Catarina

Let the amount of money Mr. Smith invested at \(8\)% be \(m\). This makes the amount of money Mr. Smith invested at \(5\)% equal to \(2m\).

Now we can create an equation:

 \($0.08m + 0.05 \cdot $2m = $720\)

Simplifying the equation: 

\(0.05 \cdot $2m = $0.1m\).

And the simplified equation: 
\($0.08m + $0.1m = $720\)

Adding like terms: 

\($0.18m = $720\)

Solving for \(m\):

\({0.18m \over 18} = {720 \over 18}\)

\(0.01m = $40\)

\(0.01m \cdot 100 = 40 \cdot 100\)

\(m = 4000\)

Mr. Smith invested \($4000\) at \(8\)%, so he invested \($8000\) at \(5\)%.

ANSWER: \($8000\).

I hope this is helpful,

Catarina

You can do anything to a fraction as long as you do it to the numerator and the denominator and the fraction will still represent the same amount, so let's make 4.5 fifths into a clearer number by multiplying the numerator and denominator by 2.

\({4.5 \over 5} = {{4.5 \cdot 2} \over {5 \cdot 2}} = {9 \over 10}\)

ANSWER: 

in fraction form: \(9 \over 10\)

in decimal form: \(0.9\)

in percent form: \(90%\)%

I hope this is helpful,

Catarina

a) A starting lineup of five players that includes Corey and Peter is: Corey, Pater, and three other players. These three must be chosen from the eight that are not Corey and Peter. 

Choosing three people out of eight when order doesn't matter makes: \({{8 \cdot 7 \cdot 6} \over {3 \cdot 2 \cdot 1}} = 56\) starting lineups. 

a) ANSWER: \(56\) starting lineups. 

b) A starting lineup that includes Corey but not Peter is: Corey and four other players who are not Peter. These four must again be chosen from the eight that are not Corey and Peter. 

Choosing four people out of eight where order doesn't matter makes: \({{8 \cdot 7 \cdot 6 \cdot 5} \over {4 \cdot 3 \cdot 2 \cdot 1}} = 70\) starting lineups. 

A starting lineup that includes Peter but not Corey is: Peter and four other players who are not Corey. There is the same number of these lineups as there are lineups with Corey but not Peter: \(70\) starting lineups. 

 The number of lineups with both Corey and Peter has already been calculated: \(56\).

So the total number of lineups with Corey, Peter, or both is: \(70 + 70 + 56 = 196\).

b) ANSWER: \(196\) starting lineups. 

I hope this is helpful,

Catarina 

Division by \(0\) is undefined, so \(1\) cannot be divided by \(0\).

1 divided by 0 = infinity. (There is some LaTeX trouble today )

I hope this is helpful,

Catarina

Let's start by assuming that everyone who went to the premiere was an adult (\(174 \) adults and \(0\) children).Then how much money would Mega Movies have made?

\(174 \cdot $8 = $1392\).

That is too much, so there must have been some children. For every adult replaced by a child, Mega Movies lost \($2\) from their total, since a child ticket cost \($2\) less than an adult ticket. 

They lost a total of \($1392 - $1342 = $50\). 

\({$50 \over $2} = 25\), so \(25\) children went to the film premiere. 

\(174 - 25 = 149\), so the answer is: 
 

\(149 \) adults and \(25\) children. 

I hope this is helpful,

Catarina 

IF THE ORDER OF THE JOKERS AND FRUIT DOES NOT MATTER: 

1. All you have to do is choose the fruit to appear, so there are \(5\) possible ways to win. 

2. You have to choose the first fruit (five ways) and choose the second fruit (five ways) so there are \(25\) possible ways to win. 

IF THE ORDER OF THE JOKERS AND FRUIT DOES MATTER:

1. Choose the slot that will not have a joker (three ways) and the fruit to appear in it (five ways), so there are \(15\) possible ways to win. 

2. Two cases. 

Case 1: Joker and two different fruits. 

Choose the slot that will have a joker (three ways) and the slot that will have one of the fruits (two ways) and then that fruit (five ways) and then the second fruit (four ways), so there are \(120\) possible ways to win.

Case 2: Joker and two of the same fruit.

Choose the slot that will have a joker (three ways) and then the type of fruit (five ways), so there are \(15\) possible ways to win.

ANSWER: \(135\) ways to win. 

I hope this is helpful,

Catarina

I tried this and it's not correct. I don't think the answer is supposed to be this complicated. Maybe we're missing something obvious here?

Thank you for the help!

--Catarina

The equation \(y = x - 3\) is in the form \(y = mx + b\). 

In this form, \(m\) represents the slope of the line and \(b\) represents the y-intercept, which in this case is \(-3\). Since \(m\) is not visible here, \(m = 1\), because the equation \(y = 1x - 3\) is the same as the equation \(y = x - 3\).

The slope of this line is \(1\). 

I hope this is helpful.

--Catarina

The correct answer is D: (-2,1) because the position of the point before the reflection is 2 units to the right of the y-axis and 1 unit above the x-axis. Reflecting the point across the y-axis does not change the position relative to the x-axis. It also does not change the number of units away from the y-axis, only the direction away, so instead of 2 units to the right, the point will be 2 units to the left, at the position (-2,1). 

It is false that perpendicular lines hae the same slopes. Parallel lines have the same slope, perpendicular lines have opposite slopes. The slopes of perpendicular lines are negative reciprocals. For example, if line A has a slope of 2, then line B, which is perpendicular to line A, will have a slope of \(-{1 \over 2}\). 

I hope this is helpful to you.

--Catarina.