Meet #2

AlgebraAlgebra

Self-study Packet

Math League SCASD

2019-20

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

1. Mystery: Problem solving

2. Geometry: Area and perimeter of polygons

3. Number Theory: Divisibility GCF, LCM, prime factorization

4. Arithmetic: Fractions, terminating and repeating decimals, percents

5. Algebra: Word problems with 1 unknown; working with formulas; reasoning in number sentences

Meet #2, Algebra

Meet #2 – Algebra Ideas you should know:

! Common Fraction:

! Money Answers: “What is One-Quarter of a dollar, minus two cents?”

23¢ $0.23 $.23 Not 0.23¢ Not 23

! Square Root of a Product: “What is the square root of 14x21 x 6?”

! Slow hard way – “Multiply it out first”:

14x21=294, 294x6=1764, “Um, Are we allowed to use calculators? How are we supposed to do this?”

" Faster way – “Factor it first”:

From question: 14 21 6

Factor: 2 x 7 3 x 7 2 x 3

Regroup factors: 2 x 2 3 x 3 7 x 7

Now do ! =3 =7

Answer: 2x3x7=42

! “Five consecutive multiples of 5 have a sum of 250 …” problems:

If 5 numbers add to 250, the average is 50. That’s also the middle number, so the five numbers are 40,45,50,55,60. Often the problem will ask for the 2nd number times the 4th number (45 x 55 here).

50 55 60 45 40

Average=50

Meet #2, Algebra

! “Four times the sum of a number and one is two more than seven times an amount one less than the number. Find the number.”

So confusing!

Make N be the number you don’t know, then translate the words to Algebra:

Four times the sum of a number and one is two more than

4 x ( N + 1 ) - 2 =

Seven times an amount one less than the number

7 x ( N - 1 )

Or: 4 (N+1) – 2 = 7 (N-1)

Find the number.

Solve for N.

First, distribute, then combine like terms:

4N + 4 – 2 = 7N – 7 (Add 7 – 4N to each side)

4 – 2 + 7 = 7N – 4N

9 = 3N

3 = N

Then check this answer in the original problem!

“Four times (3+1) is two more than 7(3-1)” Yes, 14=14

Meet #2, Algebra

P3

! Relatively Prime (review): Having no common factors besides 1. 100 and 99 are relatively prime, since the only prime factors of 100 are 2 and 5, and 99 has 3 and 11.

Are 6 and 10 relatively prime? No, both share 2 as a factor.

Are 27 and 111 relatively prime? No, both share 3 as a factor.

Are 35 and 66 relatively prime? Yes, 35 is 5x7, 66 is 2x3x11.

How many natural numbers less than 10 are relatively prime to 10, counting 1 as relatively prime to everything? Answer: 4: 1,3,7,9

How many whole numbers less than 17 are relatively prime to 17?

! Subscripts like P3 : pronounced “P sub three”

If P = {2, 3, 5, 7, 11, 13, 17, 19, 23, …} then

P1 = 2 P2 = 3 P3 = 5 If Pn=19, what is n?

P3 just means the 3rd P in a list. For example, if M is the set of how much money Anna, Bridget, and Caroline have, MAnna (pronounced M sub Anna) is how much money just Anna has.

If T is the set of multiples of 3: T = {3, 6, 9, 12, …} then T2=6, and Tn=3n

Answer to Pn=19: n=8.

! Time for some real problems from previous meets.

Category 5 Algebra Meet #2 - November, 2017 1) Tammy the turkey laid some eggs in September. She laid seven fewer in October than she laid in September. She laid four more eggs in November than she laid in September. She laid 162 eggs in all. How many eggs did Tammy lay in September? 2) The formula gives the value, V, of any term in the following sequence: 9 15 25 39 57 . . . where n is the number of the term. For example, for the first term, 9, n = 1. For the second term, 15, n = 2, and so on. What is the number of the term whose value is 457 ? 3) If 2c + 5d = 38 and 3e - 7f = 81, then what is the value of 28f - 12e - ( 10c + 25d ) ?

