Numbers and Operations Flashcards
(39 cards)
Which of the following is equal to 4(5−2)^2−2^3 ? A. 16 B. 28 C. 76 D. 136
B. 28
Correct Answer: B
Option (B) is correct. The question requires an understanding of the order of operations. The first step to simplify the expression 4(5−2)2−23 is the evaluation of the subtraction within the parentheses; the expression is equivalent to 4×32−23. The second step is the evaluation of the powers; the expression is equivalent to 4×9−8. The third step is the evaluation of the multiplication; the expression is equivalent to 36−8. The final step is the evaluation of the subtraction; the expression is equivalent to 28.
Which of the following expresses 3/16 as a percent?
A. 0.1875%
B. 1.875%
C. 5.33%
D. 18.75%
D. 18.75%
Option (D) is correct. The question requires an understanding of percent and percentages. To convert a fraction to a percent, it is necessary to multiply the fraction by 100 and add a percent symbol. Since 3/16×100=18.75, 3/16 is equivalent to 18.75%.
Which of the following is demonstrated by the figure shown?
A. When the numerator stays the same and the denominator increases, the fraction increases.
B. When the numerator stays the same and the denominator increases, the fraction decreases.
C. When the denominator stays the same and the numerator increases, the fraction increases.
D. When the denominator stays the same and the numerator increases, the fraction decreases.
B. When the numerator stays the same and the denominator increases, the fraction decreases.
Option (B) is correct. The question requires an understanding of fractions and how to use geometrical representations to compare fractions. All of the fractions shown are unit fractions; that is, fractions in which the numerator is 1. This is a pictorial representation that demonstrates that if the numerator of a set of fractions is fixed and does not change, the size of the number represented by the fractions will decrease as the denominators are increased.
Which of the following is equal to 5(10^0)? power 0?
A. 0
B. 1
C. 5
D. 50
C. 5
Option (C) is correct. The question requires an understanding of the place value system and powers of 10. Since 10^0=1, the expression 5(10^0) is equal to 5×1, or 5.
Elisa’s number: 5,723,683
Troy’s number: 2,678,533
Both Elisa and Troy wrote a number, as shown. The digit 7 in Elisa’s number represents how many times what the digit 7 represents in Troy’s number?
A. 10
B. 100
C. 1,000
D. 10,000
A. 10
Option (A) is correct. The question requires an understanding of place value systems. The value of the number represented by the 7 in Elisa’s number is 700,000, or 7×105, while the value of the number represented by the 7 in Troy’s number is 70,000, or 7×104. Therefore, the value of the former is 10 times as great as the value of latter.
Sara went to the store to buy some clothes. She bought six shirts, half as many pairs of pants as shirts, and a fourth as many sweaters as shirts. How many pieces of clothing did Sara buy?
Which of the following statements about the solution to the word problem shown must be true?
A. Because of the real-world context, the solution must belong to the set of all rational positive numbers; therefore, the solution is acceptable.
B. Because of the real-world context, the solution must belong to the set of all rational positive numbers; therefore, the solution is not acceptable.
C. Because of the real-world context, the solution must belong to the set of all natural numbers; therefore, the solution is acceptable.
D. Because of the real-world context, the solution must belong to the set of all natural numbers; therefore, the solution is not acceptable.
D. Because of the real-world context, the solution must belong to the set of all natural numbers; therefore, the solution is not acceptable.
Option (D) is correct. The question requires an understanding of how to evaluate the reasonableness of a solution to a contextual word problem. The solution to the given word problem must belong to the set of all natural numbers because the unit is pieces of clothing, which is a positive and discrete unit
Tom’s company has to ship 1,944 boxes of shoes. If a truck can hold 450 boxes, how many trucks does he need to ship all the boxes?
_________
The correct answer is 5. The question requires an understanding of rounding strategies for reasonableness of results. The word problem can be solved by performing the division 1,944 ÷ 450, which yields 4.32. Since the result represents the number of trucks needed, the result must be rounded to an integer. Rounding down to 4 would yield the wrong answer since 4 × 450 = 1,800, so Tom would be able to ship up to 1,800 boxes of shoes with 4 trucks. Rounding up to 5 would yield the right answer since 5 × 450 = 2,250. Since with 5 trucks Tom would be able to ship up to 2,250 boxes of shoes, he would also be able to ship 1,944.
In the number 567,894, what is the value of the digit 6 ?
A. Sixty
B. Ten thousand
C. Six thousand
D. Sixty thousand
D. Sixty thousand
Option (D) is correct. The question requires an understanding of place value systems. Reading the number from left to right, the digit 4 is in the ones position and has a value of 4. The digit 9 is in the tens position and has a value of 90. The digit 8 is in the hundreds position and has a value of 800. The digit 7 is in the thousands position and has a value of 7,000. The digit 6 is in the ten-thousands position and has a value of 60,000.
