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MATHCOUNTS 1112
Warm-Up 1
Weishan Xia
[email protected]
Question #1
• How many months are there in 35 years?
How many months are there in 1 year?
12
How many months are there in 35 years?
12 x 35
= 420
= (10 + 2) x 35
= 10 x 35 + 2 x 35
= 350 + 70
= 420
Question #1.2
• How many minutes are there in 7 days?
How many minutes are there in 1 hour?
60
How many hours are there in 1 day?
24
How many minutes are there in 1 day?
60 x 24
= 1440
How many minutes are there in 7 days?
1440 x 7
= 10080
Question #2
• A function exists of the form y = x + a that models the data
shown. What is the value of a?
X
y
2
5
3
6
4
7
y=x+a
y – x = (x + a) – x
y–x =a
a=y-x
=5–2=3
Question #3
• How many odd numbers are there between 20 and 158?
How many numbers between 20 and 22?
22 – 20 – 1 = 1
How many numbers between 20 and 24?
24 – 20 – 1 = 3
How many numbers between 20 and 158?
158 – 20 – 1 = 137
{21, 22, 23, 24, … , 155, 156, 157}
How many odd numbers?
{(21, 22), (23, 24), … , (155, 156), 157}?
136 / 2 + 1 = 68 + 1 = 69
Question #4
• The figure shown is a square. What is the value of x?
x-2
3
All sides of a square have equal length
x–2=3
x–2+2=3+2
x=5
Question #5
• A bowl contains 50 grapes of different colors. If
20% of the grapes in the bowl are red, how many
grapes are not red?
Percent means parts per 100
The symbol is %
Example: 20% means 20 per 100
20% = 20/100
= 1/5
= 0.2
Question #5
• A bowl contains 50 grapes of different colors. If 20% of
the grapes in the bowl are red, how many grapes are
not red?
How many red grapes?
50 x 20%
= 50 x 20 / 100
= 10
How many grapes not red?
50 – 10
= 40
Question #6
• In the figure, a circle with radius 4 ft is inscribed in a
square. What is the area of the square?
L
d
r
Diameter of the circle (d) = radius (r) x 2 = 4 x 2 = 8 ft
Length of the square (L) = Diameter of the circle (d) = 8 ft
Area of the square = L x L = 8 x 8 = 64 square ft
Question #7
• A triangle with an area of 120 mm2 has a height of 10 mm.
What is the area of a similar triangle with a height of 20
mm?
20
10
b
2b why?
b x 10 / 2 = 120
b = 120 / 5 = 24
Big triangle area = 2b x 20 / 2 = 48 x 20 / 2
= 480
Question #8
• A bookshelf is to hold 6 literature books, 3
geometry books and 7 algebra books,
arranged in random order. What is the
probability that the books are arranged so
that the first book on the shelf is a math
book? Express your answer as a common
fraction.
What is probability?
Probability
"Probability" (or "Chance") is how likely something is to happen.
Example 1:
What is the probability of getting a "Head" when tossing a coin?
So how do we calculate probability?
(Number of ways it can happen) / (Total number of outcomes)
Number of ways it can happen: 1 (Head)
Total number of outcomes: 2 (Head and Tail)
So the probability = 1/2
= 0.5
Probability
Example 2:
What is the probability of getting a "5" or "6" when rolling a die?
Number of ways it can happen: 2 ("5" and "6")
Total number of outcomes: 6 ("1", "2", "3", "4", "5" and "6")
So the probability = 2/6
= 1/3
= 0.333...
Question #8
• A bookshelf is to hold 6 literature books, 3 geometry books
and 7 algebra books, arranged in random order. What is the
probability that the books are arranged so that the first book
on the shelf is a math book? Express your answer as a
common fraction.
Number of ways it can happen:
10 (3 geometry books and 7 algebra books)
Total number of outcomes:
16 (All the books: 6 + 3 + 7)
So the probability is,
= 10 / 16
=5/8
Question #9
• The volume of a rectangular prism is 120 in3. Its
length and width are 5 in and 4 in, respectively, as
shown. What is the height of the prism?
Height
4 in
5 in
Volume = Length x Width x Height
120 = 5 x 4 x Height = 20 x Height
Height = 120 / 20 = 6 in
Question #10
• What is the greatest prime factor of 96?
A Prime Number can be divided evenly only by 1, or itself
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37…
A prime factor is a factor that is a prime number
E.g. For 24, we can find the prime factorization by taking 40 and
dividing it by the smallest prime number that goes into
40 ÷ 2 = 20
20 ÷ 2 = 10
10 ÷ 2 = 5
Since 5 is prime, we're done factoring,
and the prime factorization is 2×2×2×5.
Question #10
• What is the greatest prime factor of 96?
For 96, we can find the prime factorization by taking 96 and
dividing it by the smallest prime number that goes into,
96 ÷ 2 = 48
48 ÷ 2 = 24
24 ÷ 2 = 12
12 ÷ 2 = 6
6÷2=3
Since 3 is prime, we're done factoring,
and the prime factorization of 96 is 2×2×2×2x2x3
So the greatest prime factor of 96 is 3