Answers

1)

2)

3)

Solutions to Category 5 Algebra Meet #2 - November, 2017 1) Let E = the number of eggs laid in September. Answers E - 7 = the number of eggs laid in October E + 4 = the number of eggs laid in November. 1) 55 Then E + (E - 7) + (E + 4) = 162 3e - 3 = 162 2) 15 3E = 165 E = 55 3) - 514 Since E represents the number of eggs laid in September, and E = 55, then Tammy laid 55 eggs in September. 2) Therefore, 457 is the 15th term in the sequence, and n = 15. 3) This problem involves the distributive property and its implications in values of literal expressions and their numeric multiples. If 2c + 5d = 38, then 5(2c + 5d) = 5(38), or 190. If 3e - 7f = 81, then 7f - 3e = - 81 and 4(7f - 3e) = 4(- 81), or - 324. Then 28f - 12e - (10c + 25d) = 4(7f - 3e) - 5(2c + 5d) = 4(- 81) - 5(38) = - 324 - 190 = - 514.

Category 5 Algebra Meet #2 - November, 2015 1) If 3N + 2Y = 25, then what is the value of 12N + 8Y - 19 ? 2) Romeo has two more 4-pound cantaloupes than 6-pound cantaloupes. The cantaloupes weigh 158 pounds in all. How many cantaloupes does Romeo have? 3) If and then what is the value of N ? The first cash register was patented on November 4, 1880.

Answers

1)

2)

3)

Solutions to Category 5 Algebra Meet #2 - November, 2015 1) 3N + 2Y = 25, so 4(3N + 2Y) = 12N + 8Y = 100, Answers so 12N + 8Y - 19 = 100 - 19 = 81. 1) 81 2) Let C = the number of 6-pound cantaloupes C + 2 = the number of 4-pound cantaloupes 2) 32 6C = the weight, in pounds, of the 6-pounders 4(C + 2) = the weight, in pounds, of the 4-pounders 3) - 6 6C + 4(C + 2) = 158 6C + 4C + 8 = 158 10C + 8 = 158 10C = 150 C = 15 So, there are 15 6-pounders. C + 2 = 17 and there are 17 4-pounders and there are 15 + 17, or 32 cantaloupes in all. 3) - 0.5 (2 - N) = 2 + N - 1 + 0.5N = 2 + N - 3 = 0.5N N = - 6

Category 5AlgebraMeet #2 - November, 2013

1) Sam has three more eggs than Sham but twice as many eggs as Faro. If there are 62 eggs in all, then how many eggs does Sham have?

2) If X + 3Y - 4A = 17 and X + 7Y - 4A = 45, then what is the value of 3X - 12A ?

3) The formula that converts a temperature in Celsius (C) degrees to Fahrenheit (F) degrees is

A roasted turkey is sufficiently cooked when a thermometer inserted into the thickest part of the thigh registers 180 degrees Fahrenheit. Priscilla only has a Celsius thermometer. How many minutes longer must she roast the turkey if its current temperature is 70 degrees Celsius and every five minutes in the oven produces a rise in temperature of one degree Celsius? Round your answer to the nearest minute.

ANSWERS

1) ______

2) ______

3) ______

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1)

3)

Answers

Solutions to Category 5AlgebraMeet #2 - November, 2013

Answers

1) 23 1) Let X = the number of Faro's eggs 2X - 3 = the number of Sham's eggs

2) - 12 2X = the number of Sam's eggs

3) 61 X + (2X - 3) + 2X = 62 5X - 3 = 62

5X = 65 X = 13 (Faro)

2X - 3 = 23 (Sham)

2) Compare the two equations and find their difference:

X + 3Y - 4A = 17X + 7Y - 4A = 45

Difference: 4Y = 28 Y = 7

Substituting 7 for Y into the first equation: X + 3(7) - 4A = 17 X + 21 - 4A = 17 X - 4A = - 4

Multiply both sides by 3: 3X - 12A = -12 Done!!

