What's on it:
- Crowd size (from quiz #1)
- Tennis balls (from quiz #1)
- Phone numbers / license plates (from quiz #1)
- TV's (from quiz #2)
- Tires (from quiz #2)
- Grade weighting (new; Wed/Thurs)
- Baseball (new; Wed/Thurs)
- Football (new; Wed/Thurs)
- UPC's (new; Fri/Tues/Wed)
- Credit Cards (new; Fri/Tues/Wed)
- You should also be prepared to write sentences to discuss
- the real-world context of any of these problems, and
- assumptions made when trying to answer a big Fermi type question
I think I now have an example and a practice problem for each of these.
You can use this to study and practice.
Please ask me or Ms. Johnson for help - in class, after school, or on group me!
Also, I'm only human - if you think you've found a mistake, please let me know so I can fix it!
Examples:
Crowd size (from quiz #1)
Step 1: Find the total area:
square = 500 ft * 500 ft = 250,000 ft^2
MPC = 15 ft * 25ft = 375 ft^2
+ _________________________________
250,375 ft^2
Step 2: Set up the ratios and cross-multiply to solve for x, the total number of people
1 person / 3 ft^2 = X people / 250,375 ft^2
3 ft^2 * X people = 1 person * 250,375 ft^2
X people = 1 person *250,375 ft^2 / 3 ft^2 = 83458.333 people, which rounds to
83,458 people
You Try:
The MHS homecoming parade comes down Polk St. People crowd both sides of the street for a length of 0.5 miles, 5 feet back. There are 5280 feet in a mile! If each person has about 2.2 square feet of space, estimate the total number of people. This one is like the one on the first quiz!
Tennis balls (from quiz #1)
step 1: the volume of the room (in cubic INCHES - convert if and ONLY IF needed)
length = 40 ft (12 inches/ft) = 480 inches
width = 30 ft (12 inches/ft) = 360 inshes
height = 9.25 ft (12 inches/ft) = 111 inches
volume = length * width * height = 19180800 inches^3
step 2: the volume used by one ball (we basically assume a small cube holding one ball)
length = width = height = 3.5 inches (the diameter of one ball)
volume = 3.5 in * 3.5 in * 3.5 in = 42.875 inches^3
step 3: set up the ratio and cross-multiply:
1 ball / 42.875 in^3 = X balls / 19,180,800 in^3
42.875 in^3 * X balls = 1 ball * 19,180,800 in^3
X balls = 1 ball * 19180800 in^3 / 42.875 in^3 = 447,365.59 balls, which rounds to
447,366 balls.
You Try:
Sara's bedroom is 20 ft wide, 15 feet long, and 9.5 feet high. One balloon has a diameter of 10 inches. How many balloons could her annoying little sister fill her room with (assuming the balloons stack straight up in rows, the walls are flat, and the room is empty)?
Phone numbers / license plates (from quiz #1)
H is set, not changing. So we don't care.
I is set, not changing. So we don't care
5 is set, not changing. So we don't care.
{ } has 9 possibilities (1,2,3,4,5,6,7,8,9)
# has 10 possibilities (0,1,2,3,4,5,6,7,8,9), AND there are 5 of them!
[] has 26 possibilities (A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z)
total possibilities = 9*10*10*10*10*10*26 = 23,400,000
You Try:
Phone numbers in Switzerland follow a pattern: 78 () # # # # # @, where () is any number 2-9 (not 0 or 1), # is any number 0-9, and @ is a 4 or a 5. How many different combinations are there?
TV's (from quiz #2)
Find the area of a 60 inch TV with a 16:9 aspect ratio.
60 inches is the diagonal. The width is 16x and the height is 9x
(from the aspect ratio we know that they are 16 and 9 times some unknown amount : x)
step 1: Set up Pythagorean Theorem and solve for x:
(16x)^2 + (9x)^2 = (60in)^2
256 x^2 + 81 x^2 = 3600 in^2 (square EVERYTHING)
377 x^2 = 3600 in^2 (combine like terms)
x^2 = 9.549 (divide both sides by the 377)
x = 3.090 (take the square root)
step 2: Plug back in and get the width and the height:
width = 16(3.090) = 49.44 INCHES (REMEMBER TO PUT UNITS!!!)
height = 9(3.090) = 27.81 INCHES
step 3: Multiply to get the area (A = W*H)
Area = 49.44 inches * 27.81 inches = 1374.926 inches^2
You Try: Find the area of a 48 inch TV with an aspect ratio of 4:3.
