Chapter 6 Review

Remember the principle of 100 when dealing with percentages. If you see the percent sign %, then divide by 100 and remove the sign. If you don’t see a percent sign and want to put one on, then multiply by 100 and put on the % symbol.

Algebra can be used to solve many different types of problems. If we have a formula that applies to a problem, we plug in what we know and solve for what we do not know. Here are some of the formulas we talked about in chapter 6.

$\frac{\text{Percent}}{100}=\frac{\text{Amount}}{\text{Total}}$ This formula is used to solve general percent problems. We can find the percent, amount, or total by setting the cross products equal and solving. Remember the percent is already converted because it is over 100.

$C=T\times r$ This formula is used to calculate the amount of commission made when a person sells T amount of money in merchandise and r is the commission rate percent. Remember to convert r into a decimal or fraction before plugging it in. If we solve for r , we will need to convert our answer back to a percent.

$I=P\times r\times t$ This formula is used to calculate the amount of simple interest made when a person invests P amount of money in an account at an interest rate r percent for t number of years. Remember to convert r into a decimal or fraction before plugging it in. If we solve for r , we will need to convert our answer back to a percent.

$T=A+rA$ This is the classic percent of increase formula that can be used both for taxes and for markup problems. The A is the amount before tax or markup. The T is the total after tax or markup and r is the tax rate or markup rate percent. Remember to convert r into a decimal or fraction before plugging it in. If we solve for r, we will need to convert our answer back to a percent.

$T=A-rA$ This is the classic percent of decrease formula that can be used for discount sales price problems. The A is the amount before the discount. The T is the total after the discount and r is the discount rate percent. Remember to convert r into a decimal or fraction before plugging it in. If we solve for r, we will need to convert our answer back to a percent.

To solve classic algebra problems, remember the steps given below. Also remember if you are dealing with money, the number of bills or coins has to be multiplied by the value of the bill or coin to find the amount of money. 5 dimes does not mean 5 dollars. 5 dimes means 5 x 0.10 = $0.50

Steps to solving problems with algebra

1. Read and reread the problem and write down all of the unknowns in the problem on a piece of paper.
2. Let your variable (x) stand for the unknown that you have the least amount of information about.
3. Use the information given in the problem to write algebraic expressions for all of your other unknowns.
4. Once you have algebraic expressions for all your unknowns, then you can use them to write an equation.
5. Solve the equation for the variable (x).
6. Go back and plug in the value of the variable (x) into all the algebraic expressions and find all the unknowns.
7. Check and see if your answers make sense in the context of the problem.

Review Problems Chapter 6

1. Look at the formula $V=\pi r^{2}h$ which gives the volume of a circular cylinder propane tank. If we use 3.14 as an approximation of $\pi$ and we know that the radius needs to be 3 feet, then how tall should the propane tank be if it needs to hold 310.86 cubic feet of propane?

   Check answer

   11 ft

2. We used regression theory in statistics to find a formula that will predict the weight of a black bear based on its age. The formula was $W=65.2+2.7A$ where A is the age of the bear in months and W is the weight. Forest rangers caught a black bear in the wild and found the weight to be 227.2 pounds. How old do we predict the bear to be?

3. The volume of a box is given by the formula $V=L\times W\times H$ where V is the volume, L is the length, W is the width, and H is the height. A toy container needs to have a length of 15 cm and a width of 9 cm. How tall should the container be if it needs to hold 3105 cubic cm of toys?

   Check answer

   23 cm

4. We used regression theory in statistics to find the following formula F = 0.028C - 1.98 where C is the number of calories a breakfast cereal has and F is the grams of fat the breakfast cereal has. Suppose a cereal has 3.06 grams of fat. How many calories do we expect the cereal to have?

5. The formula for compound interest that banks use is $A=P{(1+\frac{r}{n})}^{nT}$ where P is the principal (amount initially invested), A is the future amount in the account, T is the number of years the money is in the account, n is how many times it compounds per year (how many times they put the interest back in the account) and r is the interest rate. John and Lacy want to buy a car in two years and will need $22,898. How much money (P) should they invest in their savings account if the interest rate is 7% (0.07), compounded annually (n=1)? Round your answer to the nearest dollar.

   Check answer

   $20,000

Convert the following fractions into percentages. Write your answers as mixed numbers when appropriate.

6. $$\frac{1}{4}$$

7. $$\frac{4}{5}$$

   Check answer

   80 %

8. $$\frac{5}{8}$$

9. $$\frac{4}{7}$$

   Check answer

   $57⁤\frac{1}{7}$ %

Convert the following decimals into percentages.

10. 0.69

11. 0.127

    Check answer

    12.7 %

12. 0.014

13. 0.0063

    Check answer

    0.63 %

Convert the following percentages into fractions. Write your answer in lowest terms.

