Part-to-Part and Part-to-Whole Relationships: Extended Worked Examples

Grade 8 focus

This record is for learners in Grade 8 studying ratios, rates, and proportions in the Canadian curriculum. It does not reteach the full concept from the linked lesson; instead, it gives a carefully sequenced set of worked examples that show how to make correct choices in real ratio questions.

Use this record with the linked materials

Before working here, read the conceptual explanations in:

• rea.m08.number.ratios-rates-proportions.ratio-concepts.part-whole.overview
• rea.m08.number.ratios-rates-proportions.ratio-concepts.part-whole.lesson

If you want broader ratio-comparison practice beyond part-to-part versus part-to-whole, also use:

• rea.m08.number.ratios-rates-proportions.ratio-concepts.worked-examples

Fast decision rule

When reading a ratio question, ask:

1. What are the two quantities being compared?
2. Are both quantities parts of the group?
3. Or is one quantity the entire group?

If both are parts, the ratio is part-to-part. If one quantity is the whole group, the ratio is part-to-whole.

Worked Example 1: Identify the type of ratio

A class has 12 girls and 8 boys.

Find:

1. the ratio of girls to boys
2. the ratio of girls to students

Solution

1. Girls to boys means compare one part to another part.

• girls = 12
• boys = 8
• ratio = 12:8
• simplify by dividing both numbers by 4: 3:2

So the ratio of girls to boys is 3:2.

2. Girls to students means compare one part to the whole group.

• girls = 12
• total students = 12 + 8 = 20
• ratio = 12:20
• simplify by dividing both numbers by 4: 3:5

So the ratio of girls to students is 3:5.

Commentary

The important choice is not the arithmetic. It is deciding whether the second number is another part or the whole set. In the second question, using 8 would be a meaning error because 8 names boys, not all students.

Worked Example 2: Move from part-to-part to part-to-whole

At a pet store, the ratio of cats to dogs is 4:7. There are no other pets in this section.

Find:

1. the ratio of cats to all pets
2. the ratio of dogs to all pets

Solution

The ratio 4:7 is part-to-part.

• cats = 4 parts
• dogs = 7 parts
• whole = 4 + 7 = 11 parts

1. Cats to all pets:

• cats : all pets = 4:11

2. Dogs to all pets:

• dogs : all pets = 7:11

Commentary

When converting a part-to-part ratio into a part-to-whole ratio, add the parts first to create the whole. A common mistake is to keep the 7 as the second term in the first answer, but that would still be cats to dogs, not cats to all pets.

Worked Example 3: Move from part-to-whole to another part

In a basket, the ratio of red apples to all apples is 5:9. The rest are green apples.

Find the ratio of red apples to green apples.

Solution

The ratio 5:9 means:

• red apples = 5 parts
• all apples = 9 parts

So green apples must be:

• 9 - 5 = 4 parts

Now compare red to green:

• red : green = 5:4

Commentary

Here the whole is already given, so you do not add. You subtract the known part from the whole to find the missing part. The direction matters: the question asks for red to green, not green to red.

Worked Example 4: Find missing numbers from a part-to-whole ratio

At camp, the ratio of students wearing hats to all students is 3:8. There are 24 students in total.

How many students are wearing hats? How many are not wearing hats?

Solution

3:8 means:

• 3 equal parts correspond to students wearing hats
• 8 equal parts correspond to all students

Since 8 parts = 24 students,

• 1 part = 24 / 8 = 3 students

Then:

• wearing hats = 3 parts = 3 x 3 = 9
• not wearing hats = 24 - 9 = 15

So 9 students are wearing hats and 15 are not.

Commentary

Using the whole first is the cleanest choice because the ratio directly connects a part to the whole. Another valid path would be to find the missing part as 8 - 3 = 5 parts, then scale, but using the given total immediately is more efficient.

Worked Example 5: Find missing numbers from a part-to-part ratio

The ratio of blue marbles to yellow marbles is 2:5. There are 35 marbles altogether.

How many blue marbles and how many yellow marbles are there?

Solution

The ratio 2:5 is part-to-part.

• blue = 2 parts
• yellow = 5 parts
• whole = 2 + 5 = 7 parts

Since 7 parts = 35 marbles,

• 1 part = 35 / 7 = 5 marbles

Then:

• blue marbles = 2 x 5 = 10
• yellow marbles = 5 x 5 = 25

Check:

• 10 + 25 = 35

So there are 10 blue marbles and 25 yellow marbles.

Commentary

This example looks similar to Example 4, but the structure is different. Because the original ratio is part-to-part, you must first build the whole by adding the ratio parts.

Worked Example 6: Decide which statement matches a ratio

A school club has 18 returning members and 12 new members.

Which ratio matches each statement?

• 18:12
• 18:30
• 12:30
• 3:5

Statements:

1. returning members to new members
2. returning members to all members
3. new members to all members
4. simplified returning members to all members

Solution

First find the whole:

• all members = 18 + 12 = 30

Now match each statement.

