Math in Our World Section 8.1 Percents

Learning Objectives Convert between percent, decimal, and fraction form. Perform calculations involving percents. Solve real-world problems involving percents. Find percent increase or decrease. Evaluate the validity of claims based on percents.

Percent Percent means hundredths, or per hundred. That is, 1% = Converting Percents to Decimals In order to change a percent to a decimal, drop the % sign and move the decimal point two places to the left.

EXAMPLE 1 Changing Percents to Decimals Change each percent to a decimal. (a) 84% (b) 5% (c) 37.5% (d) 172% SOLUTION (Drop the % sign and move the decimal point two places to the left.) (a) 84% = 84.% = 0.84 (b) 5% = 5.% = 0.05 (c) 37.5% = 0.375 (d) 172% = 172.% = 1.72

Percent to Fractions Converting Percents to Fractions A percent can be converted to a fraction by dropping the percent sign and using the percent number as the numerator of a fraction whose denominator is 100.

EXAMPLE 2 Changing Percents to Fractions Change each percent to a fraction. (a) 42% (b) 6% (c) (d) 15.8% SOLUTION

EXAMPLE 2 Changing Percents to Fractions SOLUTION (c) When converting fractional or decimal percents, it might be helpful to multiply by some number over itself to clear fractions or decimals. In this case, multiplying by 2/2 is helpful:

Decimals to Percents Converting a Decimal to a Percent To change a decimal to a percent, move the decimal point two places to the right and add a percent sign.

EXAMPLE 3 Changing Decimals to Percents Change each decimal to a percent. (a) 0.74 (b) 0.05 (c) 1.327 (d) 5.463 SOLUTION 0.74 = 074.% = 74% 0.05 = 005.% = 5% 1.327 = 132.7% = 132.7% 5.463 = 546.3% = 546.3%

Fractions to Percents Converting a Fraction to a Percent To change a fraction to a percent, first change the fraction to a decimal, and then change the decimal to a percent.

EXAMPLE 4 Changing Fractions to Percents Change each fraction to a percent. (a) (b) (c) (d) SOLUTION

Problems Involving Percents The most common calculations involving percents involve finding a percentage of some quantity. 50% of 10 is 5 0.5 x 10 = 5 When writing a percentage statement in symbols, the word “of” becomes multiplication, and the word “is” becomes an equal sign. Also, we must change the percent into decimal or fractional form.

EXAMPLE 5 Finding a Certain Percentage of a Whole In a class of 66 students, 32% got a B on the first exam. How many students got a B? SOLUTION First write 32% in decimal form, as 0.32. The question is “what is 32% of 66?” which we translate into symbols: 32% of 66 is _____ 0.32 x 66 = 21.12 We can’t have 0.12 students, so we interpret the answer as 21 students got a B.

EXAMPLE 6 Finding a Certain Percentage of a Portion Of 60 runners who started a 5k race, 45 finished the race in under 40 minutes. What percent is that? SOLUTION Write the statement in the form we discussed above: 45 is what percent of 60? 45 = x  60

EXAMPLE 6 Finding a Certain Percentage of a Portion SOLUTION This is the equation 60x = 45, which we solve for x. The decimal 0.75 corresponds to 75%, so 75% finished in under 40 minutes.

EXAMPLE 7 Finding a Whole Amount Based on a Percentage A medium-sized company reported that it had to cut its work force back to 70% of what it was last year. If it has 63 workers now, how many did it have a year ago? SOLUTION Convert 70% to a decimal: 70% = 0.70. Now write as a question and translate to symbols: 70% of what number is 63? 0.70  x = 63

EXAMPLE 7 Finding a Whole Amount Based on a Percentage SOLUTION This is the equation 0.70x = 63, which we solve for x. The company had 90 workers a year ago.

EXAMPLE 8 Calculating Sales Tax The sales tax in Allegheny County, Pennsylvania, is 7%. What is the tax on a calculator that costs $89.95? What is the total amount paid? SOLUTION Find 7% of $89.95: Write 7% as 0.07 in decimal form, then multiply. 0.07 x $89.95 = $6.30 (rounded) The sales tax is $6.30. The total amount paid is $89.95 + $6.30 = $96.25

EXAMPLE 9 Calculating Cost of Sale from Commission A real estate agent receives a 7% commission on all home sales. How expensive was the home if she received a commission of $5,775.00? SOLUTION In this case, the problem can be written as $5,775.00 is 7% of what number? The home was purchased for $82,500.00.

Problems Involving Percents Procedure for Finding Percent Increase or Decrease Step 1 Find the amount of the increase or the decrease. Step 2 Make a fraction as shown: Step 3 Change the fraction to a percent.

EXAMPLE 10 Finding a Percent Change A large latte at the Caffeine Connection sells at a regular price of $3.50. Today it is on sale for $3.00. Find the percent decrease in the price. SOLUTION The original price is $3.50. Step 1 Find the amount of decrease. $3.50 - $3.00 = $0.50. Step 2 Make a fraction as shown Step 3 Change the fraction to a percent. The decrease in price is 14.3%.

EXAMPLE 11 Recognizing Misuse of Percents in Advertising A department store advertised that certain merchandise was reduced 25%. Also, an additional 10% discount card would be given to the first 200 people who entered the store on a specific day. The advertisement then stated that this amounted to a 35% reduction in the price of an item. Is the advertiser being honest?

EXAMPLE 11 Recognizing Misuse of Percents in Advertising SOLUTION Let’s say that an item was originally priced at $50.00. (Note: any price can be used.) First find the discount amount. Discount = rate x selling price = 25% x $50.00 = 0.25 x $50.00 = $12.50

EXAMPLE 11 Recognizing Misuse of Percents in Advertising SOLUTION Then find the reduced price. Reduced price = original price – discount = $50.00 – $12.50 = $37.50 Next find 10% of the reduced price. Discount = rate x reduced price = 10% x $37.50 = $3.75

EXAMPLE 11 Recognizing Misuse of Percents in Advertising SOLUTION Find the second reduced price. Reduced price = $37.50 – $3.75 = $33.75 Now find the percent of the total reduction. The total percent of the reduction was 32.5%, and not 35% as advertised.