Park Forest Math Team

Meet #44

Self-study Packet

Problem Categories for this Meet (in addition to topics of earlier meets):

1. Mystery: Problem solving 2. Geometry: Properties of Circles 3. Number Theory: Modular Arithmetic, Series and Sequences 4. Arithmetic: Percent Applications 5. Algebra: Word Problems (linear, including direct proportions or systems)

Important information you need to know about ALGEBRA: Word Problems

• Formula to know: distance = rate • time • Set up equations by trying to write everything in terms of one variable.

Example Melinda has 4 more green M&Ms than Logan. Adrianna has 1 fewer than double the number of green M&Ms Logan has. Between the 3 of them, they have a total of 43 green M&Ms. How many green M&Ms does each person have? Solution (Everyone is being compared to Logan, so make Logan your standard variable). Logan = x Melinda = x + 4 Adrianna = 2x – 1 Logan + Melinda + Adrianna = 43 green M&Ms, so… x + (x + 4) + (2x – 1) = 43 4x + 3 = 43 4x = 40 x = 10 x represents Logan, so Logan has 10, Melinda has 14 , Adrianna has 19. Double check—is 10 + 14 + 19 equal to 43? Yes!

• If all else fails, guess and check!

Double check your work! ! Are answers rounded properly, as instructed? ! Is your answer in the requested form? (Mixed number, decimal, improper

fraction, whole number)

! Does your answer make sense?

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Answers

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Solutions to Category 5 Algebra Meet #4, February 2009

1. Since it took Joey 6 days to build 32 houses that gives us the

ratio !"#$%&

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the number of days, call that -. We can write a second ratio and

form a proportion now: !'( .

/00( 1

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'230!23 .

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So - . 45 days.

2. "Math" has 3 consonants and 1 vowel, while "league" has 2 consonants and 4 vowels. Using 6 as the value of a consonant, and 7 as the value of a vowel we can write these two equations: 86 9 57 . 8: 46 9 ;7 . <= Multiplying the first equation by 4 results in these two equations: 546 9 ;7 . 5;= 46 9 ;7 . <= Subtracting the two equations we get: 5=6 . >= which tells us that a consonant is worth 8 points. Using the original equation for "math" we have 8?>@ 9 7 . 8: 1 7 . 55 so vowels are worth 11 points. "Intermediate" has 6 consonants and 6 vowels for a value of <?>@ 9 <?55@ . ";> 9<< . 55; points

3. Using A for Pat's weight and B for Snog's weight we get the equations: 5C4:?A 9 4=@ . B 9 4= and 5C:?A D 4=@ . B D 4= """""5C4:A 9 4: D 4= . B and 5C:A D 8= 9 4= . B """""""""""""""""""""5C4:A 9 : . B and 5C:A D 5= . B Substituting for B we get: 5C4:A 9 : . 5C:A D 5= 5: . C4:A <= . A Plugging 60 in for A in the first equation we get 5C4:?<= 9 4=@ . B 9 4= 5C4:?>=@ . "B 9 4= 5== . B 9 4= 1 >= . B So the combined weight is <= 9 >= . 5;= lbs

Answers 1. 21 2. 114 3. 140

www.imlem.org

Category 5 Algebra Meet #4, February 2007 1. On her most recent birthday, Emily became 1

3 of her father’s age. In six years,

she will be 512

of her father’s age. How old was Emily’s father when she was

born?

2. Dennis usually averages 70 miles per hour on his trip to his grandmother’s house. When he was towing a trailer he averaged only 60 miles per hour, and the trip took half an hour longer than usual. How many miles is it to his grandmother’s house? 3. In a warehouse there are three kinds of boxes. If you stack two A boxes, one B box, and two C boxes, the stack is 88 inches tall. If you stack one A box, two B boxes, and one C box, the stack is 77 inches tall. If you stack three A boxes, one B box, and one C box, the stack is 83 inches tall. If you make a stack of C boxes that is as close as possible to eight feet tall, how many inches less than eight feet tall is the stack?

