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.
In the 2004 vice-presidential debates, Edwards 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? How can we make sense of these numbers?
The word Percent literally means "per 100" or "parts per hundred." If we have a part that is some percent of a whole, then:
Equivalently:
In a study, 243 people out of 400 people state that they like dogs. What percent of people like dogs?
Write each of the following as a percentage:
Part 1: To convert 1/4 to a percentage, we first have to turn it into a decimal, which is 0.25. Then, we multiply the decimal by 100% to get:
$$0.25 \times 100\% = 25\%$$Part 2: To convert a decimal to a percentage we multiply by 100%:
$$0.02 \times 100\% = 2\%$$Part 3:
$$2.35 \times 100\% = 235\%$$Write each percentage as a decimal:
To convert a percentage to a decimal, we divide each percentage by 100%:
$$\frac{1.25\%}{100\%} = 0.0125$$ $$\frac{0.02\%}{100\%} = 0.0002$$ $$\frac{52.3\%}{100\%} = 0.523$$The sales tax in a town is 9.4%. How much tax will you pay on a $140 purchase?
We are looking to find the part that makes up 9.4% of the whole of $140. First, we will need to convert our percent to a decimal: 9.4% ÷ 100% = 0.094. Then, we multiply the whole by the decimal to get the part.
$$\$140 \times 0.094 = \$13.16$$In the News you hear, "tuition is expected to increase by 7% next year." If tuition this year was $1,200 per quarter, what will the cost of tuition be next year?
First we need to find the annual tuition. If the tuition is $1,200 a quarter, that means it is 4($1,200) = $4,800 a year. Next to find a 7% increase, it is really 107% because you already pay 100% of the tuition and an increase would be added to that amount. Thus next year's annual tuition would be:
$$\$4,800(1.07) = \$5,136$$Which would be an increase of $5,136 - $4,800 = $336 annually or $336 ÷ 4 = $84 quarterly.
Alternative Method: This could also be solved by finding the increase first and then adding it to the initial amount:
$$\text{Increase: } \$4,800 \times \frac{7\%}{100\%} = \$336$$ $$\text{Total: } \$4,800 + \$336 = \$5,136$$A TV originally priced at $799 is on sale for 30% off. What is the price after the discount?
Given two quantities:
Absolute change has the same units as the original quantity, and should always be positive. Relative change is the percent change. It is a percent increase if the ending quantity is larger than the starting quantity, and a percent decrease if the starting number is lower than the ending quantity. The starting quantity is called the base of the percent change.
The value of a car dropped from $7,400 to $6,800 over the last year. What is the percent decrease?
Using the relative change formula:
$$\frac{|\$7,400 - \$6,800|}{\$7,400} \times 100\%$$ $$\frac{|\$600|}{\$7,400} \times 100\%$$ $$0.081 \times 100\%$$ $$8.1\%$$The value of the vehicle has decreased by 8.1% over the last year.
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?
To answer this question, suppose the value started at $100. After one week the value dropped 60%:
$$\$100 - \$100(0.60) = \$100 - \$60 = \$40$$In the next week, notice that the base of the percentage has changed from $100 to $40. Computing the 75% increase:
$$\$40 + \$40(0.75) = \$40 + \$30 = \$70$$In the end, the stock is lower than where it started. To be exact, it is $\frac{|100 - 70|}{100} \times 100\% = \frac{30}{100} \times 100\% = 30\%$. Thus the total change over time was a 30% decrease.
The U.S. federal debt at the end of 2001 was $5.77 trillion, and grew to be $6.20 trillion by the end of 2002. At the end of 2005 it was $7.91 trillion, and grew to be $8.45 trillion by the end of 2006. Calculate the relative change for 2001-2002 and 2005-2006. Which year(s) saw the larger increase in federal debt?