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Grade 8 Algebra1 Review of Proportion and Percent

Grade 8 Algebra1 Review of Proportion and Percent. Warm Up. 1) Find each percent change from 25 to 45. 1) 80% increase. 2) Find the result when 40 is increased by 40%. 2) 56. 3) Find the result when 70 is increased by 9%. 3) 76.3.

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Grade 8 Algebra1 Review of Proportion and Percent

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  1. Grade 8 Algebra1Review of Proportion and Percent CONFIDENTIAL

  2. Warm Up 1) Find each percent change from 25 to 45. 1) 80% increase. 2) Find the result when 40 is increased by 40%. 2) 56 3) Find the result when 70 is increased by 9%. 3) 76.3 4) What is the final price on a $165 leather jacket that is on sale for 30% off? 3) $115.5 CONFIDENTIAL

  3. Rates, Ratios, and Proportions A ratio is a comparison of two quantities by division. The ratio of a to b can be written a:b or a , b where b ≠ 0. Ratios that name the same comparison are said to be equivalent. A statement that two ratios are equivalent, such as 1 = 2 , is called a proportion. 12 24 Read the proportion 1 = x15675 “1 is to 15 as x is to 675.” CONFIDENTIAL

  4. 1 = x 15 675 675 675 Using Ratios The ratio of faculty members to students at a college is 1:15. There are 675 students. How many faculty members are there? Faculty= 1 Students15 Write a ratio comparing faculty to students. 1= x 15675 Write a proportion. Let x be the number of faculty members. Since x is divided by 675, multiply both sides of the equation by 675. x = 45 There are 45 faculty members. CONFIDENTIAL

  5. A rate is a ratio of two quantities with different units, such as or 34mi , 2gal Rates are usually written as unit rates. A unit rate is a rate with a second quantity of 1 unit, such as 34mi , 2gal or 17 mi/gal. You can convert any rate to a unit rate. Conversion factor A rate such as 12in. , 1 ft in which the two quantities are equal but use different units, is called a conversion factor. To convert a rate from one set of units to another, multiply by a conversion factor. CONFIDENTIAL

  6. Finding Unit Rates Garryate 53.5 hot dogs in 12 minutes to win a contest. Find the unit rate. Round your answer to the nearest hundredth. 53.5= x 12 1 Write a proportion to find an equivalent ratio with a second quantity of 1. 4.46 ≈ x Divide on the left side to find x. The unit rate is approximately 4.46 hot dogs per minute. CONFIDENTIAL

  7. Conversion factor The dwarf sea horse Hippocampus zosterae swims at a rate of 52.68 feet per hour. What is this speed in inches per minute? Step1:Convert the speed to inches per hour. To convert the first quantity in a rate, multiply by a conversion factor with that unit in the second quantity. 52.68ft× 12 in 1h1ft 632.16 in. 1h The speed is 632.16 inches per hour. Next page  CONFIDENTIAL

  8. Step2:Convert this speed to inches per minute. 632.16 in× 1 h 1h60 min To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity. 10.536 in 1 min The speed is 10.536 inches per minute. • Check that the answer is reasonable. • The answer is about 10 in./min. • There are 60 min in 1 h, so 10 in./min is 60 (10) = 600 in./h. • • There are 12 in. in 1 ft, so 600 in./h is • 600= 50 ft/h. This is close to the rate • 12 • given in the problem, 52.68 ft/h. CONFIDENTIAL

  9. Solving Proportions Solve the proportion. A) 5 = 3 9 w 5 = 3 9 w 5 (w)=9 (3) Use cross products. 5w = 27 5w = 27 5 5 Divide both sides by 5. w = 27 5 CONFIDENTIAL

  10. A scale is a ratio between two sets of measurements, such as 1 in : 5 mi. A scale drawing or scale model uses a scale to represent an object as smaller or larger than the actual object. A map is an example of a scale drawing. A scale written without units, such as 32 : 1, means that 32 units of any measure correspond to 1 unit of that same measure. CONFIDENTIAL

  11. Scale Drawings and Scale Models A) On the map, the distance from Houston to Beaumont is 0.8 in. What is the actual distance? 1 in : 100 mi Solution: map = 1 in actual 100 mi Write the scale as a fraction. 0.8 in = 1 in x 100 mi Let x be the actual distance. x · 1 = 100 (0.8) Use cross products to solve. x = 80 The actual distance is 80 mi. CONFIDENTIAL

  12. B) The actual distance between Bryan- College Station and Galveston is 127 mi. What is this distance on the map? 1 in : 100 mi Solution: map = 1 in actual 100 mi Write the scale as a fraction. x = 1 in 127 100 mi Let x be the distance on the map. 127 = 100x Use cross products to solve. 127 = 100x 100 100 Since x is multiplied by 100, divide both sides by 100 to undo the multiplication. 1.27 = x The actual distance is 80 mi. CONFIDENTIAL