Answers 1. _______________ 2. _______________ 3. _______________

You may use a calculator today!

www.imlem.org

Solutions to Category 5 Algebra Meet #4, February 2007

1. Let Emily’s father’s age now be x years. Then Emily’s age now is 1

3x

years. In six years, Emily’s father will be x + 6. We can calculate

Emily’s age six years from now as 13

x + 6 and as 512

x + 6( ). These are,

of course, equal, so we can write the following equation and solve for x: 13

x + 6 = 512

x + 6( ). Multiplying both sides by 12, we get

4x + 72 = 5 x + 6( ). Now we can distribute on the right, which give us 4x + 72 = 5x + 30. Finally, subtracting 4x and 30 from both sides, we get

x = 42. That’s Emily’s fathers age now. Emily must be 13

! 42 = 14 now,

so her father must have been 42 – 14 = 28 years old when she was born.

2. Let the distance to Dennis’s grandmother’s house be d miles. If Dennis travels

at 60 mph instead of 70 mph, he is traveling 67

of his normal speed, so it will take

76

the amount of time. We know that the extra sixth of the time is half an hour.

This means it usually takes six half hours, or three hours. It must be 3 hrs " 70 mph = 210 miles to Dennis’s grandmother’s house. 3. From the information given, we can write the following system of equations.

2A + B + 2C = 88A + 2B + C = 773A + B + C = 83

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If we double the second equation and subtract the first, we get 3B = 66, so B = 22. Subtracting the second equation from the third, we get 2A – B = 6. If we substitute B = 22 into this equation, we get 2A – 22 = 6. Adding 22 to both sides, we find that 2A = 28, so A = 14. Now we can solve any of the three equations for C and find that C = 19. If we stack 5 of these 19-inch boxes, our stack would be 95 inches tall, which is 1 inch less than eight feet.

Answers 1. 28 2. 210 3. 1

www.Imlem.org

Category 5 Algebra Meet #4, February 2005 Average number of correct answers: 1.60 out of 3 1. What negative value of x will make the following proportion true?

121x

= x144

2. In parallelogram MATH, MA = 168 mm, AT = 15x + 3 mm, TH = 3xy3 mm, and MH = 108 mm. What is the numerical value of yx ? 3. Together Jim and Bob weigh 357 pounds. Together Jim and Larry weigh 393 pounds. The combined weight of all three men is 565 pounds. How much do Bob and Larry weigh together?

Answers 1. _______________ 2. _______________ 3. _______________

You may use a calculator

www.Imlem.org

Solutions to Category 5 Average team got 19.15 points, or 1.6 questions correct Algebra Meet #4, February 2005

1. Cross multiplying, we can turn the proportion into the equation 121 !144 = x2. Each of the numbers on the left is a perfect square, so we can rearrange factors without ever multiplying 121 by 144: 121 !144 = 11 !11 !12 !12 = 11 !12 !11 !12 = 132 !132 = x2 The value of x could be positive or negative. The question asks for a negative value that will make the proportion true, so the answer is –132.

2. Since opposite sides of a parallelogram have the same length, we can write two equations from the information given: 108 = 15x + 3 and 168 = 3xy3 . Solving the first equation for x, we get 105 = 15x, so x = 105 ÷ 15 = 7. Substituting 7 in place of x in the second equation, we get 168 = 21y3 . This means y3 = 168 ÷ 21 = 8, so y = 2. Finally, the numerical value of yx is 27 = 128. 3. There are many ways to solve this system of equations. One clever way is to double the combined weight of all three men and subtract the two paired weights. Since Jim is included in both the paired weights, his weight will be subtracted twice and each of Bob’s and Larry’s weights will be subtracted once. This leaves the combined weight of Bob and Larry, which is what we want. Thus, the answer is 2 " 565 – 357 – 393 = 1130 – 357 – 393 = 380 pounds. Notice that we did not need to find the individual weights, but, for those interested, Larry weighs 208 pounds, Bob weighs 172 pounds, and Jim weighs 185 pounds.

Answers 1. –132 2. 128 3. 380

M A

T H

168

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