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## MTEL General Curriculum Mathematics Practice

Question 1 |

#### The chairs in a large room can be arranged in rows of 18, 25, or 60 with no chairs left over. If C is the smallest possible number of chairs in the room, which of the following inequalities does C satisfy?

\( \large C\le 300\) Hint: Find the LCM. | |

\( \large 300 < C \le 500 \) Hint: Find the LCM. | |

\( \large 500 < C \le 700 \) Hint: Find the LCM. | |

\( \large C>700\) Hint: The LCM is 900, which is the smallest number of chairs. |

Question 2 |

#### How many factors does 80 have?

\( \large8\) Hint: Don't forget 1 and 80. | |

\( \large9\) Hint: Only perfect squares have an odd number of factors -- otherwise factors come in pairs. | |

\( \large10\) Hint: 1,2,4,5,8,10,16,20,40,80 | |

\( \large12\) Hint: Did you count a number twice? Include a number that isn't a factor? |

Question 3 |

#### What is the greatest common factor of 540 and 216?

\( \large{{2}^{2}}\cdot {{3}^{3}}\) Hint: One way to solve this is to factor both numbers: \(540=2^2 \cdot 3^3 \cdot 5\) and \(216=2^3 \cdot 3^3\). Then take the smaller power for each prime that is a factor of both numbers. | |

\( \large2\cdot 3\) Hint: This is a common factor of both numbers, but it's not the greatest common factor. | |

\( \large{{2}^{3}}\cdot {{3}^{3}}\) Hint: \(2^3 = 8\) is not a factor of 540. | |

\( \large{{2}^{2}}\cdot {{3}^{2}}\) Hint: This is a common factor of both numbers, but it's not the greatest common factor. |

Question 4 |

#### Exactly one of the numbers below is a prime number. Which one is it?

\( \large511 \) Hint: Divisible by 7. | |

\( \large517\) Hint: Divisible by 11. | |

\( \large519\) Hint: Divisible by 3. | |

\( \large521\) |

Question 5 |

#### There are six gumballs in a bag — two red and four green. Six children take turns picking a gumball out of the bag without looking. They do not return any gumballs to the bag. What is the probability that the first two children to pick from the bag pick the red gumballs?

\( \large \dfrac{1}{3}\) Hint: This is the probability that the first child picks a red gumball, but not that the first two children pick red gumballs. | |

\( \large \dfrac{1}{8}\) Hint: Are you adding things that you should be multiplying? | |

\( \large \dfrac{1}{9}\) Hint: This would be the probability if the gumballs were returned to the bag. | |

\( \large \dfrac{1}{15}\) Hint: The probability that the first child picks red is 2/6 = 1/3. Then there are 5 gumballs in the bag, one red, so the probability that the second child picks red is 1/5. Thus 1/5 of the time, after the first child picks red, the second does too, so the probability is 1/5 x 1/3 = 1/15. |

Question 6 |

#### Which of the following values of x satisfies the inequality \( \large \left| {{(x+2)}^{3}} \right|<3?\)

\( \large x=-3\) Hint: \( \left| {{(-3+2)}^{3}} \right|\)=\( \left | {(-1)}^3 \right | \)=\( \left | -1 \right |=1 \) . | |

\( \large x=0\) Hint: \( \left| {{(0+2)}^{3}} \right|\)=\( \left | {2}^3 \right | \)=\( \left | 8 \right | \) =\( 8\) | |

\( \large x=-4\) Hint: \( \left| {{(-4+2)}^{3}} \right|\)=\( \left | {(-2)}^3 \right | \)=\( \left | -8 \right | \) =\( 8\) | |

\( \large x=1\) Hint: \( \left| {{(1+2)}^{3}} \right|\)=\( \left | {3}^3 \right | \)=\( \left | 27 \right | \) = \(27\) |

Question 7 |

#### If two fair coins are flipped, what is the probability that one will come up heads and the other tails?

\( \large \dfrac{1}{4}\) Hint: Think of the coins as a penny and a dime, and list all possibilities. | |

\( \large \dfrac{1}{3} \) Hint: This is a very common misconception. There are three possible outcomes -- both heads, both tails, and one of each -- but they are not equally likely. Think of the coins as a penny and a dime, and list all possibilities. | |

\( \large \dfrac{1}{2}\) Hint: The possibilities are HH, HT, TH, TT, and all are equally likely. Two of the four have one of each coin, so the probability is 2/4=1/2. | |

\( \large \dfrac{3}{4}\) Hint: Think of the coins as a penny and a dime, and list all possibilities. |

Question 8 |

#### Which of the graphs below represent functions?

**I.**

**II.**

**III.**

**IV.**

## I and IV only.Hint: There are vertical lines that go through 2 points in IV . | |

## I and III only.Hint: Even though III is not continuous, it's still a function (assuming that vertical lines between the "steps" do not go through 2 points). | |

