High School Math : Number Theory

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #1 : Number Theory

Without using a calculator, which of the following is the best estimate for \(\displaystyle \sqrt{90}\)?

Possible Answers:

\(\displaystyle 8.7\)

\(\displaystyle 9.5\)

\(\displaystyle 10.2\)

\(\displaystyle 9.1\)

Correct answer:

\(\displaystyle 9.5\)

Explanation:

We know that \(\displaystyle 9^2=81\) and \(\displaystyle 10^2=100\).

\(\displaystyle 81< 90< 100\)

\(\displaystyle \sqrt{81}< \sqrt{90}< \sqrt{100}\)

\(\displaystyle 9< \sqrt{90}< 10\)

Because 90 falls is approximately halfway between 81 and 100, the square root of 90 is approximately halfway between 9 and 10, or 9.5.

Example Question #2 : Number Theory

Place in order from smallest to largest:

\(\displaystyle \frac{3}{4},\; \; \frac{1}{8},\; \; \frac{5}{8},\; \; \frac{1}{6},\; \; \frac{1}{2},\; \; \frac{1}{5},\; \; \frac{1}{3}\)

Possible Answers:

\(\displaystyle \frac{1}{8},\; \; \frac{1}{5},\; \; \frac{1}{2},\; \; \frac{3}{4},\; \; \frac{1}{6},\; \; \frac{1}{3},\; \; \frac{5}{8}\)

 

\(\displaystyle \frac{1}{8},\; \; \frac{1}{6},\; \; \frac{1}{5},\; \; \frac{1}{3},\; \; \frac{1}{2},\; \; \frac{5}{8},\; \; \frac{3}{4}\)

\(\displaystyle \frac{1}{2},\; \; \frac{1}{3},\; \; \frac{3}{4},\; \; \frac{1}{5},\; \; \frac{1}{6},\; \; \frac{1}{8},\; \; \frac{5}{8}\)

\(\displaystyle \frac{1}{5},\; \; \frac{3}{4},\; \; \frac{1}{3},\; \; \frac{1}{2},\; \; \frac{5}{8},\; \; \frac{1}{8},\; \; \frac{1}{6}\)

\(\displaystyle \frac{1}{2},\; \; \frac{3}{4},\; \; \frac{1}{6},\; \; \frac{5}{8},\; \; \frac{1}{3},\; \; \frac{1}{5},\; \; \frac{1}{8}\)

Correct answer:

\(\displaystyle \frac{1}{8},\; \; \frac{1}{6},\; \; \frac{1}{5},\; \; \frac{1}{3},\; \; \frac{1}{2},\; \; \frac{5}{8},\; \; \frac{3}{4}\)

Explanation:

To place in order, first we must find a common denominator and convert all fractions to that denominator.

\(\displaystyle 2, \;4,\; 8\) have a common denominator of \(\displaystyle 8\).

\(\displaystyle 6,\; 3\) have a common denominator of \(\displaystyle 6\).

\(\displaystyle 6, 8, 5\) have a common denominator of \(\displaystyle 120\).

Therefore we can use common denominators to make all of the fractions look similar.  Then the ordering becomes trivial.

\(\displaystyle \frac{1}{8}=\frac{15}{120},\; \; \frac{1}{6}=\frac{20}{120},\; \; \frac{1}{5}=\frac{24}{120}\)

\(\displaystyle \frac{1}{3}=\frac{40}{120},\; \; \frac{1}{2}=\frac{60}{120}\)

\(\displaystyle \frac{5}{8}=\frac{75}{120},\; \; \frac{3}{4}=\frac{90}{120}\)

Example Question #1 : Understanding Real Numbers

What number is \(\displaystyle 20\%\) of \(\displaystyle 15\) ?

Possible Answers:

\(\displaystyle 3\)

\(\displaystyle 6\)

\(\displaystyle 4\)

\(\displaystyle 2\)

\(\displaystyle 5\)

Correct answer:

\(\displaystyle 3\)

Explanation:

For percent problems there are verbal cues:

"IS" means equals and "OF" means multiplication.

Then the equation to solve becomes:

\(\displaystyle n=0.20\cdot 15=3\)

Example Question #4 : Number Theory

Which of the following is NOT a real number? 

Possible Answers:

\(\displaystyle 4\pi\)

\(\displaystyle \sqrt{84}\)

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

\(\displaystyle 6e\)

\(\displaystyle \sqrt{-9}\)

Correct answer:

\(\displaystyle \sqrt{-9}\)

Explanation:

We are looking for a number that is not real. 

\(\displaystyle 4\pi\)\(\displaystyle 6e\), and \(\displaystyle \sqrt{84}\) are irrational numbers, but they are still real. 

 

Then, \(\displaystyle 4i^{2}\) is equivalent to \(\displaystyle -4\) by the rules of complex numbers. Thus, it is also real. 

That leaves us with: \(\displaystyle \sqrt{-9}\) which in fact is imaginary (since no real number multiplied by itself yields a negative number) and simplifies to \(\displaystyle 3i\)

Example Question #2 : Understanding Real Numbers

Which of the following are considered real numbers? 

Possible Answers:

\(\displaystyle 1.9344\)

\(\displaystyle \pi\)

\(\displaystyle \textup{All of the above}\)

\(\displaystyle \frac{3}{2}\)

\(\displaystyle 23\)

Correct answer:

\(\displaystyle \textup{All of the above}\)

Explanation:

Real numbers can be found anywhere on a continuous number line ranging from negative infinity to positive infinity; therefore, all of the numbers are real numbers. 

Example Question #2 : Number Theory

If a card is drawn randomly from a regular shuffled 52 card deck, what is the probability that the card is either a spade or a 3?

Possible Answers:

\(\displaystyle \frac{17}{52}\)

\(\displaystyle \frac{2}{13}\)

\(\displaystyle \frac{4}{13}\)

\(\displaystyle \frac{5}{52}\)

\(\displaystyle \frac{7}{52}\)

Correct answer:

\(\displaystyle \frac{4}{13}\)

Explanation:

How many cards in the deck are either a spade or a 3?

There are thirteen spades, including a 3 of spades.

There are four 3's, including a 3 of spades. 

Since we are counting the same card (3 of spades) twice, there are actually

\(\displaystyle 13+4-1\) 

\(\displaystyle =16\) distinct cards that fit the criteria of being either a spade or a 3.

Since any of the 52 cards is equally likely to be drawn, the probability that it is a spade or a 3, is

\(\displaystyle \frac{16}{52}\)\(\displaystyle =\frac{4}{13}\)

Example Question #3 : Number Theory

Find the distance between \(\displaystyle 11\) and \(\displaystyle -4\) on a number line.

Possible Answers:

\(\displaystyle 7\)

\(\displaystyle 7.5\)

\(\displaystyle 15\)

\(\displaystyle 10.25\)

Correct answer:

\(\displaystyle 15\)

Explanation:

To find the distance on a number line:

\(\displaystyle d=x_{2}-x{{_1}}\)

\(\displaystyle d=11-(-4)\)

\(\displaystyle d=11+4\)

\(\displaystyle d=15\)

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