Title: M2.0 Fundamental Concepts and Operations of Algebra Quiz
Course: Technical Math for Industry > Section: Module 2.0: Fundamental Concepts and Operations of Algebra

Q: Margaret wants to call her friend overseas. There is a connection fee of and a charge of per minute. So the cost of the call is dollars for minutes.

Can she talk for 15 minutes without going over a budget of ?
- No, she would need . (Incorrect. You may have forgotten the order of operations and added before multiplying by 15. Multiplication comes before addition.)
- Yes, she would spend . (Incorrect. You may have forgotten to add the $2 and just found .)
- Yes, she would spend . (Correct. After talking for 15 minutes she would have spent:)
- No, she would need . (Incorrect. You may have added and then multiplied by 1.5. You need to multiply first in the expression .)
A: Yes, she would spend .

Q: Nathan is building a toolshed with a rectangular floor. The area of a rectangle is given by the formula , where l represents the length, and w represents the width. The floor of the shed will have measurements of either and or and . What is the area of the larger floor?
- 49 (Correct. First substitute 8 for l and 6 for w in the expression and then multiply: . Next substitute 7 for l and 7 for w in the expression and then multiply: . The larger floor area is 49.)
- 56 (Incorrect. You may have used the wrong combination of the given numbers. You should substitute 8 for l and 6 for w for the first rectangle, and then 7 for l and 7 for w for the second rectangle. Then determine which area is larger.)
- 42 (Incorrect. You may have used the wrong combination of the given numbers. You should substitute 8 for l and 6 for w for the first rectangle, and then 7 for l and 7 for w for the second rectangle. Then determine which area is larger.)
- 48 (Incorrect. You may have only found the area of the first rectangle by substituting 8 for l and 6 for w and multiplying. You need to also find the area of the rectangle that has a length of 7 and a width of 7 and determine which rectangle is larger.)
A: 49

Q: Which name describes only those numbers highlighted on the number line?
- whole numbers (Incorrect. Whole numbers include the number 0, which is not graphed on the number line.)
- natural numbers (Correct. Natural numbers are the numbers that we use when we count. They are the numbers 1, 2, 3, and so on.)
- negative numbers (Incorrect. Negative numbers are found to the left of 0 on the number line. The numbers graphed here are to the right of 0.)
- integers (Incorrect. Integers include 0 and numbers to the left and right of 0. The numbers graphed here are just to the right of 0.)
A: natural numbers

Q: Find when and .
- and (Incorrect. The second statement is true, but the first statement is false. Absolute value will change a negative to a positive and leave a positive number alone. It is the distance between a number's place on the number line and 9, so absolute value is always positive or 0.)
- Both values are 45. (Correct. Both 45 and are 45 units from zero on the number line. The absolute value is equal to this distance.)
- and (Incorrect. The first statement is true, but the second is false. The absolute value of a number can never be negative. It is the distance between the number's place on the number line and 0. Distance is always positive or 0.)
- Both values are . (Incorrect. The absolute value of a number can never be negative. It is the distance between the number's place on the number line and 0. Distance is always positive or 0.)
A: Both values are 45.

Q: What is the opposite of 45?
- 45 (Incorrect. Opposites have the same absolute value and different signs. The numbers are the same, so they have the same absolute value. However, they have the same sign.)
A: (a choice with no text)
Feedback: Correct. Opposites have the same absolute value and different signs. and are both equal to . The numbers and also have different signs.
Feedback: Incorrect. Opposites have the same absolute value and different signs. , so they have the same absolute value. However, they have the same sign.
Feedback: Incorrect. Opposites have the same absolute value and different signs. , so they have the same absolute value. However, they have the same sign.

Q: Which rational number also belongs to the set of natural numbers?
- 0.1666... (Incorrect. Since the number 0.1666... is between 0 and 1 on the number line, it cannot be a natural number.)
- 15 (Correct. The number 15 is in the set of natural numbers.)
A: 15
Feedback: Incorrect. The set of natural numbers does not include any negative numbers.
Feedback: Incorrect. Since the number is between 0 and 1 on the number line, it cannot be a natural number.

