods. Mastering these skills is key to solving many real-world problems, like adjusting recipes or understanding map scales.

❓ Before you read — have a guess: If the ratio of apples to oranges is 1:2, and you double the number of both fruits, what is the new ratio?

The new ratio will still be 1:2, because it is an equivalent ratio.

⚡ Key points — the 5 things to remember
  • Equivalent ratios represent the same comparison between quantities.
  • You generate equivalent ratios by multiplying or dividing all terms by the same non-zero whole number.
  • Simplest form means the ratio terms have no common factors other than 1.
  • To find a missing term, identify the scaling factor between the known terms.
  • Always ensure quantities are in the same units before forming a ratio.

What you must be able to do

  • Identify and create equivalent ratios.
  • Find the missing number in a pair of equivalent ratios.

What are Equivalent Ratios?

An equivalent ratio is a ratio that has the same value or comparison as another ratio, even though the numbers used might be different. Think of it like equivalent fractions.

  • You can get an equivalent ratio by multiplying each term in the ratio by the same whole number (e.g., 2:3 = 4:6).
  • You can also get an equivalent ratio by dividing each term in the ratio by the same common whole number (e.g., 6:9 = 2:3).
  • A ratio is in its simplest form when its terms have no common factors other than 1.
a:b = ka:kb
aFirst quantity in the ratio
bSecond quantity in the ratio
kCommon non-zero whole number multiplier or divisor
Worked example

Express the ratio 6:10 in its simplest form. Then, generate an equivalent ratio by multiplying by 3.

  1. Find the greatest common factor (GCF) of 6 and 10.
    Why this step? We find the GCF to simplify the ratio to its smallest whole numbers.
  2. Divide both terms of the ratio by the GCF to get the simplest form.
    Why this step? Dividing both terms by the GCF ensures the ratio remains equivalent while becoming simpler.
  3. Multiply both terms of the simplest form ratio by 3 to generate an equivalent ratio.
    Why this step? Multiplying both terms by the same number keeps the ratio equivalent, just with larger numbers.
Simplest form: 3:5. Equivalent ratio (multiplied by 3): 9:15.
✏️ Now you try

Express the ratio 12:18 in its simplest form. Then, generate an equivalent ratio by multiplying by 2.

  1. What is the greatest common factor (GCF) of 12 and 18? The GCF of 12 and 18 is 6.
  2. Divide both terms of 12:18 by the GCF to find its simplest form. 12 ÷ 6 : 18 ÷ 6 = 2:3.
  3. Multiply both terms of the simplest form ratio by 2 to generate an equivalent ratio. 2 × 2 : 3 × 2 = 4:6.
Reveal the answer Simplest form: 2:3. Equivalent ratio (multiplied by 2): 4:6.
Equivalent Ratios: 2:3 and 4:6 Part A (Original) 2 units Part B (Original) 3 units Part A (Equivalent) 4 units Part B (Equivalent) 6 units
The top two bars show a ratio of 2:3. The bottom two bars show an equivalent ratio of 4:6, where each 'unit' has been doubled in size (or multiplied by 2).
💡 Exam Tip Students often forget to divide ALL terms by the common factor when simplifying. Always check that every number in the ratio has been divided.
💡 Exam Tip To check if two ratios are equivalent, you can simplify both to their simplest form. If the simplest forms are the same, the original ratios are equivalent.
🧠 Memory Aid To keep ratios EQUAL, do the SAME to ALL!
📍 Singapore Focus When you scale a recipe, like doubling the ingredients, you are using equivalent ratios. If a recipe calls for 1 cup of sugar to 2 cups of flour (1:2), doubling it means 2 cups of sugar to 4 cups of flour (2:4), which is an equivalent ratio.

Solving for Missing Terms

Finding a missing term in a ratio means working out an unknown quantity in an equivalent ratio, using the relationship between the known quantities.

  • The Principle of Equivalent Ratios (a:b = ka:kb) is used to find missing terms.
  • Identify the 'scaling factor' (k) by comparing the two known corresponding terms.
  • Apply this same scaling factor to the other term to find the missing value.
Worked example 🧰 Look for patterns

The ratio of red marbles to blue marbles is 3:5. If there are 12 red marbles, how many blue marbles are there?