Which of the following numbers is least?
A. 0.103
B. 0.1041
C. 0.1005
D. 0.11
C. 0.1005
Option (C) is correct. The question requires an understanding of place value. To compare the four numbers in the options, the digits that determine place value must be compared moving left to right beginning with the digit in the tenths place. The digit 1 is in the tenths place of each number, so one must compare the digit in the hundredths place of each number. The digit 1 in the hundredths place of the number in (D) is greater than the digit 0 in the hundredths place of each of the numbers in (A), (B), and (C), so the number in (D) is the greatest. In the thousandths place, the number in (C) has a digit of 0, while the numbers in (A) and (B) have digits of 3 and 4, respectively. The number in (C) is thus less than the numbers in (A) and (B), making the number in (C) the least of the four numbers.
In a certain year, 5 percent of the 2,800 employees of a company had a perfect attendance record. Which of the following computations can be used to determine the number of employees with a perfect attendance record?
A. 1/40×2,800
B. 1/20×2,800
C. 1/5×2,800
D. 5×2,800
B. 1/20×2,800
Option (B) is correct. The question requires an understanding of fractions, percentages, and decimals and the ability to recognize equivalence among them. Since 5 percent means 5 one-hundredths, 5 percent is equivalent to 0.05, or 5 divided by 100, 5/100, which simplifies to 1/20. To find 5 percent of 2,800 employees, it is necessary to multiply 2,800 by either 0.05 or 1/20.
15(4+3)=15×4+15×3
The equation shown demonstrates which of the following?
A. The distributive property of multiplication over addition
B. The commutative property of multiplication
C. The associative property of multiplication
D. The additive inverse and additive identity properties
A. The distributive property of multiplication over addition
Which of the following is the product of two even numbers and an odd number, each of which is greater than 1 ?
A. 15
B. 16
C. 20
D. 21
C. 20
Option (C) is correct. The question requires an understanding of factors of natural numbers. The question requires a determination of the number that has two even factors and one odd factor. The even numbers need not be unique. In (C), 20=2×2×5; 20 can be written as the product of 2, 2, and 5, so 20 can be written as the product of two even numbers and one odd number. In (A), 15=3×5, and in (D), 21=3×7; 15 and 21 do not have any even factors. In (B), 16=2×2×2×2; 16 does not have any odd factors.
A wholesale nut company makes 10-pound and 25-pound bags of trail mix. For the 10-pound bag, the company uses 3 pounds of raisins, and the rest is nuts. If the proportion of raisins to nuts is the same in the 25-pound bag as in the 10-pound bag, how many pounds of nuts does the company need for the 25-pound bag?
A. 7.5
B. 17.5
C. 18.5
D. 22.0
B. 17.5
Option (B) is correct. The question requires an understanding of how to use proportional relationships to solve ratio and percent problems. If 3 pounds of raisins are used in the 10-pound mixture, then 7 pounds of nuts are used in the mixture, giving a ratio of pounds of nuts to pounds of total mixture of 7:10. So 70% of the total number of pounds in the mixture consists of nuts. Since the ratio of pounds of nuts to pounds of total mixture in the 25-pound mixture is the same, then 70% of 25, or 17.5, gives the number of pounds of nuts in the 25-pound mixture.
The population of a certain city was 50,000 people. One year later, the population of the same city grew to 50,600. What was the percent increase in the city’s population in that one-year period?
A. 0.6%
B. 1.2%
C. 6%
D. 12%
B. 1.2%
Option (B) is correct. The question requires an understanding of computing percent increase. The increase in the population of the city is 50,600−50,000=600 people. The value of the fraction 600/50,000 gives the percent increase based on the population before the increase occurred. The fraction is equivalent to the decimal 0.012, which is equivalent to 1.2 percent.
Which of the following is an example of the associative property of multiplication?
A. ab+c=ba+c
B. ab+c=c+ab
C. (ab)c=a(bc)
D. a(b+c)=ab+ac
C. (ab)c=a(bc)
What is the greatest odd factor of the number 2,112 ?
A. 3
B. 21
C. 33
D. 111
C. 33
Option (C) is correct. The question requires an understanding of prime factorization of a number. The prime factorization of 2,112 is 26×3×11. Since 3 and 11 are the only odd prime factors of 2,112, the greatest odd factor is given by the product of 3 and 11, or 3×11, or 33.
Which of the following are equivalent to dividing 288 by 24 ?
Select all that apply.
A. (288 ÷ 4) ÷ 6
B. 2(144 ÷ 24)
C. (144 ÷12) + (144 ÷ 12)
D. (240 ÷ 24) + (48 ÷ 24)
A, B, D
Options (A), (B), and (D) are correct. The question requires an understanding of the order of operations in basic computations. Operations within the parentheses must be solved before operations outside the parentheses. That is, the parentheses group together the expressions that should be solved first. If you divide 288 by 24, the result is 12. Each of the choices also equals 12 except (C), which equals 24.
