1. 7-4 Percents and Interest
• Fractions, decimals, and percents as representations of rational numbers with conversions from one form to another.
• Proportional relationships to solve percent problems.
• Techniques to solve problems including discounts, interest, compound interest, and percent increase and percent decrease.
• Strategies for percent mental computation and estimation.

2. Percents
Percents are special kinds of fractions, namely, fractions with a denominator of 100.
The word percent comes from the Latin phrase per centum, which means per hundred.
Definition of Percent

3. Example Write each of the following as a percent. a. 0.03 3% b. 0.3
33.3%
c. 1.2
120% d
0.042%
e. 1
100% f.
60% g.
66.6% h.

4. Example Write each of the following percents as a decimal. a. 5% 0.05
b. 6.3%
0.063
c. 100% 1
d. 250%
2.5 e.
0.006% f.
0.3%

5. Applications Involving Percent
Applications involving percent usually fall into one of the following forms:
Finding a percent of a number.
Finding what percent one number is of another.
Finding a number when a percent of that number is known.

6. Example
A house that sells for $92,000 requires a 20% down payment. What is the amount of the down payment?
We need to find 20% of $92,000.
20% of $92,000 = 0.20 · $92,000 = $18,400
The down payment is $18,400.

7. Example
If Alberto has 45 correct answers on an 80-question test, what percent of his answers are correct?
We need to find what percent of 80 is 45.
56.25% of Alberto’s answers are correct.

8. Example
Forty-two percent of the parents of the school-children in the Paxson School District are employed at Di Paloma University. If the number of parents employed by the university is 168, how many parents are in the school district?
We need to find a number such that 42% of that number is 168.

9. Example (continued)
Let n be the number of parents in the school district.
Then, 42% of n is 168.
There are 400 parents in the school district.

10. Example
Kelly bought a bicycle and a year later sold it for 20% less than what she paid for it. If she sold the bike for $144, what did she pay for it?
We are looking for the original price, P, that Kelly paid for the bike. We know that she sold the bike for $144 and that this included a 20% loss.
$144 = P − Kelly’s loss

11. Example (continued) Kelly’s loss is 20% of P, so
Kelly paid $180 for the bike.

12. Mental Math with Percents
Use fraction equivalents

13. Mental Math with Percents
Use a known percent
Frequently, we may not know a percent of something, but we know a close percent of it. For example, to find 55% of 62, do the following:

14. Estimations with Percent
Estimations with percents can be used to determine whether answers are reasonable.

15. Example
Laura wants to buy a blouse originally priced at $26.50 but now on sale at 40% off. She has $17 in her wallet and wonders if she has enough cash. How can she mentally find out? (Ignore the sales tax.)
It is easier to find 40% of $25 (versus $26.50) mentally.

16. Example (continued)
Laura estimates that the blouse will cost $26.50 − $10 = $16.50
Since the actual discount is greater than $10, Laura will have to pay less than $16.50 for the blouse, so she has enough cash.

17. Computing Interest
Suppose you borrow $6000 from a bank at simple interest at a rate of 5% to be paid off 2 years from now.
The $6000 represents the principal, the amount of the loan.
The 5% represents the interest rate.
The period of the loan is the amount of time you owe the money – two years in this case.

18. Computing Interest
Simple interest means that the interest will only be computed on the principal for the stated time of the loan.
The formula for simple interest is
I = Prt,
where I represents simple interest, P represents principal, r represents the interest rate, and t represents the number of years.

19. Computing Interest
The formula for the total amount, A, to pay off a loan is
A = P + I = P + Prt = P(1 + rt).

20. Example Vera opened a saving account that pays simple
interest at the rate of per year. If she
deposits $2000 and makes no other deposits, find the interest and the final amount for 90 days
The interest is approximately $25.89 and the final amount is approximately $

21. Example
Find the annual interest rate if a principal of $10,000 increased to $10,900 at the end of 1 year.
Let r = annual interest rate. We know that r% of $10,000 is the increase.
The increase is $10,900 − $10,000 = $900, so
The annual interest rate is 9%.

22. Compound Interest
When earned interest is added to the principal, and then interest is earned on the new amount, that is, it is earned on the previous interest as well as the principal, this is called compound interest.

23. Compound Interest
Compounding usually is done annually (once a year), semiannually (twice a year), quarterly (4 times a year), monthly (12 times a year), or daily (365 times a year). However, even when the interest is compounded, it is given as an annual rate.
The interest rate per period is the annual interest rate divided by the number of periods in a year.

24. Compound Interest
If you invest $100 at 8% annual interest compounded quarterly, how much will you have in the account after 1 year?
The quarterly interest rate is
If at the beginning of any of the four periods there are x dollars in the account, at the end of that period there will be

25. Compound Interest
Thus, to find the amount at the end of any period, we need only multiply the amount at the beginning of the period by 1.02.
The amount at the end of 1 year is 100 · 1.024, or approximately $

26. Compound Interest
The formula for computing the amount at the end of the nth compound interest period is
where A is the amount, P is the principal, r is the interest rate per period, and n is the number of periods.

27. Example
Suppose you deposit $1000 in a savings account that pays 6% annual interest compounded quarterly.
a. What is the balance at the end of 1 year?
At the end of 1 year, the balance is about $

28. Example
To save for their child’s college education, a couple deposits $3000 into an account that pays 7% annual interest compounded daily. Find the amount in this account after 8 years.
The balance is about $