  1. Identify the known ratio and the known quantity for one part.
    Why this step? This helps us set up the problem and see which parts correspond.
  2. Find the scaling factor by dividing the new known quantity by the original known quantity.
    Why this step? The scaling factor tells us how many times the original quantity has been multiplied to get the new quantity.
  3. Multiply the other term in the original ratio by the scaling factor to find the missing term.
    Why this step? Applying the same scaling factor ensures the new ratio is equivalent to the original one.
There are 20 blue marbles.
✏️ Now you try

The ratio of boys to girls in a class is 4:3. If there are 16 boys, how many girls are there?

  1. What is the original ratio of boys to girls and the new number of boys? Original ratio: 4:3 (Boys:Girls). New number of boys: 16.
  2. Find the scaling factor by comparing the number of boys. Scaling factor = New boys / Original boys = 16 / 4 = 4.
  3. Multiply the original number of girls by the scaling factor to find the new number of girls. New number of girls = Original girls × Scaling factor = 3 × 4 = 12.
Reveal the answer There are 12 girls.
Finding Missing Term: Red to Blue Marbles Red Marbles 3 units 12 marbles Blue Marbles 5 units ?
If 3 units represent 12 marbles, then 1 unit represents 12 ÷ 3 = 4 marbles. So, 5 units (blue marbles) represent 5 × 4 = 20 marbles.

When you have a ratio like A:B and you know a new value for A (let's call it A'), you can find the new value for B (B') by figuring out how A changed to become A'. This change is called the 'scaling factor'. You then apply the same scaling factor to B to find B'.

💡 Exam Tip Students often try to add or subtract to find the missing term instead of multiplying or dividing. Remember, ratios are about multiplication/division, not addition/subtraction.
💡 Exam Tip Always check that you are comparing the correct terms (e.g., first term to first term, second term to second term) to find the scaling factor.
📍 Singapore Focus Map scales use equivalent ratios. If a map scale is 1:1000, it means 1 cm on the map represents 1000 cm (or 10 m) in real life. If you measure a distance of 5 cm on the map, you can find the real-life distance using this ratio.

🪤 Watch for these traps

The wrong answers examiners plant to catch you — and how to dodge them.

Tempting answerFor 2:3 = 4:?, the missing term is 5 because 2+2=4, so 3+2=5.
Looks right because Students mistakenly apply addition or subtraction to find the missing term, seeing a simple pattern of adding a constant value.
But actually Ratios represent multiplicative relationships. To get from 2 to 4, you multiply by 2. Therefore, you must multiply 3 by 2 to get the missing term, which is 6.
✓ How to avoid it Always look for a multiplication or division factor (scaling factor) between the known corresponding terms, not an addition or subtraction difference.
Tempting answerTo simplify 12:18, divide 12 by 6 to get 2, and 18 by 3 to get 6, so the ratio is 2:6.
Looks right because Students correctly identify common factors but use different factors for different terms, or don't use the greatest common factor for both.
But actually To simplify a ratio, you must divide ALL terms by the SAME common factor. The greatest common factor of 12 and 18 is 6. So, 12 ÷ 6 = 2 and 18 ÷ 6 = 3, making the simplest ratio 2:3.
✓ How to avoid it Ensure the divisor used is the same for every term in the ratio. For simplest form, use the greatest common factor (GCF).

🔍 Question Decoder

What exam phrasings are really asking for.

When the question says… It is testing… Your move
Express the ratio of A to B in its simplest form. Your ability to find the greatest common factor and divide both terms by it. Find the GCF of the two numbers and divide both terms by it.
Find the missing number in the ratio X:Y = P:? Your understanding of equivalent ratios and proportional scaling. Find the scaling factor from X to P (P ÷ X), then multiply Y by this factor.
The ratio of red balls to blue balls is 2:3. If there are 10 red balls, how many blue balls are there? Your ability to apply proportional reasoning to a word problem. Set up an equivalent ratio (2:3 = 10:?) and find the scaling factor (10 ÷ 2 = 5). Then multiply 3 by 5.

🧭 Understand → Plan → Work → Check

How to answer: Word problems involving finding missing terms in ratios.

  1. 1Understand: Read the problem carefully. What is the ratio given? What quantity is known? What quantity needs to be found? 1 mark
  2. 2Plan: Decide on the method. Will you use the scaling factor method or the unit method? Draw a bar model if it helps visualise the parts. 1 mark
  3. 3Work: Show all your calculations clearly, step-by-step, with units. 1-2 marks
  4. 4Check: Does your answer make sense in the context of the problem? Is the new ratio equivalent to the original ratio? 1 mark
Example: The ratio of apples to oranges in a basket is 3:4. If there are 15 apples, how many oranges are there?
✅ Earns the mark

“Understand: Given ratio of Apples : Oranges = 3:4. Number of apples = 15. Need to find the number of oranges. Plan: Use the scaling factor method. I will find how many times the 'apples' part of the ratio has been multiplied, then apply the same multiplication to the 'oranges' part. Work: 1. Number of units for apples = 3 units. 2. 3 units = 15 apples. 3. 1 unit = 15 ÷ 3 = 5 apples. 4. Number of units for oranges = 4 units. 5. Number of oranges = 4 units × 5 apples/unit = 20 oranges. Check: Original ratio 3:4. New ratio 15:20. 15 ÷ 5 = 3, 20 ÷ 5 = 4. The ratios are equivalent. The answer is reasonable. Answer: There are 20 oranges.”

This answer clearly follows all steps of the framework. It states the understanding, plans the method, shows clear workings with units, and includes a check for reasonableness and equivalence. All marks would be awarded.

❌ Loses the mark

“3 units = 15 1 unit = 5 4 units = 20 oranges”

This answer shows the correct calculation but lacks the 'Understand', 'Plan', and 'Check' steps. It also doesn't explicitly state what 'units' refer to initially. While the final answer is correct, it might lose marks for incomplete working or lack of clarity, especially in more complex problems.

🗝️ Words that earn marks

Simplest form You need to reduce a ratio to its smallest whole numbers by dividing by the greatest common factor.
Equivalent ratio You are comparing ratios that represent the same proportion or need to generate a new ratio with larger or smaller terms that maintain the same comparison.
Scaling factor You need to find the multiplier or divisor that relates one term of a ratio to its corresponding term in an equivalent ratio.
Common factor You are simplifying a ratio and looking for a number that divides into all terms without a remainder.
🎯

Test Yourself

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3

Ratios and Fractions (Part-Whole Relationship)

This section teaches you how to connect ratios with fractions. You will learn how a part:part ratio (like boys to girls) can be expressed as a part:whole fraction (like boys to total students), and how to do the reverse. Understanding this link is key to solving many ratio problems.

❓ Before you read — have a guess: If the ratio of red apples to green apples is 2:3, what fraction of all the apples are red?

2/5

⚡ Key points — the 6 things to remember
  • A part:part ratio like a:b compares two different quantities.
  • To find the total number of parts, you add the terms of the ratio (a + b).
  • A part:whole fraction shows one quantity as a fraction of the total quantity.
  • If the ratio of Part A to Part B is a:b, then Part A is a/(a+b) of the total, and Part B is b/(a+b) of the total.
  • Always simplify ratios to their simplest form.
  • Ensure you are comparing parts to the same whole when converting between ratios and fractions.

What you must be able to do

  • Convert a ratio to a fraction and a fraction to a ratio.

Understanding Part:Part Ratios and Total Parts

A ratio is a way to compare two or three whole number quantities. Each number in a ratio is called a term. A part:part ratio compares different parts of a whole, for example, the number of boys to the number of girls.

  • A ratio like a:b means there are 'a' units of the first quantity and 'b' units of the second quantity.
  • The total number of units or parts is found by adding all the terms in the ratio (e.g., for a:b, total units = a + b).
  • This total represents the 'whole' when you think about fractions.
Ratio of Boys to Girls (2:3) Boys 1 unit 1 unit Girls 1 unit 1 unit 1 unit Total Students (5 units)
This bar model shows 2 units for boys and 3 units for girls, making a total of 5 units.

