A3 · Solve proportional problems using unit analysis

Alberta Grade 11 Mathematics

Algebra

Use units to choose operations, convert measurements, and check whether an answer makes sense.

What you will learn

• Identify the units in a proportional problem and decide which units the answer needs.
• Use a rate and multiplication or division to solve for an unknown amount.
• Convert measurements by multiplying by a conversion factor that equals one.
• Check that units cancel correctly and that the answer is reasonable.

Start with the units

• Write down the known quantities with their units.
• Write the unit required for the answer before choosing an operation.
• Use the meaning of the rate to decide whether to multiply or divide.

Use unit analysis to set up a calculation

• Choose a conversion factor that cancels the unwanted unit.
• Units that appear once above and once below a fraction can cancel.
• A correct number with the wrong unit is not a complete answer.

Check the answer

• State the answer with the requested unit.
• Use an estimate and the situation to check the size of the answer.
• Check whether the answer changes in the expected direction.

Worked example

Find a travel time

1. Choose the answer unit
   The question asks how long the trip takes, so the answer should be in hours. The given rate is 50 kilometres for each hour.
   $$50\,\text{km/h}$$
2. Divide distance by rate
   Dividing the distance by the speed gives the time. In the units, km divided by km/h leaves h, which confirms that the setup gives time.
   $$150\,\text{km}\div 50\,\text{km/h}=3\,\text{h}$$
3. Check the result
   At 50 km each hour, three hours covers 150 km. The answer has the requested unit and matches the situation.
   $$50\,\text{km/h}\times 3\,\text{h}=150\,\text{km}$$

Worked example

Convert a map measurement

1. Use the scale
   Each centimetre on the plan represents 2500 centimetres in reality. Multiply the plan length by this real-distance amount per plan centimetre. The plan centimetres cancel.
   $$3.2\,\text{cm}\times\frac{2500\,\text{cm}}{1\,\text{cm}}=8000\,\text{cm}$$
2. Convert to metres
   The conversion factor is provided: 100 cm equals 1 m. Use the fraction with centimetres below so those units cancel and metres remain.
   $$8000\,\text{cm}\times\frac{1\,\text{m}}{100\,\text{cm}}=80\,\text{m}$$
3. Check the size
   A plan length of a little over 3 cm represents 8000 cm, or 80 m, at this scale. The result is larger than the plan measurement because the plan is a reduced drawing.
   $$8000\,\text{cm}=80\,\text{m}$$

Worked example

Find production over part of a shift

1. Find the hourly rate
   Divide the number of boxes by the hours. This gives boxes per hour, the rate needed for the next step.
   $$84\,\text{boxes}\div 3\,\text{h}=28\,\text{boxes/h}$$
2. Scale the rate to the time
   Multiply the hourly rate by 6.5 hours. The hour units cancel, leaving boxes.
   $$28\,\text{boxes/h}\times 6.5\,\text{h}=182\,\text{boxes}$$
3. Check against the rate
   Six and a half hours is a little more than six hours. At 28 boxes per hour, the result should be a little more than 168 boxes. The calculated 182 boxes fits that estimate.
   $$28\,\text{boxes/h}\times 6\,\text{h}=168\,\text{boxes}$$

Common mistakes and how to avoid them

Lesson summary

• A rate compares quantities, often as an amount per unit of time.
• Set up the calculation so units cancel and the requested unit remains.
• Use conversion factors made from equal measurements.
• Estimate and check whether the answer is reasonable for the situation.

Check your understanding

Key terms

Unit

A label that tells what a number measures, such as hours or metres.

Rate

A comparison of quantities measured in different units, such as boxes per hour.

Unit analysis

Tracking units through a calculation to help choose operations and check the answer.

Conversion factor

A fraction made from two equal measurements in different units, used to rewrite a measurement.

Proportional relationship

A relationship in which one quantity changes at a steady rate compared with another.

Continue through Math 20-3

View the complete Math 20-3 Alberta Grade 11 Mathematics curriculum and lessons

• A2 · Interpret slope as rise over run and as a constant rate
• S1 · Create and interpret bar, histogram, line, and circle graphs
• M2 · Solve volume and capacity problems with SI and imperial units
• G1 · Solve problems involving two or three right triangles
• G2 · Solve practical scale and scale-diagram problems
• G3 · Model 3-D objects and draw their top, front, and side views

Tutor support for A3 · Solve proportional problems using unit analysis

Working on A3 · Solve proportional problems using unit analysis in Math 20-3? Bring the question and your own attempt. In a live session, you can represent the quantities clearly, explain the operation you chose, and estimate whether the answer is reasonable. Ask about the first step that does not make sense, then try a similar question yourself.

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برای مبحث A3 · مسئله‌های تناسبی را با تحلیل واحد حل کنید در درس Math 20-3، سؤال و تلاش خود را همراه بیاورید. در جلسه می‌توانید کمیت‌ها را روشن نمایش دهید، دلیل انتخاب عمل را توضیح دهید و معقول بودن پاسخ را برآورد کنید. راهنمای معلم ریاضی را ببینید یا جلسهٔ معارفهٔ رایگان ۳۰ دقیقه‌ای درخواست کنید. زمان جلسه پس از پذیرش معلم قطعی می‌شود.

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-3), expectation A3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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