Properties & Exponent Rules
Evaluate, using & naming one or more properties.
\frac{1}{7}\cdot 125000\cdot 0.1739\cdot 56
Using the commutative property:
\frac{1}{7}\cdot 125000\cdot 0.1739\cdot 56
\frac{1}{7}\cdot 56 \cdot 125000\cdot 0.1739
=8\cdot 125000\cdot 0.1739
=1000000\cdot 0.1739
=173900
Express 56.0078 as a mixed number in simple form.
56.0078
=56\frac{78}{10000}
=56\frac{39}{5000}
Write & solve a proportion:
The tax on a $24 restaurant meal was $1.44. Find the tax on a restaurant meal costing $35.
\frac{1.44}{24}=\frac{x}{35}
50.4=24x
2.1=x
The tax on a $35 meal is $2.10.
True or False? If a whole number is not prime, then it is even.
False! e.g. 1, 9, 15, etc.
Suppose you roll a fair 12-sided die. What is the probability of rolling an even number or a number less than 11?
P(even or less than 11) =\frac{6}{12}+\frac{10}{12}-\frac{5}{12}
=\frac{11}{12}
Write in scientific notation:
(3.81\times 10^11)\times(1.2\times 10^109)
(3.81\times 10^11)\times (1.2\times 10^109)
=(3.81\times1.2)(10^11\times10^109)
=4.572\times 10^20
What is the 100th digit to the right of the decimal point in the decimal representation of \frac{4}{37} ?
\frac{4}{37}=0.\bar{108}
so the 100th decimal digit is 1.
Express the following ratio in lowest terms:
0.62 : \frac{1}{500}
0.62 : \frac{1}{500}
=310:1
True or False? The product of two prime numbers is a composite number.
True! If
p,q
are (not necessarily distinct) primes, then
p|pq
is a nontrivial prime factor of
pq
You have one penny, one nickel, one dime, and one quarter in your pocket. You select two coins at random. What is the probability that have taken at least 25 cents from your pocket?
P(at least 25c)=\frac{1}{2} (several strategies)
Evaluate.
\sqrt{8}\cdot\sqrt{18}\cdot 28
\sqrt{8}\cdot\sqrt{18}\cdot 28
=\sqrt{8\cdot 18}\cdot 28
=\sqrt{144}\cdot 28
=12\cdot 28
=336
Does the fraction \frac 121{34375} have a terminating or repeating decimal representation? Be prepared to explain!
Terminating! We have
\frac{121}{34375}
=\frac{11}{3125}
=\frac{11}{5^5}
Write & solve a proportion:
The ratio of the measure of a certain angle's supplement to the measure of its complement is 11:3. What is the measure of the angle?
Let x= measure of the angle.
\frac{180-x}{90-x}=\frac{11}{3}
x=56.25^\circ
True or false? If two coplanar rays are not parallel, they intersect.
False! Can draw counterexample.
Suppose I roll two fair 6-sided dice. What is the probability that the two rolls are different?
P(different rolls) = 1 - P(same roll)
=1-\frac{6}{36}
=\frac{5}{6}
Evaluate.
99999^2+99999
99999^2+99999
=99999(99999+1)
=99999(10000)
=999,990,000
0.3\overline{24} is the reciprocal of what mixed number?
0.3\overline{24}=\frac{107}{330} is the reciprocal of
3\frac {9}{107}
Answer using a proportion:
The ratio of the sum of two numbers to the difference between them is 5:1. What is the ratio of the two numbers?
\frac{a+b){a-b}=\frac{5}{1}
5(a-b)=a+b
5a-5b=a+b
4a=6b
\frac{a}{b}=\frac{6}{4}
\frac{a}{b}=\frac{3}{2}
True or false? If j,k,l are coplanar lines with j||k, j\botl, then k\botl .
True! Property of transversals between parallel lines.
A certain box contains 4 grey books, 6 red books, and 2 blue books. If I randomly select 3 books from the box without replacement, what is the probability that they are all the same color?
P(all same) = P(all grey) + P(all red)
=(\frac{4}{12})^3+(\frac{6}{12})^3
\frac{35}{216}
Express as a power of 2:
2^20-2^19-2^18
2^20-2^19-2^18
=2^18(2^2-2-1)
=2^18(4-2-1)
=2^18(1)
=2^18
Compare
\frac{998}{999}
and
0.98\bar{3}
using >,<,=.
Since
0.98\bar{3}=\frac{59}{60}
Then,
\frac{59}{60}=1-\frac{1}{60}
\frac{998}{999} = 1-\frac{1}{999}
\frac{998}{999}>0.98\bar{3}
Daisy had $66 in savings, while her older brother Oliver had $82. One day, Daisy & Oliver each donated the same amount of money to charity. The ratio of their savings became 3:7. How much money do they each have left?
Let x= amount donated. Then,
\frac{66-x}{82-x}=\frac{3}{7}
(66-x)(7)=(82-x)(3)
462-7x=246-3x
462=246+4x
216=4x
54=x
Daisy has 66-54= $12 left, while Oliver has 82-54= $28 left.
True or False? If a proper fraction's denominator is 1728, then the fraction has a repeating decimal representation.
False! For example,
\frac{27}{1728}=\frac{1}{64}=\frac{1}{2^6}
Two-fifths of the students at Central Middle School are boys. One-third of the girls and one-quarter of the boys have blonde hair. If a student is selected at random, what is the probability that they have blonde hair?
P(boy, blonde) =
\frac{2}{5}\cdot\frac{1}{4}=\frac{1}{10}
P(girl, blonde) =
\frac{3}{5}\cdot\frac{1}{3}=\frac{1}{5}
P(blonde) =
\frac{1}{10}+\frac{1}{5}=\frac{3}{10}