  13. Now you try! Solve each proportion. 1) 3 = 1 z 8 2) f + 3 = 7 12 2 1) z = 24 2) f = 39 3) x = 1 3 5 4) -1 = 3 5 2d 3) x = 0.6 4) d = -7.5 5) The ratio of the sale price of a jacket to the original price is 5 : 7. The original price is $105. What is the sale price? 5) $75 6) 200 times/sec 6) Find the unit rate. A computer’s fan rotates 2400 times in 20 seconds. CONFIDENTIAL

  14. Similar Figures • Similar figureshave the same shape (but not necessarily the same size) and the following properties: • Corresponding sides are proportional. That is, the ratios of the corresponding sides are equal. • Corresponding angles are equal. Corresponding sides of two figures are in the same relative position, and corresponding angles are in the same relative position. Two figures are similar if and only if the lengths of corresponding sides are proportional and all pairs of corresponding angles have equal measures. CONFIDENTIAL

  15. corresponding angles 10 7.5 6 8 7.5 10 corresponding sides. X A ABC XYZ 62° 62° C Z 63° 63° 55° 55° B Y m B = m Y m C = m Z m A = m X Make sure corresponding vertices are in the same order. It would be incorrect to write ABC EFD. AB = BC = CA XYYZZX CONFIDENTIAL

  16. C D H G A B E F Scale factor Any two corresponding sides in two similar figures have a common ratio called the scale factor. The trapezoids ABCD and EFGH shown above are similar. So, AB = AD = CD = BC = k = scale factor. EF EH HG FG • The ratio of areas of two similar figures is the square of the scale factor. • The ratio of volumes of two similar figures is the cube of the scale factor. CONFIDENTIAL

  17. 3 cm 6 cm 2cm 4 cm Changing Dimensions B) Every dimension of a cone with radius 4 cm and height 6 cm is multiplied by 0.5 to form a similar cone. How is the ratio of the volumes related to the ratio of corresponding dimensions? B A CONFIDENTIAL Next page

  18. 3 cm 6 cm 2cm 4 cm Radii = 4 = 2 =2 2 1 Heights = 6 = 2 =2 3 1 Volumes = 32∏ = 8 = 23 4∏ 1 • The ratio of the volumes is thrice the ratio of the corresponding dimensions. CONFIDENTIAL

  19. 1) Parallelogram ABCD parallelogram EFGH. Find the value of x. 15 E F 12 A B x 5 D G C H Now you try! 1) 3 units 2) Sam is 7 feet tall and casts a shadow 4 feet long. At the same time, the pole outside his house casts a shadow 15 feet long. Write and solve a proportion to find the height of the pole. 2) 8.6 feet CONFIDENTIAL

  20. 3) Triangle ABC triangle ADE. Find the value of x. A 4 4 B C x 4 4 E D 10 3) 5 units 4) A rectangle has length 12 feet and width 8 feet. Every dimension of the rectangle is multiplied by 3 4 to form a similar rectangle. What is the ratio of the areas of the two triangles? 4) 1.78 units CONFIDENTIAL

  21. Percent • Percent is a ratio that compares a number to 100. For example, • 20% = 20 • 100 • To find the Fractionequivalent of a percent, write the percent as a ratio with a denominator of 100. Then simplify. • 20% =20 = 1 • 100 5 • To find the decimal equivalent of a percent, divide by 100. • 20% =20 = 0.2 • 100 CONFIDENTIAL

  22. The greatest percent shown in the table is 100%, or 1. But percents can be greater than 100% or less than 1%. For example, 130% = 130 = 1.3 and 0.6% = 0.6 = 0.006 100 100 CONFIDENTIAL

  23. Finding the Percent A) What percent of 60 is 15?. Method 1 Use a proportion. part = percentUse the percent proportion. whole 100 15 = xLet x represent the percent. 60 100 60x = 1500Find the cross products. x = 25Since x is multiplied by 60, divide both sides by 60 to undo the multiplication. 15 is 25% of 60. CONFIDENTIAL

  24. Finding the Whole B) 60 is 0.6% of what number? Method 2 Use an equation. Write an equation. Let x represent the whole. 60 = 0.6% of x 60 = 0.006x Write the percent as a decimal. Since x is multiplied by 0.006, divide both sides by 0.006 to undo the multiplication. 60 = 0.006x 0.006 0.006 10000 = x 60 is 0.6% of 10000 CONFIDENTIAL

  25. A) Jewelers use the karat system to determine the amount of pure gold in jewelry. Pure gold is 24 karat, meaning the item is 100% gold. A 14-karat gold ring contains 14 parts gold and 10 parts other metal. What percent of the ring is gold? Round your answer to the nearest percent. • part = percent • whole100 • 14 = x • 100 • 24x = 1400 • 24x = 1400 • 24 24 • x = 58.3 Use the percent proportion. Let x represent the percent. Find the cross products. Since x is multiplied by 24, divide both sides by 24 to undo the multiplication. A 14-karat gold ring is approximately 58% gold. CONFIDENTIAL

