Grade 8 Algebra1 Applications of Percents CONFIDENTIAL

Warm Up 1) Find 75% of 60. 2) Find 115% of 80. 3) What percent of 40 is 25? 4) What percent of 225 is 135? 5) 56 is what percent of 20? CONFIDENTIAL

Applications of Percent Sales representatives use percents to calculate their total pay. We can apply percent concepts to find the discount and sale price of an item. The discount rate is given as a percent. Percent concepts can be applied to compute percent increase and percent decrease of various items. We can apply percent concepts to solve problems of the form "One number is some percent of another number." by identifying the given information, setting up a proportion, and solving for the unknown value. CONFIDENTIAL

Simple Interest, I = P × r × t A commission is money paid to a person or a company for making a sale. Usually the commission is a percent of the sale amount. Interest is the amount of money charged for borrowing money, or the amount of money earned when saving or investing money. Principal is the amount borrowed or invested. Simple interest is interest paid only on the principal. Formula for Simple Interest Paid Annually Simple Interest, I = P × r × t Time in years Interest rate per year as a decimal Principal CONFIDENTIAL

total pay = base salary + commission Business Application A) Daniela earns a base salary of $40,000 plus a 2.5% commission on sales. Her total sales one year were $800,000. Find her total pay for the year. SOLUTION: total pay = base salary + commission Write the formula for total pay. = base salary + % of total sales Write the formula for commission. = 40,000 + 2.5% of 800,000 Substitute values given in the problem. Next page  CONFIDENTIAL

= 40,000 + 2.5 of 800,000 100 = 40,000 + (0.025)(800,000) Write the percent as a decimal. = 40,000 + 20,000 Add. = 60,000 Daniela total pay was $60,000. CONFIDENTIAL

Now you try! Solve: 1) A telemarketer earns $350 per week plus a 12% commission on sales. Find her total pay for a week in which her sales are $950. CONFIDENTIAL

The amount of interest is $540. Finance Application A) Find the simple interest paid annually for 3 years on a $1200 loan at 15% per year. SOLUTION: I = P r t Write the formula for simple interest. I = (1200) (0.15) (2) Substitute known values. Write the interest rate as a decimal. I = 540 The amount of interest is $540. CONFIDENTIAL

One month is 1 year. In the given problem, t =3 . 12 12 B) After 3 months the simple interest earned annually on an investment of $7000 was $63. Find the interest rate. SOLUTION: When you are using the formula I = P r t to find simple interest paid annually, t represents time in years. One month is 1 year. In the given problem, t =3 . 12 12 I = P r t Write the formula for simple interest. 63 = (700) (r) (3) 12 Substitute known values. Next page  CONFIDENTIAL

63 = 1750r 63 = 1750r 1750 1750 0.036 = r The interest rate is 3.6%. Multiply 7000 (3) . 12 63 = 1750r 63 = 1750r 1750 1750 Since r is multiplied by 1750, divide both sides by 1750 to undo the multiplication. 0.036 = r The interest rate is 3.6%. CONFIDENTIAL

Now you try! Solve: 1) Find the simple interest earned after 2 years on an investment of $3000 at 4.5% interest earned annually. 2) The simple interest paid on a loan after 6 months was $306. The annual interest rate was 8%. Find the principal. CONFIDENTIAL

Sales tax is a percent of an item’s cost. A tip is an amount of money added to a bill for service. It is usually a percent of the bill before sales tax is added. Sales tax is a percent of an item’s cost. Sales tax and tips are sometimes estimated instead of calculated exactly. When estimating percents, use percents that you can calculate mentally. Find 10% of a number by moving the decimal point one place to the left. Find 1% of a number by moving the decimal point two places to the left. Find 5% of a number by finding 1 of 10% of the number. 2 CONFIDENTIAL

A) The dinner check for Maria’s family is $67.95. Estimate a 15% tip. Estimating with Percents A) The dinner check for Maria’s family is $67.95. Estimate a 15% tip. SOLUTION: Step 1: First round $67.95 to $70. Step 2: Think: 15% = 10% + 5% 10% of $70 = $7.00 Move the decimal point one place left. Step 3: Think: 5% = 10% ÷ 2 = $7.00 ÷ 2 = $3.50 Step 4: 15% = 10% + 5% = $7.00 + $3.50 = $10.50 The tip should be about $10.50. CONFIDENTIAL

Estimating with Percents B) The sales tax rate is 6.25%. Estimate the sales tax on a shirt that costs $29.50. SOLUTION: Step 1: First round 6.25% to 6% and $29.50 to $30 Step 2: Think: 6% = 6 (1%) 1% of $30 = $0.30 Move the decimal point two places to the left. Step 3: 6% = 6 (1%) = 6 ($0.30) = $1.80 The sales tax is about $1.80. CONFIDENTIAL

1) Estimate a 15% tip on a check for $21.98. Now you try! Solve: 1) Estimate a 15% tip on a check for $21.98. 2) Estimate the tax on shoes that cost $68.50 when the sales tax rate is 8.25%. CONFIDENTIAL

A gets 46% of the valid votes. Some more examples A) Two candidates A and B contest an election. A gets 46% of the valid votes and is defeated by 1600 votes. Find the total number of valid votes cast in the election? SOLUTION: A gets 46% of the valid votes. B gets (100 – 46)% = 54% of the valid votes. Therefore, % difference if the votes by which A is defeated is = 54% - 46% = 8%. 8% of the total valid votes cast = 1600. i.e. 8 of the total votes = 1600. 100 Total number of valid votes cast = 100 ×1600 = 20,000 8 Total number of valid votes cast is 20,000 CONFIDENTIAL

