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MATHCOUNTS 1112
Warm-Up 2
Weishan Xia
[email protected]
Question #1
• What is the positive difference between the
value of 2 × (3 + 4) and the value of 2 × 3 + 4?
What is the order of operations?
Order of Operations
The rules of which calculation comes first in an expression
1. do everything inside parentheses first: ()
2. then do exponents, like 52, 103, 84, x10, etc.
3. then do multiplies and divides from left to right
4. lastly do the adds and subtracts from left to right
Example:
5 × (3 + 4) - 8 ÷ 2
=5×7-8÷2
= 35 - 4
= 31
Question #1
• What is the positive difference between the
value of 2 × (3 + 4) and the value of 2 × 3 + 4?
2 × (3 + 4)
= 2 x 7 = 14
2×3+4
= 6 + 4 = 10
14 – 10
=4
Question #2
• Manny has 5 shirts, 3 pairs of pants, 2 ties and
4 pairs of shoes. If Manny’s school uniform
consists of a shirt, a pair of pants, a tie and a
pair of shoes, how many different uniforms
can he wear to school?
We can use ‘Rule of Product’ to solve this problem.
What is ‘Rule of Product’?
Rule of Product
• If you have a ways to select something from a set, and
b ways to select another from another set, you have
a⋅b ways to select one element from each set.
• This also applies if you have more than 2 sets!!!
• Example:
• When you decide to order pizza, you must first choose
the type of crust: thin or deep dish (2 choices).
• Next, you choose the topping: cheese, pepperoni, or
sausage (3 choices).
• How many possible combinations of ordering a 1topping pizza?
Rule of Product
• Using the rule of product, we know that there
are 2 × 3 = 6 possible combinations of
ordering a 1-topping pizza.
• We can verify this by listing all the
combinations,
• {thin, cheese}
{deep dish, cheese}
• {thin, pepperoni} {deep dish, pepperoni}
• {thin, sausage}
{deep dish, sausage}
Question #2
• Manny has 5 shirts, 3 pairs of pants, 2 ties and 4
pairs of shoes. If Manny’s school uniform consists
of a shirt, a pair of pants, a tie and a pair of
shoes, how many different uniforms can he wear
to school?
Let’s use ‘Rule of Product’
Total number of different combinations is
5x3x2x4
= 120
Question #3
• A circular pizza was cut into 12 congruent slices, as
shown. If 2 slices were eaten, what is the sum of the
central angles of the slices that were not eaten?
What is an angle?
Angle
• An angle is the amount of turn between two straight
lines that have a common end point (the vertex).
Type of Angle
Type of Angle
Acute Angle
Description
an angle that is less than 90°
an angle that is 90° exactly
Right Angle
Obtuse Angle an angle that is greater than 90° but less than 180°
an angle that is 180° exactly
Straight Angle
Reflex Angle
an angle that is greater than 180°
Note: A full rotation is a 360 degree angle
Question #3
• A circular pizza was cut into 12 congruent slices, as shown. If 2
slices were eaten, what is the sum of the central angles of the
slices that were not eaten?
What is the sum of the total central angles?
360 degree
What is the central angle of each slice?
360 / 12 = 30 degree
What is the sum of the central angles of the slices not eaten?
30 x (12 – 2)
= 30 x 10
= 300 degree
Question #4
• A tennis court with a length of 78 ft is 6 ft longer
than twice its width. What is the width of the
tennis court?
78 – 6
= 72
72 / 2
= 36 ft
Question #4
• A tennis court with a length of 78 ft is 6 ft longer
than twice its width. What is the width of the
tennis court?
Another way to solve this problem (Equation!!!)
Let a be the width of the tennis court
2a + 6 = 78
a=?
What is an Equation?
• An equation says that two things are equal. It will
have an equals sign "=" like this:
• a-2=6
• This equations says: what is on the left (a - 2) is
equal to what is on the right (6)
• So an equation is like a statement "this equals
that“
• The letter (in this case an a) just means "we don't
know this yet", and is often called the
unknown or the variable.
How to Solve an Equation?
• Rule #1: We may add the same number to
both sides of an equation. Why?
