Classifying and Solving Percent Problems

Math 131

Spring 04

Percentages: Setting Up and Solving Problems

Many simple mathematical statements involving percentages can be reworded so that they follow the format given below:

__________ is ____________ % of ________________

Here are some examples:

20 is 50% of 40

60 is 150% of 40

40 is 100% of 40

In class we have labeled the parts of the statements as follows:

__________ is ____________ % of ________________

the “part” the percentage the whole

Note that the word part is in quotations because sometimes the “part” is bigger than the “whole” (although sometimes it makes no sense for the “part” to be bigger than the “whole”).

In class we’ve discussed various examples. In each, we first need to decide what we are missing -- the “part,” the whole, or the percentage. Then we can use the rest of the data to estimate or solve for what we are missing. (note: this is an old handout; some of the numbers are off a bit).

Here are some examples:

A) The U.S. population is about 270 million and the world population is about 6 billion. What percentage of the people in the world live in the U.S.?

B) In 1996, 23.3% of all people in Massachusetts were children, and there were 1,421,929 children in Massachusetts in 1996. How many people were there in Massachusetts in 1996?

C) In 1996 in Massachusetts 9.3% of all children didn’t have health insurance, and there were 1,421,929 children in Massachusetts then. How many children didn’t have health insurance?

D) A C.D. player was on sale for 20% off. The sale price was $200. What was the regular price?

E) The bottom 40% of U.S. households in terms of wealth, saw their average net worth shrink from $4,400 to $900 between 1983 and 1995. What percentage decrease was this?

F) Michael Eisner was the highest paid CEOs in 1998, with a total compensation package of $576 million. Eisner was also the highest paid executive 10 years earlier, when his pay package topped $40 million. By what percentage did his compensation increase in 10 years?

G) The number of foster children in Illinois increased by 130.6% from 1990 to 1995. In 1990 there were 20,753 foster children in Illinois. How many were there in 1995?

Take a few minutes and try to classify these problems. For each one, decide what is missing: the whole, the “part,” or the percent.

Missing Percentages

Questions A, E, and F are missing percentages. In question A, the whole is the world population of 6 billion. The part is the U.S. population of 270 million. We can start by estimating our answer. This is always a good idea. Sometimes an estimate is all we need, and even when we need a more precise answer, an estimate can help us decide whether our answer is reasonable.

We know that

100% of the world’s population = 6 billion, so

50% = 3 billion

10% = 600 million

5% = 300 million

So the U.S. population of 270 million is a little less than 5%. Of course, there are also other strategies for making this estimate.

Now to compute exactly, we can express the fraction of the world’s population that lives in the U.S. as:

270,000,000 = 270 = .045 = .045 = 4.5 = 4.5%

6,000,000,000 6,000 1 100

Note that the last few steps explain why we “move the decimal two places to the right” to convert a decimal to a percentage.

We can also solve this using an equation: P% of 6,000,000 = 270,000,000

so P%=270,000,000/6,000,000,000 = .045 = 4.5%.

We can also use ratios: N = 270,000,000

100 6,000,000,000

so 6,000,000,000N = (270,000,000)*100, or N=270,000,000,000/6,000,000,000 = 4.5

Problem E is also missing a percentage. This problem might be confusing because there is a percentage in the problem, “the bottom 40%,” but notice that we could replace this phrase with, “the poorest people” and the rest of the problem would stay the same. Thus, this 40% doesn’t really have to do with the question.

In both problems E and F, the whole refers the quantity that is earliest in time. When there is a comparison over time, the reference is almost always the earlier number; the increase or decrease is compared to it. In problem E we have the following data:

1983 Change 1995

Median

Wealth $4,400 -$3500 $900

Percentage

of 1983

Median 100% - P% (100-P)%

Wealth

We see that P% = $3500, and that 100% = $4,400. Now this problem is like problem A.

We can estimate:

100% of 1983 Median Wealth is $4400

50% 2200

25% 1100

75% 3300

So our answer should be a little more than 75%.

As a fraction, we have, 3500/4400 = 35/44 = .795 = 79.5%.

As an equation: P * 4400 = 3500, and then it is the same as above.

