- Simplify: \( 8-15/3+1 \)
- In Atlanta, GA the record low temperature is \( -9^{\circ}\text{F} \) and the record high is \( 105^{\circ}\text{F} \). What is the temperature difference between these records?
- Simplify: \( \frac{1}{2}+\frac{2}{3}-\frac{3}{4} \)
- Mikkel bakes a pie. He then eats \( \frac{1}{3} \) of the pie and puts the rest in the fridge. The next day he and his friends eat \( \frac{3}{4} \) of what is left of the pie. What amount of the original pie is still leftover?
- Simplify: \( \frac{4}{5}\div\frac{2}{3}-\frac{7}{10} \)
- Alan works for 4.6 hours at $4.75/hour. During this time he also makes $82.63 in tips. How much money has Alan earned?
- In April 2010, the population of India was estimated at 1.18 billion. Express this number in scientific notation.
- Simplify: \( \sqrt{3^{4}} \)
- What value of \( x \) solves the proportion \( \frac{x}{9}=\frac{5}{15} \) ?
- If 9 sweet potatoes costs $6.75, how much do 5 sweet potatoes cost?
- 22,000 people voted in an election where the winning candidate won with 54% of the votes. How many votes did the winning candidate receive?
- 60% of the students in a college class are over 25 years old. Of the remain students, 50% are younger than 21. What percentage of the students are between 21 and 25?
- Four people are in a room. If you add up all of their ages and divide this number by 10 you get 14.8. What is the average age of the people in the room?
- A total of 40 math and physics majors were given a test. The average score of the 15 physics majors was 66. The average score of the 25 math majors was 62. What was the average score of the entire group?
- If \( x=4 \), what is the value of \( \frac{x^{3}+1}{x+1} \) ?
- Given \( a=1, b=-2, \text{ and } c=5 \), what is the value of \( \frac{(c-a)^{2}-b}{bc-2a} \) ?
- In a certain ecosystem the population of sheep (\( s \)) is \( 15% \) greater than \( 500 \) minus the population of wolves (\( w \)). Express \( s \) in terms of \( w \).
- One student scored 1 on a test, two scored 2, three scored 3, four scored 4, and x scored 5. If the average score was 3.75, what is x?
- What is the sum of the polynomials \( 4a^{2}b^{2}-3a^{2}b+ab \) and \( ab+a^{2}b+5ab^{2}-3a^{2}b^{2} \) ?
- If \( 5x+1=2(x-4) \), then \( x=? \)
- Factor the polynomial: \( x^{2}-6x+8 \)
- If \( \frac{1}{3}x-\frac{1}{2}=\frac{1}{4} \), then \( x=? \)
- For nonzero values of \( x \), \( y \), and \( z \), simplify: \( \frac{-2x^{3}y^{2}z}{6x(yz)^{6}} \)
- For all \( x\neq-1 \), simplify the expression: \( \frac{x^{2}+6x+5}{x+1} \)
- For \( x,y>0 \), rationalize the denominator of \( \frac{-\sqrt{x}}{2\sqrt{x}-\sqrt{y}} \)
- For all \( x \) values in the domain, simplify the expression \( \frac{16-x^{2}}{x^{2}+9x+20} \)
- What is the y-intercept of the line with the equation \( 3x+2y+6=0 \) ?
- Carlos has worked at a company five years longer than Katie. Katie has worked at the company half the time of Carlos. How long have the both worked at the company for?
- At what point do the lines \( y=2x-1 \) and \( x+y=5 \) intersect?
- What is equation of the line parallel to \( y=\frac{1}{2}x \) that passes through the point \( (2,2) \) ?
Solutions
- \( 8-15/3+1=8-5+1=4 \)
- \( 105-(-9)=105+9=114^{\circ}\text{F} \)
- Solution: $$
\begin{align*}
& \frac{1}{2}+\frac{2}{3}-\frac{3}{4} = \\[5pt]
& \frac{6}{12}+\frac{8}{12}-\frac{9}{12} = \\[5pt]
& \frac{5}{12}
\end{align*}
$$
- After the first day \( \frac{2}{3} \) of the pie is leftover. On the second day \( \frac{3}{4} \) of this is eaten, leaving \( \frac{1}{4} \) of \( \frac{2}{3} \) of the original pie, or: \( (\frac{2}{3})(\frac{1}{4})=\frac{1}{6} \).
- Solution: $$
\begin{align*}
& \frac{4}{5}\div\frac{2}{3}-\frac{7}{10} = \\[5pt]
& \frac{4}{5}\cdot\frac{3}{2}-\frac{7}{10} = \\[5pt]
& \frac{12}{10}-\frac{7}{10} = \\[5pt]
& \frac{5}{10}=\frac{1}{2}
\end{align*}
$$
- \( 4.6\cdot4.75+82.63=\$104.48 \)
- \( 1.18 \text{ billion}=1,180,000,000=1.18\cdot10^{9} \)
- \( \sqrt{3^{4}}=\sqrt{(3^{2})^{2}}=3^{2}=9 \)
- Solution: $$
\begin{align*}
& \frac{x}{9}=\frac{5}{15} \rightarrow \\[5pt]
& \frac{x}{9}=\frac{1}{3} \rightarrow \\[5pt]
& 3x=9 \rightarrow \\[5pt]
& x=3
\end{align*}
$$
- Solution: $$
\begin{align*}
& \frac{x}{5}=\frac{6.75}{9} \rightarrow \\[5pt]
& 9x=5(6.75) \rightarrow \\[5pt]
& x=\$3.75
\end{align*}
$$
- \( 0.54(22,000)=11,880 \text{ votes} \)
- If 60% are over 25, then 40% are under 25. Since half of this 40% are under 21, \( (0.5)(0.4)=0.2=20\% \) are between 21 and 25.
