Math Elementary Education: Multiple Subjects Mathematics (5003)

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.

Correct Answer: D 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. According to the word problem, Sara bought 6 shirts, 3 pairs of pants, and 112 sweaters, totaling 1012 pieces of clothing. The solution is, therefore, not acceptable since it is not a natural number.

Kyle's father set up a savings account for him with an initial balance of $100. Since then, Kyle has been depositing $28 into the account each week. Kyle represents the amount of money he has saved after x weeks by the expression 28x+100. Which of the following is equivalent to Kyle's expression? A.4(7x+25) B.7(4x+100) C.7(4x+25) D.4(7x+100)

Option (A) is correct. The question requires an understanding of algebraic expressions and the ability to manipulate them. Since both 28 and 100 are divisible by 4, 4 is a factor of the expression 28x+100. To arrive at the equivalent expression given in (A), the distributive property was applied to 28x+100. In fact, 28x+100=4×7x+4×25=4(7x+25).

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

Option (A) is correct. The question requires an understanding of the basic properties of real numbers. The distributive property of multiplication over addition states that for any real numbers a, b, and c, a(b+c) is equal to ab+ac. In other words, adding the numbers within the parentheses and then multiplying by the number outside the parentheses yields the same result as multiplying each term within the parentheses by the number outside the parentheses and then adding the two products together.

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. 140×2,800 B. 120×2,800 C. 15×2,800 D. 5×2,800

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, 5100, which simplifies to 120. To find 5 percent of 2,800 employees, it is necessary to multiply 2,800 by either 0.05 or 120.

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

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 problem could also be solved by setting up a proportion using x to represent the number of pounds of nuts in the 25-pound mixture. Then 710=x25, and solving for x yields 17.5.

In the formula d=r×t, if d equals 60 and t remains constant, which of the following is equivalent to r ? A.60t B.60t C.t60 D.d60t

Option (B) is correct. The question requires an understanding of simple formulas and the ability to work with them. If d is equal to 60, d can be replaced with 60 in the equation, which will result in 60=r×t. The question asks for determining which option is equivalent to r, so it is necessary to solve the equation for r. Since r is multiplied by t, both sides of the equation must be divided by t to isolate r. The result is 60t=r.

In which quadrant is the point (−8,2) located? A.Quadrant I B.Quadrant II C.Quadrant III D.Quadrant IV

Option (B) is correct. The question requires an understanding of the coordinate plane. Since points in the second quadrant have a negative x-coordinate and a positive y-coordinate, the point with coordinates (−8,2) is located in quadrant II.

Which of the following is equal to 4(5−2)2−23 ? A. 16 B. 28 C. 76 D.136

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.

A two-dimensional net of a certain three-dimensional figure includes five faces, nine edges, and six vertices. Which of the following three-dimensional figures is represented by the net? A. Triangular pyramid B. Triangular prism C. Rectangular pyramid D. Rectangular prism

Option (B) is correct. The question requires an understanding of three-dimensional geometry. Triangular pyramids and rectangular prisms have four and six faces, respectively. Triangular prisms have five faces: two are triangles, and three are quadrilaterals. Rectangular pyramids also have five faces: one is a rectangle, and four are triangles. Triangular prisms have nine edges, while rectangular pyramids only have eight edges.

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

Option (C) is correct. The question requires an understanding of algebraic properties. The associative property of multiplication concerns the order in which the multiplications are performed when three or more numbers are multiplied, which can be changed by inserting or removing grouping symbols such as parentheses. If the same factors are present, and in the same order, but the grouping symbols dictate that the order in which the multiplications are performed has changed, then the associative property of multiplication has been applied. In ab+c=ba+c, ab+c=c+ab, and a(b+c)=ab+bc, there are only products with two factors in each, so the associative property of multiplication cannot be applied. In (ab)c=a(bc), the parentheses dictate that on the left side, a and b are multiplied first, and then the result is multiplied by c, and that on the right side, b and c are multiplied first, and then a is multiplied by this result. The fact that the end results on each side are equal is guaranteed by the associative property of multiplication.

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

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.

What is the greatest odd factor of the number 2,112 ? A. 3 B. 21 C. 33 D.111

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.

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

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): L37=52. 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.

If 125+4x=7y, what is x in terms of y ? A.x=4(7y−125) B.x=4(7y+125) C.x=(7y−125)÷4 D.x=(7y+125)÷4

Option (C) is correct. The question requires an understanding of solving an equation for a given variable. To find x in terms of y, the equation 125+4x=7y must be transformed so that the variable x is isolated, or by itself, on one side of the equal sign. Subtracting 125 from both sides of 125+4x=7y yields 4x=7y−125. Dividing both sides of 4x=7y−125 by 4 yields x=(7y−125)÷4. The variable x is now expressed in terms of y.

Step 1: Select a numerical value for the variable k. Step 2: Add 3 to k. Step 3: Multiply the result by 8. Step 4: Cube the result. Step 5: End. Executing the algorithm shown is equivalent to evaluating which of the following algebraic expressions? A. (8k+3)3 B. 8(k+3)3 C. 83(k+3) D. 83(k+3)3

Option (D) is correct. The question requires an understanding of algorithms and their algebraic representations. The expression can be found by following the instructions in each of the steps given. The key steps in the algorithm are steps 3 and 4. When multiplying the binomial k+3 obtained in step 2, it is necessary to multiply both terms. The result of step 3 is, therefore, 8(k+3). When raising a product to the power of 3, it is necessary to raise each factor in the product to the power of 3. The factors are 8 and (k+3). The result of step 4 is, therefore, 83(k+3)3.

On Greg's map, 1 inch represents 30 miles, and on Lori's map, 1 inch represents 20 miles. The area of a 1-inch by 1-inch square represents how many more square miles on Greg's map than on Lori's map? A.100 B.250 C.400 D.500

Option (D) is correct. The question requires an understanding of calculating areas using standard, real-world miles on a map. Area is a two-dimensional representation of a surface (length times width, base times height, etc.). A 1-inch by 1-inch square on Greg's map represents a square 30 miles on each side. The area of this square is 30 miles multiplied by 30 miles, or 900 square miles. On Lori's map, the 1-inch by 1-inch square represents a square 20 miles on each side. The area of this square is 20 miles multiplied by 20 miles, or 400 square miles. The difference between these is 500 square miles.

Which of the following expresses 316 as a percent? A. 0.1875% B. 1.875% C. 5.33% 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 316×100=18.75, 316 is equivalent to 18.75%.