## 10 Random Questions

Hints will display for most wrong answers; explanations for most right answers. You can attempt a question multiple times; it will only be scored correct if you get it right the first time. To see ten new questions, reload the page.

I used the official objectives and sample test to construct these questions, but cannot promise that they accurately reflect what’s on the real test. Some of the sample questions were more convoluted than I could bear to write. See terms of use. See the MTEL Practice Test main page to view questions on a particular topic or to download paper practice tests.

## MTEL General Curriculum Mathematics Practice

Question 1 |

#### What set of transformations will transform the leftmost image into the rightmost image?

## A 90 degree clockwise rotation about (2,1) followed by a translation of two units to the right.Hint: Part of the figure would move below the x-axis with these transformations. | |

## A translation 3 units up, followed by a reflection about the line y=x.Hint: See what happens to the point (5,1) under this set of transformations. | |

## A 90 degree clockwise rotation about (5,1), followed by a translation of 2 units up. | |

## A 90 degree clockwise rotation about (2,1) followed by a translation of 2 units to the right.Hint: See what happens to the point (3,3) under this set of transformations. |

Question 2 |

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

#### \( \large \dfrac{\left( 7,154 \right)\times \left( 896 \right)}{216}\)

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

## 2,000Hint: The answer is bigger than 7,000. | |

## 20,000Hint: Estimate 896/216 first. | |

## 3,000Hint: The answer is bigger than 7,000. | |

## 30,000Hint: \( \dfrac{896}{216} \approx 4\) and \(7154 \times 4\) is over 28,000, so this answer is closest. |

Question 3 |

#### 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 4 |

#### Given that 10 cm is approximately equal to 4 inches, which of the following expressions models a way to find out approximately how many inches are equivalent to 350 cm?

\( \large 350\times \left( \dfrac{10}{4} \right)\) Hint: The final result should be smaller than 350, and this answer is bigger. | |

\( \large 350\times \left( \dfrac{4}{10} \right)\) Hint: Dimensional analysis can help here: \(350 \text{cm} \times \dfrac{4 \text{in}}{10 \text{cm}}\). The cm's cancel and the answer is in inches. | |

\( \large (10-4) \times 350
\) Hint: This answer doesn't make much sense. Try with a simpler example (e.g. 20 cm not 350 cm) to make sure that your logic makes sense. | |

\( \large (350-10) \times 4\) Hint: This answer doesn't make much sense. Try with a simpler example (e.g. 20 cm not 350 cm) to make sure that your logic makes sense. |

Question 5 |

#### A map has a scale of 3 inches = 100 miles. Cities A and B are 753 miles apart. Let d be the distance between the two cities on the map. Which of the following is not correct?

\( \large \dfrac{3}{100}=\dfrac{d}{753}\) Hint: Units on both side are inches/mile, and both numerators and denominators correspond -- this one is correct. | |

\( \large \dfrac{3}{100}=\dfrac{753}{d}\) Hint: Unit on the left is inches per mile, and on the right is miles per inch. The proportion is set up incorrectly (which is what we wanted). Another strategy is to notice that one of A or B has to be the answer because they cannot both be correct proportions. Then check that cross multiplying on A gives part D, so B is the one that is different from the other 3. | |

\( \large \dfrac{3}{d}=\dfrac{100}{753}\) Hint: Unitless on each side, as inches cancel on the left and miles on the right. Numerators correspond to the map, and denominators to the real life distances -- this one is correct. | |

\( \large 100d=3\cdot 753\) Hint: This is equivalent to part A. |

Question 6 |

#### 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 7 |

#### What is the length of side \(\overline{BD}\) in the triangle below, where \(\angle DBA\) is a right angle?

\( \large 1\) Hint: Use the Pythagorean Theorem. | |

\( \large \sqrt{5}\) Hint: \(2^2+e^2=3^2\) or \(4+e^2=9;e^2=5; e=\sqrt{5}\). | |

\( \large \sqrt{13}\) Hint: e is not the hypotenuse. | |

\( \large 5\) Hint: Use the Pythagorean Theorem. |

Question 8 |

#### Which of the following is equivalent to

#### \( \large A-B+C\div D\times E\)?

\( \large A-B-\dfrac{C}{DE}
\) Hint: In the order of operations, multiplication and division have the same priority, so do them left to right; same with addition and subtraction. | |

\( \large A-B+\dfrac{CE}{D}\) Hint: In practice, you're better off using parentheses than writing an expression like the one in the question. The PEMDAS acronym that many people memorize is misleading. Multiplication and division have equal priority and are done left to right. They have higher priority than addition and subtraction. Addition and subtraction also have equal priority and are done left to right. | |

\( \large \dfrac{AE-BE+CE}{D}\) Hint: Use order of operations, don't just compute left to right. | |

\( \large A-B+\dfrac{C}{DE}\) Hint: In the order of operations, multiplication and division have the same priority, so do them left to right |

Question 9 |

#### Some children explored the diagonals in 2 x 2 squares on pages of a calendar (where all four squares have numbers in them). They conjectured that the sum of the diagonals is always equal; in the example below, 8+16=9+15.

#### Which of the equations below could best be used to explain why the children’s conjecture is correct?

\( \large 8x+16x=9x+15x\) Hint: What would x represent in this case? Make sure you can describe in words what x represents. | |

\( \large x+(x+2)=(x+1)+(x+1)\) Hint: What would x represent in this case? Make sure you can describe in words what x represents. | |

\( \large x+(x+8)=(x+1)+(x+7)\) Hint: x is the number in the top left square, x+8 is one below and to the right, x+1 is to the right of x, and x+7 is below x. | |

\( \large x+8+16=x+9+15\) Hint: What would x represent in this case? Make sure you can describe in words what x represents. |

Question 10 |

#### Elena is going to use a calculator to check whether or not 267 is prime. She will pick certain divisors, and then find 267 divided by each, and see if she gets a whole number. If she never gets a whole number, then she’s found a prime. Which numbers does Elena NEED to check before she can stop checking and be sure she has a prime?

## All natural numbers from 2 to 266.Hint: She only needs to check primes -- checking the prime factors of any composite is enough to look for divisors. As a test taking strategy, the other three choices involve primes, so worth thinking about. | |

## All primes from 2 to 266 .Hint: Remember, factors come in pairs (except for square root factors), so she would first find the smaller of the pair and wouldn't need to check the larger. | |

## All primes from 2 to 133 .Hint: She doesn't need to check this high. Factors come in pairs, and something over 100 is going to be paired with something less than 3, so she will find that earlier. | |

## All primes from \( \large 2\) to \( \large \sqrt{267}\).Hint: \(\sqrt{267} \times \sqrt{267}=267\). Any other pair of factors will have one factor less than \( \sqrt{267}\) and one greater, so she only needs to check up to \( \sqrt{267}\). |

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