Grade 8 lesson

This lesson belongs to Grade 8 Number: Ratios, Rates and Proportions.

For the broader place of this topic in the unit, see rea.m08.number.ratios-rates-proportions.ratio-concepts.part-whole.overview and rea.m08.number.ratios-rates-proportions.ratio-concepts.overview.

For general ratio meaning, wording, and notation, see rea.m08.number.ratios-rates-proportions.ratio-concepts.ratio-meaning.

For additional comparison problems after learning this lesson, see rea.m08.number.ratios-rates-proportions.ratio-concepts.worked-examples.

Learning goal

By the end of this lesson, you should be able to:

• identify what each quantity in a ratio refers to
• decide whether a ratio is part-to-part or part-to-whole
• write both kinds of ratios from the same situation
• convert between part-to-part and part-to-whole ratios when the whole is known or can be found
• explain why two ratios with the same numbers can mean different things

1. Intuition: what is being compared?

A ratio is not just two numbers. A ratio tells which quantity is being compared to which quantity.

Suppose a box contains 3 red tiles and 5 blue tiles.

A simple visual model is:

R R R B B B B B

There are two parts:

• red tiles: 3
• blue tiles: 5

And one whole:

• all tiles: 8

Now there are different correct ratios, depending on the question.

• red to blue = 3:5 because it compares one part to another part
• red to all tiles = 3:8 because it compares a part to the whole
• blue to all tiles = 5:8 because it compares a part to the whole

These are all about the same set, but they do not mean the same thing.

2. Visual meaning

Part-to-part compares two pieces inside the same whole.

Example picture:

AAA BBBB

• part A = 3
• part B = 4
• whole = 7

Then:

• A:B = 3:4 is part-to-part
• A:whole = 3:7 is part-to-whole
• B:whole = 4:7 is part-to-whole

A reliable habit is to name the quantities in words before writing numbers.

Do not start by grabbing numbers first. Start by asking:

1. What is the first quantity?
2. What is the second quantity?
3. Is the second quantity another part, or the whole set?

3. Precise notation

If a set has:

• part A = a
• part B = b

then the whole is:

• whole = a + b

Common ratios are:

• part-to-part: a:b
• part-to-whole: a:(a+b)
• other part-to-whole: b:(a+b)

You may also see ratios written as:

• a to b
• a/b

In this lesson, a:b is used most often because it makes the two compared quantities easy to track.

4. Key relationship

If there are only two parts, then:

• whole = part 1 + part 2

So if a part-to-part ratio is known, you can build a related part-to-whole ratio by adding the parts.

Example:

If boys:girls = 2:3, then total parts = 2 + 3 = 5.

So:

• boys:whole = 2:5
• girls:whole = 3:5

This works because the whole is made from all parts together.

5. Step-by-step procedures

Procedure A: Decide what kind of ratio you have

1. Read the words carefully.
2. Identify the first quantity.
3. Identify the second quantity.
4. Ask whether the second quantity is another part or the whole.
5. Label the ratio as part-to-part or part-to-whole.

Procedure B: Write all useful ratios from a situation with two parts

1. Count or identify part 1.
2. Count or identify part 2.
3. Add to find the whole.
4. Write part 1 : part 2.
5. Write part 1 : whole.
6. Write part 2 : whole.
7. State in words what each ratio means.

Procedure C: Convert a part-to-part ratio to a part-to-whole ratio

1. Add the ratio parts.
2. Use that sum as the whole number of parts.
3. Keep the chosen part as the first term.
4. Write chosen part : total parts.

Example pattern:

If a:b, then part-to-whole for the first part is a:(a+b).

Procedure D: Find missing counts when one total is known

1. Add the ratio parts.
2. Match that sum to the total number of equal ratio parts.
3. Divide the actual total by the ratio total to find the scale factor.
4. Multiply each ratio term by the scale factor.
5. Check that the parts add to the whole.

6. Fully worked examples

Example 1: Basic identification

A fruit bowl has 4 apples and 6 oranges.

Write:

• the apples-to-oranges ratio
• the apples-to-fruit ratio
• the oranges-to-fruit ratio

Step 1: Identify the parts and whole.

• apples = 4
• oranges = 6
• total fruit = 4 + 6 = 10

Step 2: Write the requested ratios.

• apples to oranges = 4:6
• apples to fruit = 4:10
• oranges to fruit = 6:10

Step 3: Classify them.

• 4:6 is part-to-part
• 4:10 is part-to-whole
• 6:10 is part-to-whole

Answer:

• apples:oranges = 4:6
• apples:fruit = 4:10
• oranges:fruit = 6:10

Example 2: Converting part-to-part into part-to-whole

In a class, the ratio of students wearing glasses to students not wearing glasses is 3:11.

Find the ratio of:

• glasses to whole class
• no glasses to whole class

Step 1: Interpret the given ratio.

• wearing glasses = 3 parts
• not wearing glasses = 11 parts

This is part-to-part.

Step 2: Find the total number of parts.

3 + 11 = 14

So the whole class is 14 ratio parts.

Step 3: Write each part-to-whole ratio.

• glasses : whole = 3:14
• no glasses : whole = 11:14

Check:

The whole uses all parts, and 3 + 11 = 14, so the conversion is consistent.

