Problem Solving
In previous math courses, you’ve no doubt run into the infamous “word problems.” Unfortunately, these problems rarely resemble the type of problems we actually encounter in everyday life. In math books, you usually are told exactly which formula or procedure to use, and are given exactly the information you need to answer the question. In real life, problem solving requires identifying an appropriate formula or procedure, and determining what information you will need (and won’t need) to answer the question. In this chapter, we will review several basic but powerful algebraic ideas: percents, rates, and proportions. We will then focus on the problem solving process, and explore how to use these ideas to solve problems where we don’t have perfect information.
Percents
In the 2004 vice-presidential debates, Edwards's claimed that US forces have suffered "90% of the coalition casualties" in Iraq. Cheney disputed this, saying that in fact Iraqi security forces and coalition allies "have taken almost 50 percent" of the casualties.[footnote]http://www.factcheck.org/cheney_edwards_mangle_facts.html[/footnote] Who is correct? How can we make sense of these numbers? Percent literally means “per 100,” or “parts per hundred.” When we write 40%, this is equivalent to the fraction or the decimal 0.40. Notice that 80 out of 200 and 10 out of 25 are also 40%, since .Example 1
243 people out of 400 state that they like dogs. What percent is this?Solution
. This is 60.75%. Notice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right.Example 2
Write each as a percent:- 0.02
- 2.35
Solutions
- = 25%
- 0.02 = 2%
- 2.35 = 235%
Percents
If we have a part that is some percent of a whole, then , or equivalently,. To do the calculations, we write the percent as a decimal.Example 3
The sales tax in a town is 9.4%. How much tax will you pay on a $140 purchase?Solution
Here, $140 is the whole, and we want to find 9.4% of $140. We start by writing the percent as a decimal by moving the decimal point two places to the left (which is equivalent to dividing by 100). We can then compute: tax = 0.094(140) = $13.16 in tax.Example 4
In the news, you hear “tuition is expected to increase by 7% next year.” If tuition this year was $1200 per quarter, what will it be next year?Solution
The tuition next year will be the current tuition plus an additional 7%, so it will be 107% of this year’s tuition: $1200(1.07) = $1284. Alternatively, we could have first calculated 7% of $1200: $1200(0.07) = $84. Notice this is not the expected tuition for next year (we could only wish). Instead, this is the expected increase, so to calculate the expected tuition, we’ll need to add this change to the previous year’s tuition: $1200 + $84 = $1284.Try It Now
A TV originally priced at $799 is on sale for 30% off. There is then a 9.2% sales tax. Find the price after including the discount and sales tax.Example 5
The value of a car dropped from $7400 to $6800 over the last year. What percent decrease is this?Solution
To compute the percent change, we first need to find the dollar value change: $6800 – $7400 = –$600. Often we will take the absolute value of this amount, which is called the absolute change: |–600| = 600. Since we are computing the decrease relative to the starting value, we compute this percent out of $7400: 8.1% decrease. This is called a relative change.Absolute and Relative Change
Given two quantities,Absolute change =
Relative change:
Absolute change has the same units as the original quantity. Relative change gives a percent change. The starting quantity is called the base of the percent change.Example 6
There are about 75 QFC supermarkets in the United States. Albertsons has about 215 stores. Compare the size of the two companies.Solution
When we make comparisons, we must ask first whether an absolute or relative comparison. The absolute difference is 215 – 75 = 140. From this, we could say “Albertsons has 140 more stores than QFC.” However, if you wrote this in an article or paper, that number does not mean much. The relative difference may be more meaningful. There are two different relative changes we could calculate, depending on which store we use as the base:Using QFC as the base, .
This tells us Albertsons is 186.7% larger than QFC.
Using Albertsons as the base,.
This tells us QFC is 65.1% smaller than Albertsons.