How to Solve an Equation?
a-2=6
a–2+2=6+2
a= 6+2
a=8
• Or you can move something to the other side
of the equation by changing its sign
• In this case, we moved -2 to the right side by
changing it to +2
How to Solve an Equation?
• Rule #2: We may subtract the same number from both
sides of an equation. Why?
• Example:
b+2=6
b+2–2=6–2
b=6-2
b=4
• Or you can move something to the other side of the
equation by changing its sign
• In this case, we moved +2 from the left side to the right
side by changing it to -2
How to Solve an Equation?
• Rule #3: We may multiple the same number to
both sides of an equation. Why?
• Example:
𝐜
=6
𝟐
𝑐
x2=6x2
2
c=6x2
c = 12
• Or you can move a denominator of one side to
the other side by changing it into a numerator
How to Solve an Equation?
• Rule #4: We may divide both sides of an equation
by the same number. Why?
• Example:
2d = 6
2𝑑
2
=
𝟔
d=
𝟐
6
2
d=3
• Or you can move a numerator of one side to the
other side by changing it into a denominator
Question #4
• A tennis court with a length of 78 ft is 6 ft longer than
twice its width. What is the width of the tennis court?
Another way to solve this problem (Equation!!!)
Let a be the width of the tennis court
2a + 6 = 78
2a = 78 – 6
2a = 72
a = 72 / 2
a = 36 ft
Which Way is Better?
• Step by step calculation?
• Or using equation?
• Why?
Question #5
• Sam wishes to contribute a total of $2500 to Charity A
and Charity B, in the ratio of 2:3. How many dollars
should Sam contribute to Charity B?
In the ratio 2:3 means if you divide the total contribution
into 5 (2 + 3) equal parts, Charity A will have 2 parts and
Charity B will have 3 parts.
2500 / (2 + 3)
= 500
500 x 3
= 1500
Question #5
• Sam wishes to contribute a total of $2500 to Charity A and
Charity B, in the ratio of 2:3. How many dollars should Sam
contribute to Charity B?
Another way to solve this problem (Equation!!!)
Let b be the dollars to Charity B
2500 −b 2
=
b
3
2b
2500 – b =
3
3 (2500 – b) = 2b
7500 – 3b = 2b
7500 = 2b + 3b
7500 = 5b
b = 7500 / 5
b = 1500
Which Way is Better?
• Step by step calculation?
• Or using equation?
• Why?
Question #6
• A number is selected at random from the first 20 positive
integers. What is the probability the number selected is an
odd prime number? Express your answer as a percent.
What are the prime numbers between 1 and 20?
2, 3, 5, 7, 11, 13, 17, 19
What are the odd prime number between 1 and 20?
2, 3, 5, 7, 11, 13, 17, 19
How many odd prime number between 1 and 20?
7
What is the probability the number selected is an odd prime?
7/20
= 35/100 = 35%
Question #7
• What is the value of 5 × (11 + 4 ÷ 4)?
5 × (11 + 4 ÷ 4)
= 5 x (11 + 1)
= 5 x 12
= 60
Question #8
• The ratio of the lengths of corresponding sides of two similar
decagons is 1:2. If the perimeter of the smaller decagon is 76
cm, what is the perimeter of the larger decagon?
A decagon is a 10-sided polygon
(A polygon is a flat shape with straight sides)
Perimeter is the distance around a two-dimensional shape.
Question #8
• The ratio of the lengths of corresponding sides of two similar
decagons is 1:2. If the perimeter of the smaller decagon is 76
cm, what is the perimeter of the larger decagon?
Let the lengths of the sides of the smaller decagon be,
a, b, c, d, e, f, g, h, i, j
The lengths of the sides for the larger decagon will be,
2a, 2b, 2c, 2d, 2e, 2f, 2g, 2h, 2i, 2j
The perimeter of the smaller decagon is,
a+b+c+d+e+f+g+h+i+j
= 76
The perimeter of the larger decagon is,
2a + 2b + 2c + 2d + 2e + 2f + 2g + 2h + 2i + 2j
= 2 x (a + b + c + d + e + f + g + h + i + j)
= 2 x 76 = 152 cm
Question #9
• Each day a man floating on a raft paddles 3 mi north, but each
night while he rests, the current of the river carries the raft 2
mi south. How many days will it take him to first reach a
location 50 mi north of his starting location?
