Algebraic Thinking Flashcards
(30 cards)
If the geometric sequence below continues to increase in the same way, what is the next number in the sequence?
2, 6, 18, 54, 162, …
A.243
B.324
C.486
D.729
C.486
The ratio is 3 since 2 x 3 = 6 and 6 x 3 = 18, and 18 x 3 = 54. Then 162 x 3 = 486.
3x5+7x3−5
Which of the following lists the coefficients and degrees of the terms in the polynomial shown?
A. Coefficients: 3, 7, −5; Degrees: 5, 3, 0
B. Coefficients: 3, 7, −5; Degrees: 5, 3, 1
C. Coefficients: 5, 3, 0; Degrees: 3, 7, −5
D. Coefficients: 5, 3, 1; Degrees: 3, 7, −5
A. Coefficients: 3, 7, −5 ; Degrees: 5, 3, 0
Option (A) is correct. The question requires an understanding of algebraic terminology. The coefficients of the terms of a polynomial are the numbers by which the variables are multiplied. The degrees of the terms of a polynomial are the exponents to which the variables are raised. Therefore, the coefficients are 3, 7, and −5 and the degrees are 5, 3, and 0.
3y+2y
What is the value of the algebraic expression shown when y=5 ? A. 15 B. 25 C. 60 D. 85
B. 25
Option (B) is correct. The question requires an understanding of algebraic expressions and the ability to manipulate them. To find the value of the given algebraic expression when y=5, 5 must be substituted for y in the expression. Therefore, 3y+2y = 3×5 + 2×5, Then add 15 + 10 = 25.
Which of the following graphs in the xy-plane could be used to solve graphically the inequality x−2
C.
Option (C) is correct. The question requires an understanding of inequalities and the ability to solve them graphically. The given inequality is equivalent to x−2
Mr. Smythe has asked his students to come up with function rules for the following table of data.
x y
–1 1
0 0
Adam says that the function rule is y=−x, Belinda says that the function rule is y=−x2, and Chandra says that the function rule is y=−x2−2x. Which of the three equations could be the function rule for the table?
A. Adam’s only
B. Adam’s and Chandra’s only
C. Belinda’s and Chandra’s only
D. Adam’s, Belinda’s, and Chandra’s
B. Adam’s and Chandra’s only
Option (B) is correct. The question requires an understanding of the concept of function and its definition. A function is a rule that establishes a relationship between two quantities: the input and the output. To find out whether or not Adam’s function could be the rule for the given table, it is necessary to substitute each pair of values, (0,0) and (−1,1), into y=−x to verify whether or not the substitution gives the correct output. This process must be repeated for both Belinda’s function and for Chandra’s function. It is easy to see that (0,0) satisfies all three rules.
Since 1=−(−1), the pair (−1,1) satisfies Adam’s function rule.
Since 1 =−(−1)2−2(−1)=−1+2=1, the pair (−1,1) also satisfies Chandra’s function rule. The pair (−1,1) does not satisfy Belinda’s rule, since 1≠−(−1)2=−1.
The community pool has a capacity of 50,000 gallons. It is leaking at a rate of 450 gallons per day. The equation g = 50,000 – 450d can be used to find the number of gallons g remaining in the pool after d days. Which of the following statements is true?
A. g is the dependent variable because the volume is dependent on the number of days d.
B. g is the independent variable because it is what needs to be found.
C. d is the dependent variable because it is being multiplied by the independent rate of 450.
D. Dependent and independent variables cannot be determined in this situation because the equation is linear.
A. g is the dependent variable because the volume is dependent on the number of days d.
Option (A) is correct. The question requires an understanding of dependent and independent variables within various formulas. The input of a function is referred to as the independent variable because the input can be any number. In this instance, the output, referred to as the dependent variable, is the number of gallons g remaining in the pool, because the volume depends on the input variable d, or the number of days since the pool started to leak.
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)
A. 4(7x+25)
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).
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. 60/t
C. t60
D. d60t
B. 60/t
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 60/t=r.
Consider the algorithm shown. 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
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.
2, 3, a3 , a4 , a5 , … The numbers 3, 4, and 5 are below a.
The first two terms of the sequence shown are 2 and 3. Each subsequent term, beginning with the third, is found by adding the two preceding terms and multiplying the sum by −2. What is the value of a5, the fifth term of the sequence?
