ISEE Upper Level Math : ISEE Upper Level (grades 9-12) Mathematics Achievement

Study concepts, example questions & explanations for ISEE Upper Level Math

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Example Questions

Example Question #133 : Algebraic Concepts

Solve for \(\displaystyle x\):
\(\displaystyle x^2-4=5\)

Possible Answers:

\(\displaystyle x=\pm3\)

\(\displaystyle x=\pm1\)

\(\displaystyle x=1\)

\(\displaystyle x=3\)

Correct answer:

\(\displaystyle x=\pm3\)

Explanation:

\(\displaystyle x^2-4=5\)

\(\displaystyle x^2-4+4=5+4\)

\(\displaystyle x^2=9\)

\(\displaystyle \sqrt{x^2}=\sqrt9\)

\(\displaystyle x=\pm3\)

Example Question #141 : Algebraic Concepts

What is the value of the expression \(\displaystyle \sqrt{169-144}\) ?

Possible Answers:

\(\displaystyle 1\)

\(\displaystyle 5\)

\(\displaystyle 6\)

\(\displaystyle 25\)

\(\displaystyle 4\)

Correct answer:

\(\displaystyle 5\)

Explanation:

The first step here is to recognize you need to simplify within the radical before moving outside of the radical (avoid a common mistake of getting the square root of 169 and 144 before subtracting). Thus, you subtract 144 from 169, which gives you a difference of 25:

\(\displaystyle \sqrt{169-144}\) = \(\displaystyle \sqrt{25}\) = \(\displaystyle 5\)

Then you can take the square root of 25, which gives you the answer of 5.

Example Question #142 : Algebraic Concepts

Solve for \(\displaystyle x\):

\(\displaystyle \sqrt{x}+7=9\)

Possible Answers:

\(\displaystyle x=4\)

\(\displaystyle x=2\)

\(\displaystyle x=\sqrt2\)

\(\displaystyle x=-2\)

Correct answer:

\(\displaystyle x=4\)

Explanation:

\(\displaystyle \sqrt{x}+7=9\)

\(\displaystyle \sqrt{x}+7-7=9-7\)

\(\displaystyle \sqrt{x}=2\)

\(\displaystyle (\sqrt{x})^2=2^2\)

\(\displaystyle x=4\)

Example Question #141 : Equations

A sweater was originally $25, but then marked down to $22. What is the percent decrease?

Possible Answers:

14%

12%

25%

19%

3%

Correct answer:

12%

Explanation:

To find the percent decrease, remember this formula: difference/original. First, find the difference between the prices. This is $3. Then divide that by the original price, which is $25. \(\displaystyle \frac{3}{25}=.12\). Since the question is asking for a percentage, move the decimal over to the right two places and you get 12%.

Example Question #841 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Solve for x.

\(\displaystyle \frac{x}{3}+0.4=1.2\)

Possible Answers:

\(\displaystyle x=.24\)

\(\displaystyle x=1.3\)

\(\displaystyle x=8\)

\(\displaystyle x=2.4\)

\(\displaystyle x=24\)

Correct answer:

\(\displaystyle x=2.4\)

Explanation:

To solve for x, first subtract 0.4 from both sides. This gives you \(\displaystyle \frac{x}{3}=0.8\). Then, multiply both sides by 3. This yields 2.4 for x.

Example Question #843 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Simplify:

\(\displaystyle \frac{\sqrt{24}}{144}\)

Possible Answers:

\(\displaystyle \frac{\sqrt{2}}{36}\)

\(\displaystyle 36\)

\(\displaystyle \sqrt{6}\)

\(\displaystyle \frac{1}{}6\)

\(\displaystyle \frac{\sqrt{6}}{72}\)

Correct answer:

\(\displaystyle \frac{\sqrt{6}}{72}\)

Explanation:

First, simplify the radical on the numerator. \(\displaystyle \sqrt{24}\) can be simplified to \(\displaystyle \sqrt{4} \cdot \sqrt{6}\), which is also \(\displaystyle 2\sqrt{6}\). Then, put that over the denominator: \(\displaystyle \frac{2\sqrt{6}}{144}\). Then, simplify like terms. 2 goes into both the numerator and denominator, so that final answer is \(\displaystyle \frac{\sqrt{6}}{72}\).

Example Question #146 : Algebraic Concepts

Solve for x in the equation below:

\(\displaystyle \sqrt{x}=\frac{4+5\cdot2-18\div2}{5}\)

Possible Answers:

\(\displaystyle 1\)

\(\displaystyle 5\)

\(\displaystyle 25\)

\(\displaystyle 15\)

Correct answer:

\(\displaystyle 1\)

Explanation:

The first step is to simplify the numerator in the right side of the equation. 

\(\displaystyle \sqrt{x}=\frac{4+5\cdot2-18\div2}{5}\)

\(\displaystyle \sqrt{x}=\frac{4+10-9}{5}\)

\(\displaystyle \sqrt{x}=\frac{5}{5}\)

\(\displaystyle \sqrt{x}=1\)

Next, we square each side of the equation. 

\(\displaystyle x=1\)

 

Example Question #147 : Algebraic Concepts

Which of the answer choices below has a value that is different from the others?

 

Possible Answers:

\(\displaystyle \sqrt{16}\)

\(\displaystyle \frac{8}{2}\)

\(\displaystyle 2^{2}\)

 

\(\displaystyle 4\div \frac{1}{4}\)

Correct answer:

\(\displaystyle 4\div \frac{1}{4}\)

Explanation:

\(\displaystyle 4\div \frac{1}{4}=16\)

\(\displaystyle \sqrt{16}=4\)

\(\displaystyle \frac{8}{2}=4\)

\(\displaystyle 2^{2}=4\)

Therefore, \(\displaystyle 4\div \frac{1}{4}\) has a value that is different from the other expressions. 

 

Example Question #148 : Algebraic Concepts

Solve for \(\displaystyle x\) in this equation:

\(\displaystyle \frac{5x}{2}=\frac{8}{x}\)

Possible Answers:

\(\displaystyle \frac{5}{4}\)

\(\displaystyle \frac{\sqrt{5}}{4}\)

\(\displaystyle \frac{4}{5}\)

\(\displaystyle \frac{4\sqrt{5}}{5}\)

Correct answer:

\(\displaystyle \frac{4\sqrt{5}}{5}\)

Explanation:

In order to solve for \(\displaystyle \frac{5x}{2}=\frac{8}{x}\), cross multiplication must be used. Applying this, the equestion to be solved will be:

\(\displaystyle 5x^{2}=16\)

Next, each side must be divided by 5. This results in:

\(\displaystyle x^{2}=\frac{16}{5}\)

The square root of each side is now taken. This gives us:

\(\displaystyle x=\sqrt{}\frac{16}{5}\)

This reduces to:

\(\displaystyle \frac{4}{\sqrt{5}}\)

 

Finally, we rationalize the denominator:

\(\displaystyle \frac{4}{\sqrt{5}}\cdot\frac{\sqrt{5}}{\sqrt{5}}=\frac{4\sqrt{5}}{5}\)

Example Question #32 : Problem Solving

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?

 

Possible Answers:

\(\displaystyle 25\)

\(\displaystyle 100\)

\(\displaystyle 75\)

\(\displaystyle 50\)

Correct answer:

\(\displaystyle 75\)

Explanation:

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

\(\displaystyle 900\cdot \frac{1}{9}=100\)

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. 

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