Notice both of these are showing percent differences. We could also calculate the size of Albertsons relative to QFC:, which tells us Albertsons is 2.867 times the size of QFC. Likewise, we could calculate the size of QFC relative to Albertsons:, which tells us that QFC is 34.9% of the size of Albertsons.Example 7
Suppose a stock drops in value by 60% one week, then increases in value the next week by 75%. Is the value higher or lower than where it started?Solution
To answer this question, suppose the value started at $100. After one week, the value dropped by 60%: $100 – $100(0.60) = $100 – $60 = $40. In the next week, notice that base of the percent has changed to the new value, $40. Computing the 75% increase: $40 + $40(0.75) = $40 + $30 = $70. In the end, the stock is still $30 lower, or = 30% lower, valued than it started.Try It Now
The US federal debt at the end of 2001 was $5.77 trillion, and grew to $6.20 trillion by the end of 2002. At the end of 2005 it was $7.91 trillion, and grew to $8.45 trillion by the end of 2006.[footnote]http://www.whitehouse.gov/sites/default/files/omb/budget/fy2013/assets/hist07z1.xls[/footnote] Calculate the absolute and relative increase for 2001–2002 and 2005–2006. Which year saw a larger increase in federal debt?Example 8
A Seattle Times article on high school graduation rates reported “The number of schools graduating 60 percent or fewer students in four years—sometimes referred to as “dropout factories”—decreased by 17 during that time period. The number of kids attending schools with such low graduation rates was cut in half.”- Is the “decrease by 17” number a useful comparison?
- Considering the last sentence, can we conclude that the number of “dropout factories” was originally 34?
Solution
- This number is hard to evaluate, since we have no basis for judging whether this is a larger or small change. If the number of “dropout factories” dropped from 20 to 3, that’d be a very significant change, but if the number dropped from 217 to 200, that’d be less of an improvement.
- The last sentence provides relative change, which helps put the first sentence in perspective. We can estimate that the number of “dropout factories” was probably previously around 34. However, it’s possible that students simply moved schools rather than the school improving, so that estimate might not be fully accurate.
Example 9
In the 2004 vice-presidential debates, Edwards's claimed that US forces have suffered "90% of the coalition casualties" in Iraq. Cheney disputed this, saying that in fact Iraqi security forces and coalition allies "have taken almost 50 percent" of the casualties. Who is correct?Solution
Without more information, it is hard for us to judge who is correct, but we can easily conclude that these two percents are talking about different things, so one does not necessarily contradict the other. Edward’s claim was a percent with coalition forces as the base of the percent, while Cheney’s claim was a percent with both coalition and Iraqi security forces as the base of the percent. It turns out both statistics are in fact fairly accurate.Try It Now
In the 2012 presidential elections, one candidate argued that “the president’s plan will cut $716 billion from Medicare, leading to fewer services for seniors,” while the other candidate rebuts that “our plan does not cut current spending and actually expands benefits for seniors, while implementing cost saving measures.” Are these claims in conflict, in agreement, or not comparable because they’re talking about different things?Example 10
A politician’s support increases from 40% of voters to 50% of voters. Describe the change.Solution
We could describe this using an absolute change: . Notice that since the original quantities were percents, this change also has the units of percent. In this case, it is best to describe this as an increase of 10 percentage points. In contrast, we could compute the percent change: increase. This is the relative change, and we’d say the politician’s support has increased by 25%.Example 11
A basketball player scores on 40% of 2-point field goal attempts, and on 30% of 3-point of field goal attempts. Find the player’s overall field goal percentage.Solution
It is very tempting to average these values, and claim the overall average is 35%, but this is likely not correct, since most players make many more 2-point attempts than 3-point attempts. We don’t actually have enough information to answer the question. Suppose the player attempted 200 2-point field goals and 100 3-point field goals. Then they made 200(0.40) = 80 2-point shots and 100(0.30) = 30 3-point shots. Overall, they made 110 shots out of 300, for a overall field goal percentage.Proportions and Rates
If you wanted to power the city of Seattle using wind power, how many windmills would you need to install? Questions like these can be answered using rates and proportions.Rates
A rate is the ratio (fraction) of two quantities. A unit rate is a rate with a denominator of one.Example 12
Your car can drive 300 miles on a tank of 15 gallons. Express this as a rate.Solution
Expressed as a rate, . We can divide to find a unit rate:, which we could also write as, or just 20 miles per gallon.Proportion Equation