  26. Now you try! 1) 1 is what percent of 15? 1) 6.67% 2) 48 is 40% of what number? 2) 120 3) 13 is 20% of what number? 3) 65 4) On average, sloths spend 16.5 hours per day sleeping. What percent of the day do sloths spend sleeping? Round your answer to the nearest percent. 12) 68.75 CONFIDENTIAL

  27. Applications of Percent 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

  28. 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

  29. = 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 Danielatotal pay was $60,000. CONFIDENTIAL

  30. 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

  31. 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

  32. 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

  33. 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 castis 20,000 CONFIDENTIAL

  34. 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

  35. Now you try! 1) $2824.5 1) A sales representative earns a 3.5% commission on sales. Find the commission earned when the total sales are $80,700. 2) $42575 2) Karen earns a salary of $32,500 per year plus a 3.25% commission on sales. Find her total salary for a year when the sales are $310,000. 3) Find the total amount owed after 6 months on a loan of $900 at an annual interest rate of 8.5%. 3) $229.5 4) Estimate the tip on a $42.65 check using a tip rate of 15%. 4) $6.45 CONFIDENTIAL

  36. Applications of Percent A percent change is an increase or decrease given as a percent of the original amount. Percent increase describes an amount that has grown and percent decrease describes an amount that has been reduced. Percent Change percent change = amount of increase or decrease original amount , expressed as a percent CONFIDENTIAL

  37. Finding Percent Increase or Decrease Find each percent change. Tell whether it is a percent increase or decrease. A) from 25 to 56 SOLUTION: percent change = amount of increase original amount = 56 - 25 25 = 31 25 Simplify the numerator. = 0.96 = 96% Write the answer as a percent. CONFIDENTIAL

  38. Finding the Result of a Percent Increase or Decrease A) Find the result when 20 is increased by 40%. Find 40% of 20. This is the amount of the increase. 0.40 (20) = 8 It is a percent increase, so add 8 to the original amount. 20 + 8 = 28 20 increased by 40% is 28. B) Find the result when 75 is decreased by 60%. Find 60% of 75. This is the amount of the decrease. 0.60 (75) = 45 It is a percent decrease, so subtract 45 from the original amount. 75 - 45 = 30 75 decreased by 60% is 30. CONFIDENTIAL

  39. Common applications of percent change are discounts and markups. discount = % of original price A discount is an amount by which an original price is reduced. final price = original price - discount markup = % of wholesale cost A markup is an amount by which a wholesale cost is increased. final price = wholesale cost + markup CONFIDENTIAL

  40. Markups A) Kale buys necklaces at a wholesale price of $48 each. He then marks up the price by 75% and sells the necklaces. What is the amount of the markup? What is the selling price? Solution: Method 1: A markup is a percent increase. So find $48 increased by 75%. Find 75% of 48. This is the amount of the markup. 0.75 (48) = 36 Add to 48. This is the selling price. 48 + 36 = 84 Next Page CONFIDENTIAL

  41. B) Lars purchased a daily planner for $32. The wholesale cost was $25. What was the percent markup? Solution: Find the amount of the markup. 32 - 25 = 7 7 is what percent of 25? Let x represent the percent. 7 = x (25) 7 = x . (25) 25 25 Since x is multiplied by 25, divide both sides by 25 to undo the multiplication. 0.28 = x 28% = x Write the answer as a percent. The markup was 28%. CONFIDENTIAL

  42. Some more examples A) A trader marks his goods 40% above cost price and allows a discount of 25% .What gain percent does he make? SOLUTION: Let the cost price be $100. Then marked price = $140. Discount = 25% of marked price = $140 × 25 = $35 100 Therefore, net selling price = Marked price – Discount = $(140 – 35) = $105 Therefore, gain = 5% Hence, the trader gains 5%. CONFIDENTIAL

  43. Finding Percent Increase or Decrease Find each percent change. Tell whether it is a percent increase or decrease. A) from 25 to 56 SOLUTION: percent change = amount of increase original amount = 56 - 25 25 = 31 25 Simplify the numerator. = 0.96 = 96% Write the answer as a percent. CONFIDENTIAL

  44. B) from 25 to 17 SOLUTION: percent change = amount of decrease original amount = 25 - 17 25 = 8 25 Simplify the numerator. = 0.32 = 32% Write the answer as a percent. CONFIDENTIAL

  45. Now you try! Find each percent change. Tell whether it is a percent increase or decrease 1) 35 to 45 1) 28.57% increase 2) 11 to 8 2) 27.27% decrease 3) A dealer marked his goods 45% above the cost price and allows a discount of 20% on the marked price. Find his gain or loss percent. 3)Gain = 16% 4) What is the final price on a $175 leather jacket that is on sale for 40% off? 4) $105 CONFIDENTIAL

  46. You did a great job today! CONFIDENTIAL

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