Tires (from quiz #2)
REMEMBER: There are always 25.4 mm per inch.
original tires: new tires:
width: 250 mm 270 mm
aspect ratio: 70%=0.70 73% = 0.73
height of the rubber: 250*0.70/25.4 = 6.890 in 270*0.73/25.4=7.760 in
diameter of the rim: 17 in 19 in
total diameter: 17 + 6.890 + 6.890 = 30.780 in 19 + 7.760 + 7.760 = 34.520 in
circumference: 30.780*pi = 96.698 inches 108.448 inches
ratio of the bigger to the smaller circumference: 108.448 in/ 96.698 in = 1.122
(NO UNITS for a ratio)
CORRECTED odometer reading = 14,000 miles * 1.122 = 15,708 miles
CORRECTED speedometer reading = 60 mph * 1.122 = 67.320 mph
You Try:
Your car came with P225/72R16 tires. You decide to replace them with P250/74R20. If your odometer reads 25,000 miles (the total distance you think you have driven) and your speedometer reads 75 miles per hour (the speed you think you are driving), what are your ACTUAL miles driven and speed?
Grade weighting (new; Wed/Thurs)
step 1: test average: 70+80+82+60 = 292. 292/4 = 73 (divide by 4 because there are 4 of them)
(Any time you're given a list of scores, average them)
step 2: quiz average: 54+80 = 134 134/2 = 67
Step 3: multiply by the weights (as decimals, so divide by 100%!!), and add up the grade so far
73(0.60) = 43.8
67(0.15) = 10.05
75(0.05) = 3.75
+____________
57.6
Step 4: Add in the Final Exam (x) times the weight for that, set equal to the goal grade, and then solve for x
57.6 + 0.20x =73
-57.6 -57.6
___________________
0.20x = 15.4
divide both sides by 0.20: x = 77
You Try:
Geometry grades are 50% for tests, 20% for quizzes, 10% for homework, and 20% for the final exam. Kim has 3 test grades: 73, 85, 100. She has 4 quizzes: 88, 92, 99, 85. She has a 80 average for her homework. What does she need on the final exam to get a B (80 average) ?
Baseball (new; Wed/Thurs)
My nephew is in junior baseball. He has the following stats for this season:12 Doubles 4 Triples 53 Singles 8 Home runs 300 At bats
Find the Slugging Percentage.
SLG = (1*S) + (2*D) + (3*T) + (4*HR)
AB
= (1*53) + (2*12) + (3*4) + (4*8) = 121/300 = 0.40333333
300
To get in as a percent, multiply by 100%: 0.40333333 * 100% = 40.333%
You Try: His older brother is in the same league. He has the following stats out for this season:
12 Doubles 4 Triples 53 Singles 8 Home runs 300 At bats
12 Doubles 4 Triples 53 Singles 8 Home runs 300 At bats
Find the Slugging Percentage.
Football (new; Wed/Thurs)
Find
the quarterback rating of a player with the following stats:
- 250 completed passes in 420 attempts
- Total of 2200 yards
- 18 touchdowns and
- 15 interceptions
Percent of total completions
(%COMP)= 250/420 * 100% = 59.524%
Percent of total touchdowns
(%TD)= 18/420 * 100% = 4.286%
Percent of interceptions per
attempt (%INT)= 15/420 * 100% = 3.571%
Average yards gained per
attempt (YD)= 2200/420 = 5.238 (yds)
Quarterback rating: (It's a long formula, but then you just plug in!)
QR = 25 + 10(%COMP) + 40(%TD) - 50(%INT) + 50(YD)
12
= 25 + 10(59.524) + 40(4.286) - 50(3.571) + 50(5.238) = 875.03/12 = 72.919
12
You Try:
Find the quarterback rating of a player with the following stats:
- 325 completed passes in 510 attempts
- Total of 2805 yards
- 22 touchdowns and
- 17 interceptions
UPC's (new; Tues/Wed)
3(a1) + a2 + 3(a3) + a4 + 3(a5) + a6 + 3(a7) + a8 + 3(a9) + a10 + 3(a11) + d
3(8) + 3 + 3(2) + 9 + 3(4) + 6 + 3(7) + 3 + 3(2) + 0 + 3(8) + d
= 24 + 3 + 6 + 9 + 12 + 6 + 21 + 3 + 6 + 0 + 24 + d
= 114 + d
the total has to end in 0, so 114 + d = 120 (the next number that ends in 0)
d = 120 - 114 = 6.
You Try:
What is the last digit of the UPC 9 28363 45821 [d] ?
Credit Cards (new; Tues/Wed)
Find the last digit (the check digit) of the Visa number below so that it is valid:
2314 2463 8536 225[d]
step 1: add the digits in the ODD positions (1st, 3rd, 5th, 7th ... 15th)
2 + 1 + 2 + 6 + 8 + 3 + 2 + 5 = 29
step 2: double that number:
29*2 = 58
step 3: COUNT how many of those ODD positioned numbers are more than 4, and add that:
There are 3 that are more than 4: 6, 8, and 5. 58 + 3 = 61
step 4: add the digits in the EVEN positions (2nd, 4th, 6th, 8th ... 14th) to that total:
61 + 3 + 4 + 4 + 3 + 5 + 6 + 2 = 88
step 5: (just like UPC) find the digit you can add to make a total that ends in 0:
88 + d = 90
-88 -88
___________
d = 2
You Try:
Find the last digit (the check digit) of the Mastercard number below so that it is valid:
8234 3248 3255 324[d]
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