Chapter 2 Discussion/Participation

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Using & Understanding Mathematics: A Quantitative Reasoning Approach

Eighth Edition

Chapter 2

Approaches to Problem Solving

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Unit 2A Understand, Solve and Explain

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Three-Step Problem Solving: Understand-Solve-Explain (U-S-E) (1 of 3)

Step 1: Understand the problem.

Think about what the problem asks you to do.

Draw a picture or diagram to help make sense of the problem.

Ask yourself what the solution should look like.

Try to map a path (either mentally or in writing) that will lead you from your understanding of the problem to its solution.

Continually revisit your understanding of the problem.

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Three-Step Problem Solving: Understand-Solve-Explain (U-S-E) (2 of 3)

Step 2: Solve the problem.

Obtain any needed information or data.

For multi-step problems, be sure to keep an organized, neatly written record of your work.

Double-check each step as you work to avoid carrying errors through to the end of your solution.

Constantly reevaluate your plan as you work.

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Three-Step Problem Solving: Understand-Solve-Explain (U-S-E) (3 of 3)

Step 3: Explain your result.

Be sure that your result makes sense.

Recheck your calculations once more or, even better, find an independent way to check your result.

Identify and understand any potential sources of uncertainty in your result. If you made assumptions, were they reasonable?

Write your solution clearly and concisely, using complete sentences to make sure the context and meaning are clear.

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Units

The units of a quantity describe what that quantity measures or counts.

Unit analysis is the process of working with units to help solve problems.

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Example 1: Using Key Words (1 of 2)

You are buying 30 acres of farm land at $12,000 per acre. What is the total cost?

Solution

Understand. The question asks about total cost, and one of the given units is dollars, so we expect an answer in dollars.

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Example 1: Using Key Words (2 of 2)

Solve: We carry out the calculation; note that the price is given in dollars per acre, so we write the division by acres in fraction form. That allows “acres” to cancel, leaving the final answer in dollars:

Explain: We have found that purchasing 30 acres of farmland at a price of $12,000 per acre will cost a total of $360,000.

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Unit Analysis in Problem Solving

Remember:

You cannot add or subtract numbers with different units, but you can combine different units through multiplication, division, or raising to powers.

It is easier to keep track of units if you replace division with multiplication by the reciprocal. For example, instead of dividing by 60 s/min, multiply by 1 min/60 s.

When you complete your calculations, make sure that your answer has the units you expected. If it doesn’t, then you’ve done something wrong.

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Key Words and Operations with Units

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Example: Using Units to Find a Pathway

How many crates do you need to hold 2000 apples if each crate holds 40 apples?

Solution

The question asks “how many crates,” so the answer should have units of crates. Notice that the statement “each crate holds 40 apples” implies that we can fit 40 apples per crate, which we write as 40 apples/crate.

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Conversion Factors

A conversion factor is a statement of equality that is used to convert between units.

Some conversion factors:

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Example: Feet to Inches

Convert a distance of 9 feet into inches.

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Example: Using a Chain of Conversions

How many seconds are in 94 days?

Solution

1 day = 24 hours

1 hour = 60 minutes

1 minute = 60 seconds

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Conversions with Units Raised to Powers

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Example: Carpeting a Room

You want to carpet a room that measures 12 feet by 15 feet, making an area of 180 square feet. But carpet is usually sold by the square yard. How many square yards of carpet do you need?

Solution

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Example: Filling a Planter

How many cubic yards of soil are needed to fill a planter that is 20 feet long by 3 feet wide by 4 feet tall?

The volume is

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Currency Conversions

Converting between currencies is a unit conversion problem in which the conversion factors are known as the exchange rates. The table represents some typical currency exchange rates:

Sample Currency Exchange Rates

Currency	Dollars per Foreign	Foreign per Dollar
British pound	1.624	0.6158
Canadian dollar	1.005	0.9950
European euro	1.320	0.7576
Japanese yen	0.0120	83.33
Mexican peso	0.07855	12.73

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Example: Gas Price per Liter (1 of 2)

A gas station in Canada sells gasoline for C A D 1.34 per liter. (C A D is an abbreviation for Canadian dollars.) What is the price in dollars per gallon? Use the currency exchange rate in Table 2.4.

Solution

We use a chain of conversions to convert from C A D to dollars and then from liters to gallons. From Table 2.4, the currency conversion is $1.005 per C A D, and from Table 2.3, there are 3.785 liters per gallon.

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Example: Gas Price per Liter (2 of 2)