Therefore, it would take an additional 61 minutes to roast the turkey to perfection! Scrumptious!

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Meet #2 December 2011

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Category 5 – Algebra

1. If you add to the number , you’d get a number that’s times one third of

. What is ?

2. Inheriting a large sum of money, Mr. Lazy decided he does not need to work

and can simply live off his fortune. He spends the same amount of money

every year. After years, he realized he has of the money he had years

earlier. How many years will his fortune last overall?

3. A car drives from point to point , then turns around and drives back to point

at twice the original speed.

The average speed for the round trip was mph (miles-per-hour).

What was the car’s original speed?

Answers

1. _______________

2. __________ years

3. __________ mph

Meet #2 December 2011

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Solutions to Category 5 - Algebra

1. Writing this algebraically:

. To solve we can multiply both sides by :

or

Note:

The original problem said “ times greater than one third of .”

Some students observed that this could be interpreted as “4/3 of A MORE than

1/3 of A,” in other words, 5/3 of A. In this case, A+30 = 5 A / 3, giving A=45.

This was judged a reasonable interpretation of the problem and so both answers

were allowed.

One former mathlete wrote: The issue is whether "4 times greater" means 4 times as big or 5 times

as big. The 5 interpretation makes sense since "400% greater" means 500% as much.

2. If we note the number of years the money will suffice for as , then each year

he spends

of his initial fortune, and we know that:

(

) . Multiplying by we get:

or

3. If we note the car’s original speed as , and the distance between points and

as , then the trip from to took time

and the trip back took time

.

Overall the round-trip took time

to cover a distance of , and

so the average speed is

and so

Note that the distance D does not affect the answer: For any distance, if we

drive it once at mph, and then back at mph, the average speed is mph.

Answers

1.

2.

3.

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Category 5 - Algebra

Meet #2, December 2009

1. During a basketball game, your team scored three times as many 2-point field goals

than it did 3-point field goals, and scored a total of 90 points.

How many field goals did your team score?

(There were no 1-point free-throws.)

2. The force by which an object in space is pulled by the Earth’s gravity is proportional

to 𝑚

𝑅2 where m is the mass of the object, and R its distance from the center of the

Earth.

Satellite #1 is orbiting Earth at a distance (from its center) of 5,200 miles.

Satellite #2 is orbiting Earth at a distance of 20,800 miles, and has half the mass of

satellite #1.

What is the ratio of the gravitational force on satellite #1 to that on satellite #2?

3. The product of 3 consecutive even natural numbers divided by their sum is 64.

What is the middle number?

Answers

1. _______________

2. _______________

3. _______________

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Solutions to Category 5 - Algebra

Meet #2, December 2009

1. If we call the number of 3-point goals made G, then there were (3 ∙ 𝐺) 2-point goals

made, and the total score would be:

3𝑝𝑜𝑖𝑛𝑡𝑠 ∙ 𝐺 + 2𝑝𝑜𝑖𝑛𝑡𝑠 ∙ 3 ∙ 𝐺 = 9 ∙ 𝐺 = 90 𝑝𝑜𝑖𝑛𝑡𝑠.

So 𝐺 = 10, and the total number of field goals made is 4 ∙ 𝐺 = 40 (Ten 3-pointers

and thirty 2-pointers).

2. The ratio we seek is

𝑚1𝑅1

2

𝑚2𝑅2

2 , or (

𝑅2

𝑅1)2 ∙

𝑚1

𝑚2 and we know that

𝑚1

𝑚2= 2 𝑎𝑛𝑑

𝑅2

𝑅1= 4

so that the ratio is 42 ∙ 2 = 32

3. If we call the middle number x then the problem is 𝑥−2 ∙𝑥∙(𝑥+2)

𝑥−2+𝑥+𝑥+2= 64

If we simplify this a bit we get 𝑥∙(𝑥2−4)

3∙𝑥= 64 =

𝑥2−4

3 Or 𝑥2 = 3 ∙ 64 + 4 = 196

Therefore 𝑥 = 14. (𝑥 = −14 is a solution too, but we’re looking for a natural

number).