A proportion equation is an equation showing the equivalence of two rates or ratios.Example 13
Solve the proportion for the unknown value x.Solution
This proportion is asking us to find a fraction with denominator 6 that is equivalent to the fraction. We can solve this by multiplying both sides of the equation by 6, giving .Example 14
A map scale indicates that ½ inch on the map corresponds with 3 real miles. How many miles apart are two cities that are inches apart on the map?Solution
We can set up a proportion by setting equal two rates, and introducing a variable, x, to represent the unknown quantity—the mile distance between the cities.Multiply both sides by x and rewriting the mixed number | |
Multiply both sides by 3 | |
Multiply both sides by 2 (or divide by ½) | |
Example 15
Your car can drive 300 miles on a tank of 15 gallons. How far can it drive on 40 gallons?Solution
We could certainly answer this question using a proportion: . However, we earlier found that 300 miles on 15 gallons gives a rate of 20 miles per gallon. If we multiply the given 40 gallon quantity by this rate, the gallons unit “cancels” and we’re left with a number of miles: Notice if instead we were asked “how many gallons are needed to drive 50 miles?” we could answer this question by inverting the 20 mile per gallon rate so that the miles unit cancels and we’re left with gallons:Unit Conversions
Length
1 foot (ft) = 12 inches (in) | 1 yard (yd) = 3 feet (ft) |
1 mile = 5,280 feet | |
1000 millimeters (mm) = 1 meter (m) | 100 centimeters (cm) = 1 meter |
1000 meters (m) = 1 kilometer (km) | 2.54 centimeters (cm) = 1 inch |
Weight and Mass
1 pound (lb) = 16 ounces (oz) | 1 ton = 2000 pounds |
1000 milligrams (mg) = 1 gram (g) | 1000 grams = 1kilogram (kg) |
1 kilogram = 2.2 pounds (on earth) |
Capacity
1 cup = 8 fluid ounces (fl oz)[footnote]Fluid ounces are a capacity measurement for liquids. 1 fluid ounce ≈ 1 ounce (weight) for water only.[/footnote] | 1 pint = 2 cups |
1 quart = 2 pints = 4 cups | 1 gallon = 4 quarts = 16 cups |
1000 milliliters (ml) = 1 liter (L) |
Example 16
A bicycle is traveling at 15 miles per hour. How many feet will it cover in 20 seconds?Solution
To answer this question, we need to convert 20 seconds into feet. If we know the speed of the bicycle in feet per second, this question would be simpler. Since we don’t, we will need to do additional unit conversions. We will need to know that 5280 ft = 1 mile. We might start by converting the 20 seconds into hours: Now we can multiply by the 15 miles/hr Now we can convert to feet We could have also done this entire calculation in one long set of products:Try It Now
A 1000 foot spool of bare 12-gauge copper wire weighs 19.8 pounds. How much will 18 inches of the wire weigh, in ounces?Example 17
Suppose you’re tiling the floor of a 10 ft by 10 ft room, and find that 100 tiles will be needed. How many tiles will be needed to tile the floor of a 20 ft by 20 ft room?Solution
In this case, while the width the room has doubled, the area has quadrupled. Since the number of tiles needed corresponds with the area of the floor, not the width, 400 tiles will be needed. We could find this using a proportion based on the areas of the rooms:Example 18
Suppose a small company spends $1000 on an advertising campaign, and gains 100 new customers from it. How many new customers should they expect if they spend $10,000?Solution
While it is tempting to say that they will gain 1000 new customers, it is likely that additional advertising will be less effective than the initial advertising. For example, if the company is a hot tub store, there are likely only a fixed number of people interested in buying a hot tub, so there might not even be 1000 people in the town who would be potential customers.Example 19
Compare the 2010 U.S. military budget of $683.7 billion to other quantities.Solution
Here we have a very large number, about $683,700,000,000 written out. Of course, imagining a billion dollars is very difficult, so it can help to compare it to other quantities. If that amount of money was used to pay the salaries of the 1.4 million Walmart employees in the U.S., each would earn over $488,000. There are about 300 million people in the U.S. The military budget is about $2,200 per person. If you were to put $683.7 billion in $100 bills, and count out 1 per second, it would take 216 years to finish counting it.Example 20
Compare the electricity consumption per capita in China to the rate in Japan.Solution
To address this question, we will first need data. From the CIA[footnote]https://www.cia.gov/library/publications/the-world-factbook/rankorder/2042rank.html[/footnote] website we can find the electricity consumption in 2011 for China was 4,693,000,000,000 KWH (kilowatt-hours), or 4.693 trillion KWH, while the consumption for Japan was 859,700,000,000, or 859.7 billion KWH. To find the rate per capita (per person), we will also need the population of the two countries. From the World Bank,[footnote]http://data.worldbank.org/indicator/SP.POP.TOTL[/footnote] we can find the population of China is 1,344,130,000, or 1.344 billion, and the population of Japan is 127,817,277, or 127.8 million. Computing the consumption per capita for each country:China: ≈ 3491.5 KWH per person