A. –14
B. –12
C. –10
D. –8
D. –8
Option (D) is correct. The question requires an understanding of following a rule to continue a sequence of numbers. The first term is 2, the second term is 3, and the third term is found by adding 2 and 3 and multiplying the result by −2. So the third term is −10. The fourth term is found by adding 3 and −10 and multiplying the sum by −2. The fourth term is, therefore −7 times −2, or 14. The fifth term is found by adding −10 and 14 and multiplying the sum by −2. The fifth term is, therefore, 4 times −2, or −8.
x f(x) 100 60 24 45 18 30 12 Some values of the linear function f are given in the table shown. What is the value of f(100) ? A. 30 B. 36 C. 40 D. 48
C. 40
Option (C) is correct. The question requires an understanding of finding patterns in data. Since 18−12/45−30=6/15=2/5, the value of f(x) increases by 2 units for every 5-unit increase in the value of x. The value of x increases by 40 units from 60 to 100, so the value of f(x) must increase by 16 units between f(60) and f(100). Since f(60) = 24, f(100) = 24+16 = 40.
At a yard sale, Tenille sold drinking glasses for $2.00 each and plates for $3.50 each. Nicholas spent a total of $18.00 on drinking glasses and plates at Tenille’s yard sale. If Nicholas bought at least one glass and one plate, how many drinking glasses did he buy?
A.1
B.2
C.3
D.4
B.2
Option (B) is correct. The question requires an understanding of equations and the ability to translate a word problem into an equation. If x represents the number of glasses and y represents the number of plates that Nicholas bought, then 2x+3.5y=18. Both x and y must be integers. Therefore, 18−2x must be a multiple of 3.5. The possible multiples of 3.5 for this problem are 3.5, 7, 10.5, 14, and 17.5. The only multiple of 3.5 that is equivalent to 18−2x as an integer is 14. Thus, x=2.
If one cup of soup costs 2 dollars, which of the following represents the number of cups of soup that can be purchased with d dollars?
A. 2d
B. d/2
C. d/100×2
D. 100×d/2
B. d/2
Option (B) is correct. This question requires an understanding of algebraic representations. The number of dollars spent divided by the number of dollars per cup of soup gives the number of cups of soup purchased. Therefore, the number of cups of soup that can be purchased with d dollars is d/2.
x y −10 −5 −5 0 5 10 10 15 Which of the following graphs best represents the relationship between x and y in the table shown?
A.
B.
C.
D.
A.
Option (A) is correct. The question requires an understanding of representing points in the xy-coordinate plane. The ordered pairs in the table can be represented in the xy-coordinate plane as four points on a line. The point (−5,0) is one of the points on the line and indicates that the line crosses the x-axis at −5. The only one of the graphs shown that is a line that meets this condition is the graph in option (A).
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
C. 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.
Which of the following is an algebraic expression?
Answer the question by selecting the correct response.
A. 6x−4
B. 6y<4
C. 6z=4
D. 6+4
A. 6x−4
Option (A) is correct. The question requires an understanding of how to differentiate between algebraic expressions and equations. An algebraic expression is made of constants, variables, and algebraic operations.
The following figures are the first three figures in a pattern.
Figures. (detailed description follows)
Three figures are shown. From left to right, the figures are labeled Figure 1, Figure 2, and Figure 3. Each figure consists of connected line segments that make up squares and triangles.
Figure 1.
Figure 1 consists of eight line segments that make up one square and two triangles. The square is at the center of the figure, and is comprised of four line segments. Two line segments extend outward from the left side of the square to form the first triangle. Two line segments extend outward from the right side of the square to form the second triangle.
Figure 2.
Figure 2 consists of eleven line segments that make up two squares and two triangles. The squares are positioned side by side at the center of the figure, such that they share one side. Two line segments extend outward from the left side of the square on the left to form the first triangle. Two line segments extend outward from the right side of the square on the right to form the second triangle.
Figure 3.
Figure 3 consists of fourteen line segments that make up three squares and two triangles. The squares are positioned side by side at the center of the figure, such that each adjacent square shares one side. Two line segments extend outward from the left side of the square on the left to form the first triangle. Two line segments extend outward from the right side of the square on the right to form the second triangle.
End figure description.
Figure 1 is composed of two triangles and one square. Each figure after figure 1 is composed of two triangles and one square more than the preceding figure.
How many line segments are in figure 10 of the pattern?
Answer the question by selecting the correct response.
A. 35
B. 38
C. 41
D. 44
A. 35
I drew the figure and then counted all on the line segments.
Which of the following inequalities is equivalent to the inequality 4x+4≤9x+8 ?
Answer the question by selecting the correct response.
A. x≥−4/5
B. x≤−4/5
C. x≥−12/5
D. x≤−12/5
A. x≥−4/5
4x(3x+2y)
What does 2 y2y represent in the expression shown?
Answer the question by selecting the correct response.