Answers

1. 40

2. 32 or 32:1

3. 14

Category 5 Algebra Meet #2, November 2007

1. Two years ago, Bob was 2

3as old as he will be in 6 years. In how many years

from now will Bob be 40 years old?

2. The formula for Volume of a sphere is 34

3V rπ= and the formula for Surface

Area of a sphere is 24SA rπ= . If the Volume of a given sphere is 972π , what

is the Surface Area of the same sphere? Express your answer in terms of π .

(note: 972π is an example of a number given "in terms of π ".) 3. The sum of seven consecutive multiples of 7 is 1078. What is the sum of the second smallest and the second largest of these seven numbers?

Answers

1. _______________

2. _______________

3. _______________

Solutions to Category 5 Algebra Meet #2, November 2007

1. So if Bob is 18 years old now, he will be 40 in 22 more years. 2.

SO

24 9

4 81 324

SA

SA

π

π π

=

= =

3. If we call the middle of the seven numbers x, we could use this equation :

( ) ( ) ( ) ( ) ( ) ( )21 14 7 7 14 21 1078

7 1078

154

x x x x x x x

x

x

− + − + − + + + + + + + =

=

=

Since we want the sum of the 2nd largest and 2nd smallest, we are looking for :

( ) ( )14 14 2 2(154)x x x− + + = = =308

Answers 1. 22

2. 324π

3. 308

22 ( 6)

3

22 4

3

16

3

18

b b

b b

b

b

− = +

− = +

=

=

3

3

3

3

4972

3

4972

3

3972

4

729

9

r

r

r

r

r

π π=

=

⋅ =

=

=

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Category 5 Algebra Meet #2, December 2005 1. Sammy Squirrel, Sally Squirrel, and Sydney Squirrel have stored a total of 219 acorns for the winter. Sammy then gives 12 acorns to Sally and 12 acorns to Sydney. Now Sammy and Sally have the same number of acorns, but Sydney has 18 fewer than each of them. How many acorns did Sydney have originally? 2. Each shape in the picture below represents a number. Same shapes have the same number. If the triangle has a positive value, what number goes in the hexagon? 3. The formula for the sum of the squares of the numbers from 1 to n is:

��

12 + 22 + 32 +�+ n2 =n n + 1( ) 2n + 1( )

6

Find the sum of the squares of the numbers from 1 to 25.

Answers 1. _______________ 2. _______________ 3. _______________

÷ 5 =

3 × – =

(-2)3 =

12 + =

× = 49

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Solutions to Category 5 Algebra Meet #2, December 2005

1. Let x, y, and z represent the original number of acorns stored by Sammy, Sally, and Sydney respectively. Then x + y + z = 219. After Sammy gives 12 acorns to Sally and 12 acorns to Sydney, he has x – 24 acorns, Sally has y + 12 acorns, and Sydney has z + 12 acorns. Let’s imagine that Sydney receives another 18 acorns from an anonymous squirrel. Sydney would then have z + 12 + 18 = z + 30, which is the same amount as the other squirrels. The new total would be 219 + 18 = 237 acorns, and we would know that x – 24 = y + 12 = z + 30. Dividing 237 by 3, we find that each squirrel would have 79 acorns. Now we can solve z + 30 = 79 for z. Sydney must have had 79 – 30 = 49 acorns originally. 2. The value of the pentagon can be computed directly: −2( )3 = −8. Also, the triangle must be 7, since 7 × 7 =

49. Now we can find that the circle equals 12 + 7, which is 19. The value of the square must then be 3 × 19 – (-8) = 57 + 8 = 65. Finally, the value of the hexagon is 65 ÷ 5 = 13.

3. Substituting 25 in place of n in the formula, we get

��

12 + 22 + 32 +�+ 252 =25 25 + 1( ) 2 ⋅ 25 + 1( )

6

= 25 ⋅ 26 ⋅ 516

= 25 ⋅13 ⋅17 = 5525

Answers 1. 49 2. 13 3. 5525