Writing a Proportion

1. Writing a Proportion To write a proportion, you set one ratio equal to another ratio. The only rule is this: Your left side and right side must be set up the same way. So, to see the Eiffel Tower was really 1064 feet tall, you measured it’s shadow to be 798 feet long. Then, you held up a 4 foot broom, and the broom’s shadow was 3 feet long. Write 2 different proportions that compare height and shadow. EF shadow broom shadow broom height broom shadow EF height EF shadow EF height broom height 798 ft 1064 ft 4 ft 1064 ft 3 ft 4 ft 3 ft 798ft 266 266 1.333... 1.333... When you write a proportion, your left side and right side must be set up the same way.

2. Is it a Proportion? Dana earned a salary of $27,000/mo, and Teresa earned a salary of $6,500/mo. Dana paid $10,530 in income taxes, but Teresa paid $1,625 in income taxes. Did they pay a proportional amount of taxes? Step 1: Make a ratio (fraction) with Dana’s Step 2: Set that equal to Teresa’s Step 3: Cross-multiply. Are the cross products equal? Dana Teresa Since cross products are NOT equal, they do NOT pay a proportional amount of taxes. 43,875,000 10,530 1,625 10,530 1,625 • = 27,000 6,500 27,000 6,500 68,445,000 When cross products are EQUAL, it IS a proportion When cross products are NOT equal, it is NOT a proportion

3. Solving a Proportion Drew was bragging that he could bench press 245 newtons (N). To convert that weight to pounds (lb), use the chart to the right: 100 ounces (oz) = 6.25 pounds (lb) 8 newtons (N) = 1.8 pounds (lb) 25 ounces (oz) = 6.95 newtons (N) Step 1: Write your empty proportion. 8 N Step 2: Set up your left side anyway you want. * I’d recommend starting with “ 245 N ” x lb 1.8 lb Step 3: Find the correct conversion... • Since newtons (N) is on top on the left, then put newtons (N) on top on the right. ... then write it as your right side ratio. = 8 ● x 245● 1.8 ... since you’re trying to find weight in pounds, call that x. 8x 441 = Step 4: Cross-multiply, then set the cross-products equal to each other 8 8 x = 55.125 Step 5: Solve the equation you made. Drew bench pressed, about, 55 pounds (lb). To solve a proportion, cross multiply, set the products equal to each other, then solve.

4. Writing & Solving Proportions Jack’s recipe makes 2.25 pounds (lb) of chili, and it calls for 1.5 tablespoons of lime juice. If he wanted to make 5.1 lb of chili, how many tablespoons of lime juice should he use? Chili #1 Chili #2 To write a proportion, each ratio must be set up the same way. 2.25 5.1 • = 1.5 l chili Let’s set up both ratios as juice 2.25 • l = 5.1 • 1.5 Now, find the cross products. 2.25l = 7.65 Simplify, then solve 2.25 2.25 Jack should use 3.4 tablespoons of lime juice. l = 3.4

5. Scale Models Harmony MS has the longest bus rides in the county. Here’s a map showing the route from Loudoun Heights (next to Harper’s Ferry) to Harmony. If their map length is 1 3 inches apart, then how far apart are they actually apart? 4 Step 1: Write an empty proportion. Step 2: On the left side, write the map scale as a fraction. Step 3: On the right side, fill in the distance given in the problem. Let’s change those fractions to decimals. m Step 4: Call the missing distance, m. 0.25 1.75 2 Step 5: Cross-multiply, then solve for m. 0.25 • m 1.75 • 2 = map scale = 0.25m 3.5 The bus ride would be, about, 14 miles. 0.25 0.25 m = 14

6. Finding a Scale Ratio The longest drive in the contiguous U.S. is from Key West, FL to Seattle, WA. If the actual distance of the drive is 3,450 miles, what is the scale of the map. You’ll need a ruler with inches for this one. Set up your proportion. Usually, scale ratios set the first value as 1 Measure to find the mapdistancebetween them. Solve. scale ratio = distance ratio 1 in in = a = so, the scale is 1 in = mi 3450 = a

7. Using a Map’s Scale Ratio José and Rosa, from Lisbon, Portugal, are going to visit the grandparents, in Vladivostok, Russia, for the holidays.. What will be the actual distance of their drive? scale ratio = distance ratio Set up your proportion. cm cm Measure the distancein cm. = a Now, measure the map’s scaleratioin cm... ... and fill it in your scaleratio. Solve. a = The distance between Lisbon, Portugal and Vladivostok, Russia is miles.

