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1)

Natural and Whole Numbers.

A student at St. F. X. decided to become his own employer by using his car as a taxi for the summer. It costs the student $693.00 to insure his car for the 4 months of summer. He spends $452.00 per month on gas. If he lives at home and has no other expenses for the 4 months of summer and charges an average of $7.00 per fare, how many fares will he have to get to be able to pay his tuition of $3280.00?

Hints: 

1) Let the number of fares=x

2) 

If twenty people in a factory make 43 shoes in [(3)^.5] / 2 hours, then how many shoes do they make in an eight hour day?

Hints: 

1) The number of shoes have to be an integer.

3) 

Brendan went to Toronto to see a Blue Jays game. He had $650.00 for the trip. It cost him $40.00 for his ticket to the game, $216.00 to stay 2 nights at a hotel, $329.00 for his flight and $29.00 for food. He met a man on Young Street who was running a small casino. Brendan bet $5.00 in Black Jack and won an amount of money that was triple what he had left after making the bet. He spent half of this at the ball game. How much money did Brendan take home?

Hints: 

1) He had 31 dollars left after making the bet.

4) 

The area of a trapezoid is given by the formula A = 1/2 (a + b) h. If h = 1/2b and a = 1/3b.

a) What will be the value of b when a = 26 ?

b) What will be the value of h if a = 6? What will be the area of the trapezoid?

Hints: 

1) 3a = b, h = .5 * 3a

5) 

Joe was driving on the highway. A car ahead of him was driving far below the speed limit so he decided to pass. In the first second he gained 5m on the car and as he accelerated he gained 1.5 times as much distance in each second as he had the second before. If there was 30m between Joe and the car he was passing, then how long did it take him to pass?

6) 

If a turntable rotates through 720 degrees in one second, how many revolutions does it make in one minute.

Hints: 

1) there are 360 degrees in one revolution

7) 

Two girls agree to go on a road trip together. They travel (x + 5)km on the first day. On the second day they travel 2km more than half of the distance they travelled on the first day. On the third day they drove 3 times as far as they did on the second day. If they drove 5000km total, find the value of x.

Hints: 

1) Let (x+5)= distance on day1

2) Let (x+5)/2=distance on the second day

8) 

A rectangle has the following sides. One side must have an odd integer length. Find the perimeter.

Hints: 

1) Multiplying by an even number makes an odd number even, so 4y + 8 and 20x - 16 are even  numbers.

9) 

If a tire rotates at 150 revs/min when the truck is travelling 40km/h, what is the circumference of the tire?

Hints: 

1) Fix a time - say 1 min.

10) 

From the rectangle DEFG the square ABCD is removed, leaving an area of 92 square units. If AE is 4 and GC is 8 units, then use a polynomial expression‎ to find the original area of DEFG.

Hints: 

1) Set area of ABCD to x^2

11) 

Points A, B, C, D, are arranged in order on a line so that AB = 3, BC = 2CD what is BD as a fraction of AD?  

Hints: 

1) Let segment BC have length x

12) 

If (-2 + x^2)^5 = 1 then what can x be equal to in the set of positive irrational numbers?

Hints: 

1) 1^5=1

2) ((x^2-2)^5)^(1/5)=1^(1/5)=1

13) 

Given the symmetric shape below with a known perimeter of 77, find the area of the shape.

Hints: 

1) Solve for x using perimeter.

2) Use Pythagoras to find ??????

14) 

A man in a speed boat which could only travel at 50km/h was driving his boat up a river which flowed with a constant speed. As he was driving his spare life jacket fell of the boat, 20 minutes later the man noticed the jacket was gone and turned back to get it. He found it 3km away from where he lost it. How fast was the river flowing?

Hints: 

1)

                      boat at

                      40 min

         <------------- .

         . -----------> .       3 km     .

      boat at        jacket           jacket

      20 min         lost             found

15) 

Find an odd number with 3 digits such that all the digits are different and add up to 15. The difference between the first two digits equals the difference between the last two digits. The hundreds digit is greater than the sum of the tens and ones digits.

Hints: 

1) Let the number be represented by abc, (a-b)^2=(b-c)^2

16) 

The Jones family had a square patch of lawn in their backyard. Its original area was x^2. They increased its length by 3 and its width by 5. Write a polynomial to describe the new area.

Hints: 

1) Use the diagram

17) 

Write a polynomial to describe the area of the triangle ABC.  

Hints: 

1) Use similar triangles

18) 

How many minutes is it until 5:00 if forty minutes ago it was four times as many minutes past 2 o'clock?

Hints: 

1) Represent current time as 5:00-x(minutes)

19) 

Given a trapezoid of area 126 and a = 6, b = 8, use the formula 1/2 (ah) + h(a + b) / 2 + 1/2 (bh) = 126 to solve for variable h, then find the perimeter of the trapezoid.

Hints: 

1) Use the pythagorean theorem to the other two sides of the trapezoid.

20) 

A jeweler sells all of his merchandise for double what it cost him but he charges no tax to his customers. Instead he pays it out of his profits. A woman comes in to buy a ring one day and she notices a scratch on the band. The jeweler takes 1/4 of the price and sells the ring for $159.99.