10% increase means that my income becomes $110. B) My income was increased by 10% and later decreased by 10%. What is the total change in the percent in my income? SOLUTION: Let my income be $100. 10% increase means that my income becomes $110. Decreased income = $110 - $ 10 × 110 100 = $110 - $11 = $ 99. % change in income = Change in income ×100% original income = $100 - $99 ×100% = 1% $100 The percent in my income is 1%. CONFIDENTIAL

Now you try! Solve: 1) What percent of votes are in favor of a candidate if 25 persons vote for him and 45 persons vote against him? CONFIDENTIAL

Assessment 1) A sales representative earns a 2.5% commission on sales. Find the commission earned when the total sales are $80,700. 2) A man spends 45% of his income and saves $2640 in a month. What is his monthly income? 3) Karen earns a salary of $28,600 per year plus a 4.25% commission on sales. Find her total salary for a year when the sales are $310,000. CONFIDENTIAL

7) Estimate the tip on a $42.65 check using a tip rate of 15%. 4) Find the amount of simple interest earned after 2 years on $480 invested at a 7% annual interest rate. 5) Find the number of years it would take for $1200 to earn simple interest of $324 at an annual interest rate of 6% per year. 6) Find the total amount owed after 6 months on a loan of $900 at an annual interest rate of 8.5%. 7) Estimate the tip on a $42.65 check using a tip rate of 15%. CONFIDENTIAL

8) Estimate the tax on a $198 stereo when the sales tax is 5.25%. 9) The weight of the box by mistake was written as 21 kg, while it was 25 kg. What is the error %. 10) A boy remains awake for 16 hrs a day. For what percentage of the time does he sleep? CONFIDENTIAL

Simple Interest, I = P × r × t Let’s review A commission is money paid to a person or a company for making a sale. Usually the commission is a percent of the sale amount. Interest is the amount of money charged for borrowing money, or the amount of money earned when saving or investing money. Principal is the amount borrowed or invested. Simple interest is interest paid only on the principal. Formula for Simple Interest Paid Annually Simple Interest, I = P × r × t Time in years Interest rate per year as a decimal Principal CONFIDENTIAL

total pay = base salary + commission review Business Application A) Daniela earns a base salary of $40,000 plus a 2.5% commission on sales. Her total sales one year were $800,000. Find her total pay for the year. SOLUTION: total pay = base salary + commission Write the formula for total pay. = base salary + % of total sales Write the formula for commission. = 40,000 + 2.5% of 800,000 Substitute values given in the problem. Next page  CONFIDENTIAL

review = 40,000 + 2.5 of 800,000 100 = 40,000 + (0.025)(800,000) Write the percent as a decimal. = 40,000 + 20,000 Add. = 60,000 Daniela total pay was $60,000. CONFIDENTIAL

The amount of interest is $540. review Finance Application A) Find the simple interest paid annually for 3 years on a $1200 loan at 15% per year. SOLUTION: I = P r t Write the formula for simple interest. I = (1200) (0.15) (2) Substitute known values. Write the interest rate as a decimal. I = 540 The amount of interest is $540. CONFIDENTIAL

A) The dinner check for Maria’s family is $67.95. Estimate a 15% tip. Estimating with Percents A) The dinner check for Maria’s family is $67.95. Estimate a 15% tip. SOLUTION: Step 1: First round $67.95 to $70. Step 2: Think: 15% = 10% + 5% 10% of $70 = $7.00 Move the decimal point one place left. Step 3: Think: 5% = 10% ÷ 2 = $7.00 ÷ 2 = $3.50 Step 4: 15% = 10% + 5% = $7.00 + $3.50 = $10.50 The tip should be about $10.50. CONFIDENTIAL

Estimating with Percents review Estimating with Percents B) The sales tax rate is 6.25%. Estimate the sales tax on a shirt that costs $29.50. SOLUTION: Step 1: First round 6.25% to 6% and $29.50 to $30 Step 2: Think: 6% = 6 (1%) 1% of $30 = $0.30 Move the decimal point two places to the left. Step 3: 6% = 6 (1%) = 6 ($0.30) = $1.80 The sales tax is about $1.80. CONFIDENTIAL

A gets 46% of the valid votes. Some more examples review A) Two candidates A and B contest an election. A gets 46% of the valid votes and is defeated by 1600 votes. Find the total number of valid votes cast in the election? SOLUTION: A gets 46% of the valid votes. B gets (100 – 46)% = 54% of the valid votes. Therefore, % difference if the votes by which A is defeated is = 54% - 46% = 8%. 8% of the total valid votes cast = 1600. i.e. 8 of the total votes = 1600. 100 Total number of valid votes cast = 100 ×1600 = 20,000 8 Total number of valid votes cast is 20,000 CONFIDENTIAL

10% increase means that my income becomes $110. review B) My income was increased by 10% and later decreased by 10%. What is the total change in the percent in my income? SOLUTION: Let my income be $100. 10% increase means that my income becomes $110. Decreased income = $110 - $ 10 × 110 100 = $110 - $11 = $ 99. % change in income = Change in income ×100% original income = $100 - $99 ×100% = 1% $100 The percent in my income is 1%. CONFIDENTIAL

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