Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 8 chapters and video availability01 · Read and understand
What you will learn
- Explain how to choose the denominator in a two-way table.
- Solve the two practice problems and explain how the assumptions affect the answers.
Before you start
Fractions and category totals.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A club survey records 12 bus riders and 8 walkers in Grade 7, and 18 bus riders and 12 walkers in Grade 8. What fraction of Grade 8 students ride the bus, and what fraction of all respondents are Grade 8 bus riders?
Why this math matters
A conditional relative frequency uses the size of the named group as its denominator, not automatically the whole sample. Name the reference group in a sentence whenever reporting a percentage from a two-way table.
Set up the model
A useful answer starts with clear assumptions:
- Each respondent belongs to exactly one grade and one travel category.
- The four counts include all survey respondents.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Choose the denominator in a two-way table
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A club survey records 12 bus riders and 8 walkers in Grade 7, and 18 bus riders and 12 walkers in Grade 8. What fraction of Grade 8 students ride the bus, and what fraction of all respondents are Grade 8 bus riders?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the quantities
Grade 8 total = 18 + 12 = 30; overall total = 12 + 8 + 18 + 12 = 50
Compute both possible reference groups before choosing a denominator.
Apply the relationship
among Grade 8: 18/30 = 3/5 = 60%
The phrase among Grade 8 restricts the denominator to that grade.
Check and interpret
of all respondents: 18/50 = 9/25 = 36%
The same count becomes a different proportion when compared with the whole survey.
The result
of all respondents: 18/50 = 9/25 = 36%
The same count becomes a different proportion when compared with the whole survey.
Common mistakes to catch
- A joint proportion of the whole sample differs from a conditional proportion within a subgroup.
- Words such as among, of, and given identify the denominator.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What fraction of all bus riders in the original survey are Grade 7 students?
Show a hint
Restrict the denominator to bus riders.
Reveal answer and explanation
2/5
There are 12 + 18 = 30 bus riders; 12/30 = 2/5.
Practice 2
What fraction of Grade 7 respondents walk?
Show a hint
Use the Grade 7 row total.
Reveal answer and explanation
2/5 or 40%
8/(12 + 8) = 8/20 = 0.40.
Take the idea with you
Name the reference group in a sentence whenever reporting a percentage from a two-way table.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Build a cylinder from circular layers
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.