ISEE Upper Level Quantitative Reasoning - How to find the solution to an equation

Which of the following makes the equation true:

$\frac{x}{5} +9 = 12$

Explanation

To find the answer, we will solve for x. So, we get

$\frac{x}{5} +9 = 12$

$\frac{x}{5} +9 -9 = 12-9$

$\frac{x}{5} + 0 = 3$

$\frac{x}{5} = 3$

$\frac{x}{5} \cdot 5 = 3 \cdot 5$

Solve the equation for :

$\frac{e^{3x}}{e^x}=3$

Explanation

First, simplify the equation as much as possible:

$\frac{e^{3x}}{e^x}=3\Rightarrow e^{(3x-x)}=3\Rightarrow e^{2x}=3$

Now we can take the of both sides:

$e^{2x}=3\Rightarrow ln\ e^{2x}=ln\ 3$

$\Rightarrow 2x=ln\ 3$

$\Rightarrow x=\frac{ln\ 3}{2}=\frac{1.098}{2}$

$\Rightarrow x=0.549$

A class has 25 students. If 60% of them are boys, how many students are girls?

Explanation

We know that 40% of the class are girls because 60% are boys. To express 40% as a fraction of the 25 total students, we set up a proportion to find the number of girls in the class.

$\frac{40}{100} = \frac{x}{25}$

$\frac{2}{5}=\frac{x}{25}$

When solving for , we find that there are 10 girls in the class.

Adam has 3 siblings. When his mother bakes a cake, each of the 4 children is given ¼ of the cake. Adam only finished 1/3 of his share of cake. If the original cake had 12 slices, how many slices did Adam eat?

Explanation

Each of the children gets 1/4 of the cake. Taking 1/4 of 12 shows that each child gets 3 slices. If Adam finishes 1/3 of his share, then he has eaten one out of the 3 slices.

A small bakery sells two flavors of muffins: chocolate and banana.

At the end of the day, there are 13 muffins left.

There are chocolate muffins and banana muffins.

$x=\frac{7}{2}\cdot \frac{4}{7}-1$

how many banana muffins are left?

Explanation

The first step is to solve for x.

$x=\frac{7}{2}\cdot \frac{4}{7}-1$

If there are 1x chocolate muffins and 2y banana muffins, for a total of 13 muffins then:

Since the number of banana muffins remaining is 2y, there are 12 banana muffins left.

On average, 1 in 9 peanuts that are harvested will only have one pod instead of two pods. Of those one-pod peanuts, three quarters of them will be smaller than the average peanut.

If 900 peanuts are harvested, how many will be one-pod peanuts that are smaller than average in size?

Explanation

If there are 900 peanuts and 1 in 9 will be one-podded, that means that 100 will be one-podded.

$900\cdot \frac{1}{9}=100$

Three quarters of these 100 one-podded peanuts will be of smaller than average size. Three quarters of of 100 is 75. Therefore, 75 is the correct answer.

Julie sells cars. She has a monthly salary of $1000 and a commission of $500 for each car sold. How much money will she make this month if she sells 7 cars?

$4500

$3500

$7000

$8000

$1500

Explanation

To find out how much Julie makes from selling the cars, we first multiply the 7 cars she sells by the $500 that she receives for each car. This gives us $3,500. We now add the $1,000 that she gets for her monthly salary for a total of $4,500.

Fifteen percent of the students in the classroom brought their lunch from home. If 9 students brought their lunch from home, how many students are in the classroom total?

135

Explanation

You can think of the total number of students in the classroom as an unknown variable, .

You can set up an equation that includes and the information given. Normally in a percent problem, you would start with the total amount of something and multiply it by the percent to get the part of that total amount. You can set this up the same way, but the total amount is the missing information: