Fundamentals of Ratio Division

Expressing Quantities as a Ratio

Every time you mix squash with water, you are applying a ratio in real life.
A ratio is a way of comparing one part of a whole to another part (part-to-part) or to the whole itself (part-to-whole).
We use colon notation (such as $a : b$ ) to write them, and the order of the numbers must strictly match the order of the text.
Quantities must be converted to the same units before a ratio is formed, but the final ratio itself is unitless.
You can create an equivalent ratio representing the same proportion by multiplying or dividing all parts by the same non-zero number.
Ratio simplification involves reducing a ratio by dividing all parts by their Highest Common Factor (HCF) (the largest integer that divides them without a remainder).
This results in a ratio in its simplest form, where the parts are integers with no common factors other than $1$ .

Worked Example: Expressing and Simplifying Ratios

Question: Mix $800$ g of sand and $1.2$ kg of cement. Express the division of sand to cement as a ratio in its simplest form.

Step 1: Convert to the same units so they can be compared. $1.2$ kg $= 1200$ g
Step 2: Write using colon notation. $800 : 1200$
Step 3: Perform ratio simplification by dividing both sides by the Highest Common Factor (HCF). The Highest Common Factor (HCF) of $800$ and $1200$ is $400$ . $(800 \div 400) : (1200 \div 400) = 2 : 3$
Result: This creates an equivalent ratio in its simplest form, where the parts are integers with no common factors other than $1$ .

Dividing a Quantity in a Ratio

You and a friend buy a winning lottery ticket, but you paid £1 and they paid £4. How do you split a £500 prize fairly?
You use the additive method (often called the unitary method) to share a total amount into given proportions.
First, find the total parts by summing all the numbers in the ratio to determine the total number of shares.
Next, calculate the share value (the amount one single part is worth) by dividing the total quantity by the total parts.
This share value acts as a multiplier to scale up any part of the ratio.

Worked Example: The Additive Method

Question: Divide £450 between Anna and Ben in the ratio $4 : 5$ .

Step 1: Sum the parts of the ratio to find the total parts. $4 + 5 = 9$ total parts.
Step 2: Divide the total quantity by this sum to find the share value (the value of one single share). $£450 \div 9 = £50$ per share.
Step 3: Multiply the share value by the original ratio numbers. Anna: $4 \times £50 = £200$ Ben: $5 \times £50 = £250$
Step 4: Check that the final amounts sum to the original total. $£200 + £250 = £450$

Equivalent Ratios and the 1:n Form

Ratios and fractions are closely connected ways of describing proportions; a ratio $a:b$ implies the first part represents the fraction $\frac{a}{a+b}$ of the entire amount.
AQA exam questions often require you to write a ratio in the specific form $1 : n$ or $n : 1$ .
You achieve this by dividing both sides by the number that needs to become $1$ , even if the other side becomes a decimal.
For example, to express $20 : 50$ in the form $1 : n$ , divide both sides by the first part ( $20$ ). This gives $(20 \div 20) : (50 \div 20)$ , which simplifies to $1 : 2.5$ .

Contextual and Reverse Ratio Problems

When mixing concrete on a building site, you might know exactly how much cement you have, but need to calculate the required amount of the other ingredients.
These are reverse ratio problems, where the value of one part is given instead of the overall total.
To solve them, divide the given value by its corresponding ratio part to find the unitary share value, which can then be used to find the missing parts.
In practical problems, such as buying tins of paint, you must round logically. If a ratio calculation suggests $1.3$ tins are needed, you cannot buy a partial tin, so the answer must be $2$ .

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often incorrectly divide the total quantity by the largest number in the ratio instead of summing all the numbers to find the total parts.
2
In a unit conversion question, always check units first! Writing $30:5$ for $30$ cm and $5$ m instead of $30:500$ will cost you accuracy marks.
3
Always label your ratios in your working (e.g., writing 'Anna : Ben' above the numbers) to avoid 'reversal errors' where you assign the correct final amounts to the wrong person.
4
Avoid using equals signs ( $=$ ) between different equivalent ratios in your working (e.g., do not write $12:18 = 2:3$ ); AQA examiners prefer arrows or the phrase 'is equivalent to'.
5
In calculation questions, explicitly write down your method (e.g., showing $+ 5 = 9$ or $\div 9$ ) to secure M1 (method) marks even if your final answer is incorrect.

Key Terms(10)

Ratio: A way of comparing one part of a whole to another part (part-to-part) or to the whole itself (part-to-whole).
Colon notation: The standard mathematical format for writing a ratio, using a colon between the numbers (e.g., a : b).
Ratio simplification: The process of reducing a ratio to its lowest integer values by dividing all parts by their highest common factor.
Highest Common Factor (HCF): The largest integer that divides two or more numbers without leaving a remainder.
Equivalent ratio: Ratios that represent the same proportion, found by multiplying or dividing all parts by the same non-zero number.
Simplest form: A ratio where the parts are integers with no common factors other than 1.
Additive method: The specific sequence of adding ratio parts together to determine the total number of shares.
Unitary method: A mathematical process where the value of a single unit (one share) is found first to calculate the value of multiple units.
Total parts: The sum of all the numbers in a ratio, representing the total number of shares an amount is split into.
Share value: The calculated amount that one single part of a ratio is worth.

Previous Subtopic: Ratio NotationRatio Notation Next NoteReal-Life Ratio Applications and Best-Buys

Back to Ratio Division

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(10)

Ratio: A way of comparing one part of a whole to another part (part-to-part) or to the whole itself (part-to-whole).
Colon notation: The standard mathematical format for writing a ratio, using a colon between the numbers (e.g., a : b).
Ratio simplification: The process of reducing a ratio to its lowest integer values by dividing all parts by their highest common factor.
Highest Common Factor (HCF): The largest integer that divides two or more numbers without leaving a remainder.
Equivalent ratio: Ratios that represent the same proportion, found by multiplying or dividing all parts by the same non-zero number.
Simplest form: A ratio where the parts are integers with no common factors other than 1.
Additive method: The specific sequence of adding ratio parts together to determine the total number of shares.
Unitary method: A mathematical process where the value of a single unit (one share) is found first to calculate the value of multiple units.
Total parts: The sum of all the numbers in a ratio, representing the total number of shares an amount is split into.
Share value: The calculated amount that one single part of a ratio is worth.

Fundamentals of Ratio Division

Expressing Quantities as a Ratio

Every time you mix squash with water, you are applying a ratio in real life.
A ratio is a way of comparing one part of a whole to another part (part-to-part) or to the whole itself (part-to-whole).
We use colon notation (such as $a : b$ ) to write them, and the order of the numbers must strictly match the order of the text.
Quantities must be converted to the same units before a ratio is formed, but the final ratio itself is unitless.
You can create an equivalent ratio representing the same proportion by multiplying or dividing all parts by the same non-zero number.
Ratio simplification involves reducing a ratio by dividing all parts by their Highest Common Factor (HCF) (the largest integer that divides them without a remainder).
This results in a ratio in its simplest form, where the parts are integers with no common factors other than $1$ .

Worked Example: Expressing and Simplifying Ratios

Question: Mix $800$ g of sand and $1.2$ kg of cement. Express the division of sand to cement as a ratio in its simplest form.

Step 1: Convert to the same units so they can be compared. $1.2$ kg $= 1200$ g
Step 2: Write using colon notation. $800 : 1200$
Step 3: Perform ratio simplification by dividing both sides by the Highest Common Factor (HCF). The Highest Common Factor (HCF) of $800$ and $1200$ is $400$ . $(800 \div 400) : (1200 \div 400) = 2 : 3$
Result: This creates an equivalent ratio in its simplest form, where the parts are integers with no common factors other than $1$ .

Dividing a Quantity in a Ratio

You and a friend buy a winning lottery ticket, but you paid £1 and they paid £4. How do you split a £500 prize fairly?
You use the additive method (often called the unitary method) to share a total amount into given proportions.
First, find the total parts by summing all the numbers in the ratio to determine the total number of shares.
Next, calculate the share value (the amount one single part is worth) by dividing the total quantity by the total parts.
This share value acts as a multiplier to scale up any part of the ratio.

Worked Example: The Additive Method

Question: Divide £450 between Anna and Ben in the ratio $4 : 5$ .