When you see a ratio like '2:3' for boys to girls, it means that for every 2 boys, there are 3 girls. The total number of 'parts' or 'units' is 2 + 3 = 5. So, the boys make up 2 parts out of 5 total parts, and the girls make up 3 parts out of 5 total parts.

⚠️ Common Mistake Students often think a ratio of 2:3 (e.g., boys to girls) means the first part (boys) is 2/3 of the total. — Correct: A ratio of 2:3 means the first part is 2 units and the second part is 3 units. The total is 2 + 3 = 5 units. So, the first part is 2/5 of the total.
💡 Exam Tip Always identify the 'parts' and the 'total parts' clearly before forming any fractions. A bar model can help you see this.
📍 Singapore Focus This concept is used when comparing quantities in a collection, such as the number of red balls to blue balls, or the amount of sugar to flour in a recipe.

Converting Part:Part Ratio to Part:Whole Fraction

A part:whole fraction expresses one part of a quantity as a fraction of the entire quantity. For example, if there are 2 boys and 3 girls, the fraction of boys to the total students is 2/5.

  • If the ratio of Part A to Part B is a:b, then the fraction of Part A to the total is a / (a + b).
  • The fraction of Part B to the total is b / (a + b).
Fraction_of_Part_A = a / (a + b)
aNumber of units for Part A
bNumber of units for Part B
Fraction_of_Part_AFraction of Part A out of the total
Worked example 🧰 Draw a model

The ratio of red marbles to blue marbles in a bag is 2:3. What fraction of the marbles are red?

  1. Find the total number of parts: 2 (red) + 3 (blue) = 5 parts.
    Why this step? We add the parts in the ratio to find the 'whole' or total number of units.
  2. Form the fraction for red marbles: (Number of red parts) / (Total parts) = 2/5.
    Why this step? The fraction is formed by placing the specific part's units over the total units.
2/5
✏️ Now you try

The ratio of fiction books to non-fiction books in a library is 5:4. What fraction of the books are non-fiction?

  1. First, find the total number of parts for all books. Total parts = 5 (fiction) + 4 (non-fiction) = 9 parts.
  2. Now, form the fraction for non-fiction books. Fraction of non-fiction books = (Number of non-fiction parts) / (Total parts) = 4/9.
Reveal the answer 4/9
Red to Blue Marbles (2:3) Red Marbles 1 unit 1 unit Blue Marbles 1 unit 1 unit 1 unit Total Marbles (5 units)
The bar model shows 2 units for red marbles and 3 units for blue marbles. The total is 5 units. So, red marbles are 2/5 of the total.

To convert a ratio to a fraction, you first need to determine the total number of parts by adding all the terms in the ratio. Then, for each quantity, the fraction is formed by placing its number of parts over the total number of parts. For example, if the ratio of boys to girls is 2:3, the boys are 2/(2+3) = 2/5 of the total students.

⚠️ Common Mistake Thinking that a ratio of 'boys to girls = 2:3' means 'boys to total students = 2:3'. — Correct: The ratio 'boys to girls = 2:3' is a part:part comparison. The ratio 'boys to total students' is a part:whole comparison, which would be 2:(2+3) = 2:5.
💡 Exam Tip When a question asks for a fraction 'of the total', remember to use the sum of all ratio parts as your denominator.
💡 Exam Tip Always check if the question asks for a fraction of one part to another part, or one part to the whole. These are different!
📍 Singapore Focus This conversion is useful when you need to know what proportion of a whole quantity belongs to a specific part, such as finding the fraction of boys in a class or the fraction of water in a mixed drink.

Converting Part:Whole Fraction to Part:Part Ratio

To convert a part:whole fraction back to a part:part ratio, you identify the parts represented by the numerator and denominator, then find the remaining part to form the ratio.

  • If a quantity is a/b of the total, it means 'a' units belong to that part, and 'b' units represent the total.
  • The other part can be found by subtracting the first part from the total: (b - a) units.
  • The ratio of the first part to the other part will be a : (b - a).
  • Always express the final ratio in its simplest form.
Worked example 🧰 Draw a model

3/7 of the fruits in a basket are apples. The rest are oranges. What is the ratio of apples to oranges?

