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Fractions, Percents, and Ratios Part B: Decimals and Percents (45 minutes)

In This Part: Percent as Proportion
The word “percent” means “out of 100.” See Note 6 below. For example, 49% means 49 out of 100. Percents can also be expressed as fractions or decimals, since they too can be used to imply some part of a whole. So 49% can also be written as 49/100 or 0.49.

A percent implies a ratio: It is some part “per 100.” Ratios enable us to set up a relationship between two numbers. For example, in a water molecule, there is always one oxygen atom for every two hydrogen atoms, which means that the ratio of oxygen to hydrogen is 1:2. In a percent, the second number in the ratio is always 100. Such ratios always express a number of parts per 100 parts.

You can approach any kind of percent problem if you think of it as a proportion that equates two ratios: a data ratio and a percent ratio. In other words:

Since the percent whole is always 100, we can substitute 100 for “percent whole” in this formula:

Notice that there are three different unknowns in this equation. If you know any two of them, you can easily find the third.

For example, if you want to know how much 30% of \$150 is, you’d write the proportion as follows:

From here, you can easily calculate the value you’re looking for, which in this case is \$45.

Problem B1
a. You bought a new television set at a 20% discount and saved \$80. What was the original price of the set?
b. How much did you pay for the set?

Problem B2
Jane bought a dress on a 25%-off sale for a total of \$39. What was the original pre-sale price of the dress?

“Percent Part” for this problem should not be 25%. Why?

Problem B3
The bookstore reduced all items by 20% for the spring sale. After the sale, it increased the prices to 20% above the sale price. Were these prices the same as the original prices? Explain.

Try starting with an original price of \$100. Note that when working with these types of percent problems, using 100 as a starting point can greatly simplify your calculations.

In This Part: Percents as Fractions and Decimals
As we’ve mentioned, percents can also be expressed as fractions and decimals. In this case, all three representations are used to indicate some part of a whole.

• What percent and decimal are represented by the fraction 1/8?

Using cross-multiplication (that is, multiplying both sides of the equation first by 100 and then by 8), we get 1 • 100 = 8x, so x = (1 • 100)8. One hundred divided by 8 is 12.5, so x = 12.5%, or 0.125 (i.e., 12.5/100).

• What fraction and decimal are represented by 35%?

You can use the same process as above, but in this case it is easier to remember the definition of percent. Thirty-five percent means 35 out of 100, which is the fraction 35/100 (which reduces to 7/20) and the decimal 0.35 (35 hundredths).

• What percent and fraction are represented by the decimal 1.8?

This decimal is 1 8/10, or 18/10:

Since the denominator of this fraction is 10, it’s easiest just to multiply both the top and bottom by 10, which gives us 180/100, or 180%.

Problem B4
What percent and decimal are represented by the fraction 1/200?

Problem B5
What fraction and decimal are represented by 0.2%?

Problem B6
What fraction and decimal are represented by 170%?

Problem B7
What fraction and percent are represented by the decimal 0.004?
Knowing some fraction, decimal, and percent equivalents allows you to estimate the answers for percent problems or conversions. Some critical values are shown in the following table:

Percent

Decimal

Reduced Fraction

 0.5% 1% 10% 12.5% 20% 25% 33.33% 50% 66.67% 75% 80% 100% 150%
 0.005 0.01 0.1 0.125 0.2 0.25 0.5 0.75 0.8 1.0 1.5
 1/200 1/100 1/10 1/8 1/5 1/4 1/3 1/2 2/3 3/4 4/5 1 3/2

Problem B8
If you have \$12,000, how would you use this table to compute 25%?

Problem B9
Twenty percent of an 80-meter bridge has been built. Using fractions, calculate how many more meters remain to be completed.

In This Part: Percent Models
Many tools can be used to visually represent percentages; for example, a 100-grid (a grid containing 100 squares) that is shaded to represent a percent. The grid below represents 25%, or 25 out of 100:

This grid also represents the fraction 1/4 and the decimal 0.25.