How many miles does the man go north each day and night?
3 – 2 = 1 mile
How many days will it take him to first reach a location 50 mi north of
his starting location?
50 / 1 = 50 days (Is this the correct answer?)
No (Why?)
How many days will it take him to reach a location 47 mi north of his
starting location?
47 / 1 = 47 days
The next day (before night), he will go north 3 miles to reach the
location 50 mi north of his starting location. So the correct answer is,
47 + 1 = 48 days
Question #10
• Suppose ABCDEF, shown here, is a regular hexagon with
sides of length 6 cm. What is the length of segment AD?
A hexagon is a 6-sided polygon (a flat shape with straight sides)
If all angles are equal and all sides are equal, then it is regular
Otherwise it is irregular
Triangle
• A triangle is a 3-sided polygon
• A triangle has three sides and three angles
• The three angles always add to 180° (why?)
Vertical Angles
• Vertical Angles are the angles opposite each other when two
lines cross.
• In this example, a° and b° are vertical angles
• Also called opposite angles or vertically opposite angles
• Vertical Angles are always equal. Why?
• Can you find more vertical angles in the following diagram?
Vertical Angles
•
•
•
•
Example: Find angles a°, b° and c° below:
Because b° is vertically opposite 40°, it must also be 40°
A full circle is 360°, so that leaves 360° - 2×40° = 280°
Angles a° and c° are also vertical angles, so must be equal,
which means they are 140° each. (280° / 2)
Parallel Lines
• Two lines on a plane that never meet.
• They are always the same distance apart.
• The red line is parallel to the blue line in both these cases:
Transversal
• A Transversal is a line that crosses at least two other lines.
• When a transversal crosses 2 parallel lines, it creates 8 angles:
– a, b, c, d, e, f, g, h
Corresponding Angles
•
•
•
•
•
•
•
•
•
•
•
The angles in matching corners are called Corresponding Angles.
E.g., a and e are Corresponding Angles
Can you find more Corresponding Angles?
b and f
c and g
d and h
Corresponding Angles are equal!
a=e
b=f
c=g
d=h
Alternate Interior Angles
• The pairs of angles on opposite sides of the transversal but
inside the two lines are called Alternate Interior Angles.
• E.g., c and f are Alternate Interior Angles
• Can you find more Alternate Interior Angles?
• d and e
• Alternate Interior Angles are equal!
• Why?
• Because c = b (why?)
• And because b = f (why?)
• So c = f
• Can you proof d = e?
Consecutive Interior Angles
• The pairs of angles on one side of the transversal but inside
the two lines are called Consecutive Interior Angles..
• E.g., c and e are Consecutive Interior Angles
• Can you find more Consecutive Interior Angles?
• d and f
• c + e = 180° and d + f = 180°
• Why?
• Because e = d (why?)
• And because c + d = 180° (why?)
• So c + e = 180°
• Can you proof d + f = 180°?
Triangle
•
•
•
•
•
•
•
•
Draw a parallel line at the top and it creates 2 new angles: d and e
a = d (why?)
Because a and d are Alternate Interior Angles!
b = e (why?)
Because b and e are Alternate Interior Angles!
a+b+c=d+e+c
= 180 degrees (why?)
e
d
c
Because it’s a straight angle!
a
b
Isosceles Triangle
• A triangle with two equal sides
• The angles opposite the equal sides are also
equal
Equilateral Triangle
• A triangle with all three sides of equal length.
• All the angles are equal (60°, why?)
Question #10
• Suppose ABCDEF, shown here, is a regular hexagon with sides of
length 6 cm. What is the length of segment AD?
Connect AD, BE and FC to make 6 small triangles inside the hexagon
All these small triangles are Isosceles Triangles (Why?)
Actually, all these small triangles are also Equilateral Triangles (Why?)
Equilateral Triangle is a triangle with all three sides of equal length.
The length of segment AD is 6 + 6 = 12 m