  1. Identify parts from the fraction: Apples are 3 parts, Total fruits are 7 parts.
    Why this step? The numerator represents the specific part, and the denominator represents the total.
  2. Find the number of parts for oranges: Total parts - Apple parts = 7 - 3 = 4 parts.
    Why this step? Subtracting the known part from the total gives the other unknown part.
  3. Form the ratio of apples to oranges: 3 : 4.
    Why this step? A ratio compares the units of the first item to the units of the second item.
3:4
✏️ Now you try

In a class, 5/9 of the students are boys. The rest are girls. What is the ratio of boys to girls?

  1. First, identify the parts for boys and the total parts from the fraction. Boys are 5 parts, Total students are 9 parts.
  2. Next, find the number of parts for girls. Girls parts = Total parts - Boys parts = 9 - 5 = 4 parts.
  3. Finally, form the ratio of boys to girls. Ratio of boys to girls = 5 : 4.
Reveal the answer 5:4
Apples to Total Fruits (3/7) Apples 1 unit 1 unit 1 unit Oranges 1 unit 1 unit 1 unit 1 unit Total Fruits (7 units)
The bar model shows 3 units for apples out of a total of 7 units. The remaining 4 units are oranges. So the ratio of apples to oranges is 3:4.

To convert a part:whole fraction (like 3/7) into a part:part ratio, you can think of the numerator as the number of units for one part (e.g., 3 units for apples) and the denominator as the total number of units (e.g., 7 units for all fruits). The remaining part (e.g., oranges) would then be the total units minus the first part's units (7 - 3 = 4 units). So, the ratio of apples to oranges would be 3:4.

💡 Exam Tip When given a fraction, always check if it refers to a part of the total. If it's a fraction of a remainder, you'll need to find the total first.
💡 Exam Tip After forming the ratio, always simplify it to its simplest form by dividing all terms by their highest common factor.
📍 Singapore Focus This skill is used when you know a proportion of a group and need to find the comparison between the different subgroups, such as knowing the fraction of boys in a class and needing the ratio of boys to girls.

🪤 Watch for these traps

The wrong answers examiners plant to catch you — and how to dodge them.

Tempting answerIf the ratio of A to B is 2:3, then A is 2/3 of the total.
Looks right because The numbers 2 and 3 are directly from the ratio, making it seem like a direct conversion.
But actually The denominator of a part:whole fraction must represent the total number of parts. For a 2:3 ratio, the total parts are 2 + 3 = 5. So, A is 2/5 of the total.
✓ How to avoid it Always add all the terms in the ratio to find the total number of units before forming a part:whole fraction. Draw a bar model to visualise the total.
Tempting answerIf 4/10 of the students are boys, the ratio of boys to girls is 4:6.
Looks right because You correctly identified 4 parts for boys and 6 parts for girls (10-4).
But actually Ratios should always be expressed in their simplest form. The ratio 4:6 can be simplified by dividing both terms by their highest common factor, 2, to get 2:3.
✓ How to avoid it After forming a ratio, always check if all its terms can be divided by a common factor. If they can, simplify the ratio to its simplest form.

🔍 Question Decoder

What exam phrasings are really asking for.

When the question says… It is testing… Your move
What fraction of the total ...? Your ability to convert a part:part ratio into a part:whole fraction. Add all parts in the ratio to find the total. Place the specific part's units over this total.
The ratio of A to B is X:Y. What fraction of the total is A? Your understanding of part:whole fractions from a given ratio. The numerator is X, and the denominator is X + Y.
A is X/Y of the total. What is the ratio of A to B? Your ability to convert a part:whole fraction into a part:part ratio. A has X units. The total has Y units. B has (Y - X) units. The ratio is X : (Y - X).

🧭 Understand → Plan → Work → Check

How to answer: Problems involving converting between ratios and fractions.