As a proportion: N/100 = 3500/4400; 4400N = 3500 * 100; N=79.5%.

Another way to do this problem is to calculate the fraction, 900/4400 = 20.5%, which represents the percentage of 1983 wealth that people had in 1995. This is a change from 100%, and the decrease is 100% - 20.5% = 79.5%.

Problem F is similar, but it represents an increase, not a decrease. Also, this increase is more than 100%. We cannot have a decrease of more than 100%, but we can have an increase. We have the following table:

1988 Change 1998

Eisner’s

Compensation $40 million +$536 million $576 million

Percentage

of 1988

Compensation 100% + P% (100+P)%

We start with an estimate

100% of 1988 compensation = $40 million

1000% $400 million

300% $120 million

400% $160 million

We see that the increase is between 1300% and 1400%. Note that these are very big confusing numbers. Be careful with the decimal points!

As a fraction, we have: 536/40 = 13.4 = 1340%.

As a proportion, we have 536/40 = N/100.

Missing “Parts:”

Problems C and G have missing “parts.” In problem C, we need to find 9.3% of 1,421,929. We start by estimating. This problem is “close” to the problem of finding 10% of 1,400,000, which would be 140,000, so we expect the answer to be on the order of a hundred thousand or two.

To find 9.3% of 1,421,929, we are finding the fraction 9.3/100 of 1,421,929. Remember that in the last unit we talked about why taking a fraction of a number is the same as multiplying by that number. So we want to find

(9.3/100) * 1,421,929.

This is easiest to do if we first express 9.3/100 as the decimal .093 and find

.093 * 1,421,929 = 132,239, which is in line with our estimate.

We can also use a proportion: 9.3/100 = N/1421929, (9.3)*( 1,421,929)=100N;

N = 132,239.

In problem G, the “part” is bigger than the whole. Note that in a problem like problem C, it would make no sense to have a part bigger than 100%; for example, to say 120% of the population has no health insurance makes no sense. However, in problem G, a “part” that is bigger than the whole makes sense.

We can set up a table like we did in problems E and F:

1990 Change 1995

Children in

Foster Care 20,753 + N 20,753+N

Percentage

of 1990

Number of 100% + 130.6% 230.6%

Children

We start with an estimate:

100% of children in foster care 20,000

200% 40,000

10% 2,000

30% 6,000

230% 46,000

To do this problem in one step, we know that the number of foster children in 1995 is 230.6% of the 1990 number, and we need to find 230.6% of 20,753. This is a lot like problem C now. We need to be careful converting 230.6% to a decimal, however: 230.6% = 230.6/100 = 2.306. Now we find:

2.306 * 20,753 = 47,856 children in foster care in Illinois in 1995, which is in line with our estimate.

With a proportion: 230.6/100 = N/20,753.

Note that we can also do this problem by finding 130.6% of 20,753 and adding it to 20,753.

Missing Whole

Many students find these problems to be most difficult. Problems B and D are examples with missing wholes. It is especially good to start with an estimate here. For problem B we have:

23.3% 25% of Mass population 1.5 million

50% 3 million

100% 6 million.

To compute the number directly, DO NOT take 23.3% of 1,421,929. Many students make this mistake, but we are looking for the whole, not the part. We can use an equation:

23.3% of Mass Population = 1,421,929

.233 P =1,421,929

P = 1,421,929/ .233 = 6,102,700, which is close to our estimate. Note how dividing by .233, yields a number bigger than 1,421,929; this is as we discussed in the previous unit.

We can also use a proportion: 23.3/100 = 1,421,929/N.

Problem D is the problem we discussed in-depth in class; it is also a “missing whole” problem. We discussed many other ways to solve this problem , but we can also set up a table as follows:

Regular Price Change Sale Price

Cost of CD

Player N $200

Percentage

of Regular

Price 100% -20% 80%

We see that 80% of our regular price is $200. We can divide both sides by 4, to find that 20% of the regular price is $50, and then multiply both sides by 5 to find that 100% of the regular price is $250.

We can also solve .8N = 200; N = 200/.8 =$250. As a proportion, we have 80/100 = 200/N.

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