What was the jewelers profit? (Tax is 20%)

Hints: 

159.99=(3/4)*2X 

21) 

Mary and Gary graduated from university together. Gary became a teacher and earned half what Mary earned for 5 years. Mary spent 1/3 of her money; Gary spent 1/4 every year for those 5 years. Greg has $80,000 after 5 years. How much does Mary have?

Hints: 

• 1)Let x = Gary's salary

• Let 2x = Mary's salary

• Let 1/4x = Greg's spending

• Let 1/3(2x) = Mary's spending

22) 

The cost of a lunch of 3 sandwiches, 7 cups of coffee and 1 donut is $3.15. The cost of a lunch of 4 sandwiches, 10 cups of coffee and 1 donut was $4.20 at the same cafe.

How much will 1 sandwich, 1 cup of coffee and 1 donut cost?

23) 

Brendan and Shawn were out for a bike ride when the tire fell off Brendan's bike. They were 20km from home. They decide that Brendan will walk for a while and Shawn will take the bike, leaving it further along and walking the rest of the way. When Brendan reaches the bike, he will bike home. Brendan walks at 5km/h and rides at 12km/h. Shawn walks at 4km/h and rides at 10km/h. How far should Shawn ride the bike for both to arrive home at the same time?

24) 

Four children are playing with marbles. At the end of the day one child has 4 less than half the marbles, the second child has 6 more than one fifth of the marbles, the third child has one third of what the first child had and the fourth child has 1 less than the third child.

How many marbles are there?

25) 

Thales of Miletus proved that any triangle inscribed in a semicircle will be a right triangle.

Given a circle of radius a and a smaller circle of radius b = 1/2a, find the ratio of the areas of the triangles ABC and DEC. Both triangles have altitudes intersecting their base at 1/4 the length of their bases.

h1 = (b (b + a))^.5

h2 = (1/2 (1/2 b + b))^.5

26) 

Given a circle with center A and radius 2. If ABCD is a parallelogram, find the area of the shaded region. Area parallelogram = b*h

27) 

The distance from an observation point to a comet at a given time is known to be 3.5 million kilometers. Given that a line drawn from the comet to the observation point is parallel to a line going through the center of the earth, find the values of x and y.

28) 

A pennant is made in the shape of an isosceles triangle. A baseball team wants to fly two of its pennants side by side. The pennants are of different sizes and the team wants the flag poles spaced so that the distance between them is h + g.

Given the following information...

        a = 2 * (3^0.5)         c = 2

        d = 3^0.5              b = (28^0.5)/2 = 7^0.5

How far apart will the flag poles be?

29) 

Given a figure with squares ABCD and EFGD not equal, and knowing CE = CD and angle DEC = 40 degrees, find angle BDG.

30) 

Find the area of triangle ABC.

31) 

The algebraic equation of a circle with its center at (0,0) is x^2 + y^2 = r^2. Given a circle of radius 4 and a chord of this circle which is the perpendicular bisector of a radius, find the length of a chord.

32) 

Given a triangle of area 5 perform a series of reflections to create a solid figure of area 40 (want 360 degrees about point 0) using only the x and y axes and the line y = x.

33) 

The lookout on a pirate ship is keeping his eyes open for land. From his perch 6m above the water he can only look down at a 60 degree angle. He cannot see the land (x) from his position at t = 0 but in 3 seconds it comes into view. What was the distance from the ship's position at t=0 to land?

34) 

Given the following information find the area of triangle AEB.

     Lines:

        The length of AE is not equal to the length of AD

        EB is parallel and equal in length to DC

        The length of AD=the length of DC

        The area of triangle ADC =8

        The area of triangle BDC=3

        Angle DAB is a right angle

35) 

The Egyptians used a hexagon (all sides equal) incribed in a square to approximate the area of a circle. If the radius of the circle is 2 and the circle is tangent to all four sides of the square, find the area of the circle, without the use of pi, assuming that the hexagon trisects each side of the square.

36) 

In a soccer tournament the average of 8 goals scored in Caratine's first 5 games was 6.4. The average of her next five games was 6.5. If there were 9 goals scored in the tenth game what was the overall average.

37) 

In a class of four students there are four marks given to students A, B, C, and D. The average of the first two marks is 50. The average of the second and third marks is 75 and the average of the third and fourth marks is 70. What is the average of the first and fourth marks?

38) 

A right triangular prism has edges in the ratio 3:4:5:10. If the volume is 202.5 units find the actual length of the longest side.

39) 

A woman who runs two breakfast cafes wants to know how much it will cost to supply each if one uses 1066 slices of bread per week and 48L of orange juice and the other uses 988 slices of bread per week and 56L of orange juice. One slice of bread costs 1.5 cents and 1 L of orange juice costs 99 cents. Use matrices to solve this problem.

40) 

Form a quadratic equation with rational coefficients having 2 - 4(3)^.5 as one of its roots.

41) 

The average age of a group of teachers and professors is 40. If the teachers average 35 years and the professors 50 years, then what is the ratio of the number of teachers to professors?

42) 

The graph of y = (x+2) * (x-3) intersects the x axis at points A and B. Find the length of AB.

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