  1. 1Understand: Read the question carefully. What is given (ratio or fraction)? What needs to be found (ratio or fraction)? 1 mark
  2. 2Plan: Decide if you need to find the total parts. Choose whether to convert ratio to fraction or fraction to ratio. Consider drawing a bar model. 1 mark
  3. 3Work: Show all your steps clearly. Calculate the total parts if needed. Form the fraction or ratio. Simplify the ratio if necessary. 1 mark
  4. 4Check: Does your answer make sense? Is the fraction less than 1? Is the ratio in simplest form? Does it answer the question asked?
Example: The ratio of boys to girls in a class is 3:5. What fraction of the class are boys?
✅ Earns the mark

“Understand: Given: Ratio of boys to girls = 3:5. Find: Fraction of the class that are boys. Plan: 1. Find the total number of parts (boys + girls). 2. Form the fraction: (Boys' parts) / (Total parts). Work: 1. Total parts = 3 (boys) + 5 (girls) = 8 parts. 2. Fraction of boys = 3 / 8. Check: 3/8 is less than 1, which is reasonable for a fraction of a whole. The numbers match the ratio parts and total. The answer is 3/8.”

This answer clearly shows all steps, identifies the given and required information, and includes a check for reasonableness. It would earn full marks.

❌ Loses the mark

“Boys are 3/5 of the class.”

This answer is incorrect because it confuses the part:part ratio (3:5) with a part:whole fraction. It misses the crucial step of finding the total number of parts (3+5=8) for the denominator, leading to a wrong answer and loss of marks.

🗝️ Words that earn marks

Total parts You need to find the sum of all quantities in a ratio to form a part:whole fraction.
Part:part ratio You are comparing two or three distinct groups directly (e.g., boys to girls).
Part:whole fraction You are expressing one group as a fraction of the entire collection (e.g., boys to total students).
Simplest form You are giving your final answer for a ratio, ensuring all terms are divided by their highest common factor.
🎯

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4

Solving Ratio Word Problems

Ratio word problems ask you to find unknown quantities when you are given a comparison between two or three amounts. You will learn how to divide a total amount into parts based on a ratio, and how to solve problems where you are given a difference or a specific part instead of the total. Mastering these skills will help you solve real-world problems like sharing items or mixing ingredients.

❓ Before you read — have a guess: If the ratio of red apples to green apples is 2:3, and there are 10 red apples, how many green apples are there?

15 green apples

⚡ Key points — the 6 things to remember
  • A ratio compares two or three quantities, showing how much of one there is compared to another.
  • To divide a quantity by a ratio, first find the total number of 'units' in the ratio.
  • Calculate the value of one 'unit' by dividing the total quantity by the total number of units.
  • Multiply the value of one unit by each term in the ratio to find the individual parts.
  • When solving word problems, always identify what the given number represents (total, a part, or a difference).
  • Bar models are a useful tool to visualise and solve ratio word problems.

What you must be able to do

  • Divide a given quantity into two or three parts according to a given ratio.
  • Solve word problems involving ratios, including those with two or three quantities, sharing, and distribution.

Dividing a Quantity by a Ratio

Dividing a quantity by a ratio means splitting a total amount into parts that are in a specific proportion to each other. For example, sharing money between two people in the ratio 2:3 means one person gets 2 'parts' for every 3 'parts' the other person gets.

  • The first step is to find the total number of 'units' or 'parts' in the ratio.
  • Next, find the value of one 'unit' by dividing the total quantity by the total number of units.
  • Finally, multiply the value of one unit by each term in the ratio to find the size of each part.
Worked example 🧰 Draw a model

Mr Tan shared 120 sweets between his two children, Siti and Bala, in the ratio 3:2. How many sweets did each child receive?

  1. Find the total number of units in the ratio: 3 + 2 = 5 units
    Why this step? We add the ratio terms to find the total number of equal parts the quantity is divided into.
  2. Find the value of 1 unit: 120 sweets ÷ 5 units = 24 sweets/unit
    Why this step? We divide the total quantity by the total units to find out how much each single unit is worth.
  3. Calculate Siti's share: 3 units × 24 sweets/unit = 72 sweets
    Why this step? We multiply Siti's ratio term by the value of one unit to find her share.
  4. Calculate Bala's share: 2 units × 24 sweets/unit = 48 sweets
    Why this step? We multiply Bala's ratio term by the value of one unit to find his share.
Siti received 72 sweets and Bala received 48 sweets.
✏️ Now you try

A sum of $180 was divided between Alice and Ben in the ratio 4:5. How much money did each person receive?