Q: Which of the following numbers is shown plotted on the number line?
A: (a choice with no text)
Feedback: Incorrect. The number . This number falls between and . The point shown is just to the left of , so it is between and .
Feedback: Correct. The point shown is just to the left of , so it is between and . The number is the only answer choice that falls between these two numbers.
Feedback: Incorrect. Positive numbers are to the right of 0. The number 2.75 is to the right of 0. The point shown is just to the left of , so it is between and .
Feedback: Incorrect. Positive numbers are to the right of 0. The number falls between 2 and 3. The point shown is just to the left of , so it is between and .
Feedback: Incorrect. This number falls between and . The point shown is just to the left of , so it is between and .

Q: Which statement is true?
A: (a choice with no text)
Feedback: Incorrect. Positive numbers are always to the right of negative numbers on a number line, so .
Feedback: Incorrect. Negative numbers are always to the left of positive numbers on a number line, so .
Feedback: Correct. is to the left of on a number line, so .
Feedback: Incorrect. 6.9 is to the right of 1.2 on a number line, so .

Q: Which sets does  4.333... belong to?
- integers and irrational numbers (Incorrect. 4.333... is not an integer because it has a decimal part. It is also not an irrational number because irrational numbers are nonrepeating when written in decimal form.)
- rational and real numbers (Correct. Since 4.33... is repeating when written in decimal form, it is a rational number. Also, all rational numbers are real numbers.)
- irrational and whole numbers (Incorrect. Irrational numbers are nonrepeating when written in decimal form. This number is repeating and therefore rational. Also 4.333... is not a whole number because it has a decimal part.)
- rational and natural numbers (Incorrect. Since 4.333... is repeating when written in decimal form, it is a rational number. However, it is not a natural number because it has a decimal part.)
A: rational and real numbers

Q: Find .
- 249 (Incorrect. You added the absolute values, but you omitted a negative sign in the answer. The sum must have the same sign as the numbers you are adding.)
A: (a choice with no text)
Feedback: Incorrect. You subtracted 71 and 32 from 146 and gave the result the same sign as the numbers in the problem. You need to add 71, 146, and 32, and then give the sum the same sign as the addends.
Feedback: Correct. All the addends are negative, so the sum is negative.
Feedback: Incorrect. You forgot to add a regrouped hundred when adding the absolute values.

Q: Find .
- 119 (Incorrect. You may have correctly added to get 43, and then incorrectly added . To add two numbers with different signs, you must find the difference of their absolute values (). If the positive addend has a greater absolute value, the sum is positive. If the negative addend has a greater absolute value, the sum is negative.)
- 33 (Incorrect. You may have correctly added to get 43, but then incorrectly added . When you are adding numbers with different signs, and the positive addend has a greater absolute value, the sum is positive. If the negative addend has a greater absolute value, the sum is negative.)
A: (a choice with no text)
Feedback: Incorrect. You may have correctly added to get 43, and then incorrectly added . To add two numbers with different signs, you must find the difference of their absolute values (). If the positive addend has a greater absolute value, the sum is positive. If the negative addend has a greater absolute value, the sum is negative.
Feedback: Correct. Add the first two numbers. Because they are both positive, you can add as usual (). Now add . Because the two numbers have different signs, you need to find the difference of their absolute values (). Since , the result is negative.

Q: Find .
- 91.1 (Incorrect. You added the absolute values of the decimals but forgot to give the sum the right sign. Since all the addends in the problem are negative, their sum is negative.)
A: (a choice with no text)
Feedback: Correct. All addends in the problem are negative, so their sum is negative.
Feedback: Incorrect. You did not line up the decimals before adding the absolute values. The first step is to add .
Feedback: Incorrect. You added 6.8 and 12 to . Instead, you should first add the absolute values of the decimals. Then, because all the addends in the problem are negative, their sum is negative.

Q: Find .
- 54.545 (Incorrect. You did not line up the decimals when finding the difference of the absolute values. First subtract , lining up the decimal points. Then compare and to see if the answer is positive or negative.)
- 22.622 (Correct. . The number with the greater absolute value is positive, so the sum is positive.)
A: 22.622
Feedback: Incorrect. You may have added , and then supplied the answer with a negative sign. However, you need to find the difference of . The answer will be positive, because .
Feedback: Incorrect. You correctly found the difference of the absolute values. However, you have to compare and . If the negative number has a greater absolute value, the answer is negative. If the positive number has a greater absolute value, the answer is positive.