Problem B10
a. What percent, fraction, and decimal are represented by the shaded part below?

b. How would you represent 39% on a 100-grid?

Here is another model you can make for working with percents. Get a board that has a meter stick or number line on it, and attach a wide elastic band. Then, on the elastic band, mark all the key percents up to 100. Release the elastic band. Now, if you stretch the elastic band to line up the 100% mark you made with any number — for example, 40, as shown below — the other percents will automatically line up with the correct numbers (50% will line up with 20, etc.). This is an easy way to tell how much a given percent is of a given number. (You can use this model for numbers greater than 100 as well.)

Problem B11
Complete the following: Forty percent of 80 is ______ % of 96. Try using the elastic model above to solve this problem.

Sometimes we can use an area model to represent percentages. For example, in Problem B2, Jane bought a dress marked down 25%, for a total of \$39. We can represent that as follows:

 Calculating the original price would require increasing the sale price by approximately 33.3%, rather than 25%.

In this case, a visual model can help us better understand and solve this percentage problem.

Notes

Note 6
The word percent comes from the Latin “per centum,” meaning “per 100.”

Solutions

Problem B1
a.
Set up the equation, knowing that the Data Part is \$80 and the Percent Part is 20:

Here, Data Whole is the original price of the set, not the discounted price. The fractions can be made equal by multiplying the top and bottom of the right side of the equation by 4, which makes the original price \$400. (You could also multiply 80 by 100 and then divide by 20.)
b. Since you saved \$80 off the original price, the sale price was \$320.

Problem B2
Again, we know the Data Part, but this time it represents the percentage after the discount, not the value of the discount (as it was in Problem B1). This means that the price we are given is 75% of the original price, not 25%.

You have several options at this point. You can multiply 39 by 100 and divide by 75. The original pre-sale price was \$52.

Problem B3
No, the prices are not the same, because 20% of a sale price is less than 20% of the original price. For example, suppose that a set of books costs \$100 before the sale. Reducing the items by 20% is a savings of \$20, so the new price is \$80. After the sale, the price is raised by 20%; 20% of \$80 is \$16, so the new price is \$96.

Another way to think about this is that a 20% savings is equal to multiplying by 0.8, and a 20% price increase is equal to multiplying by 1.2. Doing both is equal to multiplying by (0.8 • 1.2) = 0.96, a 4% savings, or \$96 for every \$100 of the original.

Problem B4

This gives us 1 • 100 = 200 • x, so x = 1 • 100200, which is 0.5%, or 0.005.

Problem B5
This means 0.2 out of 100, or 2 out of 1,000, which is the fraction 2/1,000 (which reduces to 1/500) and the decimal 0.002.

Problem B6
This means 170 out of 100, which is the fraction 170/100 (which reduces to 17/10, or 1 7/10) and the decimal 1.7.

Problem B7
The fraction is 4/1,000, or 1/250; 1/250 is 0.4/100, so the percent is 0.4%.

Problem B8
Using the benchmark table, 25% of 12,000 is equivalent to 1/4 • 12,000 or 0.25 • 12,000, which equals 3,000.

Problem B9
Since 20% of the bridge has been built, 80% more remains to be completed. Using the benchmark fractions, this is equivalent to 4/5 • 80 = 320/5 = 64. Sixty-four meters must still be completed.

Problem B10
a. The shaded area is 68 out of 100; this represents 68%, 68/100 (which reduces to 17/25), and 0.68.
b. Thirty-nine percent is represented below:

Problem B11
To use the elastic model, use a meter stick. Expand your marked elastic so that 100% lines up with 80 centimeters. You should find that 40% of 80 is 32.

Then expand the elastic so that 100% lines up with 96 centimeters, and look for the percentage that lines up with 32 centimeters. You should find that 32 centimeters is exactly one-third along the elastic, or 33.33…%.