A. binomial
B. factor
C. coefficient
D. monomial
D. A monomial
Option (D) is correct. The question requires an understanding of how to use mathematical terms to identify parts of expressions and describe expressions. A monomial is an algebraic expression that consists of one term that is a number, a variable, or a product of a number and a variable, where all exponents are whole numbers.
a=5,000(1+r)
The formula shown can be used to find the amount of money in dollars, a, in an account at the end of one year when $5,000 is invested at simple annual interest rate r for the year. Which of the following represents the independent variable in the formula?
Answer the question by selecting the correct response.
A. a
B. 5,000
C. r
D. 1 plus r1+r
C. r
Option (C) is correct. The question requires an understanding of how to differentiate between dependent and independent variables in formulas. In the given formula, there are two variables, a and r. The formula can be used to investigate how the amount of money a varies depending on the interest rate r. Therefore, the dependent variable is a and the independent variable is r.
Which of the following expressions is equivalent to negative 4 times, open parenthesis, 3 minus 2 x, close parenthesis−4(3−2x) ?
Answer the question by selecting the correct response.
A. −2x−12
B. 2x−12
C. −8x−12
D. 8x−12
D. 8x−12
Option (D) is correct. The question requires an understanding of how to use the distributive property to generate equivalent linear algebraic expressions. Using the distributive property of multiplication over addition, negative 4 times, open parenthesis, 3 minus 2 x, close parenthesis, equals negative 4 times 3, minus, 4 times negative 2 x−4(3−2x)=−4(3)−4(−2x); that is, negative 12 plus 8 x−12+8x. Using the commutative property of addition yields 8 x minus 128x−12.
The following table shows the cost of a membership to Gym G for the five possible membership lengths.
Membership Length, in months Cost, in dollars 1 75 3 125 6 200 12 350 24 650
Gym H has the same possible membership lengths, and the cost, y, in dollars, of a membership to Gym H for x months is given by the equation 2 y minus 50 x, equals 852y−50x=85.
Which of the following is true about the cost, in dollars, of a membership to Gym H compared with the cost of a membership to Gym G?
A. The cost of a membership to Gym G is greater than the cost of a membership to Gym H for membership lengths of 6 months or less but is greater for membership lengths of greater than 6 months.
B. The cost of a membership to Gym H includes the same initial membership fee as the cost of a membership to Gym G but a greater monthly fee.
C. The cost of a membership to Gym H includes a greater initial membership fee than the cost of a membership to Gym G but a lower monthly fee.
D. The cost of a membership to Gym G is greater than the cost of a membership to Gym H for any number of months.
D. The cost of a membership to Gym G is greater than the cost of a membership to Gym H for any number of months.
Option (D) is correct. The question requires an understanding of how to use linear relationships represented by equations, tables, and graphs to solve problems. The table describes the costs of varying lengths of membership to Gym G and can be represented by the linear equation y equals, 25 x plus 50y=25x+50, where y is the cost of a membership lasting x months. The equation that describes the cost y of a membership to Gym H lasting for x months can be rewritten as y equals, 25 x plus 42.50y=25x+42.50. The monthly fees, represented by the slopes of the two linear equations, are equal for the two memberships. However, the y-intercept of the equation representing Gym G is greater than the y-intercept of the line representing Gym H. This can be interpreted to mean that the initial fee for Gym G is greater than the initial fee for Gym H. Since the monthly memberships are the same but Gym G has a greater initial fee, the membership cost for Gym G is always more expensive than the membership cost for Gym H for any number of months.
The formula V equals I times RV=IR relates the voltage V, in volts, to the current I, in amperes, and the resistance R, in ohms, in a circuit. What is the current produced by a 9-volt battery in a circuit with 4 ohms of resistance?
Answer the question by selecting the correct response.
A. 1.50 amperes
B. 2.00 amperes
C. 2.25 amperes
D. 2.50 amperes
C. 2.25 amperes
Option (C) is correct. The question requires an understanding of how to use formulas to determine unknown quantities. Since V equals 9V=9 volts and R equals 4R=4 ohms, I equals, the fraction V over R, which equals the fraction 9 over 4, which equals 2.25I=VR=94=2.25 amperes.
If y = 2, what is the value of 4−2(4y)+5y ?
Answer the question by selecting the correct response.
A. −22
B. −2
C. 22
D. 26
B. −2
Option (B) is correct. The question requires an understanding of how to evaluate simple algebraic expressions. The first step is to substitute 2 in place of the variable y, which yields the arithmetic expression 4 minus, 2 times, open parenthesis, 4 times 2, close parenthesis, plus, 5 times 24−2(4×2)+5×2. Using the order of operations, 4 minus, 2 times, open parenthesis, 4 times 2, close parenthesis, plus, 5 times 2, equals 4 minus, 2 times 8, plus 10, which equals 4 minus 16, plus 10, which equals negative 24−2(4×2)+5×2=4−2×8+10=4−16+10=−2.
(((Use a calculator for the final part. )))