8. Using the Percent Proportion 1.8 is124% of what number? (If needed, round to the nearest tenth.) part whole part whole Proportions compare a part/whole ratio to another part/whole ratio. = This is the percent proportion. A percent proportion compares a ratio to a percent/100. p is = Which number is the “part”? It’s 1.8 of 100 What little word is next to it? “is” 1.8 124 I dunno. Let’s say x. Which number is the “whole”? = x 100 What little word is next to it? “of” 180 = 124x ____ ____ 124 124 Write and solve the proportion 1.8 is 124% of 1.5 x= 1.5

9. Using the Percent Proportion What number is 56.1%of 31? (If needed, round to the nearest tenth.) is of _p_ 100 Which number is the “percent”? it’s 56.1 = Which number is the “is”? (I dunno ?!) let’s call it x Which number is the “of”? it’s 31 x 17.4 Solve for x What percent of 47is 41 ? (If needed, round your percent to the nearest tenth.) is of _p_ 100 Which number is the “percent”? I dunno…let’s call it p = 41 Which number is the “is”? Which number is the “of”? 47 p 87.2% Solve for p

10. Using the Percent Proportion In 2009, Loudoun Valley High School had a capacity of 1350 students, but 1497 students are enrolled. What percent of capacity was Valley running? part whole _p_ 100 Which number is the “percent”? = x Which number is the “part”? it’s actually 1497 Which number is the “whole”? 1350 Solve for x x 110.9 Valley was running at 110.9% capacity. Harvard University accepted 4,983 freshman applications last year. Only 7% of all students who applied were accepted. How many students applied? part whole _p_ 100 Which number is the “percent”? it’s 7 = 4983 Which number is the “part”? (I dunno ?!) let’s call it x Which number is the “whole”? x 71,186 71,186 students applied. Solve for x

11. K J The triangles are SIMILAR Are They Similar? L PR QR ? Polygons are similar when: = Corresponding angles are congruent. JL KL * QPR KJL * PRQ JLK 2. Corresponding sides are in proportion 21.8 23.2 *Put a pair of corresponding sides in a ratio. 10.9 11.6 *Set them equal to another pair of corresponding sides. *Cross-multiply – Are they equal? 252.88 = 252.88 http://www.mathopenref.com/similartriangles.html

12. Finding Missing Measures of Similar Figures 25 W V 15 20 T U

13. Finding Missing Measures of Similar Figures 30 12 x 19

14. Writing a Ratio Write the ratio 3 feet (ft) to 8 inches (in) as a fraction in simplest form. 9 A ratio is a comparison of 2 values. 3 ft 36 in 9 in •12 = = = Compare these 2 values as a fraction. 8 in 8 in 2 in 2 Convert to the same unit. Hint: Always convert to the smaller unit. 9 in 9 9:2 9 to 2 or or or Now, reduce 2 in 2 There are other ways to write a ratio, also.

15. Finding a Unit Rate In 1987, Bo Jackson ran the 40 yd (yard) dash in 4.12 s (seconds). In 1989, Deion Sanders ran the 60 yd dash in 6.68 s. Who ran at a faster rate? To compare these runners, find each players yards seconds Bo Deion 40 yd 60 yd 9.708 9.7 yd = ≈ 8.982 ≈ 9.0 yd = 4.12 s s 6.68 s s Bo Jackson ran at the faster rate. Emerald’s 11 oz (ounce) can of almonds sells for $7.69. Planters 5 oz bag of Trail Mix sells for $3.25. Which has the lower unit rate? $7.69 $3.25 Emerald’s $0.70 Planters 0.65 ≈ $0.65 = 0.699 ≈ 11 oz 5 oz oz oz