Mathematics:

Answers and Explanations

This page includes VERY detailed explanations of the math practice problems. If you had trouble answering some of the questions, take the time to review my explanations very carefully. If you have answered any questions incorrectly, I recommend that you go back and make another attempt and then turn to this page again. You should do this until you can confidently complete all of the problems correctly.   

 

Number Sense

 

1.  Answer C — 25%

 

      Explanation — The first step in solving this problem is to subtract the old cost from the new cost,

      which  is $5.00 - $4.00 = $1.00.  This means that the ticket price increased by $1.00.  The next

      step in solving this problem is to divide the price increase by the original price, so $1.00 should

      be divided by $4.00, which equals  0.25.  To find out what percentage this is, you must multiply

      by 100%.  The final answer is 0.25 X 100% = 25%.

 

2.  Answer C — $48.00

 

      Explanation — First, remember to read the question carefully.  The question asks for the interest

      earned in three years.  In order to solve this problem you start by multiplying $200 by 8%.  In order

      to do this you need to change 8% into a decimal number.  This is found by dividing 8 by 100,

      which gives you 0.08.  Next, to find 8% of $200, you multiply $200 X 0.08 = $16.00.  Remember

      the question asks for the interest gained in three years, so you need to multiply $16.00 X 3 = $48.00. 

 

Statistics, Data Analysis, and Probability

 

1.  Answer D — 1/2

  

      Explanation — Remember that each time you flip the coin the result is independent any time you

      flipped the coin before.  So the result of flipping a coin is NOT dependent upon the previous flip. 

      If the chances of getting a tails is 50% the FIRST time that you flipped a coin, then the chances of

      getting a tails is 50% EVERY time that you flipped the coin.  Since 50% is the same as 1/2, the

      correct answer is D.

  

2.  Answer C — How many male members of the club attended the meeting?     

   

      Explanation — Remember to read the question thoroughly and underline exactly what it is asking.

      All the question tells us is that out of 36 total members, 27 attended a meeting, and 10 of those

      were female.  This question gives information about only those members of a club that attended a specific meeting,  NOT all members of the club.  In order to answer questions A, B, and D, we

      would need to know how many of the total members were male or how many were female.  In other words, we can only answer these questions if we know MORE information than what was given. 

      This leaves only answer C as the correct answer.

 

Measurement & Geometry

 

1.  Answer D — 6 ft. 6 in. 

        

      Explanation — This question tests your knowledge of conversion factors.  First you must figure out

      how many inches are in 2 meters.  In order to do this you must multiply 39 inches by 2 which  = 78 in.  Next you find out how many feet are in 78 inches.  Given that there are 12 inches in 1 ft, you need to divide 78 inches by 12 feet = 6.5, which is translated into 6 feet and 5 inches, but you do not have an answer that says 6 ft. 5 in.  The question asks about how tall, which means that this does not need

      to be an exact answer, only a close approximation.  Answer D, 6 ft. 6 in. is therefore the best answer. 

 

2.  Answer D — 54 cm

  

      Explanation — This is a proportion problem.  The question is asking for you to determine what the

      length of an actual rectangle would be if the width was 18 cm.  A scale drawing is representative of

      the actual rectangle, so you can compare the proportions and find the correct answer.  The width of

      the scale drawing is 1.2 cm.  The length of the scale drawing is 3.6 cm.  The ratio of width to length

      in the scale rectangle  is 1.2 cm divided by 3.6 cm.  Since the proportion of the scale drawing is the same as the proportion of the actual rectangle you can compare the ratios directly using a variable

      as a placeholder (let's call it X).  That  means that 1.2 cm / 3.6 cm = 18 cm / X (the placeholder).  To

      get the final step completed, you can cross multiply, which gives you 1.2X cm =  64.8 cm, and then divide 64.8 cm by 1.2 cm (to get the X by itself), which leaves you with 54 cm.  Answer D is correct. 