  1. Find the total number of units in the ratio. 4 + 5 = 9 units
  2. Find the value of 1 unit. $180 ÷ 9 units = $20/unit
  3. Calculate Alice's share. 4 units × $20/unit = $80
  4. Calculate Ben's share. 5 units × $20/unit = $100
Reveal the answer Alice received $80 and Ben received $100.
Sharing Sweets (Siti:Bala = 3:2) Siti 1 unit 1 unit 1 unit 72 sweets Bala 1 unit 1 unit 48 sweets 120 sweets
A bar model showing 120 sweets divided into 5 equal units, with Siti getting 3 units and Bala getting 2 units.
💡 Exam Tip Always add up the ratio terms first to find the total number of units. This helps you correctly find the value of one unit.
💡 Exam Tip Double-check what the question is asking for. Are you finding one person's share, or the total amount, or the difference between shares?
📍 Singapore Focus This method is useful for sharing items like sweets, money, or fruits among people according to a specific comparison.

Solving More Complex Ratio Word Problems

More complex ratio word problems might not give you the total quantity directly. Instead, they might give you the value of one part, or the difference between two parts. You still use the ratio to find the value of one 'unit' before solving for the unknown quantities.

  • Identify what the given number in the problem represents in terms of 'units' from the ratio.
  • If a difference is given, find the difference in 'units' between the relevant parts of the ratio.
  • Use this information to find the value of one 'unit'.
  • Once you know the value of one unit, you can find any other quantity mentioned in the problem, including the total.
Worked example 🧰 Draw a model

The ratio of red, blue, and green pens in a box is 2:3:5. If there are 18 more green pens than red pens, how many blue pens are there?

  1. Find the difference in units between green and red pens: 5 units (green) - 2 units (red) = 3 units
    Why this step? We subtract the units to find the 'unit difference' that corresponds to the given numerical difference.
  2. Find the value of 1 unit: 18 pens ÷ 3 units = 6 pens/unit
    Why this step? We divide the numerical difference by the unit difference to find the value of one unit.
  3. Calculate the number of blue pens: 3 units (blue) × 6 pens/unit = 18 blue pens
    Why this step? We multiply the blue pen's ratio term by the value of one unit to find the number of blue pens.
There are 18 blue pens.
✏️ Now you try

The ratio of apples to oranges to pears in a basket is 3:4:2. If there are 12 fewer pears than oranges, how many apples are there?

  1. Find the difference in units between oranges and pears. 4 units (oranges) - 2 units (pears) = 2 units
  2. Find the value of 1 unit. 12 fruits ÷ 2 units = 6 fruits/unit
  3. Calculate the number of apples. 3 units (apples) × 6 fruits/unit = 18 apples
Reveal the answer There are 18 apples.
Pens in a Box (Red:Blue:Green = 2:3:5) Red Pens 1 unit 1 unit Blue Pens 1 unit 1 unit 1 unit ? blue pens Green Pens 1 unit 1 unit 1 unit 1 unit 1 unit 18 pens (Green - Red)
A bar model showing the ratio of red, blue, and green pens. The difference between green and red pens (3 units) is equal to 18 pens.
💡 Exam Tip Always draw a bar model for complex problems. It helps you see the relationship between the quantities and the given information.
💡 Exam Tip Carefully read the question to identify what quantity is given (e.g., total, one part, difference) and what quantity you need to find.
💡 Exam Tip When dealing with three quantities, ensure you are comparing the correct parts of the ratio to the given information.
📍 Singapore Focus This approach is useful for problems involving mixing ingredients (like paint or cordial), distributing resources, or comparing collections of items where a difference or specific part is known.

🪤 Watch for these traps

The wrong answers examiners plant to catch you — and how to dodge them.

Tempting answerCalculating the total amount when the question asks for a specific part.
Looks right because You often calculate the total units and total value as an intermediate step, making it easy to stop there.
But actually The question usually asks for a specific individual quantity, not the sum of all quantities.
✓ How to avoid it Underline the exact question asked (e.g., 'How many blue pens?') and make sure your final answer directly addresses it.
Tempting answerUsing the wrong ratio terms for comparison (e.g., comparing red to blue when the problem gives information about red and green).
Looks right because The numbers in the ratio are all present, and it's easy to pick any two terms to work with.
But actually You must match the given numerical information (e.g., '18 more green pens than red pens') to the correct corresponding terms in the ratio (green units - red units).
✓ How to avoid it Circle the quantities mentioned in the problem statement and highlight the corresponding terms in the ratio to ensure they match.