## II and III only.Hint: Learn about the vertical line test. | |

## I, II, and IV only.Hint: There are vertical lines that go through 2 points in II. |

Question 9 |

#### Use the expression below to answer the question that follows.

#### \(\large \dfrac{\left( 155 \right)\times \left( 6,124 \right)}{977}\)

#### Which of the following is the best estimate of the expression above?

## 100Hint: 6124/977 is approximately 6. | |

## 200Hint: 6124/977 is approximately 6. | |

## 1,000Hint: 6124/977 is approximately 6. 155 is approximately 150, and \( 6 \times 150 = 3 \times 300 = 900\), so this answer is closest. | |

## 2,000Hint: 6124/977 is approximately 6. |

Question 10 |

#### Use the solution procedure below to answer the question that follows:

#### \( \large {\left( x+3 \right)}^{2}=10\)

#### \( \large \left( x+3 \right)\left( x+3 \right)=10\)

#### \( \large {x}^{2}+9=10\)

#### \( \large {x}^{2}+9-9=10-9\)

#### \( \large {x}^{2}=1\)

#### \( \large x=1\text{ or }x=-1\)

#### Which of the following is incorrect in the procedure shown above?

## The commutative property is used incorrectly.Hint: The commutative property is \(a+b=b+a\) or \(ab=ba\). | |

## The associative property is used incorrectly.Hint: The associative property is \(a+(b+c)=(a+b)+c\) or
\(a \times (b \times c)=(a \times b) \times c\). | |

## Order of operations is done incorrectly. | |

## The distributive property is used incorrectly.Hint: \((x+3)(x+3)=x(x+3)+3(x+3)\)=\(x^2+3x+3x+9.\) |

Question 11 |

#### What is the least common multiple of 540 and 216?

\( \large{{2}^{5}}\cdot {{3}^{6}}\cdot 5\) Hint: This is the product of the numbers, not the LCM. | |

\( \large{{2}^{3}}\cdot {{3}^{3}}\cdot 5\) Hint: One way to solve this is to factor both numbers: \(540=2^2 \cdot 3^3 \cdot 5\) and \(216=2^3 \cdot 3^3\). Then for each prime that's a factor of either number, use the largest exponent that appears in one of the factorizations. You can also take the product of the two numbers divided by their GCD. | |

\( \large{{2}^{2}}\cdot {{3}^{3}}\cdot 5\) Hint: 216 is a multiple of 8. | |

\( \large{{2}^{2}}\cdot {{3}^{2}}\cdot {{5}^{2}}\) Hint: Not a multiple of 216 and not a multiple of 540. |

Question 12 |

#### Which property is not shared by all rhombi?

## 4 congruent sidesHint: The most common definition of a rhombus is a quadrilateral with 4 congruent sides. | |

## A center of rotational symmetryHint: The diagonal of a rhombus separates it into two congruent isosceles triangles. The center of this line is a center of 180 degree rotational symmetry that switches the triangles. | |

## 4 congruent anglesHint: Unless the rhombus is a square, it does not have 4 congruent angles. | |

## 2 sets of parallel sidesHint: All rhombi are parallelograms. |

Question 13 |

## AHint: \(\frac{34}{135} \approx \frac{1}{4}\) and \( \frac{53}{86} \approx \frac {2}{3}\). \(\frac {1}{4}\) of \(\frac {2}{3}\) is small and closest to A. | |

## BHint: Estimate with simpler fractions. | |

## CHint: Estimate with simpler fractions. | |

## DHint: Estimate with simpler fractions. |

Question 14 |

#### Which of the following is equal to one million three hundred thousand?

\(\large1.3\times {{10}^{6}}\)
| |

\(\large1.3\times {{10}^{9}}\)
Hint: That's one billion three hundred million. | |

\(\large1.03\times {{10}^{6}}\)
Hint: That's one million thirty thousand. | |

\(\large1.03\times {{10}^{9}}\) Hint: That's one billion thirty million |

Question 15 |

#### In March of 2012, 1 dollar was worth the same as 0.761 Euros, and 1 dollar was also worth the same as 83.03 Japanese Yen. Which of the expressions below gives the number of Yen that are worth 1 Euro?

\( \large {83}.0{3}\cdot 0.{761}\) Hint: This equation gives less than the number of yen per dollar, but 1 Euro is worth more than 1 dollar. | |

\( \large \dfrac{0.{761}}{{83}.0{3}}\) Hint: Number is way too small. | |

\( \large \dfrac{{83}.0{3}}{0.{761}}\) Hint: One strategy here is to use easier numbers, say 1 dollar = .5 Euros and 100 yen, then 1 Euro would be 200 Yen (change the numbers in the equations and see what works). Another is to use dimensional analysis: we want # yen per Euro, or yen/Euro = yen/dollar \(\times\) dollar/Euro = \(83.03 \times \dfrac {1}{0.761}\) | |

\( \large \dfrac{1}{0.{761}}\cdot \dfrac{1}{{83}.0{3}}\) Hint: Number is way too small. |

Question 16 |

#### A homeowner is planning to tile the kitchen floor with tiles that measure 6 inches by 8 inches. The kitchen floor is a rectangle that measures 10 ft by 12 ft, and there are no gaps between the tiles. How many tiles does the homeowner need?