Answer:

• glasses:whole class = 3:14
• no glasses:whole class = 11:14

Example 3: Same numbers, different meanings

A jar contains red, green, and yellow beads. There are 5 red beads and 8 non-red beads.

A student says, "The ratio 5:8 means 5 red beads out of 8 beads in the jar."

Explain the mistake and write the correct part-to-whole ratio for red beads.

Step 1: Read what 5:8 actually compares.

• 5 = red beads
• 8 = non-red beads

So 5:8 compares one part to another part.

That makes 5:8 a part-to-part ratio.

Step 2: Find the whole jar.

5 + 8 = 13

Step 3: Write the correct red-to-whole ratio.

• red : whole = 5:13

Step 4: Explain the error clearly.

The student treated the second number, 8, as the whole. But 8 is only the number of non-red beads, not the total number of beads. The whole must include both red and non-red beads.

Answer:

The mistake is confusing "non-red" with "all beads." The correct part-to-whole ratio for red beads is 5:13.

Example 4: Find actual counts from a total

The ratio of boys to girls in a club is 4:7. There are 33 students in the club. How many boys and how many girls are there?

Then write the boys-to-whole ratio.

Step 1: Add the ratio parts.

4 + 7 = 11

So the whole club is 11 equal ratio parts.

Step 2: Find the size of one ratio part.

There are 33 actual students, so:

33 / 11 = 3

Each ratio part represents 3 students.

Step 3: Find each actual count.

• boys = 4 x 3 = 12
• girls = 7 x 3 = 21

Step 4: Check the total.

12 + 21 = 33

The total matches.

Step 5: Write boys to whole.

• boys : whole = 12:33

This can also be written in simplest form as 4:11, which matches the ratio-part form directly.

Answer:

• boys = 12
• girls = 21
• boys:whole = 12:33, or simplified 4:11

Example 5: Reverse reasoning from part-to-whole

In a survey, the ratio of students who chose music to the whole group is 3:8. There are 48 students in total. How many chose music? How many did not choose music? What is the music-to-non-music ratio?

Step 1: Interpret the ratio.

• music = 3 parts
• whole = 8 parts

This is part-to-whole.

Step 2: Find one ratio part.

48 / 8 = 6

So each ratio part is 6 students.

Step 3: Find the number who chose music.

3 x 6 = 18

So 18 students chose music.

Step 4: Find the number who did not choose music.

Whole parts = 8, music parts = 3, so non-music parts = 8 - 3 = 5.

Actual non-music students:

5 x 6 = 30

Step 5: Write the part-to-part ratio.

• music : non-music = 18:30

Simplify by dividing both terms by 6:

• 18:30 = 3:5

Answer:

• chose music = 18
• did not choose music = 30
• music:non-music = 3:5

7. Common misconceptions and repairs

Misconception 1

"The second number is always the whole."

Repair: Read the words, not just the numbers. In red:blue, the second number is another part. In red:all, the second number is the whole.

Misconception 2

"If the numbers are the same, the meaning is the same."

Repair: 3:5 could mean red:blue, but it could also mean boys:whole only if the whole really is 5. Meaning comes from the quantities named, not only from the digits.

Misconception 3

"For a part-to-whole ratio, add the numbers again."

Repair: If you already have part:whole, the whole is already included. Adding again would count one part twice.

Misconception 4

"A part-to-part ratio tells what fraction of the whole one part is."

Repair: Not directly. First find the whole by adding the parts, then write part:whole.

8. Practice

Try these before checking the answers.

1. A basket has 7 bananas and 9 pears. Write:

• bananas:pears
• bananas:fruit
• pears:fruit

2. The ratio of cats to dogs is 5:2. Write:

• cats:whole
• dogs:whole

3. The ratio of left-handed students to the whole class is 1:6. If there are 30 students, how many are left-handed? How many are not left-handed?

4. In a team, the ratio of defenders to forwards is 3:4. There are 21 defenders and forwards altogether. How many defenders are there?

9. Practice answers

1. Bananas = 7, pears = 9, whole = 16.

• bananas:pears = 7:9
• bananas:fruit = 7:16
• pears:fruit = 9:16

2. Cats:dogs = 5:2, so whole parts = 7.

• cats:whole = 5:7
• dogs:whole = 2:7

3. Left-handed:whole = 1:6, total = 30.

• one ratio part = 30 / 6 = 5
• left-handed = 1 x 5 = 5
• not left-handed = 30 - 5 = 25

4. Defenders:forwards = 3:4, so whole parts = 7.

• one ratio part = 21 / 7 = 3
• defenders = 3 x 3 = 9

10. Final check for yourself

When you write any ratio, ask:

• What does the first number represent?
• What does the second number represent?
• Is this comparing part to part, or part to whole?
• If I say the ratio aloud in words, does it still make sense?

If you can answer those four questions, you are much less likely to confuse the meaning.

11. Summary

In Grade 8 ratio work, the central idea is that a ratio must be interpreted by the quantities it compares.

• part-to-part compares one part with another part
• part-to-whole compares one part with the entire set
• with two parts, whole = part 1 + part 2
• converting correctly depends on identifying the whole before writing the new ratio

This distinction is essential for later work with fractions, percents, probability, and rates.