🔍 Question Decoder

What exam phrasings are really asking for.

When the question says… It is testing… Your move
The ratio of A to B is X:Y. Understanding how to represent quantities using a ratio. Identify the terms for A and B in the ratio.
Share [quantity] in the ratio A:B. Ability to divide a total quantity into parts according to a given ratio. Find total units, value of 1 unit, then each part.
A has [number] more/fewer than B, and their ratio is X:Y. Ability to use a difference in quantities to find the value of one unit. Find the difference in units, then the value of 1 unit, then the required quantity.
The ratio of A:B:C is X:Y:Z. Ability to work with ratios involving three quantities. Apply the same principles (total units, value of 1 unit, differences) to three terms.

🧭 Understand → Plan → Work → Check

How to answer: Structured word problems involving ratios.

  1. 1Understand: Read the problem carefully. What is given? What needs to be found? 1 mark
  2. 2Plan: Choose a strategy (e.g., draw a bar model). How will you use the ratio to find the answer? 1 mark
  3. 3Work: Show all your calculations clearly, step-by-step, with units. 2-3 marks
  4. 4Check: Is your answer reasonable? Does it make sense in the context of the problem? Does it answer the question asked? 1 mark
Example: The ratio of the number of fiction books to non-fiction books in a library is 5:3. If there are 240 fiction books, how many non-fiction books are there?
✅ Earns the mark

“Understand: Given: Ratio of fiction to non-fiction = 5:3. Number of fiction books = 240. Find: Number of non-fiction books. Plan: I will use the number of fiction books to find the value of 1 unit, then calculate the number of non-fiction books. Work: 5 units (fiction) = 240 books 1 unit = 240 books ÷ 5 = 48 books/unit Number of non-fiction books = 3 units × 48 books/unit = 144 books Check: If fiction is 240 (5 units) and non-fiction is 144 (3 units), then 240/5 = 48 and 144/3 = 48. The value of 1 unit is consistent. The answer is reasonable. Answer: There are 144 non-fiction books.”

This answer clearly follows all steps of the framework. It states what is understood, plans the approach, shows all workings with units, and includes a check for reasonableness. All marks would be awarded.

❌ Loses the mark

“5 units = 240 1 unit = 48 3 units = 144”

This answer shows the correct calculations but lacks clarity and explanation. It does not state what the numbers represent (e.g., '5 units (fiction)'), nor does it explicitly state the final answer with units. It would lose marks for presentation and lack of clarity, especially in the 'Understand' and 'Check' phases.

🗝️ Words that earn marks

Total units Calculating the sum of the terms in a ratio to represent the whole quantity.
Value of 1 unit Finding out how much each single part of the ratio is worth.
Difference in units Comparing two parts of a ratio when a numerical difference between them is given.
Number of [item] Stating the final calculated quantity for a specific item.
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Glossary

Key terms across this topic, in alphabetical order.

Equivalent ratio — A ratio that represents the same comparison as another ratio, obtained by multiplying or dividing all terms by the same non-zero whole number.
Part:part ratio — A ratio that compares one part of a whole to another part of the same whole (e.g., boys to girls).
Part:whole fraction — A fraction that expresses one part of a quantity as a proportion of the entire quantity (e.g., boys to total students).
Part:whole ratio — A ratio that compares one part of a whole to the entire whole (e.g., boys to total students).
Quantity — An amount or number of something being compared.
Ratio — A comparison of two or three whole number quantities, showing their relative sizes.
Simplest form — A ratio where the terms have no common factors other than 1 (e.g., 2:3 is the simplest form of 4:6).
Term — Each number in a ratio (e.g., in 2:3, '2' is a term and '3' is a term).
Term (in a ratio) — Each number in a ratio (e.g., in 2:3, '2' is a term and '3' is a term).
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