Step 1: Sum the parts of the ratio to find the total parts. $4 + 5 = 9$ total parts.
Step 2: Divide the total quantity by this sum to find the share value (the value of one single share). $£450 \div 9 = £50$ per share.
Step 3: Multiply the share value by the original ratio numbers. Anna: $4 \times £50 = £200$ Ben: $5 \times £50 = £250$
Step 4: Check that the final amounts sum to the original total. $£200 + £250 = £450$

Equivalent Ratios and the 1:n Form

Ratios and fractions are closely connected ways of describing proportions; a ratio $a:b$ implies the first part represents the fraction $\frac{a}{a+b}$ of the entire amount.
AQA exam questions often require you to write a ratio in the specific form $1 : n$ or $n : 1$ .
You achieve this by dividing both sides by the number that needs to become $1$ , even if the other side becomes a decimal.
For example, to express $20 : 50$ in the form $1 : n$ , divide both sides by the first part ( $20$ ). This gives $(20 \div 20) : (50 \div 20)$ , which simplifies to $1 : 2.5$ .

Contextual and Reverse Ratio Problems

When mixing concrete on a building site, you might know exactly how much cement you have, but need to calculate the required amount of the other ingredients.
These are reverse ratio problems, where the value of one part is given instead of the overall total.
To solve them, divide the given value by its corresponding ratio part to find the unitary share value, which can then be used to find the missing parts.
In practical problems, such as buying tins of paint, you must round logically. If a ratio calculation suggests $1.3$ tins are needed, you cannot buy a partial tin, so the answer must be $2$ .

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often incorrectly divide the total quantity by the largest number in the ratio instead of summing all the numbers to find the total parts.
2
In a unit conversion question, always check units first! Writing $30:5$ for $30$ cm and $5$ m instead of $30:500$ will cost you accuracy marks.
3
Always label your ratios in your working (e.g., writing 'Anna : Ben' above the numbers) to avoid 'reversal errors' where you assign the correct final amounts to the wrong person.
4
Avoid using equals signs ( $=$ ) between different equivalent ratios in your working (e.g., do not write $12:18 = 2:3$ ); AQA examiners prefer arrows or the phrase 'is equivalent to'.
5
In calculation questions, explicitly write down your method (e.g., showing $+ 5 = 9$ or $\div 9$ ) to secure M1 (method) marks even if your final answer is incorrect.

Key Terms(10)

Ratio: A way of comparing one part of a whole to another part (part-to-part) or to the whole itself (part-to-whole).
Colon notation: The standard mathematical format for writing a ratio, using a colon between the numbers (e.g., a : b).
Ratio simplification: The process of reducing a ratio to its lowest integer values by dividing all parts by their highest common factor.
Highest Common Factor (HCF): The largest integer that divides two or more numbers without leaving a remainder.
Equivalent ratio: Ratios that represent the same proportion, found by multiplying or dividing all parts by the same non-zero number.
Simplest form: A ratio where the parts are integers with no common factors other than 1.
Additive method: The specific sequence of adding ratio parts together to determine the total number of shares.
Unitary method: A mathematical process where the value of a single unit (one share) is found first to calculate the value of multiple units.
Total parts: The sum of all the numbers in a ratio, representing the total number of shares an amount is split into.
Share value: The calculated amount that one single part of a ratio is worth.

Previous Subtopic: Ratio NotationRatio Notation Next NoteReal-Life Ratio Applications and Best-Buys

Back to Ratio Division

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(10)

Ratio: A way of comparing one part of a whole to another part (part-to-part) or to the whole itself (part-to-whole).
Colon notation: The standard mathematical format for writing a ratio, using a colon between the numbers (e.g., a : b).
Ratio simplification: The process of reducing a ratio to its lowest integer values by dividing all parts by their highest common factor.
Highest Common Factor (HCF): The largest integer that divides two or more numbers without leaving a remainder.
Equivalent ratio: Ratios that represent the same proportion, found by multiplying or dividing all parts by the same non-zero number.
Simplest form: A ratio where the parts are integers with no common factors other than 1.
Additive method: The specific sequence of adding ratio parts together to determine the total number of shares.
Unitary method: A mathematical process where the value of a single unit (one share) is found first to calculate the value of multiple units.
Total parts: The sum of all the numbers in a ratio, representing the total number of shares an amount is split into.
Share value: The calculated amount that one single part of a ratio is worth.