- The average age in the room will be the sum of the ages divided by \( 4 \). Since the sum of ages divided by \( 10 \) is equal \( 14.8 \), the sum of the ages must be \( 10\cdot14.8=148 \), and therefore the average is \( 148/4=37 \).
- \( \frac{15(66)+25(62)}{40}=63.5 \)
- \( \frac{(4)^{3}+1}{(4)+1}=\frac{65}{5}=13 \)
- Solution: $$
\begin{align*}
& \frac{((5)-(1))^{2}-(-2)}{(-2)(5)-2(1)}= \\[5pt]
& \frac{4^{2}+2}{-10-2}= \\[5pt]
& \frac{18}{-12}= \\[5pt]
& -\frac{3}{2}
\end{align*}
$$
- \( s=1.15(500-w) \)
- Solution: $$
\begin{align*}
& \frac{1^{2}+2^{2}+3^{2}+4^{2}+5x}{1+2+3+4+x}=3.75 \rightarrow \\[5pt]
& 30+5x=3.75(10+x) \rightarrow \\[5pt]
& 30+5x=37.5+3.75x \rightarrow \\[5pt]
& 1.25x=7.5 \rightarrow \\[5pt]
& x=6
\end{align*}
$$
- Solution: $$
\begin{align*}
& 4a^{2}b^{2}-3a^{2}b+ab+(ab+a^{2}b+5ab^{2}-3a^{2}b^{2})= \\[5pt]
& (4-3)a^{2}b^{2}+(-3+1)a^{2}b+5ab^{2}+(1+1)ab= \\[5pt]
& a^{2}b^{2}-2a^{2}b+5ab^{2}+2ab
\end{align*}
$$
- Solution: $$
\begin{align*}
& 5x+1=2(x-4) \rightarrow \\[5pt]
& 5x+1=2x-8 \rightarrow \\[5pt]
& 3x=-9 \rightarrow \\[5pt]
& x=-3
\end{align*}
$$
- \( x^{2}-6x+8=(x-4)(x-2) \)
- Begin by multiplying both sides of the equation by \( 3 \) to get: $$
\begin{align*}
& x-\frac{3}{2}=\frac{3}{4}\rightarrow \\[5pt]
& x=\frac{3}{4}+\frac{3}{2}\rightarrow \\[5pt]
& x=\frac{9}{4}
\end{align*}
$$
- Solution: $$
\begin{align*}
& \frac{-2x^{3}y^{2}z}{6x(yz)^{6}}= -\frac{1}{3}x^{3-1}y^{2-6}z^{1-6}= \\[5pt]
& -\frac{1}{3}x^{2}y^{-4}z^{-5}= \frac{-x^{2}}{3y^{4}z^{5}}
\end{align*}
$$
- \( \frac{x^{2}+6x+5}{x+1}=\frac{(x+1)(x+5)}{x+1}=x+5 \)
- \( \frac{-\sqrt{x}}{2\sqrt{x}-\sqrt{y}}\cdot\frac{2\sqrt{x}+\sqrt{y}}{2\sqrt{x}+\sqrt{y}}=\frac{-(2x+\sqrt{xy})}{4x-y}=\frac{2x+\sqrt{xy}}{y-4x} \)
- \( \frac{16-x^{2}}{x^{2}+9x+20}=\frac{-(x+4)(x-4)}{(x+4)(x+5)}=\frac{4-x}{x+5} \)
- By converting the equation into slope intercept form \( y=-\frac{3}{2}x-3 \), we can see that the y-intercept occurs at \( (0,-3) \).
- Let c and k represent the number of years Carlos and Katie have worked at the company. From the problem we can derive the equations: \( c=k+5 \) and \( c=2k \). From these we get: \(k+5=2k \rightarrow k=5 \). Now substituting this value into either of our original equations yields \( c=10 \). Hence Carlos has worked at the company for 10 years, and Katie has worked at the company for 5 years.
- Solving the system of equations gives the point \( (x,y) \) of intersection. We can begin by substituting the first equation into the second to get: $$ x+(2x-1)=5\rightarrow \\ 3x=6 \rightarrow \\ x=2 $$ Now substituting this value into \( y=2x-1 \) gives: \( y=2(2)-1=3 \) Therefore, the two lines intersect at the point \( (2,3) \).
- Since the equation is parallel to \( y=\frac{1}{2}x \), it must have the same slope, so the equation will have the form \( y=\frac{1}{2}x+b \). We can solve for \( b \) by substituting the values of the point \( (2,2) \) into this equation: $$ (2)=\frac{1}{2}(2)+b\\\rightarrow b=2-1=1 $$ So the equation we’re looking for is \( y=\frac{1}{2}x+1 \).