Japan: ≈ 6726 KWH per person
While China uses more than 5 times the electricity of Japan overall, because the population of Japan is so much smaller, it turns out Japan uses almost twice the electricity per person compared to China.Geometry
Geometric shapes, as well as area and volumes, can often be important in problem solving.Example 21
You are curious how tall a tree is, but don’t have any way to climb it. Describe a method for determining the height.Solution
There are several approaches we could take. We’ll use one based on triangles, which requires that it’s a sunny day. Suppose the tree is casting a shadow, say 15 ft long. I can then have a friend help me measure my own shadow. Suppose I am 6 ft tall, and cast a 1.5 ft shadow. Since the triangle formed by the tree and its shadow has the same angles as the triangle formed by me and my shadow, these triangles are called similar triangles and their sides will scale proportionally. In other words, the ratio of height to width will be the same in both triangles. Using this, we can find the height of the tree, which we’ll denote by h: Multiplying both sides by 15, we get h = 60. The tree is about 60 ft tall.Areas
Rectangle
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Circle
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Volumes
Rectangular Box
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Cylinder
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Example 22
If a 12 inch diameter pizza requires 10 ounces of dough, how much dough is needed for a 16 inch pizza?Solution
To answer this question, we need to consider how the weight of the dough will scale. The weight will be based on the volume of the dough. However, since both pizzas will be about the same thickness, the weight will scale with the area of the top of the pizza. We can find the area of each pizza using the formula for area of a circle, :A 12" pizza has radius 6 inches, so the area will be = about 113 square inches.
A 16" pizza has radius 8 inches, so the area will be = about 201 square inches.
Notice that if both pizzas were 1 inch thick, the volumes would be 113 in3 and 201 in3 respectively, which are at the same ratio as the areas. As mentioned earlier, since the thickness is the same for both pizzas, we can safely ignore it. We can now set up a proportion to find the weight of the dough for a 16" pizza: Multiply both sides by 201 = about 17.8 ounces of dough for a 16" pizza. It is interesting to note that while the diameter is = 1.33 times larger, the dough required, which scales with area, is 1.332 = 1.78 times larger.Example 23
A company makes regular and jumbo marshmallows. The regular marshmallow has 25 calories. How many calories will the jumbo marshmallow have?Solution
We would expect the calories to scale with volume. Since the marshmallows have cylindrical shapes, we can use that formula to find the volume. From the grid in the image, we can estimate the radius and height of each marshmallow. The regular marshmallow appears to have a diameter of about 3.5 units, giving a radius of 1.75 units, and a height of about 3.5 units. The volume is about π(1.75)2(3.5) = 33.7 units3. The jumbo marshmallow appears to have a diameter of about 5.5 units, giving a radius of 2.75 units, and a height of about 5 units. The volume is about π(2.75)2(5) = 118.8 units3. We could now set up a proportion, or use rates. The regular marshmallow has 25 calories for 33.7 cubic units of volume. The jumbo marshmallow will have: It is interesting to note that while the diameter and height are about 1.5 times larger for the jumbo marshmallow, the volume and calories are about 1.53 = 3.375 times larger.Try It Now
A website says that you’ll need 48 fifty-pound bags of sand to fill a sandbox that measure 8ft by 8ft by 1ft. How many bags would you need for a sandbox 6ft by 4ft by 1ft?Problem Solving and Estimating
Finally, we will bring together the mathematical tools we’ve reviewed, and use them to approach more complex problems. In many problems, it is tempting to take the given information, plug it into whatever formulas you have handy, and hope that the result is what you were supposed to find. Chances are, this approach has served you well in other math classes. This approach does not work well with real life problems. Instead, problem solving is best approached by first starting at the end: identifying exactly what you are looking for. From there, you then work backwards, asking “what information and procedures will I need to find this?” Very few interesting questions can be answered in one mathematical step; often times you will need to chain together a solution pathway, a series of steps that will allow you to answer the question.Problem Solving Process
- Identify the question you’re trying to answer.
- Work backwards, identifying the information you will need and the relationships you will use to answer that question.
- Continue working backwards, creating a solution pathway.
- If you are missing necessary information, look it up or estimate it. If you have unnecessary information, ignore it.
- Solve the problem, following your solution pathway.