## 30Hint: The floor is 120 sq feet, and the tiles are smaller than 1 sq foot. Also, remember that 1 sq foot is 12 \(\times\) 12=144 sq inches. | |

## 120Hint: The floor is 120 sq feet, and the tiles are smaller than 1 sq foot. | |

## 300Hint: Recheck your calculations. | |

## 360Hint: One way to do this is to note that 6 inches = 1/2 foot and 8 inches = 2/3 foot, so the area of each tile is 1/2 \(\times\) 2/3=1/3 sq foot, or each square foot of floor requires 3 tiles. The area of the floor is 120 square feet. Note that the tiles would fit evenly oriented in either direction, parallel to the walls. |

Question 17 |

#### In the triangle below, \(\overline{AC}\cong \overline{AD}\cong \overline{DE}\) and \(m\angle CAD=100{}^\circ \). What is \(m\angle DAE\)?

\( \large 20{}^\circ \) Hint: Angles ACD and ADC are congruent since they are base angles of an isosceles triangle. Since the angles of a triangle sum to 180, they sum to 80, and they are 40 deg each. Thus angle ADE is 140 deg, since it makes a straight line with angle ADC. Angles DAE and DEA are base angles of an isosceles triangle and thus congruent-- they sum to 40 deg, so are 20 deg each. | |

\( \large 25{}^\circ \) Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. | |

\( \large 30{}^\circ \) Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. | |

\( \large 40{}^\circ \) Hint: Make sure you're calculating the correct angle. |

Question 18 |

#### The letters A, B, and C represent digits (possibly equal) in the twelve digit number x=111,111,111,ABC. For which values of A, B, and C is x divisible by 40?

\( \large A = 3, B = 2, C=0\) Hint: Note that it doesn't matter what the first 9 digits are, since 1000 is divisible by 40, so DEF,GHI,JKL,000 is divisible by 40 - we need to check the last 3. | |

\( \large A = 0, B = 0, C=4\) Hint: Not divisible by 10, since it doesn't end in 0. | |

\( \large A = 4, B = 2, C=0\) Hint: Divisible by 10 and by 4, but not by 40, as it's not divisible by 8. Look at 40 as the product of powers of primes -- 8 x 5, and check each. To check 8, either check whether 420 is divisible by 8, or take ones place + twice tens place + 4 * hundreds place = 18, which is not divisible by 8. | |

\( \large A =1, B=0, C=0\) Hint: Divisible by 10 and by 4, but not by 40, as it's not divisible by 8. Look at 40 as the product of powers of primes -- 8 x 5, and check each. To check 8, either check whether 100 is divisible by 8, or take ones place + twice tens place + 4 * hundreds place = 4, which is not divisible by 8. |

Question 19 |

#### Use the graph below to answer the question that follows:

#### The graph above best matches which of the following scenarios:

## George left home at 10:00 and drove to work on a crooked path. He was stopped in traffic at 10:30 and 10:45. He drove 30 miles total.Hint: Just because he ended up 30 miles from home doesn't mean he drove 30 miles total. | |

## George drove to work. On the way to work there is a little hill and a big hill. He slowed down for them. He made it to work at 11:15.Hint: The graph is not a picture of the roads. | |

## George left home at 10:15. He drove 10 miles, then realized he‘d forgotten something at home. He turned back and got what he‘d forgotten. Then he drove in a straight line, at many different speeds, until he got to work around 11:15.Hint: A straight line on a distance versus time graph means constant speed. | |

## George left home at 10:15. He drove 10 miles, then realized he‘d forgotten something at home. He turned back and got what he‘d forgotten. Then he drove at a constant speed until he got to work around 11:15. |

Question 20 |

#### The window glass below has the shape of a semi-circle on top of a square, where the side of the square has length x. It was cut from one piece of glass.

#### What is the perimeter of the window glass?

\( \large 3x+\dfrac{\pi x}{2}\) Hint: By definition, \(\pi\) is the ratio of the circumference of a circle to its diameter; thus the circumference is \(\pi d\). Since we have a semi-circle, its perimeter is \( \dfrac{1}{2} \pi x\). Only 3 sides of the square contribute to the perimeter. | |

\( \large 3x+2\pi x\) Hint: Make sure you know how to find the circumference of a circle. | |

\( \large 3x+\pi x\) Hint: Remember it's a semi-circle, not a circle. | |

\( \large 4x+2\pi x\) Hint: Only 3 sides of the square contribute to the perimeter. |

List |

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