1. returning members to new members:

• 18:12

2. returning members to all members:

• 18:30

3. new members to all members:

• 12:30

4. simplified returning members to all members:

• simplify 18:30 by dividing by 6
• 18:30 = 3:5

Commentary

This kind of question checks meaning more than calculation. A learner who mixes up 18:12 and 18:30 is not making a simplification error; they are confusing part-to-part with part-to-whole.

Worked Example 7: Multi-step context with leftovers

In a jar of candies, the ratio of sour candies to all candies is 7:12. There are 60 candies in the jar.

Find:

1. the number of sour candies
2. the number of non-sour candies
3. the ratio of sour candies to non-sour candies

Solution

From 7:12:

• sour = 7 parts
• whole = 12 parts

Since 12 parts = 60 candies,

• 1 part = 60 / 12 = 5

1. Sour candies:

• 7 x 5 = 35

2. Non-sour candies:

• 60 - 35 = 25

3. Sour to non-sour:

• 35:25
• simplify by dividing by 5
• 7:5

So the answers are 35 sour candies, 25 non-sour candies, and a sour-to-non-sour ratio of 7:5.

Commentary

The last part is a conversion from part-to-whole information into part-to-part information. It is often best to find actual counts first, then form the requested ratio.

Worked Example 8: Reverse reasoning from a difference

The ratio of boys to girls in a group is 4:5. There are 6 more girls than boys.

How many boys and girls are in the group?

Solution

From 4:5:

• boys = 4 parts
• girls = 5 parts

The difference is:

• 5 - 4 = 1 part

We are told girls exceed boys by 6, so:

• 1 part = 6 students

Then:

• boys = 4 x 6 = 24
• girls = 5 x 6 = 30

Check:

• difference = 30 - 24 = 6

So there are 24 boys and 30 girls.

Commentary

This is a stronger ratio problem because the total is not given. The smartest choice is to use the difference in ratio parts. When two categories differ by 1 ratio part, and the real difference is known, the scale factor becomes visible immediately.

Worked Example 9: Challenge with two linked ratios

A theatre section has adults, teens, and children. The ratio of adults to teens is 3:2, and the ratio of teens to children is 4:5. There are 20 teens.

Find:

1. the number of adults
2. the number of children
3. the ratio of adults to all people in the section

Solution

We use the actual teen count to scale each ratio.

From adults to teens = 3:2:

• 2 parts = 20 teens
• 1 part = 10
• adults = 3 x 10 = 30

From teens to children = 4:5:

• 4 parts = 20 teens
• 1 part = 5
• children = 5 x 5 = 25

Now find the whole:

• total people = 30 + 20 + 25 = 75

Adults to all people:

• 30:75
• simplify by dividing by 15
• 2:5

So there are 30 adults, 25 children, and the ratio of adults to all people is 2:5.

Commentary

The key choice is noticing that the teen category appears in both ratios. Because the actual number of teens is known, each ratio can be scaled separately. Only after finding all three counts should you build the final part-to-whole ratio.

Worked Example 10: Challenge with error analysis

A student says: “The ratio of juice boxes to all drinks is 4:9, so the ratio of juice boxes to non-juice drinks is 4:5 because 9 - 4 = 5.”

Is the student correct? Explain.

Solution

Yes, the student is correct.

Reasoning:

• juice boxes = 4 parts
• all drinks = 9 parts
• non-juice drinks = 9 - 4 = 5 parts

So:

• juice boxes to non-juice drinks = 4:5

Commentary

This is a good subtraction because the original ratio is part-to-whole. Subtracting a part from the whole gives the missing part. The same subtraction would not be valid if 4:9 had meant juice boxes to water bottles, because then both numbers would already be parts rather than part and whole.

Common misconceptions and repairs

1. Mistake: Using the other part when the whole is needed.

• Repair: Ask, “Does this number represent all items, or just one category?”

2. Mistake: Adding ratio numbers when the ratio is already part-to-whole.

• Repair: Only add when the given ratio compares two parts.

3. Mistake: Subtracting when the given ratio is part-to-part.

• Repair: In a part-to-part ratio, subtraction gives a difference, not the whole.

4. Mistake: Reversing the order of comparison.

• Repair: Read the words exactly. red:blue is not the same as blue:red.

Quick procedure checklist

• Identify each quantity named in the words.
• Mark each as part or whole.
• If both terms are parts, add them to create the whole if needed.
• If one term is the whole, subtract to find a missing part if needed.
• Scale using totals or differences.
• Check whether the final ratio matches the exact wording of the question.

Check-yourself practice

Try these without looking back.

1. A bag has 9 black socks and 15 white socks. Find the ratio of black socks to white socks and the ratio of black socks to all socks.
2. The ratio of science books to all books on a shelf is 2:7. There are 42 books in total. How many are science books?
3. The ratio of cats to dogs is 5:3. If there are 16 fewer dogs than cats, how many cats are there?

Answers

1. Black to white = 9:15 = 3:5; black to all = 9:24 = 3:8
2. Science books = 12
3. Difference in ratio parts = 5 - 3 = 2 parts, so 2 parts = 16, 1 part = 8, cats = 5 x 8 = 40

Where to go next

After this record, use the unit quiz record to test whether you can recognize and use these relationships reliably in unfamiliar wording and mixed question types.