Item Price Cheese $1.63 Milk $1.19 Juice $1.99 Cereal $1.19 Bread $0.89 Butter $1.39
A shopper purchases one of each of the items in the list above at the prices indicated. Which of the following is closest to the change the shopper would receive after paying with a $20 bill? (Assume there is no sales tax.)
A.$10
B.$11
C.$12
D.$13
C.$12
Option (C) is correct. The question requires an understanding of addition of rational numbers. You can add the prices of the groceries and get a total of $8.28. Subtracting this amount from $20 gives you the amount of change $11.72, which is closest to $12.
Which of the following has the greatest value?
A. 5 thousands
B. 53 hundreds
C. 506 tens
D. 5,100 ones
B. 53 hundreds
Option (B) is correct. The question requires an understanding of place value. By writing each of the answer choices as a numeral, you can compare the four numbers and decide which is the greatest. 5 thousands is the same as 5 times 1,000, or 5,000. 53 hundreds is the same as 53 times 100, or 5,300. 506 tens is the same as 506 times 10, or 5,060. 5,100 ones is the same as 5,100 times 1, or 5,100. 53 hundreds is the greatest of the numbers given.
The Statue of Liberty casts a shadow that is 37 meters long at the same time that a nearby vertical 5-meter pole casts a shadow that is 2 meters long. Based on shadow height, the height, in meters, of the Statue of Liberty must be within which of the following ranges?
A. 115 meters to 120 meters
B. 105 meters to 110 meters
C. 90 meters to 95 meters
D. 60 meters to 65 meters
90 meters to 95 meters
Option (C) is correct. The question requires an understanding of proportions. The ratio between the height of the Statue of Liberty and the length of its shadow is equal to the ratio between the height of the pole and the length of its shadow. The proportion will look like this (where L represents the height of the Statue of Liberty): L/37=5/2. Multiplying both sides by 37 and then simplifying both sides of the equation gives you L=92.5 m. Note that other proportions can be set up, such as statue height (L) divided by pole height (5 meters) equals statue shadow length (37 meters) divided by pole shadow length (2 meters). This will also give the correct result.
The following table shows the populations of four neighboring counties.
County Population Brookhaven 74,702 Columbus 70,472 Davidson 74,072 Washington 74,720 Quyen lives in the county with a population of 70,000 plus 4,000, plus 70, plus 270,000+4,000+70+2 . In which county does Quyen live?
C.
Davidson
2/3÷4/3+3/5×(5/3)^2
Which of the following is equivalent to the preceding expression?
B.
the fraction 13 over 6 13/6
Option (B) is correct. The question requires an understanding of how to solve problems using the order of operations. By using the order of operations and the fact that dividing is equivalent to multiplying by the inverse, the expression two thirds divided by four thirds, plus, three fifths, times open parenthesis, five thirds, close parenthesis, squared2/3÷4/3+3/5×(5/3)2 can be simplified to two thirds times three fourths, plus, three fifths times the fraction 25 over 9 2/3×3/4+3/5×2 5/9. Performing both multiplications yields one half plus five thirds12+53, which is equivalent to the fraction 13 over 6 13/6.
Two friends went out for lunch and decided to share the dessert. One of them ate one half 1/2 of the dessert, and the other ate one third 1/3 of the remaining part. What fraction of the dessert was left over?
B.
one third 1/3
Option (B) is correct. The question requires an understanding of how to solve multistep mathematical and real-world problems. The first friend ate one half 1/2 of the dessert, while the second friend ate one third 1/3 of the remaining part; that is, one third times, open parenthesis, 1 minus one half, close parenthesis13(1−1/2), or one sixth16. Altogether they ate one half plus one sixth, equals four sixths1/2+1/6=4/6, or two thirds2/3 of the dessert. Therefore, the fraction left over is 1 minus two thirds1−2/3, or one third1/3 of the dessert.
A unit square is partitioned into identical parts having equal areas. One of the parts is removed from the square, and a shape is formed by the parts that remain after the removal. For which of the following areas of the removed part will the shape that is formed have the greatest area?
D.
one seventh 1/7
Option (D) is correct. The question requires an understanding of how to recognize concepts of rational numbers and their operations. If the unit square is partitioned into n parts having equal area, the area of each part is the fraction 1 over n1n. Therefore the area of the shape that is formed when removing one of the identical parts is 1 minus the fraction 1 over n1−1n. The smaller the area of the removed part, the greater the area of the shape that is left. Since one seventh 1/7 is the smallest of the four fractions listed, the shape that has the greatest area is the one that is left by removing a part with area one seventh 1/7.