 

Algebra and Functions

 

1.  Answer D —   x-15+2y

 

       Explanation — This question is asking you understand how to translate a word problem into a  mathematical expression.  When reading a word problem like this you have to be careful to include

      all of the information given.  The original weight of tea that the shopkeeper has in stock is denoted

      by the letter x.  Since the shopkeeper then sells some of the tea, you would then subtract the sold

      weight (15 kg) from the original weight (x kg), which gives us x - 15.  Then the shopkeeper receives

      a shipment from his supplier, which will add an amount to his stock.  In this case the added weight 

      is denoted by the term 2y kg.  So if we integrate all of this information into the mathematical

      expression we get x kg (the original weight) minus 15  kg (the amount sold) plus 2y kg

      (the amount received).  This means that x - 15 + 2y = the weight of tea the shopkeeper now has

       in stock, which is answer D. 

 

2.  Answer A — 12   

 

       Explanation — With the information given in the problem, you can determine how many pages she

       can read each day, this is called the rate.  The rate is 168 pages divided by 7 days = 24 pages per day.  Using this rate you can find out how long it will take her to read the entire 456 page book. This is 456 divided by 24 pages per day = 19 days total.  The question asks "how many more days will it take her to finish reading the book?"  At first it seems like the answer is 19 days, but you must remember that she has already read for 7 days, so 19 - 7 = 12 days, and the correct answer is A.

 

Math Reasoning

 

1.    Answer B — It is an even number 

  

       Explanation — in order to answer this question correctly, you need some understanding of odd

      and even numbers.  Remember that 1,3,5 etc. are odd numbers and 2,4,6 etc. are even numbers. 

       In order to answer this question you need to plug in an odd number for n and add 1 to it, as the expression n+1 asks. For example,  1+ 1= 2, 3 + 1 = 4, 5 + 1 = 6, etc.  Notice that each time you

      add one to an odd number, the sum is an even number.  This means the correct answer is B.  You

      can also figure out this problem through the process of elimination.  Answer A is incorrect because,

      as you saw above, any odd number plus 1 equals an even number.  Answer C is incorrect because

      in can be any number and therefore it is impossible to tell whether or not n+1 would be a prime number.  Answer D is incorrect because a number plus one cannot yield the same answer as that number minus one (it defies the laws of addition and subtraction).

 

2.  Answer A — Divide 100 by 2.5    

      Explanation — This question checks your understanding of two major mathematical concepts.  The first is being able to grasp the idea of rate.  A rate is a measurement of an activity done per unit time.  For example, driving 60 miles per hour means that for every hour of driving, you will have traveled 60 miles.  The second concept is an understanding of time segments.  For example, 15 minutes is the same as 0.25 hours is the same as 1/4 hour; all of these terms are equal to each other.  In this problem you are expected to understand that 30 minutes is the same as 0.5 hours.  The problem explains that Chris drove 100 km in 2 hours and 30 minutes.  An understanding of rate tells you that the activity (driving 100 km) has the rate of 100 km / 2 hours and 30 minutes. The problem is simplified if you know that 2 hours and 30 minutes is equal to 2.5 hours.  The correct answer then, is A, or 100 km / 2.5 hours.      

 

Algebra 1

 

1.  Answer A — (0,4)      

 

      Explanation — We can test each pair of coordinates in the equation of 4x + 5y = 20 and see which set works the best.  Let’s start with answer D (5, 4) which indicates  x is equal to 5 and Y is equal

      to 4. Substituting those two numbers into the formula gives us: 4(5) + 5(4)= 20 + 20 = 40, which

      is not the correct answer.  If we try the set from answer C (4,5), we get  4(4) + 5(5)= 16 + 25 = 41, which is also incorrect.  The coordinates from set B are (0,5) give us:  4(0) + 5 (5) = 25, which is also incorrect.  Set A is the pair (0,4), which gives us the correct answer when substituting into the formula like as follows: 4(0) + 5(4) = 20, correct!

 

2.  Answer A — 4x+20-3x-6=14

 

       Explanation —The question is asking you to write out the expression 4(x+ 5) – 3(x+ 2)= 14 in a different way.  In order to do this you first need to multiply the 4 by (x + 5) using the distributive rule of algebra, which gives you 4x + 20.  Next you multiply –3 by (x + 2), again using the distributive rule, which gives you -3x -6.  You then put these terms together and form the new equation, as shown in answer A: 4x+20-3x-6 = 14.