Q: Which of the following shows an example of the identity property of 0?
A: (a choice with no text)
Feedback: Correct. The number 0 is the additive identity. The identity property says 0 any number that number.
Feedback: Incorrect. This is an example of a number being added to itself. The identity property says 0 any number that number.
Feedback: Incorrect. You may have chosen this expression because one of the addends seemed equal to the sum, but the addend is negative while the sum is positive. The identity property says 0 any number that number.
Feedback: Incorrect. The number 0 is the additive identity, but this is not an example of the identity property. The identity property says 0 any number that number.

Q: A submarine is at the surface of the ocean, preparing to dive. It submerges and dives to feet. Then, it ascends 308 feet to avoid hitting an undersea cliff. What integer represents the submarine's position in feet (depth) after it ascends to avoid the cliff?
- 664 (Incorrect. The submarine's depth is represented by . You correctly found the difference of the absolute values of the addends, but the sum should have the same sign as the number with the greater absolute value.)
- 1,280 (Incorrect. The submarine's depth is represented by . You added the absolute values of the addends. Because the addends have different signs, you must find the difference of their absolute values and give the sum the same sign as the addend with the greater absolute value.)
A: (a choice with no text)
Feedback: Correct. The submarine's depth is represented by . Find the difference of the absolute values: . Since the number with the greater absolute value is a negative number, the sum is negative.
Feedback: Incorrect. The submarine's depth is represented by . You added the absolute values of the addends, then added a negative sign. Because the addends have different signs, you must find the difference of their absolute values and give the sum the same sign as the addend with the greater absolute value.

Q: Find .
- 454 (Incorrect. You used the wrong sign. Rewrite the problem as adding the opposite: . Find the sum of the absolute values. Give the sum the same sign as both addends.)
- 196 (Incorrect. You added the opposites of and 129, instead of just adding the opposite of 129, which would be . Find the sum of the absolute values. Give the sum the same sign as both addends.)
A: (a choice with no text)
Feedback: Incorrect. You added 129 instead of adding the opposite of 129, which would be . Find the sum of the absolute values. Give the sum the same sign as both addends.
Feedback: Correct. . Now find the sum of the absolute values: . Since both addends are negative, the sum is negative.

Q: Find .
A: (a choice with no text)
Feedback: Correct. Start with . Since , and , . Next find . Since , and both numbers are negative, . Finally subtract . Since , and , the final result is .
Feedback: Incorrect. You may have correctly found that , but then subtracted incorrectly. To do this correctly, first note that . Then you should add and make the result negative.
Feedback: Incorrect. You probably subtracted 3 as your last step. You should be subtracting , which is the same as adding 3.
Feedback: Incorrect. You probably found incorrectly by adding 34 and 15 and then putting a negative sign in front. Instead, you should subtract and then compare the addends to see if the result is positive or negative.

Q: The lowest temperature ever recorded in the Southern Hemisphere was . The lowest temperature ever recorded in the Northern Hemisphere was . Was the Northern temperature warmer or colder, and by how many degrees?
- warmer (Incorrect. You added the absolute values of the temperatures instead of subtracting to find the difference in temperatures.)
- warmer (Correct. To see the difference, take the Northern temperature and subtract the Southern temperature:)
- colder (Incorrect. You probably started with the correct calculation: Since the number with the greater absolute value is positive, the result will be positive, which means warmer.)
- colder (Incorrect. You added the absolute values of the temperatures instead of subtracting to find the difference in temperatures. Also, though is warmer than , is warmer than .)
A: warmer

Q: Find .
- 60 (Incorrect. You may have only multiplied the positive factors, forgetting to multiply by the other three negative numbers. Also, the sign is wrong since the number of negative factors is odd.)
- 180 (Incorrect. You may have forgotten to multiply by the last factor in the product, which is .)
- 360 (Incorrect. You correctly multiplied the absolute values of the factors, but you gave the product the wrong sign. The number of negative factors is odd.)
A: (a choice with no text)
Feedback: Correct. Multiply the absolute values of the numbers. The number of negative factors is odd, so the product is negative.

Q: Which of the following is an example of the identity property of 1?
A: (a choice with no text)
Feedback: Incorrect. This is an example of the identity property of 0. The identity property of 1 states that .
Feedback: Incorrect. This is a true statement, but it is not an example of the identity property of 1, which states that .
Feedback: Incorrect. You may have chosen this answer because the sum is 1. The identity property of 1 states that .
Feedback: Correct. The identity property of 1 states that .