14. 36%

15. 45%

    Check answer

    $$\frac{9}{20}$$

16. 98%

17. $$5⁤\frac{1}{3}\text{\%}$$

    Check answer

    $$\frac{4}{75}$$

Convert the following percentages into decimals.

18. 13.8%

19. 2.5%

    Check answer

    0.025

20. 88.4%

21. 0.35%

    Check answer

    0.0035

Solve the following applied percent problems by using a proportion.

22. To estimate the total number of raccoons in a forest, forest rangers tagged 26 raccoons and released them into the forest. Later, they caught a few raccoons and saw that 20% of them had tags. How many total raccoons do they estimate are in this part of the forest?

23. A women’s shoe store wants to find out how many of their customers would like to buy a new style of heels. They took a random sample of 85 customers and found that 52 of them like the new style. What percent of them like the new style? Round your answer to the tenth of a percent.

    Check answer

    61.2 %

24. For this exercise, assume that 5.3% of Americans over the age of 65 live in nursing homes. If a town has a total of 8200 adults over the age of 65, how many do we expect to live in nursing homes?

Solve the following problems with the appropriate percent formula.

25. Lynn sells jewelry at the mall and is paid a commission of what she sells. If she sold a total of $4800 in jewelry and was paid a commission of $264, what is her commission rate? Write your answer as a percent.

    Check answer

    5.5 %

26. Mario sells cars and earns a 6% commission on all he sells. If he made a commission of $1,800 on one car he sold, what was the total price of the car?

27. Simon invested $4100 into a simple interest account for 3 years. If the account yielded $430.50 in simple interest at the end of three years, what was the interest rate? Write your answer as a percent.

    Check answer

    3.5 %

28. Tianna invested some money into some stocks that yielded $345 in simple interest at the end of 2 years. If the interest rate was 7.5%, how much did she originally invest?

29. Mark bought a flat screen TV for a total of $437 with tax included. What was the price of the TV before tax if Mark lives in an area with a 9.25% sales tax rate.

    Check answer

    $400

30. David bought some basketball sneakers for $49.05 with tax included. If the price of the sneakers before tax was $45, what is the sales tax rate in David’s area? Write your answer as a percent.

31. Riley works for a store that sells toys and has a 15% markup policy. If they bought a toy from the manufacturer for $14, how much will they sell it for after the markup?

    Check answer

    $16.1

32. Maria works for a clothing store. If the store buys a skirt from the manufacturer for $23 and then sells them for $28.75, what is the store’s markup rate? Write your answer as a percent.

33. If Eric bought some paint that regularly sells for $24 a gallon on sale for $19.44 a gallon, what was the discount rate? Write your answer as a percent.

    Check answer

    19 %

34. Tyra bought a beautiful red dress to wear to a New Year’s Eve party. If the dress was on sale for 35% off and the sale price was $234, what was the regular price of the dress before the sale?

Solve the following problems using classic algebraic problem solving techniques.

35. The number of part time students at a college is 52 less than twice the number of full time students. If there are a total of 10,808 students at the college, how many are part time and how many are full time?

    Check answer

    7,188 PT; 3,620 FT

36. A tutoring business offers tutoring for elementary, junior high, and high school students. There is a total of 31 students at the tutoring center right now. There are 3 times as many elementary school students as junior high students. There are 6 more high school students than junior high. How many students of each level are there?

37. A coffee house sells three types of coffee: house blend, French roast, and decaffeinated. They sold 4 times as many bags of house blend as French roast. They sold one bag less of decaffeinated than they did French roast. If they sold a total of 53 bags, how many of each type did they sell?

    Check answer

    36 bags house; 9 bags French; 8 bags decaf

In Spring of 2016, students in Math 075 classes were asked which political party they affiliate with. Here is the bar graph of the 508 students that responded.

38. Which political party had the fewest students?

39. Which political party is the most popular?

    Check answer

    Democrat

40. Approximately how many more Democrats than Republicans were there?

41. Approximately what percentage of students in the data are Republican?

    Check answer

    Approximately 19.29% (100 × 98/508), using the estimated bar height.

In Spring of 2016, students in Math 075 classes were asked which social media was their favorite to use. Here is the pie chart of the 512 students that responded.

42. What percent of students use Facebook as their favorite social media?

43. How many students chose Snapchat as their favorite social media?

    Check answer

    142

44. What is the difference between the percentage of students that chose Instagram and Twitter as their favorite social media?

45. Do more students use Snapchat or Facebook as their favorite social media?

    Check answer

    Snapchat

46. How many more students use Twitter over Other social media not listed as their favorite social media?

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.