Q: Find:
- 54 (Incorrect. It appears that you correctly turned division by into multiplication by . Remember, though, that a negative number multiplied or divided by a positive number is negative.)
A: (a choice with no text)
Feedback: Incorrect. You multiplied the reciprocal of the first number by the second number. What you should do instead is multiply the first number by the reciprocal of the second:
Feedback: Incorrect. It appears that you divided by 4 and put the answer over the denominator 9. You need to rewrite the division as multiplication by the reciprocal of the divisor:
Feedback: Correct.

Q: A group pre-orders 9 tickets for a sightseeing tour, even though they don't have the money to pay for them yet. Each ticket costs $32, so the debt for one ticket is . The tour comapny adds a $15 fee for the entire order. Including the fee, which number represents the total debt?
A: (a choice with no text)
Feedback: Incorrect. You added a $15 fee for each ticket, when the fee was for the entire order. The amount that you need to calculate is .
Feedback: Correct. The total debt for the 9 tickets is represented by .
Feedback: Incorrect. You probably correctly found , but then added the $15 to this amount. It is true that the tour company added the fee, but this amount is also a debt, so you should be adding to the cost of the 9 tickets.
Feedback: Incorrect. You found the debt for the 9 tickets without including the fee. The amount you need to calculate is .

Q: Rewrite in a different way, using the commutative property of multiplication. Note that represents a real number.
A: (a choice with no text)
Feedback: Incorrect. Some switching has been done. The variable factor is now first, but this expression is not equal to the original expression. The digits of the number 32 should not have been switched.
Feedback: Incorrect. The commutative property of multiplication allows you to change the order of 32 and . You cannot split the digits of a number, such as 32, apart and switch one of them with the other number.
Feedback: Incorrect. This expression is equal to the original expression, but the commutative property has not been used, because the order of the number and the variable factor has not been switched.
Feedback: Correct. The commutative property of multiplication states that . Therefore, you may switch the order of 32 and .

Q: Which of the following properties could be used to rewrite the expression as ?
- the associative property followed by the commutative property (Correct. You get by using the associative property to regroup. Then you get by applying the commutative property.)
- the associative property used once (Incorrect. This is what you would get by using the associative property to regroup:)
- the commutative property used twice (Incorrect. If you did this, you would be switching the order inside and outside the parentheses. You would not regroup. This would be the result:)
- the commutative property followed by the associative property (Incorrect. This is what you get when you apply these two properties in this order: or)
A: the associative property followed by the commutative property

Q: Simplify:
- 21 (Incorrect. You must combine only like terms. If the problem asked for , that would simplify to 21, as ordinary numbers are all like terms. In the given problem, and are like terms, while 4 and 5 form a separate group of like terms.)
A: (a choice with no text)
Feedback: Incorrect. You must combine only like terms. If the problem asked for , that would simplify to . In the given problem, and are like terms, while 4 and 5 form a separate group of like terms.
Feedback: Incorrect. You may have combined unlike terms: would simplify to , but would not. Likewise, , but does not simplify to 14.
Feedback: Correct. You combined the like terms: and

Q: Simplify:
- 0 (Incorrect. When simplifying the expression inside the parentheses, you may have subtracted the 2s before applying the exponent. However, exponents come before addition or subtraction.)
- 9 (Incorrect. You may have correctly simplified inside the parentheses first to get . At this point, however, you may have divided -6 by 2 before computing the exponent. Remember that exponents come before division.)
- 18 (Correct. You need to first apply the order of operations to the numerator and then divide.)
A: 18
Feedback: Incorrect. You may have correctly simplified the numerator to get . At this point, however, you may have squared the 8 and then subtracted. You must completely simplify inside parentheses before you do other operations.

Q: Simplify:
- 20 (Correct. Simplify the absolute value expression, and then use the order of operations:)
- 30 (Incorrect. You may have ignored the absolute value symbols and thought that is equal to . Remember that the absolute value of a negative number is a positive number, so .)
A: 20
Feedback: Incorrect. You may have simplified incorrectly and gotten instead of 25. Operations inside parentheses come before exponents. So if you have , you are squaring a negative number and will get a positive result.
Feedback: Incorrect. You may have simplified incorrectly by squaring the 9 and then subtracting . You must simplify inside parentheses before doing other operations. So this part of the expression would first simplify to .